Uncertainty and Change:Survey Evidence of Firms’ Subjective Beliefs

Rüdiger Bachmann Kai Carstensen Stefan LautenbacherMartin Schneider∗

November 25, 2020

Abstract

This paper provides survey evidence on firms’ subjective uncertainty about future salesgrowth from a new panel data set of the German manufacturing sector. The main findingis that uncertainty reflects change: firms report more subjective uncertainty after either highor low growth realizations. In the cross section of firms, subjective uncertainty differs fromstatistical measures of uncertainty such as volatility: fast-growing and large firms reportlower subjective uncertainty than fast-shrinking and small firms, respectively, even if theyface shocks of similar size. By contrast, the substantial time variation in firm-specific sub-jective uncertainty resembles that in conditional volatility: both measures of uncertaintyare mildly persistent and rise more when growth is temporarily low.

Keywords: expectation formation, firms, measurement, subjective uncertainty, survey data

JEL codes: C83, D22, E23

∗Respectively: University of Notre Dame, CEPR, CESifo and ifo, e-mail: rbachman@nd.edu; University of Kiel,CESifo and ifo, e-mail: carstensen@stat-econ.uni-kiel.de; ifo and LMU Munich, e-mail: lautenbacher@ifo.de; Stan-ford University, CEPR and NBER, e-mail: schneidr@stanford.edu (corresponding author). We thank Pete Klenow,Giuseppe Moscarini, Matthew Shapiro, Stephen Terry, Michael Woodford as well as conference participants atthe 2017 SED annual meeting in Edinburgh, the 2018 ASSA meetings in Philadelphia, the 2018 Spring Meeting ofYoung Economists in Mallorca, the 2018 Cowles Summer Conference on Macroeconomics, the 2018 CESifo VeniceSummer Institute, the 2018 Kiel University workshop on “Firm and Household Uncertainty, Expectation Forma-tion, and Macroeconomic Implications”, the 2018 NBER Summer Institute Meeting on Behavioral Macro, the2018 EEA meeting in Cologne, the 2018 SITE meeting in Stanford, the 2018 workshop on “Developing and UsingBusiness Expectations Data” at the University of Chicago, the 2018 European Commission Workshop on Businessand Consumer Surveys, the 2018 ifo conference on “Macroeconomics and Survey Data”, the 2019 workshop on“Expectations Surveys: A Tool for Research and Monetary Policy” at the Federal Reserve Bank of New York, andseminar participants in Kiel, LMU and ifo Munich, Nuremberg, Carlos III in Madrid, Northwestern, Emory aswell as the Federal Reserve Banks of Philadelphia, Chicago, and Cleveland for helpful comments, Simeon Häfelefor computational assistance, and the team of ifo’s Economics and Business Data Center for excellent supportand provision of the data. Financial support by the Fritz Thyssen Foundation is gratefully acknowledged.

1 IntroductionA large literature studies how firms respond to time variation in uncertainty. It has highlightedtwo key sources of such variation. First, firms respond in the short run to news about theirbusiness conditions. For example, at the onset of a recession firms become less confident intheir forecasts of future cost or demand. Heightened uncertainty then might lead them toscale back hiring or investment.1 Second, many firms face more longer term risks that theylearn about over time: for example, expanding firms find out gradually about demand innew markets. It makes sense for firms that operate in unfamiliar territory to operate morecautiously; to an observer they may then appear to adjust “too slowly.”2

Yet, to date there is little direct evidence on how decision makers within businesses perceiveand process uncertainty. Instead, subjective uncertainty is often indirectly inferred through thelens of a particular model: for example, if a model imposes rational expectations, the realizedvolatility of shocks estimated by the modeler becomes a measure of uncertainty perceived bythe firms. However, since there is no consensus on model structure, many open questionsremain. In particular, how much does subjective uncertainty fluctuate over time? How is itshaped by past firm performance, both in the short run and the longer run? And how does itrelate to both conditional and unconditional realized volatility?

This paper provides survey evidence on firms’ perceived uncertainty about future salesgrowth. We introduce a panel data set of firms’ subjective beliefs, characteristics and perfor-mance for the German manufacturing sector. It is based on a new module of the ifo BusinessSurvey, a long-established survey used to develop business sentiment indicators. The surveyis well regarded in the German business community: questions are answered mostly by seniormanagement and there is a high response rate even from large firms.3 Our data set is basedon 14 survey waves in consecutive quarters from 2013 to 2016. We use it to document howsubjective uncertainty varies not only in the cross section of firms but also over time.

The new survey module asks firms for a forecast of one-quarter-ahead sales growth to-gether with two numbers for best and worst case sales growth scenarios. We then define thedifference between the best and worst case one-quarter-ahead sales growth scenarios, that is,span, as our quantitative measure of subjective uncertainty. The idea behind the survey designis that firms can directly report scenarios developed as part of their regular planning process.Responses to a one-time meta-survey we commissioned show that a large majority of firmsengages in scenario analysis and uses results from routine quantitative planning when fillingout the survey module. Since in addition to the forecast we also retrospectively ask for firms’realized sales growth, we can further compare subjective uncertainty to firms’ realized forecasterrors.

1Bloom (2014), Fernández-Villaverde and Guerrón-Quintana (2020), and Cascaldi-Garcia et al. (2020) surveythe macroeconomic literature on fluctuations in uncertainty. One concrete workhorse model assumes that produc-tivity exhibits stochastic volatility that—exogenously or endogenously—increases at the beginning of recessions.

2Following the seminal work of Jovanovic (1982), a large literature on firm dynamics with learning studiesquestions such as the contribution of young firms to growth and the role for subsidizing such firms.

3Research studies using the standard ifo expectation data are, for instance, Bachmann et al. (2013); Buchheimand Link (2017); Massenot and Pettinicchi (2018); Bachmann et al. (2019); Enders et al. (2019b,a).

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Our main finding is that uncertainty reflects change experienced by firms. This principledescribes beliefs in both the short and medium run. On the one hand, subjective uncertaintyperceived by an individual firm varies substantially at the quarterly frequency; in particular, itis high after both very good and very bad growth realizations. This short-run pattern is sharedby the absolute value of firms’ forecast errors, our measure of firms’ conditional volatility. Onthe other hand, average subjective uncertainty over our four-year sample correlates stronglywith measures of change relating to a firm’s medium-term environment: it is higher for firmsthat consistently grow or shrink, as well as for firms with unconditionally more volatile salesgrowth. In the cross section of firms, however, subjective uncertainty behaves differently fromrealized conditional volatility. In particular, large and growing firms report relatively lowsubjective uncertainty even when they make large (in absolute value) forecast errors.

Our results thus provide direct evidence for both the short-run and medium-run timevariation in uncertainty emphasized in the literature. One key takeaway from our findings isthe importance of short-run idiosyncratic variation. Mechanisms at work when uncertainty goesup in recessions could therefore be relevant also for firm dynamics in normal times. Moreover,a model of short-term planning should draw a connection between subjective uncertainty andconditional volatility, for which our results provide an empirical basis. Traditionally, thisconnection was simply assumed through the rational expectations assumption. Indeed, wefind that firms’ planning under uncertainty appears to reflect at least in part actual volatility,perhaps because managers are quite familiar with the short-run dynamics of their business.

At the same time, our results have implications for modelling learning over the mediumrun. Indeed, fast-growing and fast-shrinking firms not only perceive higher uncertainty, butalso make forecasts that are too conservative, that is, systematically biased towards zero. Thisis true after controlling for firm size, suggesting that even some large firms find themselvesin unfamiliar territory where growth is uncertain and hard to forecast. While we document aconnection between volatility and subjective uncertainty also for longer term risk, there is animportant second force: successful firms—either growing or large—report lower uncertaintywhen faced with the same-sized shocks in absolute value. This fact is consistent with mecha-nisms that make uncertainty matter more to decision makers in bad times, so their planningconsiders a wider span of scenarios.

To provide an idea of magnitudes, the mean span between best and worst case scenarios is12.1 percentage points (pp), slightly above the mean absolute forecast error. Firms differ in av-erage subjective uncertainty: the cross sectional standard deviation of time-averaged span perfirm is 7.4 pp. At the same time, we document large time variation in subjective uncertaintyfor individual firms: the average firm’s time series standard deviation of span is 5.9 pp. Time-varying span is thus a volatile component of firms’ planning processes. In fact, it is almost asvolatile as the usual driver of firm planning in economic models, namely changes in condi-tional expectations: the average firm’s time series standard deviation of growth forecasts is 7.4pp. Most of the time series variation in subjective uncertainty is firm-specific: time-industryfixed effects explain only a negligible share.

Our cross-sectional results relate average firm-level subjective uncertainty to two measuresof change relating to a firm’s environment over the medium term. First, we define trend asa firm’s unconditional mean growth rate over our four-year sample. We show that both high

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and low trend firms are significantly more uncertain. Span in the bottom quartile by trendis 5.9 pp higher than for the “normal” firms, that is, firms within the interquartile range; itis 2.2 pp higher in the top quartile. At the same time, high and low trend firms’ forecastsare biased towards zero by about 5 pp on average. Both results are consistent with models oflearning: fast expansion or shrinkage leads firms to a less familiar, and hence more uncertain,state of business that is difficult to forecast. They are not consistent with simple models offirm dynamics in which every firm knows its trend growth.

Our second measure of medium-term change is turbulence, defined as firms’ in-samplesales growth volatility over time. High turbulence firms face larger (absolute) shocks thanthe average firm, but do not make biased forecasts. Moreover, they not only report highersubjective uncertainty on average, but also higher time series variation in subjective uncertainty.Indeed, controlling for trend as well as size, the mean span in the top quartile by turbulence is9.6 pp higher than in the bottom quartile, whereas the time series standard deviation of span is5.5 pp higher. In other words, planning at firms that face larger shocks not only uses scenariosthat are further apart but also varies those scenarios more over time as shocks arrive.4

The short-run relationship between subjective uncertainty and past growth is V-shaped,with a minimum close to zero. Firms thus become more uncertain after either negative orpositive growth. Bad quarters increase uncertainty by more: while a one percentage pointlower negative growth rate in our preferred specification is followed by 31 basis points widerspan between firms’ best and worst case scenarios, a one percentage point higher positivegrowth rate widens span by only 18 basis points; these numbers are robust to including avariety of controls for firm heterogeneity. The V-shape is perhaps surprising in light of thenegative comovement between growth and uncertainty emphasized in the literature. It isnevertheless consistent with this literature because our results suggest that individual firms’uncertainty is shaped by its individual performance, and increases when an unfamiliar eventoccurs, especially a bad one.

There are several candidate explanations for why uncertainty might reflect change in theshort run. One possibility is that the basic principle we have found in the cross section—firmsthat operate in unfamiliar territory perceive more uncertainty—is also at work in the shortrun. If this force were dominant, we should see that uncertainty is particularly related togrowth surprises. We indeed confirm a positive relationship between lagged absolute forecasterrors and span. However, we also show that, in firm quarters with negative growth, theprevious-quarter growth rate is a sufficient statistic for predicting span in a horse race betweenthe previous-quarter growth rate and forecast error. The reason behind this result is thatpredictable low growth realizations also increase uncertainty, to a similar extent as low growthsurprises. A possible explanation is that planning takes into account state variables other thangrowth. For example, in a model with customer capital, a shrinking firm might see the sizeand/or composition of its future pool of customers and thus its sales growth become more

4This distinction matters because of its implication for behavior such as factor choice: in the presence ofadjustment costs or time to build, time variation in subjective uncertainty leads firms to respond differently eachperiod. If instead, high volatility firms simply faced larger iid shocks, they might still behave differently fromlow volatility firms, but that behavior would not vary over time. Our results say that the theoretical mechanismsthat make firms respond to uncertainty generate both cross sectional and time series variation.

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uncertain even when this shrinking occurs in a predictable way.

How does firms’ subjective uncertainty compare to the volatility of their forecast errors, ameasure of uncertainty in many models? Our key cross sectional finding here is that successfulfirms—defined as either large and fast-growing—plan with narrower spans even when theyface the same magnitude of forecast errors as less successful firms. The result has two parts.First, controlling for firm size, the absolute forecast error for growing firms is 2.5 pp higherthan for firms that are neither growing nor shrinking, while span is not significantly different.By contrast, compared to stable firms both the absolute forecast errors and span increaseby about the same amount of roughly 3 pp for shrinking firms. While high and low trendfirms are both in unfamiliar territory—in the sense of facing larger shocks—growing firms donot adjust their planning. Second, large firms with more than 250 employees make similarabsolute forecast errors as smaller firms, yet plan with spans that are up to 5 pp narrower,controlling for trend and turbulence. Size by itself thus also leads firms to plan with narrowerspans.

How does time variation in subjective uncertainty compare to that in conditional volatility?This question is more difficult to answer: it is no longer sufficient to compare absolute forecasterrors to a measure of subjective uncertainty, as we did when examining the cross section.The relevant counterpart we are looking for here is the predictable component of conditionalvolatility. We want to establish whether conditional volatility is persistent and related to pastgrowth in a similar way to span. We thus utilize the time series dimension of our data setto estimate dynamic regressions for both our subjective uncertainty measure span as well aspower GARCH models for firms’ forecast errors.

The short-run dynamics of firm-specific subjective uncertainty closely resemble that ofthe conditional volatility of shocks experienced by firms: both are mildly persistent, increasewith bad past growth and increase somewhat less with good past growth. We take awaythat, at least in the short run, firms adjust their planning process based on the experiencethat high and—even more so—low growth signals larger future surprises. In applicationsthat emphasize short-run time series dynamics, an approach that equates uncertainty withconditional volatility thus describes actual firm planning quite well.

Our study is motivated by a large body of work on firm behavior under uncertainty. Theoryhas proposed a number of mechanisms through which uncertainty impacts input choices thathave to be made before cost or demand is fully known. Examples include wait-and-see effectsor financial frictions that increase the cost of capital when uncertainty is high. While therelevant theoretical concept is subjective uncertainty, empirical tests have long had to rely onproxy measures.5 Following the pioneering work of Guiso and Parigi (1999), a small numberof studies have used survey measures of uncertainty to investigate its effects on economicactivity (see in particular, Bontempi et al., 2010; Bachmann et al., 2013; Bloom et al., 2019; Altig

5Empirical work with micro data on firms tends to rely on proxy measures for uncertainty such as the volatilityand dispersion of stock returns (Leahy and Whited, 1996; Campbell et al., 2001; Bloom et al., 2007) or otherfirm-level outcomes (Bachmann and Bayer, 2013, 2014; Jurado et al., 2015; Bloom et al., 2018), implied optionsvolatility (Bloom, 2009; Barrero et al., 2017), perceived political uncertainty from quarterly earnings conferencecalls (Hassan et al., 2019), and qualitative (Bachmann et al., 2013) as well as quantitative (Bachmann et al., 2017,2019; Bloom et al., 2019; Altig et al., 2020) firm-level forecast errors obtained from surveys.

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et al., 2020). The goal of the present paper is not to study the effect of uncertainty on outcomes,but instead to characterize how uncertainty varies over time, and in particular how it relatesto past outcomes.

In particular, our short-run results provide new evidence that speaks to an active dis-cussion about the relationship between uncertainty and growth. Following Bloom (2009), agrowing literature has incorporated uncertainty shocks into macroeconomic models.6 Suchshocks are often orthogonal to first moment shocks, for example, higher uncertainty lowerscurrent growth even if it is unrelated to past growth. More recently, several papers haveconsidered feedback effects from growth to uncertainty (Bachmann and Moscarini, 2012; Fa-jgelbaum et al., 2017; Ilut and Valchev, 2020; Ilut et al., 2018; Baley and Blanco, 2019; Bergerand Vavra, 2019; Ludvigson et al., 2020). One of our main results is the strong association ofhigh uncertainty with low or high past growth. It implies that feedback effects—or possiblycorrelated shocks—are particularly important for understanding the comovement of growthand subjective uncertainty.

Firm-level idiosyncratic uncertainty is also a key building block for models of firm dynam-ics that aim to explain the size distribution of firms, the (mis)allocation of factors of produc-tion and ultimately the level and growth rates of aggregate output (see surveys by Luttmer,2010, or Hopenhayn, 2014; for recent examples, see contributions by Pugsley et al., 2018, andDavid and Venkateswaran, 2019). Our direct evidence on long-term risks is consistent withthe mechanisms explored in quantitative models of firm learning such as Abbring and Camp-bell (2003), Eaton, Eslava, Kugler and Tybout (2012) or Arkolakis et al. (2018). Moreover, ourresults on high frequency variation in subjective uncertainty suggest that even a short time-to-build friction could give rise to effects of uncertainty on factor choice. Indeed, with any typeof adjustment costs, quarterly variation in uncertainty will work like a distortion—a wedgebetween the marginal product and price of a factor (see, for example, Ilut and Saijo, 2020, fora model of firms facing idiosyncratic risk that clarifies this feature).

The new ifo survey module is one of a handful of data sources on quantitative expectationsof leading decision makers in firms about their own economic circumstances.7 Altig et al.(2020) present results from a new business survey of top managers in US businesses, adminis-tered by the Census at the annual frequency and the Federal Reserve Banks of Atlanta at themonthly frequency. They do not study time variation in subjective uncertainty, which is themain focus of our paper. Another source of data is the panel of chief financial officers’ stockreturn expectations assembled by John Graham and Campbell Harvey at Duke University.Ben-David et al. (2013) show that managers are strongly miscalibrated in that their subjec-tive forecast densities are too narrow, thus questioning rational expectations as a modelingassumption. Gennaioli et al. (2016) show that managers’ expectations are connected to actualfirms’ investment plans, thus showing that miscalibration has real effects. Our results confirmthe presence of systematic forecast errors by firms that experience a lot of change. Moreover,

6An incomplete list is: Christiano et al. (2014); Gilchrist et al. (2014); Fernández-Villaverde et al. (2015); Basuand Bundick (2017); Arellano et al. (2019).

7There is also an active literature that studies firm expectations about aggregate variables, such as inflation(see, for New Zealand, Kumar et al., 2015; Coibion et al., 2018; and for Italy, Coibion et al., 2020) and GDP (see,for Japan, Tanaka et al., 2020).

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for successful firms, the perception of uncertainty deviates from their conditional volatility.

The paper is structured as follows. Section 2 explains our new survey questions and prop-erties of the data. Section 3 introduces the raw relationship between uncertainty and changeand presents a simple organizing framework. Section 4 studies uncertainty and change inthe cross section, while Sections 5 and 6 investigate the time series variation of subjective un-certainty. Finally, Section 7 compares the dynamic properties of subjective uncertainty andconditional volatility.

2 DataThe ifo Business Survey, run by the Munich-based ifo Institute, is a long-running survey ofGerman businesses. Despite the occasional attrition, the ifo Institute maintains a sample thatis representative of the German manufacturing sector by replacing exiting firms with newrespondents (see Sauer and Wohlrabe, 2020). The responses from the survey provide input fora leading indicator of the German business cycle, the ifo Business Climate Index. The latteris part of the EU-harmonized business surveys commissioned by the Directorate General forEconomic and Financial Affairs of the European Commission.8

In 2012, we designed an online module of quantitative questions to elicit subjective firmuncertainty that were asked of all manufacturing firms in the main survey. An initial pilotwave in December 2012 was met by strong interest. Analysis of text comments submitted byfirms further showed that firms had no trouble understanding the questions. The module hasnow been in the field since 2013, with participation remaining stable between 300 and 400firms per wave.9

A firm in the survey is either a stand-alone firm or a division of a large conglomerate. Forsimplicity, we refer to “firms” throughout this paper. The survey questions are about growthin sales. Specifically, we ask them about past an expected sales growth. The German term usedin the questionnaire, “Umsatz”, is a well-defined technical term in profit and loss accounting,translated into English as “sales” or “total revenue.” It is commonly used as an accountingstatistic at the levels of both a division and an entire firm.

The survey is administered at the beginning of every quarter. Our current sample uses 14survey waves spanning 2013:Q2 through 2016:Q3. In addition, in fall 2018, the roughly 400firms from our baseline sample were sent a one-time meta-survey we fielded with questionson how firms collect information and arrive at the views expressed in our uncertainty module.191 of these firms responded. Furthermore, Sauer and Wohlrabe (2019) document the identityof the respondents in the ifo Business Survey. Finally, there was an additional general meta-survey administered by ifo in the fall of 2019 and sent out to all the participants in the ifo mainmanufacturing survey. For our purposes, this additional meta-survey provides information on

8Aggregate survey results for Germany are presented at https://www.ifo.de/en/survey/ifo-business-climate-index, the harmonized European results, including the European Economic Sentiment Indica-tor, can be found here: https://ec.europa.eu/info/business-economy-euro/indicators-statistics/economic-databases/business-and-consumer-surveys_en.

9The raw data can be found under: “IBS-IND (2016b): Ifo Business Survey Industry 1/1980 - 12/2016, LMU-ifoEconomics & Business Data Center, Munich, doi: 10.7805/ebdc-ibs-ind-2016b.”

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the regularity of respondents in the survey.

The selection of participants in the uncertainty module is similar to that of the main manu-facturing survey (which is designed to be representative of the German manufacturing sector).Indeed, Appendix A shows that it is essentially impossible to predict participation in the un-certainty module. In addition, our data contain a substantial number of large firms: when wemeasure firm size by the number of employees, the 75th percentile is at about 250 employees.The median firm employs 100 workers while the 25th percentile is at 40.

2.1 Quality of responses

In partnering with ifo, our goal was to develop a high-quality data set that reflects the assess-ment of uncertainty by key decision makers in firms and also reflects the use of quantitativeanalysis the firms already consider in their actual decision making. Our meta-survey togetherwith other meta-surveys administered independently by ifo provides evidence on the qualityof the data along these dimensions.

Before turning to the results from our own meta-survey, we note that Sauer and Wohlrabe(2019) document that, in the overwhelming majority of firms in the manufacturing sector,86%, the respondent is a member of top management: 73% of firms mention the CEO, CFOor COO, whereas 13% of units surveyed refer to a “division head”, the natural label for thetop executive if the unit surveyed is not a stand-alone firm (see Sauer and Wohlrabe, 2019,Table 2). For very large firms with more than 500 employees, the share of responses from topmanagement is only slightly lower than in the population as a whole: a bit over 65% CEO,CFO or COO, and a bit over 15% the division head (see Sauer and Wohlrabe, 2019, Figure1). These findings are consistent with an earlier metastudy conducted by ifo about the tradesector (see Abberger et al., 2011). An additional meta-survey commissioned by ifo in the fall2019 further shows that the identity of the responder within the firm changes rarely: 83% offirms indicate the responder is “always the same person”, 15% say “mostly the same person”,and less than 2% mention a team of people or that the responder “changes frequently”.

Our meta-survey from fall 2018 asks firms what type of information they use when theyfill out the questionnaire of our module. The questionnaire in the original German is shown inAppendix B. We first ask whether answers to our uncertainty questions are guided by numbersthat the firm has already developed in house as part of a regular quantitative planning process.The results are summarized in the top panel of Table 1, both for all firms and broken downby size class.10 On average, 80% of firms respond that they use results from its quantitativeplanning. The share is remarkably stable across firms, only very small firms with less than 10employees report a somewhat lower share.

We then add a follow-up question about alternative frameworks for quantitative planningunder uncertainty:

10In line with the definition by the German Statistical Office, we define firms as “tiny” if they have less than 10employees, “small” if the number of employees is between 10 and 50, “medium” if the number of employees isbetween 50 and 250, and “large” if the number of employees exceeds 250.

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Table 1: Meta-survey 2018 answers on quantitative planning

All obs. Tiny & Small Medium Large

Firms with quantitative sales planning 0.80(0.03)

0.73(0.06)

0.80(0.05)

0.80(0.04)

Results from scenario analysisvery important 0.15

(0.03)

0.57(0.08)

0.68(0.06)

0.66(0.07)important 0.49

(0.04)

Results from statistical analysisvery important 0.13

(0.03)

0.52(0.08)

0.57(0.07)

0.47(0.07)important 0.39

(0.04)

Selection of scenariosBest and worst scenarios are plausible 0.71

(0.03)0.77(0.05)

0.67(0.06)

0.70(0.06)

Best and worst scenarios are extraordinary 0.29(0.03)

0.23(0.05)

0.33(0.06)

0.30(0.06)

Notes: the numbers are from the fall 2018 meta-survey on a sample of 191 firms. The top panel presents theshare of firms that report that their answers to our uncertainty questions are guided by numbers that the firm hasalready developed in house as part of a regular quantitative planning process. Column 1 reports the overall share,while columns 2 to 4 show the share by three size groups. In line with the definition by the German StatisticalOffice, firms are “tiny” if they have less than 10 employees, “small” if the number of employees is between 10 and50, “medium” if the number of employees is between 50 and 250, and “large” if the number of employees exceeds250. The middle panel contains the results of two follow-up questions for firms that report engaging in regularscenario planning. We present the shares of firms that consider scenario and statistical analysis, respectively, as“very important” and “important” for their quantitative sales planning. The other answer options were “lessimportant” and “not important.” Columns 2 to 4 shows the sum of the shares answering with “very important”or “important” by size group. The bottom panel displays the results from a question where we asked firms abouthow they think about the best and worst case scenarios when answering them in our main survey; the optionswere plausible scenarios or possible but extraordinary scenarios.

If yes, how important were typically results from (i) a scenario analysis around a baseline forecast (ii)statistical analysis (iii) other (please name).

For each of the options (i)-(iii), firms were asked to indicate importance on a four point scale:not important, less important, important, very important. Firms which chose option (iii) wereable to fill in an alternative approach.

One goal here was to learn about the use of statistical analysis. Moreover, we were in-terested in the use of scenario analysis, that is, thinking about the future in terms of a fewconcrete—often fairly detailed—scenarios without necessarily attaching probabilities. A well-known example of scenario analysis is bank stress testing: banks are asked to forecast lossesgiven a detailed set of contingencies, but they are not asked to assign probabilities to those con-tingencies. The literature suggests that scenario analysis is common in German businesses.11

11Mietzner (2009) provides an overview of the literature on strategic planning in German firms. In manyindustries, the majority of firms engage in some sort of scenario analysis.

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The middle panel of Table 1 summarizes how German manufacturing firms approachquantitative planning. Both scenario analysis and statistical analysis are popular: both meth-ods are rated as at least important by more than half of the firms. We again break down theanswers by size. The share of firms that rates each method at least important is increasing insize. Interestingly, large firms rely more heavily on scenario analysis by a substantive marginof 20 percentage points. For firms that routinely compute adverse and favorable scenarios aspart of their planning process, filling out the survey does not impose an additional forecastingtask and is likely to generate more thought-out answers.

The bottom panel of Table 1 speaks to what leaders in German manufacturing firms thinkabout the scenarios we ask them in our regular uncertainty module. We gave them two options(plus a verbal other option): are these plausible scenarios that may well occur, or are theyscenarios that are possible but extraordinary. In technical language, we want to know whetherthey view these scenarios as (close to) support bounds. The clear majority of firms viewsscenarios as plausible scenarios rather than support bounds. At the same time, the answersfrom the middle panel suggest that this is how most firms actually think about their uncertainfuture. We thus view our approach of asking firms about their subjective uncertainty throughscenarios as both a flexible and adequate elicitation method.

In light of this finding, we would expect that realized growth often falls outside the intervalbounded by the best and worst cases. This is indeed the case in our data. In a pooled sampleof firm-quarter observations, the share of instances where growth is outside the bounds is 48%for firms that consider scenarios “plausible,” again almost independently of firm size, and still40% for those that consider them “possible but exceptional.”12 Only for firms that have a lowsample variance of sales growth rates, that is, firms that do not experience much turbulence,the fraction of firm-quarter observations outside the bounds is lower at 33%, consistent withwhat one would expect. We conclude that firms generally like to think about scenarios thatare quite likely.

Finally, we find that conditional on using scenario analysis as “very important” or “im-portant”, a majority of firms (56%) values statistical analysis as “less important” or “not im-portant.” Conversely, conditional on valuing scenario analysis as “less important” or “notimportant,” 64% of firms are more keen on statistical analysis (“very important” or “impor-tant”). This suggests a certain imperfect substitutability between the two quantitative salesplanning techniques. As for which firms tend to view scenarios as plausible as opposed topossible but extraordinary events, there does not seem to be a difference between those firmswhich employ quantitative sales planning techniques and those that do not. Similarly, forthose firms that view statistical analysis as “very important” or “important” versus those thatfind them “less important” or “not important,” we find the same approximate independence.By contrast, firms that find scenario analysis as “very important” or “important” have a veryhigh conditional probability of viewing scenarios as plausible (80%), while those firms forwhich scenario analysis is “less important” or “not important” this probability shrinks to 58%.This means, the expert firms in scenario analysis are clearly not viewing them as supportbounds.

12The numbers mentioned in this and the following paragraph are not displayed in Table 1 but based onadditional cross tabulations available on request.

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2.2 Eliciting subjective uncertainty

The uncertainty module of the ifo Business Survey asks firms, at the beginning of a quarter,a two-part question. Figure 1 displays the sample questionnaire for April 2014 in the originalGerman. In English, the questionnaire reads:

The following questions refer to changes against the previous quarter.

1. By how much in percentage terms have your sales changed in the first quarter of 2014?

2. By how much in percentage terms will your sales change in the second quarter of 2014?

a. In the best possible case:In the worst possible case:

b. Taking into account all contingencies and risks, I expect for the second quarter of 2014 all inall a change of :

The questionnaire form contains four boxes for respondents to provide their four numericalanswers. Next to every box, there is a reminder to provide positive or negative integers. Inaddition, respondents have a “don’t know-”option (“weiß nicht” in German) behind the box,as shown in the figure. Finally, underneath both questions 1 and 2, firms are invited to providefree text comments (“Anmerkungen”).

Figure 1: Original survey questionnaire in German

Notes: Original questionnaire from ifo’s online module on subjective uncertainty in German; screenshot fromApril 2014.

To clarify the timing, consider a firm responding in April 2014, that is, in the first two anda half weeks of 2014:Q2. Question 1 asks for the change in sales between 2013:Q4 and 2014:Q1.

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This is the most recent sales growth realization that the firm has experienced. Question 2 thenasks for the firm’s outlook over the current quarter 2014:Q2, as compared to the last quarter2014:Q1. This is the next growth rate realization that the firm expects. Later, we will denotethe beginning of the current quarter, that is, the quarter at the beginning of which the surveyis in the field, by t.

Our quantitative measure of subjective uncertainty is the span between the best and worstcase scenarios for sales growth that firms provide in response to question 2.a. A firm’s forecasterror is the difference between its actual sales growth in the current quarter and its expectedgrowth rate at the beginning of that quarter, that is, its answer to part 2.b. At the beginningof every quarter, firms cannot perfectly predict the flow of sales over the entire quarter; theforecast errors thus captures the mistakes they make. We note that for us to observe a forecasterror for a firm, we need to observe the firm in two consecutive survey waves.

Sample construction

We describe the construction of our baseline sample, including all the data cleaning steps,in detail in Appendix C. Briefly, in a first step, we focus on firms that have at least fivesensible firm-quarter observations of the previous-quarter sales growth rate (question 1).13

Text comments provided by firms are useful here both to assess outliers and to drop firmsunwilling or unable to provide quarterly forecasts. The five-observations threshold allows usto compute meaningful time series means and volatilities of sales growth rates for firms in thissample. We use both as firm-level control variables in our analyses.

Our baseline sample needs to have also consistent and realistic answers to the secondquestion in the survey about the sales growth scenarios and consists, in the end, of 400 firmsand 2,762 firm-quarter observations from 14 quarters. We know each firm’s industry at thetwo-digit manufacturing level and form, because some two-digit industries have very fewobservations, for our purposes 14 new, slightly more aggregated industries so that we have asufficiently large number of (at least 60) observations per industry. Appendix D explains thisin detail and presents the distribution of firms across these industries. Henceforth, when wespeak of industries in this paper, we mean these 14 industries of the manufacturing sector.

Survey questions in the ifo Business Survey that ask about realized outcomes explicitly askfirms to ignore seasonal fluctuations. Consistent with this, we observe only negligible seasonaleffects in our data. Indeed, we can compare the sales growth rates measured in our survey—and thus deseasonalized by the individual firms—with a seasonally adjusted time series ofmanufacturing sales growth rates measured by the Federal Statistical Office, Destatis, throughan unrelated survey. The time series correlation between the Destatis series and our series is0.76. We thus treat the variables below as seasonally adjusted at the individual firm level.

2.3 Span as a measure of subjective uncertainty

The premise behind our survey module is that when firms worry more about the future, theycontemplate positive and negative scenarios that are further apart, and hence exhibit higher

13A robustness check with a three-observations threshold, available upon request, yields nearly identical re-sults.

11

Table 2: Meta-survey 2018 answers on determinants of scenarios

"Very important" or "Important"Very

important Important Tiny & Small Medium Large

Sales changes last 1 to 2 years 0.21 0.37 0.58 0.71 0.40Sales changes more than 2 years ago 0.02 0.09 0.09 0.16 0.06Considerations independent of

past sales changes 0.41 0.49 0.85 0.93 0.91Our risk attitude 0.19 0.57 0.78 0.86 0.60Sales change we observe with competitors 0.04 0.26 0.29 0.30 0.32

Notes: the numbers are from the fall 2018 meta-survey on a sample of 191 firms. Respondents were are askedto assess the importance of several aspects for determining scenarios for sales growth in the best and worstcase. Columns 1 and 2 report the overall share, while columns 3 to 5 show the share by three size groups. Inline with the definition by the German Statistical Office, firms are “tiny” if the have less than 10 employees,“small” if the number of employees is between 10 and 50, “medium” if the number of employees is between 50and 250, and “large” if the number of employees exceeds 250. We present the shares of firms that consider thedeterminants, respectively, as “very important” and “important” for the choice of scenarios. The other answeroptions were “less important” and “not important.” Columns 2 to 4 shows the sum of the shares answering with“very important” or “important” by size group.

span.14 Movements in span can in principle reflect changes in either beliefs or attitude towardsuncertainty. On the one hand, a firm might worry more about the future because it has lessinformation and hence perceives a lot of uncertainty. It might then modify its planning processto consider scenarios that are further away from the baseline. On the other hand, the firm mayworry more in the sense that it becomes more cautious in its approach to planning underuncertainty. This might lead it to alter scenarios even if beliefs are the same.

The meta-survey indeed confirms that survey answers reflect both information and attitudetowards uncertainty to a significant extent. We ask firms to rate, on a four point scale, theimportance of various determinants for their choice of scenarios. The results are summarizedin Table 2. They show that the most relevant factors mentioned by firms are risk attitude,recent experience of own sales growth and news about the future unrelated to past salesgrowth. In contrast, the typical firm does not attribute an important role to sales growth morethan two years in the past as well as the observation of competitors. Again these results varylittle across size classes.

These findings clarify that span is a measure of “worry” about future uncertain outcomesthat guides firms’ planning, as opposed to, say, only a measure of perceived risk. Of course,there are conditions under which worry and perceived risk are the same. To illustrate, considera firm with decision makers who think about risk and reward in terms of mean and variance,

14We have experimented with fitting a triangular or a beta-distribution to our best and worst case scenariosas well as expected sales growth under a number of identification assumptions and then entertain the standarddeviation and the interquartile range of these fitted distributions as alternative measures of subjective uncertainty.They are, across the board, almost perfectly correlated with span. This means that, in our data for the purposesof measuring uncertainty, adding the expected sales growth rate provides little new information beyond thatwhich is provided by the best and worst case sales growth scenarios.

12

and maximize a textbook objective that is linear in both moments, with a fixed coefficient onvariance capturing risk aversion. For such a firm, changes in worry about the future that arerelevant for actions come only from changes in conditional variance. We would thus expectspan to reflect the dispersion of the firm’s subjective conditional distribution.

More generally, our focus on worry means that our measure reflects changes in perceivedrisk only to the extent that they are actually relevant to the firm’s planning process. Forexample, an increase in risk will have a smaller effect on firm planning if the firm’s objectivedoes not strongly respond to risk. We would thus expect a smaller change in span. In fact, it isplausible to have two firms that face the same change in risk, but see span move more for oneof them because it plans more cautiously. Put differently, span is best viewed as the outcomeof a change in risk (or risk attitude): it captures how the planning process of the firm changes.

How can span be used to quantify models of the firm? In economic models, worry aboutfuture uncertain outcomes is usually captured by a certainty equivalent function. For example,in a standard model of firm dynamics, we can use the value function of the firm together withits conditional distribution of shocks to ask how much a firm would be willing to pay toremove the uncertainty. The answer would generally depend not only on the firm’s perceivedrisk, but also on the curvature of the objective function. The latter might be driven by variousfeatures of the firm’s environment such as technology, managerial risk aversion, or financialfrictions. Since planning for scenarios takes these features into account, we think of span as aproxy for worry that is measured using the certainty equivalent approach.

An additional advantage of our focus on worry as opposed to risk is that positive andnegative scenarios—and hence span—are meaningful numbers for a firm whether or not itroutinely reasons in terms of probabilities. As we have seen in Section 2.1, about half of firmsconsider statistical analysis unimportant for answering our survey. At the same time, 80%of firms routinely rely on some kind of quantitative analysis, in particular scenario analysis.Our question is designed to make sense to all firms, to understand better how actual decisionmakers think about uncertainty, and to encourage them to use data from routine quantitativeanalysis. Firms that develop probabilistic forecasts can provide quantiles from their subjectivedistribution. Firms that only assess the effect of scenarios without assigning probabilities canreport what those scenarios are.15

2.4 Properties of subjective uncertainty

In this section, we present stylized facts on span, forecast errors and their absolute value.Detailed tables of summary statistics for answers to the uncertainty module questions are pro-vided in Appendix E; here we discuss the key facts from the first two tables in this appendix.16

Sales growth is hard to predict

Realized firm sales growth has a standard deviation of 14.7 percentage points and an in-

15The same applies for firms that do not think in terms of probabilities. For example, firms might maximizean objective function that exhibits a concern for robustness or aversion against Knightian uncertainty.

16The rest of the tables in Appendix E provide a breakdown of these summary statistics by firm size, trend,and turbulence.

13

terquartile (IQ) range from −5% to 10% (see the first row of Table 16 in Appendix E). Relativeto this variation, the distribution of forecasts is compressed, with an IQ range from zero to5%. The variance of forecasts is about half that of the realizations. Forecasts display littlebias on average: the average forecast is only slightly higher than the average realization. Foran average firm, the standard deviation of forecast errors is 10.2 percentage points, similarin magnitude to the standard deviation of its sales growth of 11.4 percentage points (see thelower panel of Table 17 in Appendix E). Together, these moments indicate that predicting salesgrowth is difficult: unpredictable variation is close to total variation.

One might suspect that firms provide forecasts in a mechanical way by simply using pastgrowth or some constant baseline growth rate. In our data, both hypotheses are false. Indeed,the difference between a firm’s forecast and its last realization of growth has a standard de-viation of 17.2 percentage points, larger than that of the forecast itself at 14 percentage points(see the “random walk model” in Table 16 in Appendix E). At the same time, the differencebetween a firm’s forecast and its firm level mean growth rate has a standard deviation of10.8 percentage points (see the “iid model” in Table 16 in Appendix E). The results show thatthese simple models generate growth predictions that deviate substantially from firms’ actualforecasts. We conclude that firms’ forecasts are nontrivial functions of past growth.

Best and worst case scenarios and the magnitude of subjective uncertainty

Firms’ best and worst case scenarios bracket forecasts close to but not quite symmetrically.The average worst and best case scenarios are −4.8% and 7.4%, respectively (see rows threeand four of Table 16 in Appendix E). The midpoint between these bounds is 1.3% and henceless than one percentage point below the average forecast of 2.2%. The scenarios have slightlyhigher standard deviations and wider IQ ranges than forecasts. A key difference between thevariables is that the distribution of the lower (upper) bound is negatively (positively) skewed.

Our measure of subjective uncertainty is similar in magnitude to firm-level unconditionalvolatility. Indeed, the mean span for the average firm is 12.3 percentage points, while itstime series standard deviation of growth rates is 11.4 percentage points. Since growth is hardto predict, the span reported by the average firm is also similar in magnitude to the typicalabsolute forecast error experienced by a firm, 9.4 percentage points (see, respectively, row fiveupper panel, row one lower panel, and the last row upper panel, of Table 17 in Appendix E).

Subjective uncertainty varies in the cross section

To assess the variation of subjective uncertainty in the cross section, we compute the aver-age span for each firm. The cross sectional standard deviation of average span is 7.4 percentagepoints. It is similar in magnitude to the cross sectional standard deviation of the average abso-lute forecast error of 9.6 percentage points (see, respectively, rows five and nine of the upperpanel of Table 17 in Appendix E). Firms thus differ substantially in both the size of the typicalshock they experience and in the way their planning deals with perceived uncertainty. Bothvariables are positively, if imperfectly correlated in the cross section of firms: 0.43. Firms thatmake larger forecast errors on average thus tend to perceive more uncertainty on average.

14

Subjective uncertainty varies in the time series at the firm level

Our data also show substantial time variation in subjective uncertainty at the firm level. Thetime series standard deviation of span for the average firm is 5.9 percentage points (see rowfive of the lower panel of Table 17 in Appendix E) and hence more than half of the standarddeviation of span in the pooled sample. Time series variation in subjective uncertainty is alsosubstantial compared to other changes in firms’ beliefs. For example, the cross sectional meanof firms’ time series standard deviation of forecasts is 7.4 percentage points (and thus only1.5 percentage points higher than the aforementioned number for span), and numbers forbest and worst scenarios are only slightly higher (in both cases approximately 8.1 percentagepoints).

What do changes in uncertainty look like? On average, they consist of moves in both thebest and worst case scenarios. In particular, for all instances where a firm increases its spanfrom one quarter to the next, the mean change in the worst case scenario is −4.7 percentagepoints whereas the mean change in the best case scenario is +2.6 percentage points, see Figure2. In other words, the average increase in uncertainty thus consists of an outward expansionof span that is slightly asymmetric. The average decrease in span is a symmetric downwardcompression: conditional on a decrease in span, the worst case increases by 3.9 percentagepoints and the best case decreases by 3.4 percentage points.17

17These numbers are similar for the cross section of firms, as opposed to the cross section of firm-quarterobservations.

15

Figure 2: Changes in subjective uncertainty

-15

-10

-5

0

5

10

15

increase of span

change of best case

decrease of span

change of worst case

𝑏∗

𝑏

𝑤∗

𝑤

span

𝑏

𝑏∗

𝑤

𝑤∗

change of best case

change of worst case

be

st / w

ors

t ca

se

fo

reca

st sa

les g

row

thin

%+2.56

─4.70

─3.41

+3.89

Notes: The figure illustrates how, on average in the cross section of firm-quarter observations, changes in thebest and worst case scenarios generate increases (left part of the figure) and decreases (right part of the figure) ofsubjective uncertainty (span). The plot shows the pooled averages of span as well as the pooled averages of thebest and worst case sales growth rates, which are denoted by b and w, respectively. b∗ and w∗ are the best andworst case scenarios after the average changes of span. The figure is based on the 2,762 firm-quarter observationsof baseline sample span.

Finally, the variation of subjective uncertainty in both the time series and the cross sectionis overwhelmingly firm-specific. Indeed, the R-squared of a regression of span on time fixedeffects is 0.006, on time and industry fixed effects 0.030, and on time-industry fixed effects itis merely 0.084.18 This fact does not imply that we cannot uncover patterns in the variationof span, as we will see below. It simply means that the cross sectional patterns are not drivenby the industry, but rather by differences in firm perceptions within industries. Similarly, thetime series patterns are largely driven by individual firm experiences as opposed to, say, thestate of the business cycle.

3 Uncertainty and changeHow does firms’ subjective uncertainty relate to their experience? In this section, we presenta first key stylized fact on subjective uncertainty and change: subjective uncertainty has anasymmetric V-shaped nexus with previous-quarter sales growth. We then lay out a simpleorganizing framework that guides our subsequent analysis.

18The variation in firms’ forecast errors and absolute forecast errors is also overwhelmingly firm-specific. Forforecast errors, a regression on time fixed effects yields an R-squared of 0.010, a regression on time and industryeffects an R-squared of 0.023, and a regression on time-industry fixed effects an R-squared of 0.12. For absoluteforecast errors, a regression on time fixed effects yields an R-squared of 0.014, a regression on time and industryeffects an R-squared of 0.033, and a regression on time-industry fixed effects an R-squared of 0.12.

16

3.1 Uncertainty and past growth: an asymmetric V

Figure 3 displays a scatter plot of responses, with span at t, that is, the beginning of the quarterin which the survey is in the field along the vertical axis, and quarter-to-quarter sales growthrealized in the quarter prior to t along the horizontal axis. The vertical gray lines indicate theinterdecile range which reaches from −15% to +15% as reported in Table 16 in Appendix E.

Figure 3: Uncertainty and past sales growth

050

100

150

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 -15 0 15 50 100Sales growth rate in the previous quarter

Data Nonparametric reg.Predicted by past sales growth

Notes: Every dot represents a firm-quarter observation. The solid line is the prediction from a kernel-weightedlocal polynomial regression of degree zero with an Epanechnikov kernel where the bandwidth was selectedbased on the rule of thumb suggested by Fan and Gijbels (1996). The dashed line depicts the predicted valuesfrom a piecewise linear regression of subjective uncertainty on past sales growth, with a break at zero. The thinvertical lines mark the interdecile range that extends from −15% to 15%, see Table 16 in Appendix E.

Firms that have experienced larger changes are more uncertain. In particular, the rela-tionship between subjective uncertainty and past sales growth looks like the letter V with aminimum near zero. This is illustrated in the figure by two lines. The solid line is a nonpara-metric regression line. The dashed line is from a simple piecewise linear regression with abreakpoint at zero.19 The two lines are very similar, and they virtually coincide in the relevantrange where most observations are located.

19We have compared the in-sample fit of a piecewise linear regression model with breakpoint at zero with thatof a simple quadratic model. Both the Akaike and Bayesian information criteria favor the piecewise linear model.

17

Firms perceive higher uncertainty after negative change than after positive change. Indeed,the slope of the left hand branch of the letter V is about twice as large in absolute value as theslope of the right hand branch. After a one percentage point lower negative sales growth, nextquarter’s span is 50 basis points wider. In contrast, after one percentage point higher positivesales growth, span is wider by slightly more than 25 basis points. The regression coefficientsare reported in column (2) of Table 5, discussed further below.

The V-shaped regression line relates uncertainty to change; it stands in contrast to thesimple linear (often negative) relationship between uncertainty and growth emphasized inthe literature. At the same time, asymmetry implies that uncertainty and growth are in factnegatively correlated. Indeed, a linear regression returns a small but significantly negativecoefficient of -.06, shown below in column (1) of Table 5. However, ignoring the V-shapedrastically lowers the explanatory power of past sales growth rates for uncertainty from anR-squared of 0.19 for a piecewise linear regression to an R-squared of just below 0.01 for thesimple linear regression. In other words, a simple linear framework between uncertainty andsales growth appears to be misspecified.

3.2 Uncertainty and change: an organizing framework

Our organizing framework relates a firm’s subjective uncertainty to the distribution of growthmeasured by an econometrician. We use it in later sections to guide our detailed discussion ofuncertainty and change in both the cross section and the time series. For simplicity, we assumethat firms have probabilistic beliefs. As will become clear, this feature is not essential for thepoints we make here, but it allows us to express those points in simple familiar notation.

Let git+1 denote firm i’s sales growth in the quarter between t and t + 1 relative to the

previous quarter, that is, the growth rate that firm i forms beliefs about when it answersour survey questions in t. Firm i’s information set at that point in time includes gi

t, the lastobserved growth rate in the quarter between t − 1 and t. It may also include other signalsthat represent news arrived before or at t, which we collect in a vector zi

t. We then use thevector si

t to represent all information from past growth rates or other signals that is relevantfor forecasting the future dynamics of growth.

We represent firm i’s belief about its sales growth by the state space system

git+1 = f

(si

t, xi)+ σ

(si

t, xi)

εit+1 (1)

sit = S

(si

t−1, git, zi

t; xi)

(2)

where xi is a vector of firm characteristics, which we think of as fixed in the medium run, andεi

t+1 is an error that has mean zero and variance one under the firms’ subjective belief. Theobservation equation allows firm i’s forecast f (si

t, xi) to depend on the state as well as its fixedcharacteristics. The state is updated every period to incorporate new information in gi

t and zit

according to the function S.

When firm i answers our survey questions at t, it provides its forecast f (sit, xi) as well as

best and worst case scenarios. We also observe the subsequent realization git+1 and hence the

18

firm’s subjective forecast error. We further identify span, the difference between firm i’s bestand worst case scenarios, with firm i’s subjective conditional volatility σ(si

t, xi). This connec-tion is exact if firm i reports quantiles as scenarios and appropriate distributional assumptionsare in place.20 More generally, we expect firm i’s answer to the survey question to reflect somemeasure of dispersion in its forecast error.

Examples

The state space system (1)-(2) nests many models used to describe firms’ subjective uncer-tainty in economic models. As a simple example, consider the case of iid growth together withan orthogonal uncertainty shock:

git+1 = f + σi

tεit+1 (3)

σit = S

(σi

t−1, zit

)(4)

Here the only relevant state is stochastic volatility σit . Rational expectations models with

uncertainty shocks often assume that σit is correlated across firms and high in recessions,

which helps generate the observed heightened dispersion of firm growth rates in bad times.

The system (1)-(2) also nests many popular learning rules. Examples include Bayesian mod-els where firms track some latent state such as a regime, or constant gains learning where firmsrecursively estimate parameters of the one-step-ahead predictive distribution while down-weighting past observations. The common denominator of all these setups is that the statevector contains statistics of the empirical distribution that are relevant for predicting the fu-ture dynamics of growth. A natural property in many settings is that high growth gi

t increasesthe forecast f and that a large absolute value of the forecast error increases subjective uncer-tainty σ.

Comparing beliefs and the true data generating process

We would like to distinguish firms’ subjective uncertainty from actual volatility, as reflectedin the size of innovations measured by an econometrician. We thus consider a change ofmeasure from the firm’s belief to the “econometrician’s belief,” that is, the probability measurethat characterizes the true data generating process. We assume that under the econometrician’sbelief the distribution of growth rates has the alternative representation

git+1 = f

(si

t, xi)+ b

(si

t, xi)+ σ

(si

t, xi)

εit+1 (5)

sit = S

(si

t−1, git, zi

t

)(6)

where again the error has mean zero and variance one, now under the econometrician’s belief.

20Specifically, in the broad class of distributions which belong to the location-scale family—the normal, Laplaceand t-distributions as well as their generalizations such as the exponential power distribution and the asymmetricpower distribution (see Komunjer, 2007)—quantile differences are simply a multiple of the distribution’s standarddeviation.

19

The new observation equation allows for two key differences between firms’ belief andthe true data generating process. First, firms might have biased forecasts, represented bythe function b. Second, the size of the typical innovation σ might be different from firms’subjective uncertainty captured by σ. Both differences may vary either in the cross sectionwith firms’ fixed characteristics xi or over time with the information set captured by si

t. In thespecial case of rational expectations, there is no bias (b = 0) and subjective uncertainty mirrorsactual volatility, that is, σ = σ.

4 Uncertainty and change in the cross sectionIn this section, we ask what type of firms perceive more subjective uncertainty on average. Inother words, we now relate average firm-level subjective uncertainty to measures of changeover the medium term in a firm’s environment. We compute, for each firm, its average span,that is, the time series mean of all observations of span for the firm. We then regress averagespan on a number of firm-level characteristics. In terms of the framework of Section 3.2,we thus characterize the dependence of subjective uncertainty σ on fixed characteristics xi,assuming that time averaging removes the effects of information si

t. We also compare the crosssectional properties of average span with those of firms’ average absolute forecast errors. Oursecond key result is: there is substantial but different heterogeneity in subjective uncertaintyand absolute forecast errors across firm types and firm environments.

4.1 Change in firms’ environment

We define two variables that measure the medium-term dynamics in a firm’s environment,based on its realized sales growth rates (that is, answers to question 1 of our survey module).First, we refer to a firm’s sample average sales growth as its trend. Second, the turbulence expe-rienced by a firm is measured by the sample standard deviation of its sales growth rates. Weemphasize that turbulence differs from span for two reasons: First, it is purely based on real-ized growth rates. Second, it is an unconditional volatility measure over three years, whereasspan measures conditional uncertainty one quarter ahead. The cross-sectional properties ofboth trend and turbulence can be seen in the first rows, respectively, of the upper and lowerpanel in Table 17 in Appendix E.

To tractably account for potentially nonlinear effects of these firm characteristics on averagespan, we code the firm characteristics as dummies. In particular, we use turbulence dummiesthat indicate quartiles of the distribution of firm-level standard deviations of realized salesgrowth rates with the lowest quartile as the baseline. We proceed similarly for trend. However,since the middle two quartiles for trend turn out to be very similar, we introduce dummiesonly for a low trend (bottom 25%) as well as a high trend (top 25%), treating the middle groupas the baseline.

Finally, we divide firms into four size categories, with size measured as average employ-ment over our sample. Here we follow the German Statistical Office in their definition of tiny,small, medium-sized, and large firms; lower bounds for the latter three groups are at 10, 50,and 250 employees, respectively. We work with three dummies, with tiny firms as the baseline.

20

Figure 4 provides a scatter plot of trend and turbulence, defined above as the firm-levelmean and standard deviation of realized sales growth rates, respectively. Every dot representsa firm, and the color of the dot indicates firm size, as measured by the number of employees.Size increases from light blue to pink according to the color bar provided on the right handside of the figure.

The main takeaway from Figure 4 is that while trend and turbulence vary substantially,they are not particularly correlated. Firms that grow or shrink along strong trends need nottypically experience large shocks and vice versa: The correlation between a firm’s averagesales growth rate and its standard deviation of those sales growth rates is at a statisticallyinsignificant −0.046. Moreover, the correlation of either environment variable with size isalso rather weak. While the very largest firms (identified by bright pink dots) do tend tocluster where turbulence is low (correlation is −0.107), we observe firms of all sizes spreadout over the plane. The correlation between size and the average sales growth rates is indeeda statistical zero.

Figure 4: Scatter plot of trend and turbulence with firm size

-15 -10 -5 0 5 10 15

trend = mean (sales growth)

0

5

10

15

20

25

30

turb

ulen

ce =

std

dev

(sa

les

grow

th)

10

100

1000

10000

Notes: Every dot represents a firm identified by its trend (time series average realized sales growth rate by firm)and turbulence (time series standard deviation of realized sales growth rates by firm). Color indicates number ofemployees according to the color bar on the right hand side.

21

4.2 Subjective uncertainty, size, trend and turbulence

Table 3 presents regression results on the cross sectional relations between uncertainty andfirm characteristics. The first three columns ask how much variation in span can be explainedby each fixed characteristic—size, trend and turbulence—separately. All three characteristicsshow a statistically and economically strong association with span. Column (1) says thatlarger firms perceive less uncertainty. Average span in the entire population of firms is about12 percentage points, and it falls monotonically from 18 percentage points for very small firms(the omitted category) to 10 percentage points for large firms.

Columns (2) and (3) show that cross sectional variation in trend and turbulence—eachby itself—is enough to induce a V-shaped relationship between growth and uncertainty, asobserved in the pooled scatter plot. On the one hand, trend and span are directly related byan asymmetric V: quickly shrinking or growing firms report higher average spans than firmswith normal growth, by 6 and 2 percentage points, respectively. On the other hand, moreturbulent firms also report monotonically higher spans. Since more turbulent firms’ growthrate realizations fall more into the tails of realized sales growth rates, this effect also generatesa V-pattern.

Each of the three firm characteristics has independent effects on the average subjectiveuncertainty of firms. This is established in column (4) where we consider all three in the sameregression. The positive turbulence gradient is qualitatively and quantitatively unchangedcompared to the results in column (3). For trend, the interaction with other characteristicsis more subtle. In particular, once size and turbulence are controlled for, growing firms nolonger perceive higher uncertainty. At the same time, the negative branch of the V remainslarge and statistically significant.

While trend and turbulence are correlated with size, controlling for them does not removean independent role for size in explaining subjective uncertainty. Indeed, comparing columns(1) and (4), the negative size gradient is quantitatively reduced, but remains in place qualita-tively. Column (5) further shows that our three firm characteristics are not simply reflectiveof industry characteristics: including industry dummies neither changes significantly the R-squared compared to column (4) nor the coefficient estimates.

22

Tabl

e3:

Reg

ress

ions

ofti

me

seri

esav

erag

esof

subj

ecti

veun

cert

aint

y,of

abso

lute

fore

cast

erro

rs,a

ndof

fore

cast

erro

rsby

firm

onfir

mch

arac

teri

stic

s(1

)(2

)(3

)(4

)(5

)(6

)(7

)D

epen

dent

vari

able

:av

g.sp

anav

g.sp

anav

g.sp

anav

g.sp

anav

g.sp

anav

g.sp

anav

g.sp

an

Dum

my

smal

lfirm

s-3

.267∗

-1.6

10-1

.936

1.21

74.

147

(1.9

23)

(1.6

55)

(1.7

29)

(2.2

55)

(2.8

00)

Dum

my

med

ium

-siz

edfir

ms

-6.4

02∗∗∗

-3.7

11∗∗

-4.3

08∗∗∗

-0.1

651.

557

(1.7

94)

(1.5

50)

(1.6

23)

(2.0

67)

(2.5

08)

Dum

my

larg

efir

ms

-8.8

34∗∗∗

-5.0

51∗∗∗

-5.7

05∗∗∗

-0.5

683.

036

(1.8

27)

(1.6

03)

(1.7

16)

(2.0

56)

(2.4

86)

Dum

my

’bad

’sal

esgr

owth

tren

d5.

940∗∗∗

3.23

3∗∗∗

3.20

9∗∗∗

2.66

3∗∗∗

-5.3

22∗∗∗

(0.9

36)

(0.8

21)

(0.8

50)

(0.9

34)

(1.2

21)

Dum

my

’goo

d’sa

les

grow

thtr

end

2.17

7∗∗∗

0.44

40.

190

2.48

9∗∗

5.35

8∗∗∗

(0.8

00)

(0.7

30)

(0.7

39)

(1.1

26)

(1.3

64)

Dum

my

med

ium

low

turb

ulen

ce2.

287∗∗∗

1.73

1∗∗∗

1.75

1∗∗∗

2.81

6∗∗∗

-0.0

552

(0.6

23)

(0.6

13)

(0.6

64)

(0.5

15)

(0.7

33)

Dum

my

med

ium

high

turb

ulen

ce6.

028∗∗∗

5.05

2∗∗∗

4.98

5∗∗∗

5.25

9∗∗∗

0.05

25(0

.725

)(0

.701

)(0

.723

)(0

.561

)(0

.891

)D

umm

yhi

ghtu

rbul

ence

11.2

8∗∗∗

9.62

5∗∗∗

9.21

6∗∗∗

13.3

3∗∗∗

0.06

24(0

.892

)(0

.865

)(0

.898

)(1

.393

)(1

.640

)C

onst

ant

18.1

6∗∗∗

10.3

2∗∗∗

7.45

6∗∗∗

10.6

1∗∗∗

10.9

6∗∗∗

2.73

4-2

.741

(1.7

31)

(0.4

56)

(0.3

66)

(1.5

67)

(1.7

90)

(2.0

82)

(2.4

98)

Indu

stry

dum

mie

sY

ES

No.

ofob

serv

atio

ns40

040

040

040

040

038

938

9N

o.of

firm

s40

040

040

040

040

038

938

9N

o.of

para

met

ers

(exc

l.in

terc

ept)

32

38

218

8R

-squ

ared

0.10

0.11

0.34

0.41

0.43

0.35

0.14

Not

es:a

vg.s

pan

deno

tes

the

tim

e-se

ries

aver

age

offir

m-l

evel

span

,avg

.abs

.FE

deno

tes

the

tim

e-se

ries

aver

age

ofth

efir

m-l

evel

abso

lute

fore

cast

erro

r,an

dav

er.F

Ede

note

sth

eti

me-

seri

esav

erag

eof

the

firm

-lev

elfo

reca

ster

ror.

Res

ults

from

OLS

regr

essi

ons.

Indu

stry

dum

mie

sar

ede

fined

inA

ppen

dix

D.R

obus

tst

anda

rder

rors

inpa

rent

hese

s;*

p<

0.10

,**

p<

0.05

,***

p<

0.01

.

23

4.3 Subjective uncertainty, forecast errors and bias in the cross section

How does firms’ perceived uncertainty relate to the size of the fluctuations they experience?Column (6) of Table 3 reports a regression of firms’ average absolute value of its subjectiveforecast error, a measure of the size of innovations experienced by the firm. Along all threecross sectional dimensions we consider, subjective uncertainty is significantly different fromthe size of shocks experienced by the typical firm. First there is no independent effect ofsize once we control for trend and turbulence. It is true that unconditionally larger firms ex-perience smaller shocks (see the first column of Table 40 in Appendix F, where we documentfurther cross sectional results for (absolute) forecast errors and unconditional volatility). How-ever, this relationship is entirely explained by their trend and turbulence. We conclude thatthe additional effect of size on span is a subjective phenomenon: large firms’ perceive loweruncertainty even if they face the same size of shocks as smaller firms.21

A second special feature of subjective uncertainty is its asymmetric dependence on trend.For the same size of shocks, shrinking firms perceive higher subjective uncertainty; growingfirms do not. By contrast, while it is true that growing and shrinking firms experience largershocks, they do so symmetrically. Summarizing both results with respect to size and trend,this means that successful firms—either growing or large—report lower uncertainty whenfaced with the same-sized absolute shocks. This fact is consistent with mechanisms that makeuncertainty matter more to decision makers in bad times, so their planning considers a widerspan of scenarios. In other words, successful firms exhibit less worry.

A final property concerns the turbulence gradient. While more turbulent firms—whichexperience larger shocks by construction—also perceive higher uncertainty, their average fore-cast error (and their unconditional sales growth volatility) is even higher compared to lowturbulence firms.

As discussed in Section 3.2, forecast errors experienced by firms may in part reflect system-atic bias in firms’ forecasts. Column (7) of Table 3 shows a regression of the mean forecast erroron characteristics. For the size and turbulence categories, the coefficients on the dummies arenot statistically significant (this is true for the joint regression as presented here, or separatelyfor each of the three firm characteristics, as columns 4 to 6 of Table 40 in Appendix F show).Consistent with this result, group means all lie around zero when firms are sorted into size orturbulence categories (see Appendix E, Tables 18 to 25 and Tables 32 to 39).

At the same time, there is evidence that firms on trends make biased forecasts. In particular,growing firms make large positive forecast errors, defined above as realized growth minusforecast. In other words, growing firms are regularly positively surprised; their forecasts arebiased towards zero. Analogously, shrinking firms make large negative forecast errors: againthe forecast is biased towards zero—firms do not sufficiently anticipate the trend they are on.

21Columns 7 to 10 of Table 40 in Appendix F shows that a similar finding holds for the unconditional samplevolatility of realized growth rates: on their own, they are negatively correlated with firm size, but the relationvanishes after controlling for the trend and the turbulence dummies. Importantly, this means that both uncondi-tional and conditional realized volatilities have a different size gradient than subjective (conditional) uncertainty.

24

4.4 Heteroscedasticity and firm characteristics

In the previous section, we have shown that there is substantial heterogeneity in firms’ sub-jective uncertainty along observable firm characteristics: size, trend growth, and turbulence,the latter two being measures of the intensity of change in the firm’s environment. We havealso shown that heterogeneity in firms’ subjective uncertainty is substantially different fromthat in conditional and unconditional volatility. In this section, we argue that heterogeneityof subjective uncertainty in the time series dimension, that is, subjective heteroscedasticity, isrelated to firms’ characteristics in very similar ways as subjective uncertainty.

Specifically, we compute, for each firm, its standard deviation of span over all time seriesobservations. This yields a cross section of subjective heteroscedasticity measures. We thenregress these subjective heteroscedasticities on the same firm-level characteristics we have usedfor average span.

For example, column (1) of Table 4 shows that smaller firms have more fluctuations intheir subjective uncertainty. Column (2) finds that firms on a substantial positive or negativetrend have more fluctuations in their subjective uncertainty; and column (3) shows a posi-tive correlation between volatilities of sales growth and of subjective uncertainty: firms in themost volatile environments are also the most volatile in terms of their perceived uncertainty.Column (4), finally, shows the individual effects displayed in columns (1) to (3) largely sur-vive their joint inclusion in the regression, and that the cross sectional structure of subjectiveheteroscedasticity is similar to that of subjective uncertainty itself. These results mean thatthe heteroscedastic structure of subjective uncertainty can be partly explained by firm-levelcharacteristics but is also complex: it requires a substantial amount of heterogeneity to bemodelled.

25

Table 4: Regressions of time series standard deviations of subjective uncertainty by firm onfirm characteristics

(1) (2) (3) (4)Dependent variable: std. span std. span std. span std. span

Dummy small firms -3.053∗∗ -2.084∗

(1.485) (1.254)Dummy medium-sized firms -4.252∗∗∗ -2.655∗∗

(1.470) (1.292)Dummy large firms -5.094∗∗∗ -2.798∗∗

(1.481) (1.294)Dummy ’bad’ sales growth trend 3.603∗∗∗ 2.164∗∗∗

(0.802) (0.684)Dummy ’good’ sales growth trend 1.672∗∗∗ 0.771∗

(0.475) (0.450)Dummy medium low turbulence 1.254∗∗∗ 1.000∗∗

(0.384) (0.387)Dummy medium high turbulence 2.907∗∗∗ 2.369∗∗∗

(0.321) (0.353)Dummy high turbulence 6.364∗∗∗ 5.463∗∗∗

(0.797) (0.717)Constant 9.767∗∗∗ 4.543∗∗∗ 3.237∗∗∗ 5.337∗∗∗

(1.415) (0.208) (0.184) (1.251)

No. of observations 397 397 397 397No. of firms 397 397 397 397No. of parameters (excl. intercept) 3 2 3 8R-squared 0.052 0.086 0.22 0.27

Notes: std. span denotes the time-series standard deviation of firm-level span. Results from OLS regressions.Standard errors in parentheses, clustered by firm; * p < 0.10, ** p < 0.05, *** p < 0.01.

26

5 Uncertainty and change over timeWe have seen in the previous section that the V-shaped relationship between growth anduncertainty in Figure 3 in part reflects fixed differences between firms. We now turn to timeseries variation: we ask how much of a V remains once we control for fixed characteristics.In terms of the organizing framework of Section 3.2, we ask whether variation of span σ withfirms’ information si

t also contributes to the V-shape, via the correlation of sit with past growth.

The third key finding is that the V-shaped nexus between subjective uncertainty and past salesgrowth is the result of both time series and cross sectional forces.

Formally, all our regressions specifications take the basic form:

spanit = β−gi,−

t + β+gi,+t + γ′xi + εi

t, (7)

where gi,−t = gi

t I(git<0), gi,+

t = git I(gi

t ≥ 0), I(·) is the indicator function, and xi is a vector offixed firm characteristics that do not depend on time.

We include the three characteristics studied in the previous section: trend, turbulenceand size. Trend and turbulence are again coded as time-invariant dummies. As the unitof observation is now a firm-quarter pair, we measure the size of the firm as the number ofemployees at the end of the previous calendar year. We then form three size dummies: smallfirms have 10-50 employees, medium-sized firms 51-250 employees and large firms more than250 employees. The baseline “tiny” firm has fewer than 10 employees.22

5.1 Time variation in subjective uncertainty and growth

Table 5 reports the regression results. As a benchmark, we start in columns (1) and (2) witha simple linear regression and a piecewise linear regression with a break at zero, respectively.The two columns provide formal counterparts to the scatter plot Figure 3. The next fourcolumns augment the piecewise linear specification with dummies for fixed characteristics,first adding size, trend and turbulence separately, and then in column (6) adding all charac-teristics together.

The main result from Table 5 is that a strongly significant asymmetric V remains even ifwe control for fixed characteristics. Indeed, the coefficients on both negative and positivepast sales growth are statistically significant and quantitatively relevant in all specifications.Column (6) says that, holding fixed all characteristics, after a one percentage point lowernegative sales growth rate, next quarter’s span is 31 basis points wider. Similarly, a onepercentage point higher positive sales growth rate is followed by a 18 basis points wider span.This translates into a 4.6 (2.6) percentage points increase in span for a one-standard-deviationdecrease (increase) in previous-quarter sales growth rates. Responses to past growth thusaccount for a considerable part of time variation in subjective uncertainty.

22While size therefore does vary over time, change is so slow that the size dummies are essentially time-invariant. We observe only 56 jumps from one size category to another in our sample.

27

Tabl

e5:

Reg

ress

ions

ofsu

bjec

tive

unce

rtai

nty

onpa

stsa

les

grow

than

dfir

mch

arac

teri

stic

sD

epen

dent

vari

able

:spa

nbe

twee

nbe

stan

d(1

)(2

)(3

)(4

)(5

)(6

)(7

)(8

)w

orst

case

sale

sgr

owth

rate

for

quar

ter

tPO

LSPO

LSPO

LSPO

LSPO

LSPO

LSFE

POLS

Sale

sgr

owth

rate

inqu

arte

rt−

1-0

.059

8∗∗

(0.0

259)

Neg

ativ

esa

les

grow

thra

tein

quar

ter

t−1

-0.4

98∗∗∗

-0.4

70∗∗∗

-0.4

36∗∗∗

-0.3

51∗∗∗

-0.3

06∗∗∗

-0.2

72∗∗∗

-0.3

04∗∗∗

(0.0

612)

(0.0

602)

(0.0

626)

(0.0

661)

(0.0

675)

(0.0

786)

(0.0

686)

Posi

tive

sale

sgr

owth

rate

inqu

arte

rt−

10.

279∗∗∗

0.26

6∗∗∗

0.28

0∗∗∗

0.16

6∗∗∗

0.18

0∗∗∗

0.15

9∗∗∗

0.17

3∗∗∗

(0.0

329)

(0.0

317)

(0.0

327)

(0.0

335)

(0.0

314)

(0.0

319)

(0.0

303)

Dum

my

smal

lfirm

s-4

.480∗

-3.9

59∗

-3.5

08∗

(2.5

60)

(2.1

78)

(1.8

97)

Dum

my

med

ium

-siz

edfir

ms

-6.6

77∗∗∗

-5.4

52∗∗

-5.1

57∗∗∗

(2.5

16)

(2.1

41)

(1.9

72)

Dum

my

larg

efir

ms

-7.8

58∗∗∗

-6.2

95∗∗∗

-5.9

70∗∗∗

(2.5

70)

(2.1

70)

(2.0

14)

Dum

my

’bad

’sal

esgr

owth

tren

d3.

711∗∗∗

2.24

8∗∗∗

2.30

0∗∗∗

(0.9

51)

(0.8

56)

(0.8

58)

Dum

my

’goo

d’sa

les

grow

thtr

end

0.41

0-0

.434

-0.6

18(0

.658

)(0

.645

)(0

.667

)D

umm

ym

ediu

mlo

wtu

rbul

ence

1.69

9∗∗∗

1.38

8∗∗

1.34

0∗∗

(0.5

78)

(0.5

91)

(0.6

49)

Dum

my

med

ium

high

turb

ulen

ce5.

157∗∗∗

4.56

0∗∗∗

4.57

2∗∗∗

(0.7

52)

(0.7

64)

(0.7

72)

Dum

my

high

turb

ulen

ce7.

702∗∗∗

6.74

8∗∗∗

6.52

5∗∗∗

(1.0

35)

(0.9

69)

(0.9

79)

Inte

rcep

t12

.22∗∗∗

8.42

8∗∗∗

14.7

6∗∗∗

7.69

5∗∗∗

6.20

6∗∗∗

11.3

7∗∗∗

10.0

6∗∗∗

9.77

4∗∗∗

(0.3

92)

(0.4

35)

(2.5

32)

(0.4

35)

(0.4

25)

(2.1

54)

(0.4

80)

(3.0

14)

Tim

e-in

dust

rydu

mm

ies

YES

No.

ofob

serv

atio

ns2,

762

2,76

22,

762

2,76

22,

762

2,76

22,

762

2,76

2N

o.of

firm

s40

040

040

040

040

040

040

040

0N

o.of

para

met

ers

(exc

l.in

terc

ept)

12

54

510

401

199

R-s

quar

ed0.

0079

0.19

0.22

0.21

0.26

0.29

0.57

0.34

Not

es:

the

regr

essi

ons

are

base

don

the

sam

ple

offir

ms

wit

hat

leas

tfiv

ean

swer

sto

ques

tion

1.In

addi

tion

,w

ere

quir

ean

answ

erto

both

ques

tion

s2.

afo

rth

efir

m’s

span

,lea

ding

usto

the

base

line

sam

ple

span

wit

h2,

762

obse

rvat

ions

(see

Tabl

e14

inA

ppen

dix

C).

Stan

dard

erro

rsin

pare

nthe

ses,

clus

tere

dby

firm

.*p

<0.

10,*

*p

<0.

05,*

**p

<0.

01.“

POLS

”st

ands

for

“poo

led

ordi

nary

leas

tsqu

ares

regr

essi

on”;

“FE”

stan

dsfo

ra

fixed

-eff

ect

regr

essi

on.S

eeA

ppen

dix

Dfo

rth

ede

finit

ion

ofin

dust

ries

.

28

The impact of firm characteristics is also significant. First, introducing firm characteristicsdummies improves the fit of the regression: for example, the R-squared improves from 0.19 incolumn (2) to 0.29 in column (6). Coefficients on the dummies reproduce the cross sectionaleffects discussed in the previous section. For example, firms with more than 250 employees aremore than 6 percentage points less uncertain on average than tiny firms. Firms that experiencemore than median turbulence are at least 4.5 percentage points more uncertain than thosewith low turbulence. The impact of trend is asymmetric: firms on a bad trend are more than2 percentage points more uncertain than those on a normal trend, whereas a good trend hasno significant effect on span.

It is natural to conjecture that fixed characteristics other than size, trend and turbulencematter for subjective uncertainty. We thus re-estimate the regression in column (7) with firmfixed effects. As expected, we find a large increase in R2. Remarkably, however, there isvirtually no change in the coefficients on past growth. We can thus conclude that size, trendand turbulence dummies exhaustively control for the impact of firm characteristics on theuncertainty-growth relationship.

In column (8), we include time-industry dummies. This neither alters our coefficient esti-mates nor markedly improves the fit of the regression. This finding is consistent with the factthat variation in subjective uncertainty is largely firm-specific. We conclude that our resultsare neither driven by industry-composition effects, industry-specific or aggregate trends andcycles.

We finally note that a comparison between column (1) and column (2) shows that the dataclearly prefer a piecewise linear specification, a V-shape, to model the uncertainty-growthnexus. A linear specification, the traditional focus of the literature, as in column (1) finds theusual negative correlation between growth and uncertainty but the fit of this regression is tinycompared to the V-shaped regression in column (2).

5.2 Comparing cross section and time series variation in subjective uncer-tainty

The coefficients on positive and negative growth in column (6) in Table 5 effectively isolatelarge and asymmetric time series responses of firms to past growth. Perhaps interestingly,the asymmetric V-shaped response induced by time series responses is rather similar to thatinduced purely by cross sectional heterogeneity. To see this, consider Figure 5, where we showthe nonparametric regression line from Figure 3 along with several regression lines motivatedby the findings in Table 5, in order to compare the two forces.

As a benchmark, the solid blue line is the nonparametric regression line fitted to the data,and the dotted red line is the fitted line from the regression model in column (2) of Table 5,both already shown in Figure 3. The dashed yellow and the dash-dotted green nonparametricregression lines are fitted not to the data, but to clouds of predicted values from two paramet-

29

Figure 5: Cross-sectional and time series relationships between uncertainty and past salesgrowth

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 -15 0 15 50 100Sales growth rate in the previous quarter

Data Predicted, model (6)Predicted, model (2) Predicted, model with dummies

Notes: Besides a linear fitted line with break at zero that corresponds to column (2) of Table 5 (dotted red line), thechart presents three nonparametric regression lines. Respectively, the nonparametric regression lines are basedon the full sample (solid blue line), the cloud of predicted values of column (6) of Table 5 (dashed yellow line),and the cloud of predicted values from a model with size, trend, and turbulence dummies as regressors (dash-dotted green line). The nonparametric regressions are the predictions from kernel-weighted local polynomialregressions of degree zero with an Epanechnikov kernel where the bandwidth was selected based on the rule ofthumb suggested by Fan and Gijbels (1996). The thin vertical lines mark the interdecile range that extends from−15% to 15%, see Table 16 in Appendix E.

ric regressions.23 Specifically, the dashed yellow line is fitted to the predicted values from theregression model in column (6) of Table 5. The dash-dotted green line is fitted to the predictedvalues from a model with only the three classes of dummies, thus reflecting only cross sec-tional variation. We use the nonparametric regressions simply as a convenient device to makethe predictive essence of the various components of our baseline regression in column (6) ofTable 5 visible.

The main takeaway from Figure 5 is that all lines lie effectively on top of each other, es-pecially within the interdecile range. In other words, time series and cross sectional variation

23These predicted values form clouds rather than (piecewise) straight lines because growth is not the onlyregressor; there are also the fixed firm characteristics. For example, for two firms that experienced the same salesgrowth rate in the previous quarter, the model in column (6) of Table 5 will predict different spans depending onthe firms’ size, trend, and turbulence.

30

induces the same V-shape, albeit through very different mechanisms. For the time seriesresponse, the V follows directly from the difference in coefficients on positive and negativegrowth. For the cross section, the effect is more subtle and comes from the comovement ofspan with turbulence and trend growth, documented in Table 3: firms with higher span alsosee higher absolute values of their growth rates (due to differences in trend or turbulence).24

5.3 Subjective uncertainty and volatility in the time series

Section 4.4 showed that subjective uncertainty and conditional volatility vary differently in thecross section of firms. How do subjective and conditional volatility compare in the within-firm time series dimension? Table 6 compares our baseline regression of span on past growthand fixed characteristics—column (1) here reproduces column (6) of Table 5—to an analogousregression for the absolute value of the firm’s forecast error, shown in column (2).

Controlling for fixed characteristics, a firm that observes one percent worse negative growthin the previous quarter not only increases its span by 31 basis points, but also experiences, onaverage, a forecast error that is 34 basis points higher in absolute value. In contrast, onepercent higher positive growth increases span by 18 basis points and the absolute value of theforecast error by 12 basis points. The asymmetric V that emerges in the time series of firms’uncertainty is thus also present in firms’ experience of shocks. Columns (3) and (4) of Table 6show that this asymmetric V persists for both subjective uncertainty and conditional volatility,even when we control for firms’ sales growth expectations, that is, a forward-looking firstmoment.25

The differences between subjective uncertainty and conditional volatility observed in thecross section appear to be largely orthogonal to the time series nexuses of uncertainty and pastgrowth, which are similar. Indeed, the regression coefficients in columns (1) and (2) of Table 6display the same patterns as columns (4) and (6) of Table 3: large firms and very turbulentfirms are less uncertain than one might expect given the size of the shocks they face, and theasymmetric relationship between trend growth and uncertainty is a subjective phenomenon.

Finally, we show that subjective uncertainty and conditional volatility in fact move togetheras the regressors change: they jointly increase after high or low growth in the time series. Thisresult does not follow from the regressions in columns (1)-(4) of Table 6 alone. The latter onlyrelate span and the absolute forecast error separately to the regressors. It is then possible,for example, that sales growth consists of two orthogonal components that each co-move withonly one of the uncertainty measures. A firm might thus experience some episodes with largenegative growth and high span, and other episodes with large negative growth followed byhigh forecast errors, without a connection between the two.

To rule out such cases, columns (5) and (6) of Table 6 show versions of columns (1) and(2) for subjective uncertainty and conditional volatility (absolute forecast errors), respectively,but with the (contemporaneous) other uncertainty measure added to the regression. If in

24In Appendix G we also show that the V-shaped relationship between sales growth and subjective uncertaintyholds separately, and in a quantitatively similar manner, for all firm-level subgroups: the four firm size groups,the four turbulence groups, and the three growth trend groups.

25We note that only 13% of changes in span are not accompanied by a revision of the sales growth forecast.

31

fact span were correlated with one component of sales growth and the absolute forecast errorwith another, then the absolute forecast error should be conditionally correlated with spancontrolling for sales growth (as well as our other regressors). In effect, including the absoluteforecast error purifies sales growth into the component that is related to span only. However,the coefficients on all regressors in columns (5) and (6) are essentially the same as in columns(1) and (2), respectively, while the other uncertainty measure does not matter significantly. Weconclude that our results reflect comovement of subjective uncertainty, conditional volatilityand the regressors, as opposed to the composition of separate forces as in the example above.

32

Table 6: Regressions of subjective uncertainty and the absolute forecast error on past salesgrowth, firm characteristics, and additional controls

(1) (2) (3) (4) (5) (6)

Dependent variable: spanfirms’

abs(FE) spanfirms’

abs(FE) spanfirms’

abs(FE)

Negative sales growth rate in quarter t− 1 -0.306∗∗∗ -0.337∗∗∗ -0.283∗∗∗ -0.345∗∗∗ -0.364∗∗∗ -0.298∗∗∗

(0.0675) (0.0689) (0.0691) (0.0679) (0.0772) (0.0775)Positive sales growth rate in quarter t− 1 0.180∗∗∗ 0.123∗∗ 0.172∗∗∗ 0.126∗∗ 0.221∗∗∗ 0.105∗∗

(0.0314) (0.0529) (0.0318) (0.0536) (0.0351) (0.0494)Dummy small firms -3.959∗ -0.632 -4.251∗ -0.493 -3.418 -1.142

(2.178) (1.545) (2.175) (1.544) (2.310) (1.496)Dummy medium-sized firms -5.452∗∗ -1.672 -5.745∗∗∗ -1.553 -5.604∗∗ -1.516

(2.141) (1.460) (2.149) (1.465) (2.260) (1.422)Dummy large firms -6.295∗∗∗ -1.810 -6.571∗∗∗ -1.691 -6.170∗∗∗ -1.598

(2.170) (1.598) (2.174) (1.581) (2.332) (1.647)Dummy ’bad’ sales growth trend 2.248∗∗∗ 1.340∗ 2.380∗∗∗ 1.218 1.528 1.297∗

(0.856) (0.811) (0.858) (0.865) (1.004) (0.785)Dummy ’good’ sales growth trend -0.434 1.417∗ -0.560 1.558∗ -1.044 1.171

(0.645) (0.826) (0.643) (0.864) (0.710) (0.831)Dummy medium low turbulence 1.388∗∗ 1.802∗∗∗ 1.514∗∗ 1.770∗∗∗ 0.706 1.797∗∗∗

(0.591) (0.356) (0.596) (0.354) (0.643) (0.355)Dummy medium high turbulence 4.560∗∗∗ 4.160∗∗∗ 4.868∗∗∗ 4.151∗∗∗ 4.279∗∗∗ 3.706∗∗∗

(0.764) (0.472) (0.770) (0.473) (0.971) (0.557)Dummy high turbulence 6.748∗∗∗ 9.161∗∗∗ 6.899∗∗∗ 9.153∗∗∗ 5.655∗∗∗ 8.184∗∗∗

(0.969) (0.953) (0.978) (0.954) (1.261) (0.959)Forecast sales growth rate for quarter t 0.0510 -0.0357

(0.0321) (0.0458)Absolute forecast error in quarter t 0.0788

(0.0499)span 0.102

(0.0667)Constant 11.37∗∗∗ 3.782∗∗ 11.56∗∗∗ 3.702∗∗ 11.06∗∗∗ 3.135∗

(2.154) (1.484) (2.173) (1.478) (2.377) (1.616)

No. of observations 2,762 1,664 2,710 1,664 1,621 1,621No. of firms 400 389 399 389 381 381No. of parameters (excl. intercept) 10 10 11 11 11 11R-squared 0.29 0.25 0.29 0.25 0.32 0.25

Notes: span is our measure of subjective uncertainty and firms’ abs(FE) denotes firms’ absolute forecast error.All equations are estimated by pooled OLS. Standard errors in parentheses, clustered by firm; * p < 0.10, ** p< 0.05, *** p < 0.01. The forecast sales growth rate for quarter t is the answer to question 2.b of the survey (seeSection 2.2). The regressions in columns (1) and (3) are based on the baseline sample span with 2,762 observations;for column (3) we additionally need the aforementioned contemporaneous forecast. Columns (2) and (4) arebased on the baseline sample forecast with forecast errors, leading to 1,664 observations (see Table 14 in AppendixC). Finally, columns (5) and (6) start from the same baseline sample forecast with forecast errors, and, in addition,we require a contemporaneous span observation, leading to 1,621 observations.

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6 The dynamics of subjective uncertaintyIn this section, we further explore the dynamics of subjective uncertainty. Our approach is mo-tivated by two properties of many common learning rules. First, changes in uncertainty tendto propagate over time. Second, we would expect higher absolute forecast errors to increaseuncertainty. In principle, this property alone could induce a V-shaped relationship betweenuncertainty and growth, because large absolute growth rates tend to go hand in hand withlarge absolute forecast errors, which in turn raises uncertainty in the subsequent quarter. Be-fore we study the dynamics of subjective uncertainty, however, we present in Section 6.1 thedynamics of subjective sales growth expectations as a backdrop for comparison with the re-sults for subjective uncertainty. In Section 6.2, we then ask whether controlling for both laggedforecast errors and lagged uncertainty affects the V-shaped relationship between growth anduncertainty. The answer will help guide the choice among alternative models of belief updat-ing. In Section 6.3, we check whether replacing forecast errors with an alternative measure ofsales growth surprises based on deviations from the best (worst) case sales growth scenarioalters our conclusions. Or fourth key finding is that it is previous-quarter sales growth thatgenerates the V-shaped nexus with subjective uncertainty, not previous-quarter sales growthsurprises.

6.1 What moves subjective expectations?

We start by regressing expected sales growth rates for the next quarter on previous-quartersales growth rates, controlling for the usual firm characteristics. Column (1) of Table 7 showsthat there is no statistically significant linear relationship between the two. We therefore moveto piecewise linear specifications in the spirit of Chapter 5. Our first such specification re-gresses expected sales growth rates for the next quarter on negative and positive previous-quarter sales growth rates and thus presents the analogue of column (6) of Table 5, wherewe used subjective uncertainty as the dependent variable. Column (2) of Table 7 shows thatthere is, similar to the asymmetric V in the subjective uncertainty regression, a V-shaped nexusbetween previous-quarter sales growth rates and sales growth expectations. Our second spec-ification reported in column (3) replicates this asymmetric V using previous-quarter forecasterrors as explanatory variables. These estimates mean that firms expect a reversion after neg-ative sales growth experiences or surprises. A natural question that follows from these tworesults is: is the asymmetric V driven by negative surprises which are correlated with negativesales growth experiences, or is there something in negative sales growth experiences that willlead to the updating of sales growth expectations even when forecast errors are small? Theanswer to this question can be seen in column (4) of Table 7: Belief updates for sales growthexpectations occur in response to negative forecast errors. As we will show in the next section,this is different for the updating of subjective uncertainty. We finally note that the updatingworks in the following way: After brief and large negative surprises, firms, apparently view-ing such negative surprises as temporary, update their expectations upward somewhat: aftera one percentage point negative sales growth surprise, firms update their expectations by 20basis points.

In terms of the cross sectional influences on expected sales growth rates, we note that

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turbulence has little impact. Firms on low and high growth trends update their expectationstowards this trend. Finally, tiny firms have on average the most subdued growth expectations(consistent with the facts documented in Hurst and Pugsley, 2011), while small and medium-sized firms expect to grow the most.

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Table 7: Regressions of sales growth forecasts on past sales growth and past forecast errors

Dependent variable:forecast sales growth rate for quarter t (1) (2) (3) (4)

Sales growth rate in quarter t− 1 -0.0475(0.0497)

Negative sales growth rate in quarter t− 1 -0.243∗∗ -0.0993(0.104) (0.126)

Positive sales growth rate in quarter t− 1 0.0893 0.113(0.0740) (0.0780)

Negative forecast error in quarter t− 1 -0.247∗∗∗ -0.201∗∗

(0.0685) (0.0779)Positive forecast error in quarter t− 1 0.0562 -0.0182

(0.0708) (0.0726)Dummy small firms 5.702∗∗∗ 5.697∗∗∗ 5.903∗∗∗ 5.701∗∗∗

(1.774) (1.780) (1.657) (1.626)Dummy medium-sized firms 3.814∗∗ 3.723∗∗ 4.108∗∗ 3.822∗∗

(1.766) (1.777) (1.641) (1.624)Dummy large firms 3.972∗∗ 3.930∗∗ 4.290∗∗ 4.027∗∗

(1.792) (1.805) (1.669) (1.648)Dummy ’bad’ sales growth trend -2.789∗∗∗ -3.440∗∗∗ -3.291∗∗∗ -3.442∗∗∗

(0.892) (0.936) (0.871) (0.887)Dummy ’good’ sales growth trend 3.871∗∗∗ 3.458∗∗∗ 3.630∗∗∗ 3.326∗∗∗

(0.883) (0.834) (0.763) (0.787)Dummy medium low turbulence -0.675 -1.060∗∗ -1.063∗∗ -1.132∗∗

(0.481) (0.519) (0.487) (0.502)Dummy medium high turbulence -0.553 -1.279∗ -1.313∗∗ -1.481∗∗

(0.617) (0.695) (0.659) (0.683)Dummy high turbulence 2.037∗ -0.132 0.211 -0.531

(1.039) (1.111) (1.017) (1.095)Constant -2.278 -2.767 -3.066∗ -3.025∗

(1.728) (1.753) (1.642) (1.627)

No. of observations 1,531 1,531 1,531 1,531No. of firms 376 376 376 376No. of parameters (excl. intercept) 9 10 10 12R-squared 0.068 0.088 0.096 0.10

Notes: results from OLS regressions. They are based on the sample of firms with at least five answers to question1. In addition, we require an answer to question 2.b for the firm’s forecast, leading us to the baseline sample forecastwith 2,778 observations (see Table 14 in Appendix C). Finally, for columns (3) and (4) we require a lagged forecasterror, leading to 1,531 observations. For comparability we estimate the regressions in columns (1) and (2) on thatsame sample. Standard errors in parentheses, clustered by firm. * p < 0.10, ** p < 0.05, *** p < 0.01.

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6.2 What moves subjective uncertainty?

Turning now to the driving forces behind subjective uncertainty, column (1) of Table 8 repli-cates column (6) of Table 5, our baseline result, now on the somewhat smaller sample forwhich we observe firm forecast errors. The V-shape of subjective uncertainty in previous-quarter sales growth is again present. Then, in column (2), we replace sales growth withprevious-quarter forecast errors in sales growth: we find again a V-shape with somewhatsmaller coefficients. We note that the R-squared of this regression declines slightly comparedto column (1), indicating that previous-quarter sales growth has a somewhat stronger explana-tory power than previous-quarter sales growth surprises. Column (3) presents the results froma regression specification where both sales growth and the forecast error, in both cases allow-ing for asymmetry, are included. It is now the asymmetric V in sales growth that wins thehorse race between change and unanticipated change, the latter losing asymmetry and statis-tical significance. Notice that the R-squared in column (3) basically does not improve relativeto column (1).

Columns (4) to (6) repeat the same steps, but this time with lagged span included inthe regression, in order to quantify propagation of subjective uncertainty. We first note thatsubjective uncertainty displays a mild persistence because it depends on its own lag in all threespecifications. With respect to the relevance of sales growth versus forecast error, the result isthe same: in a horse race between these two regressors to determine subjective uncertainty, itis sales growth that enters with an asymmetric V, whereas the data do not ask for the forecasterror over and above sales growth.

To understand why past growth “drives out” the past forecast error in these regressions,we compare group averages of span in a two-by-two table of high versus low absolute growthrates and high versus low absolute forecast errors. We compute these group averages asfollows. To control for firm characteristics we first partial out the size, trend and turbulencedummies from span, absolute growth rates and absolute forecast errors leaving the conditionallinear relationship between the latter three variables unchanged.26 Since our partialling-outregressions include an intercept, the resulting adjusted variables have mean zero.

Subsequently, we split the adjusted absolute growth rates and the adjusted absolute fore-cast errors into observations above and below zero (their mean), thereby defining the fourquadrants shown in the upper left panel of Table 9, and compute average span for each quad-rant. For example, the upper left value of -1.82 means that a firm which, after controlling forfirm characteristics, experiences a below-average absolute growth rate and a below-averageabsolute forecast error, reports a 1.82 percentage points smaller span than the average firm atthe beginning of the subsequent quarter. In addition, we report the number of observationsthat fall in each quadrant (in square brackets below average span).

The first takeaway from the upper left panel of Table 9 is that uncertainty as measured byspan is relatively low (high) if both absolute growth and absolute forecast errors are relatively

26Technically, we invoke the Frisch-Waugh theorem which says that there are two equivalent ways to controlfor some variables z (here: the dummies) in an OLS regression of y (here: span) on x (here: past sales growthand past forecast error). Either regress y on x and z and take the coefficient of x. Alternatively, first regress y onz and x on z and then regress the residuals of these two regressions on each other.

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Table 8: Regressions of subjective uncertainty on past sales growth and past forecast errors,with dynamic modelsDependent variable: span for quarter t (1) (2) (3) (4) (5) (6)

Subjective uncertainty in quarter t− 1 0.273∗∗∗ 0.271∗∗∗ 0.276∗∗∗

(0.0706) (0.0748) (0.0695)Negative sales growth rate in quarter t− 1 -0.358∗∗∗ -0.314∗∗∗ -0.322∗∗∗ -0.333∗∗∗

(0.0755) (0.0825) (0.0754) (0.0910)Positive sales growth rate in quarter t− 1 0.148∗∗∗ 0.0990∗∗ 0.138∗∗∗ 0.0936∗

(0.0394) (0.0450) (0.0381) (0.0478)Negative forecast error in quarter t− 1 -0.232∗∗∗ -0.0599 -0.166∗∗∗ 0.0167

(0.0573) (0.0567) (0.0586) (0.0607)Positive forecast error in quarter t− 1 0.130∗∗∗ 0.0747 0.116∗∗∗ 0.0641

(0.0392) (0.0488) (0.0363) (0.0498)Size, trend, and turbulence dummies YES YES YES YES YES YESConstant 11.70∗∗∗ 11.61∗∗∗ 11.48∗∗∗ 8.587∗∗∗ 8.719∗∗∗ 8.465∗∗∗

(2.562) (2.610) (2.544) (2.055) (2.039) (2.008)

No. of observations 1,520 1,520 1,520 1,489 1,489 1,489No. of firms 373 373 373 367 367 367No. of parameters (excl. intercept) 10 10 12 11 11 13R-squared 0.35 0.32 0.35 0.40 0.38 0.41

Notes: results from OLS regressions. They are based on the sample of firms with at least five answers to question1. In addition, we require an answer to both questions 2.a for the firm’s span, leading us to the baseline samplespan with 2,762 observations. We further need the lag of the forecast error for columns (2) and (3), leading to1,520 observations. For reasons of comparability we estimate the regression in column (1) on that same sample.In columns (4) to (6), we additionally require the lag of span leading to 1,489 observations. See also Table 14 inAppendix C. Standard errors in parentheses, clustered by firm. * p < 0.10, ** p < 0.05, *** p < 0.01.

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low (high). Hence, a firm is relatively certain if it experiences a small absolute growth ratenear its expectations and it is relatively uncertain if it experiences a large absolute growth ratefar away from what it expected. Most observations (665+410=1075 of 1520 and thus 71%) fallin these two cells reflecting that sales growth is difficult to predict, so that low (high) absoluteforecast errors and low (high) absolute growth go hand in hand.

Table 9: Sales growth versus forecast errors as predictors of subjective uncertainty (two-by-twotables)

Full sample Only neg. growth and neg. FElow abs. FE high abs. FE low abs. FE high abs. FE

After partialling out size, trend, and turbulence dummieslow abs. growth −1.82∗∗∗

[obs: 665]−0.19

[obs: 188])−2.04∗∗∗[obs: 127]

−0.02[obs: 68]

high abs. growth 0.64[obs: 257]

2.63∗∗∗[obs: 410]

2.01∗[obs: 62]

3.29∗∗∗[obs: 170]

After partialling out size, trend, and turbulence dummies, and lagged spanlow abs. growth −1.38∗∗∗

[obs: 637]−0.82

[obs: 188])−1.71∗∗∗[obs: 125]

−0.20[obs: 68]

high abs. growth 0.52[obs: 242]

2.14∗∗∗[obs: 422]

1.51[obs: 61]

3.01∗∗∗[obs: 166]

Notes: the cells show group-specific means of adjusted span and, in brackets below, the number of observationsper cell. The adjustment in the two upper panels is based on a regression of span on size, trend, and turbulencedummies (1,520 observations). The groups in the four cells are, respectively, defined by the mean values ofthe residuals of the absolute sales growth and the absolute forecast error regressions on the same variablesas span. For the left panels, we use the whole sample, for the right panels a subsample with only negativesales growth rates and negative sales growth forecast errors. The lower panels mirror the upper panels, but,additionally, control for lagged span in the adjustment regressions (1,489 observations). The upper/lower panelsrefer, respectively, to the samples used in columns (1) to (3) and (4) to (6) of Table 8. * p < 0.10, ** p < 0.05, *** p< 0.01.

What happens, however, in the upper right and lower left cells? Concerning the upperright cell, a small absolute growth rate that comes as a large surprise basically does not alterspan. This suggests that firms which incorrectly expected something “big” to happen do notexperience heightened uncertainty even though the size of their forecast error is large. Inother words, firms tend not to update their subjective uncertainty after “unforced errors”,because the signal they receive tells them that they are in calm territory. By contrast, the lowerleft cell tells us that a large absolute growth rate increases uncertainty to a noticeable, if notstatistically significant amount even if it comes more or less expectedly. Altogether, it thusappears to be the signal conveyed by the absolute growth rate which shapes uncertainty andnot the expectational error.27

27This interpretation is supported by comparing the fit of two simple models, again after partialling out size,trend and turbulence dummies from span, absolute growth and absolute forecast errors. Model 1 forces the groupmeans of adjusted spans to be the same across rows, only distinguishing between low and high absolute forecasterrors. Model 2 forces the group means of adjusted spans to be the same across columns, only distinguishingbetween low and high absolute sales growth. Hence, model 1 reflects the column dimension of the upper leftquadrant of Table 9 while model 2 reflects its row dimension. The fit of model 1 is worse than the fit of model 2in all four quadrants, particularly so in the lower left cell.

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To better understand this finding, we next distinguish between positive and negativegrowth/forecast errors, that is, the sign of the experience the firm has gone through duringthe quarter beginning in t − 1, and how this experience subsequently shapes firms’ subjec-tive uncertainty. In particular, we ask whether a firm having gone through a bad experiencewill be more uncertain compared to after a good experience. The upper right panel of Table9 reports average span using only those observations that exhibit both negative growth andnegative forecast errors. By comparing the upper left and right panels, we can thus assess theasymmetry of the relationships between span and its drivers. It turns out that the differencesare moderate in all cells except for the lower left one where average span is much larger thanin the full sample: an (almost expected) large negative growth rate leads to a strong increase inspan by 2 percentage points. Hence, it is firms in a gloom situation—expecting an unusuallybad outcome and experiencing an even slightly worse realization—which drive the lower leftcell even for the full sample.28 This means, that it is firms in a gloom situation that make salesgrowth dominate sales forecast errors in the determination of firms’ subjective uncertainty.

The lower panels of Table 9 show that these results are robust to including lagged spanin the set of controls. To summarize, we find that subjective forecast errors are driven bypast sales growth rather than past forecast errors. While these two regressors are correlated,suggesting that sales forecasts, as a rule, are difficult, there is a sizeable number of observationsexhibiting small (large) absolute sales growth combined with large (small) absolute forecasterrors. In these cases, sales growth is the better predictor of span particularly for firms in agloomy situation.

What is the economic mechanism that might make firms more uncertain especially aftera negative previous-quarter sales growth rather than a negative sales growth surprise? Apossible interpretation is that a large negative sales growth could indicate a large loss in thecustomer base of a firm. Wether this loss was predicted or not, in an environment wherebuilding up customer relationships is costly and the success of it uncertain, affected firmsdo not know whether and which new customers can be found in the months going forward,making them more uncertain with respect to future sales growth.

6.3 An alternative surprise measure

As a robustness check, we ask whether the results in the previous section remain unchangedif we replace the previous-quarter forecast error by an alternative measure of sales growth sur-prise. To this end, we construct a variable that measures the distance of the realized previous-quarter sales growth rate to the sales growth forecast interval from one quarter before. Thisalternative surprise measure takes on the value of zero if the growth rate falls into the forecastinterval; the distance of the growth rate to the upper interval limit (the best case) if the growthrate falls above the forecast interval; and the distance to the lower interval limit (the worst case)if the growth rate falls below the forecast interval. We thus measure the surprise intensity forthose firms that see their previous-quarter growth rates outside their forecast interval.

28In fact, the average adjusted span in the lower left cell of the complementary group of firms that do notexhibit both negative growth and negative forecast error is 0.2 and thus almost indistinguishable to the averageof zero.

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Table 10 reports the results of regressions of span on the previous-quarter growth rate andthis alternative sales growth surprise measure. To account for potential asymmetries, we splitagain both regressors into their positive and negative parts. Note that a positive (negative)surprise is the distance of the growth rate to the upper (lower) interval limit if the growth ratefalls above (below) the interval.

A regression of span on the surprise measure with only our usual firm-specific dummyvariables shows there is again an asymmetric V-type relationship where the negative arm isstatistically significant (column 1). However, adding the sales growth rate yields the same“driving out” result reported in Section 6.2: both parameters of the surprise measure becomequantitatively small and statistically insignificant while the parameters of the sales growthrate exhibit the asymmetric V (column 2). Also, the R-squared of the regression improvessubstantially. Columns (3) and (4) replicate this outcome when lagged span is included as anadditional regressor.

Table 10: Regressions of subjective uncertainty on past sales growth and past outside devia-tions from worst or best case, with dynamic modelsDependent variable: span for quarter t (1) (2) (3) (4)

Subjective uncertainty in quarter t− 1 0.322∗∗∗ 0.282∗∗∗

(0.0614) (0.0750)Negative sales growth rate in quarter t− 1 -0.386∗∗∗ -0.304∗∗∗

(0.0882) (0.0884)Positive sales growth rate in quarter t− 1 0.180∗∗∗ 0.121∗∗

(0.0478) (0.0523)Deviation from worst case forecast in quarter t− 1 -0.227∗∗ 0.0249 -0.271∗∗∗ -0.0639

(0.102) (0.105) (0.101) (0.109)Deviation from best case forecast in quarter t− 1 0.0585 -0.0408 0.104∗∗∗ 0.0382

(0.0422) (0.0606) (0.0400) (0.0616)Size, trend, and turbulence dummies YES YES YES YESConstant 12.85∗∗∗ 11.83∗∗∗ 8.825∗∗∗ 8.525∗∗∗

(2.715) (2.616) (1.969) (2.020)

No. of observations 1,513 1,513 1,513 1,513No. of firms 372 372 372 372No. of parameters (excl. intercept) 10 12 11 13R-squared 0.30 0.35 0.37 0.40

Notes: results from OLS regressions. They are based on the sample of firms with at least five answers to question1.In addition, we require an answer to both questions 2.a for the firm’s span, leading us to the baseline samplespan with 2,762 observations. We further need the lag of span and/or of the best/worst case scenario in allregressions, leading to 1,513 observations. See also Table 14 in Appendix C. The variable “Deviation from worst(best) case forecast in quarter t− 1” has value zero if the past realized sales growth rate was inside the forecastinterval. Standard errors in parentheses, clustered by firm. * p < 0.10, ** p < 0.05, *** p < 0.01.

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7 The dynamics of subjective uncertainty and conditional vola-tility

The goal of this final section is to compare the dynamics of subjective uncertainty perceivedby firms with the dynamics of conditional volatility experienced by firms. We have alreadyseen in Section 5.3 that the projection of the absolute size of forecast errors on past growth—controlling for fixed characteristics—yields coefficients that are quite similar to the coefficientsof our baseline span regression. We now ask to what extent firms’ updating of subjectiveuncertainty studied in Section 6 resembles the dynamic behavior of the conditional volatilityof unpredictable shocks. In terms of the framework of Section 3.2, we now ask how similarthe dynamics of σ and σ are.

To this end, we take two preliminary steps, the details of which are documented in Ap-pendix H. As a first preliminary step, we distinguish between the two sources of firm forecasterrors discussed in Section 3.2: bias and conditional volatility. We “clean” subjective forecasterrors by removing the firm-specific forecast bias. To do so, we estimate a LASSO regressionto select among the many possible predictors for this bias—we consider our three firm char-acteristics and their interactions—and then subtract the estimated bias from firms’ subjectiveforecast errors. This approach based on expectational survey data putatively captures the in-formation of actual decision makers but it could also capture predictable errors. In addition,we therefore construct an alternative measure of unbiased forecast errors that an econome-trician would compute using a statistical model with only firm-level sales growth data andno expectational data available. This approach based on outcome data only is a conventionaleconometric forecasting exercise although it may fail to fully capture the information used bydecision makers.

As a second preliminary step, we then estimate dynamic models of conditional volatilityfor both the cleaned subjective forecast errors and for the statistical forecast errors to providea counterpart to the previous regressions of span on lagged span, past growth, past forecasterrors as well as fixed firm characteristics. Note that we have operationalized subjective uncer-tainty with span, the difference between best and worst case scenarios. A natural “objective”counterpart would be the length of a forecast interval constructed by the econometrician, forexample, the difference between an upper and lower quantile of the conditional distributionof forecast errors. In the broad class of distributions which belong to the location-scale fam-ily, the forecast interval length is simply a multiple of the distribution’s standard deviation.We, therefore, choose the conditional standard deviation of forecast errors as our measure of“objective” uncertainty and model it in a power GARCH framework. We select and estimatea power GARCH specification that optimally describes the data as indicated by informationcriteria. Our choice of explanatory variables turns out to mirror our analysis of subjectiveuncertainty in Section 6.2: we include past sales growth rates, past forecast errors, and fixedfirm characteristics in the power GARCH equation.

Here we compare the dynamics of all three measures of firm uncertainty, namely, subjectiveuncertainty, conditional volatility based on subjective forecast errors and conditional volatilitybased on statistical forecast errors. To preview the results, we show, as our fifth key fact, that

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the dynamics of subjective uncertainty and “objective” conditional volatility are similar. Theyinclude mild but statistically significant persistence, irrelevance of lagged forecast errors, andpredictive importance of lagged absolute sales growth, especially if it is negative. Managersunderstand the changing uncertainty environment their firms operate in and update theirsubjective uncertainty accordingly.

We start our comparison by reporting descriptive summary statistics for all three measuresin Table 11. A first notable result is that the distributions of the predicted conditional standarddeviations of the firms’ subjective and the statistical forecast errors are remarkably similar. Infact, the sample correlation of the two measures is 0.97. Moreover, the distribution of subjec-tive uncertainty as measured by span is also similar to the two distributions for conditionalvolatility, with a slightly higher mean and dispersion. The sample correlations between subjec-tive uncertainty and the firms’ subjective forecast error is 0.53, and the one with the statisticalforecast error is 0.51.

Table 11: Summary statistics for measures of subjective uncertainty and predicted conditionalvolatility

Variable #obs Mean Std. Dev. P10 P25 P50 P75 P90

Span between worst and best case forecast 932 12.1 9.8 4 5 10 15 25

Predicted conditional volatility of firms’subjective forecast errors

932 9.7 7.2 3.5 4.8 7.5 12.3 18.3

Predicted conditional volatility of statis-tical forecast errors

932 9.7 7.3 3.4 4.2 7.2 12.2 19.8

Notes: The number of usable observations shrinks from 949 to 932 here because 17 quarter-firm obser-vations we used to construct forecast errors have either a missing upper or lower interval bound, orboth, in the data, and thus we cannot compute a span for these observations (see Appendix C). Themodels used to calculate the predicted conditional standard deviations (volatilities) of subjective andstatistical forecast errors, can be found in columns (1) and (3), respectively, of Table 43 in Appendix H.P10 to P90 denotes the corresponding percentiles of the distribution.

In a final step, we compare the dynamics of subjective uncertainty to the dynamics ofconditional volatility. Since the volatility models link the conditional standard deviation topast sales growth and the dummies via an exponential function that ensures nonnegativity,we base our comparison on average partial effects. We report them in columns (2) and (3)of Table 12, while column (1) replicates the coefficient estimates of the dynamic linear modelfor span reported in column (6) of Table 8 with the only difference that we replace the twoinsignificant regressors “positive past forecast error” and “negative past forecast error” by thesingle regressor “absolute forecast error” to conform with the symmetric specification of the

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volatility models.29

29Our results are robust to two modelling decisions, one with respect to the sample choice, another one withrespect to the symmetric specification regarding the absolute forecast error in the power GARCH equation. Asto the first decision, the regression for span, as elsewhere in the paper, is based on the baseline sample span,whereas the power GARCH estimations are based on the baseline sample forecast as explained in Table 14 inAppendix C. If we had estimated the regression in column (1) on the appropriate restriction of the baseline sampleforecast, the results would be essentially the same. As to the second decision about symmetry, we note thatour information criteria do not provide us with clear guidance but the asymmetry parameter turns out to bestatistically insignificant and the other estimated power GARCH coefficients are nearly identical (see Table 43 inAppendix H).

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Table 12: Comparison of subjective uncertainty and predicted conditional volatility(1) (2) (3)

Dependent variable: Subjective uncertaintyConditional volatility

of firms’ subjectiveforecast errors

Conditional volatilityof statistical forecast

errorsUncertainty/volatility in t− 1 0.270∗∗∗ 0.235∗∗∗ 0.236∗∗∗

(0.0699) (0.0874) (0.0921)Absolute forecast error in t− 1 0.0373 0.085∗ 0.008

(0.0383) (0.051) (0.068)Negative sales growth in t− 1 -0.285∗∗∗ -0.223∗∗∗ -0.220∗∗∗

(0.0768) (0.073) (0.064)Positive sales growth in t− 1 0.112∗∗∗ 0.102∗∗ 0.060

(0.0422) (0.043) (0.044)Dummy small firms -3.507∗ -1.227 -1.316

(2.066) (1.068) (0.811)Dummy medium-sized firms -3.920∗ -1.891 -1.656∗∗

(2.056) (0.947) (0.709)Dummy large firms -4.562∗∗ -2.073∗∗ -1.550∗

(2.022) (1.006) (0.790)Dummy ’bad’ sales growth trend 2.219∗∗∗ 1.498∗∗ -0.234

(0.780) (0.706) (0.540)Dummy ’good’ sales growth trend -0.432 1.542∗∗ 0.565

(0.530) (0.756) (0.727)Dummy medium low turbulence 0.660 2.216∗∗∗ 2.294∗∗∗

(0.523) (0.459) (0.361)Dummy medium high turbulence 3.714∗∗∗ 4.094∗∗∗ 4.714∗∗∗

(0.773) (0.634) (0.525)Dummy high turbulence 4.528∗∗∗ 9.462∗∗∗ 12.057∗∗∗

(0.883) (1.314) (1.552)

No. of observations 1,489 949 949

Notes: In the first column, pooled OLS regression coefficients are displayed. Note that linear regression coeffi-cients are the same as average partial effects. The second and third column show average partial effects. Theregression in column (1) is based on baseline sample span with 2,762 observations, where, in addition, we needobservations on the lag of the forecast error and the lag of span, leading to 1,489 observations. The averagepartial effects in column (2) and (3) are based on baseline sample forecast with 2,778 observations, where we need,in addition, observations on the forecast error and its lag, leading to 949 observations (see also Table 14 in Ap-pendix C). They are from the power GARCH models shown in columns (1) and (3), respectively, of Table 43 inAppendix H. All standard errors below the coefficients and the average partial effects are clustered by firm. * p< 0.10, ** p < 0.05, *** p < 0.01.

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The results indicate that the dynamics of subjective uncertainty and “objective” conditionalvolatility are remarkably similar. There is mild but statistically significant persistence; laggedforecast errors are largely irrelevant; and lagged absolute sales growth has an asymmetriceffect with large negative realizations being roughly twice as important than large positiverealizations. Hence, when forming uncertainty beliefs firms appear to have a pretty realisticimpression of the dynamics that characterize the underlying data.

By contrast, subjective uncertainty and conditional volatility differ markedly in a numberof cross sectional dimensions, similar to our findings in Section 4. Most importantly, firmsoperating in a highly turbulent environment report, relative to firms operating in a calm en-vironment, a subjective uncertainty that is on average 4.5 percentage points higher while thedifference in terms of the conditional volatility of their forecast errors and the statistical fore-cast errors amounts to 9.5 and 12.1 percentage points, respectively. Hence, volatile firms aremore uncertain than others but not quite enough—they seem to underestimate the volatilitythey face. We also observe an opposite overestimation effect: firms on a bad sales growth trendare more uncertain than conditional volatility suggests, while larger firms feel more certainthan justified by conditional volatility.

To summarize, an average firm’s updating of subjective uncertainty over time closely re-sembles the dynamics of conditional volatility, while, in some cases, the firm’s subjective levelof uncertainty fails to adequately reflect the environment the firm is operating in.

46

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51

Appendix A Representativeness of the sampleIn this appendix, we investigate whether participation in the uncertainty module is selec-tive conditional on participation in the main manufacturing survey. We base our analysis onall 34,684 complete firm-quarter responses available in the main survey for the months theuncertainty module was executed. We then ask whether firm size, time dummies, industrydummies, and interacted time-industry dummies are able to predict participation in the un-certainty module. To this end, we run a probit regression of a participation dummy that is 1for the 5,564 observations of the uncertainty module and zero otherwise, on these predictorsand report the estimated coefficients in column (1) of Table 13. We find that there is no statisti-cally significant selectivity with respect to quarter/survey wave and industry suggesting thatthe uncertainty sample does not misrepresent specific quarters or industries. While firm sizeturns out to be significantly negative indicating that large firms are slightly underrepresentedin the uncertainty module compared to the main manufacturing survey, the pseudo R-squaredof 0.016 shows that this selectivity is quantitatively irrelevant. This is also reflected by a ROCcurve which differs only slightly from the diagonal that indicates no discriminatory power,see Figure 6.30

We repeat the analysis starting from the subset of 23,486 complete firm-quarter responsesavailable in the online part of the main survey. We thus account for the fact that some firms re-ply by fax and thus do not participate in the uncertainty module which is solely implementedonline. The results of an analogous probit regression are reported in column (2) of Table 13.Again, the very low pseudo R-squared suggests that selectivity is not a relevant issue for theuncertainty module. This conclusion is supported by a largely unaltered ROC curve near thenon-discriminatory diagonal, see Figure 6.

30The receiver operating characteristic (ROC) visualizes the discriminatory power of a binary classifier asfollows: By varying the classification threshold—here: the probability above which an observation is predictedto participate in the uncertainty module—the classifier can produce any true positive rate (type II error). TheROC curve plots the true positive rate so obtained against its corresponding false positive rate (type I error). Inthe case of no discriminatory power, true and false positive rates are always the same, the ROC curve equals thediagonal, and the area under the ROC curve (AUC) is 0.5. A good classifier has a ROC curve well above thediagonal and an AUC that is near the maximum of 1.0.

52

Table 13: Probit regression of a dummy indicating uncertainty module participation(1) (2)

probit vs. ifo all probit vs. ifo online

Log of number of employees -0.0753∗∗∗ -0.134∗∗∗(0.0127) (0.0139)

Dummy survey wave 1 -0.0441 0.0707(0.221) (0.242)

Dummy survey wave 2 0.0647 0.235(0.260) (0.289)

Dummy survey wave 3 0.248 0.452∗(0.232) (0.258)

Dummy survey wave 4 0.276 0.439∗(0.217) (0.240)

Dummy survey wave 5 0.126 0.253(0.204) (0.225)

Dummy survey wave 6 0.317 0.490∗∗(0.195) (0.211)

Dummy survey wave 7 0.224 0.387∗(0.206) (0.225)

Dummy survey wave 8 -0.244 -0.150(0.247) (0.268)

Dummy survey wave 9 0.110 0.228(0.235) (0.255)

Dummy survey wave 10 0.224 0.361∗(0.186) (0.201)

Dummy survey wave 11 -0.0333 0.0732(0.179) (0.196)

Dummy survey wave 12 0.154 0.275(0.182) (0.202)

Dummy survey wave 13 0.321 0.334(0.218) (0.232)

Dummy industry 1 0.130 0.415(0.245) (0.262)

Dummy industry 2 -0.301 -0.153(0.271) (0.289)

Dummy industry 3 -0.0686 0.154(0.229) (0.242)

Dummy industry 4 -0.0697 0.0116(0.250) (0.264)

Dummy industry 5 0.136 0.335(0.237) (0.250)

Dummy industry 6 -0.129 0.000307(0.240) (0.253)

Dummy industry 7 -0.0828 0.185(0.250) (0.267)

Dummy industry 8 0.109 0.379(0.257) (0.275)

Dummy industry 9 -0.360 -0.184(0.232) (0.245)

Dummy industry 10 0.120 0.282(0.259) (0.276)

Dummy industry 11 -0.0901 0.0704(0.240) (0.254)

Dummy industry 12 -0.0935 0.0302(0.222) (0.232)

Dummy industry 13 -0.240 -0.0867(0.283) (0.301)

Constant -0.605∗∗∗ -0.256(0.214) (0.224)

Additional time-industry dummies YES YES

No. of observations 34,684 23,486No. of firms 3,428 2,416No. of parameters (excl. intercept) 196 196Pseudo R-squared 0.016 0.030

Notes: Standard errors in parentheses, clustered by firm; * p< 0.10, ** p < 0.05, *** p < 0.01. For the definition of industriessee Appendix D.

53

Figure 6: ROC curves for probit estimations0.

000.

250.

500.

751.

00Se

nsiti

vity

0.00 0.25 0.50 0.75 1.001 - Specificity

Area under ROC curve = 0.5885

probit vs. ifo all

0.00

0.25

0.50

0.75

1.00

Sens

itivi

ty

0.00 0.25 0.50 0.75 1.001 - Specificity

Area under ROC curve = 0.6183

probit vs. ifo online

Notes: The two plots depict ROC curves that correspond to the probit estimations in columns 1 and 2 of Table13, respectively.

54

Appendix B Questionnaire for the one-time meta-survey fromfall 2018

Figure 7: Original survey questionnaire in German

Notes: Original questionnaire from ifo’s one-time online meta-survey on its “uncertainty module” in German,from fall 2018

55

Figure 8: Original survey questionnaire in German

Notes: Original questionnaire from ifo’s one-time online meta-survey on its “uncertainty module” in German,from fall 2018 56

Figure 9: Original survey questionnaire in German

Notes: Original questionnaire from ifo’s one-time online meta-survey on its “uncertainty module” in German,from fall 2018

57

Appendix C Sample creationIn this appendix, we describe the construction of our baseline sample and also explain thenumber of observations from specific regressions in this paper. The observation numbers thatremain after each step are listed in Table 14. Our starting sample from 14 survey waves consistsof 5,564 firm-quarter observations. A firm-quarter observation is in the starting sample if thefirm at that point in time answered at least one question of our online survey module andpressed the send button to return it.

Table 14: Sample creation

firm-quarter firm-quarter firms firmsobs. in sample obs. excluded in sample excluded

Original sample 5,564 1,426

Require response to question 1Response to question 1 exists 5,194 370 1,378 48

Text commentWrong reference time excluded 5,095 99 1,368 10Uncertain data quality excluded 5,067 28 1,367 1

Outliers to question 1Outliers in question 1 responses excluded 5,045 22 1,365 2

Number of observations by firmAt least 5 clean responses to question 1 3,094 1,951 401 964

Outliers and inconsistencies to question 2Inconsistent & outlier responses to question 2 excluded 2,945 149 401 0

Require responses to question 2.aBaseline sample span: Responses to question 2.a both exist 2,762 183 400 1

Lag of forecast error exists 1,520 1,242 373 27Lag of forecast error and lag of span exist 1,489 31 367 6

Lag of span exists 1,513 1,249 372 28

Require response to question 2.bBaseline sample forecast: Response to question 2.b exists 2,778 167 400 1

Forecast error exists 1,664 1,114 389 11Lag of forecast error exists 949 715 292 97Lag of forecast error, and span exist 932 17 289 3

We start by excluding 370 firm-quarter observations that were lacking an answer to ques-tion 1, realized sales growth. Then we carefully read the free text comments respondents cangive below each of the questions (see Figure 1 in the main text for the questionnaire). Weexclude 99 observations for which a comment indicates that the respondent was not able tocalculate sales growth rates on a quarterly basis. For example, some firms in some quartersstated that they use annual growth rates instead. Moreover, we drop 28 observations for whichthe comment raises doubts about the validity and quality of the answer. For example, somefirms in some quarters were not able to state realized past growth rates and used estimatesinstead. Overall, we exclude 497 firm-quarter observations based on missing or low-qualityanswers to question 1, leaving us with 5,067 firm-quarter observations.

Next, we exclude outliers which we define for question 1 (sales growth rate realizations)

58

as lying outside the interval [-100%, 100%]. 22 firm-quarter observations are thus excluded.We have also experimented with a [-15%, 15%] cutoff, that is the bottom and top deciles ofthe final baseline sample, and found very similar results. We set the upper bounds quite highbecause large (two-digit) growth rates typically appear to be deliberate responses as manytext comments reveal. Firms give explanations such as “Many projects were moved into thisquarter” and “Invoice of a major project.” This leaves us with 5,045 firm-quarter observations.After these cleaning steps we require for the firm-quarter observations of a firm to remain inthe sample that it have at least five clean firm-quarter observations on question 1, leaving uswith 3,094 firm-quarter of 401 firm observations. It is this sample that we base the calculationof the trend and turbulence dummies on.

Subsequently, we exclude, respectively, outliers and inconsistencies related to question 2.Question 2-outliers were excluded according to the following two criteria:

1. The best case and worst case sales growth rates elicited in question 2.a lie outside theintervals [-100%; 300%] and [-100%; 100%], respectively.

2. The forecast growth rate elicited in question 2.b lies outside the interval [-100%; 100%].

Then we check whether respondents order the numbers in question 2 consistently, that is, asworst case < forecast < best case. We exclude firm-quarter observations with the orderingsworst case ≥ forecast ≤ best case or worst case ≤ forecast ≥ best case because it is unclearwhat the respondents had in mind with these answers. However, we keep those firm-quarterobservations with the inverse ordering worst case ≥ forecast ≥ best case and simply swap theworst case and best case numbers; we do this for 76 firm-quarter observations. Most likelyinverse orderings were not intended by the respondent and rather a simple clerical error.Altogether, we eliminate 149 firm-quarter observations in this step.

In a final step, we eliminate those 183 firm-quarter observations which do not have answersto question 2.a, the best or worst case scenarios for sales growth, leaving us with our baselinesample of 2,762 of firm-quarter observations for 400 firms (baseline sample span). For some otherexercises, for which we do not need span observations but the answer to question 2.b, that is,the forecast growth rate, we use a slightly bigger sample of 2,778 firm-quarter observations(baseline sample forecast).

Starting from the baseline sample span, for some further exercises we additionally needa lag of the forecast error, which leaves us with 1,520 firm-quarter observations. For thesubsample of 1,489 observations, we also have the lag of span. For another exercise, we againstart from the baseline sample span and require a lag of span. This subsample contains 1,513firm-quarter observations.

Finally, we use the slightly larger baseline sample forecast to analyze forecast errors, whichis possible for 1,664 firm-quarter observations. For some exercises, we use consecutive forecasterrors. We, therefore, have to eliminate isolated forecast errors that have no forecast errorsurrounding them. This reduces the sample to 1,329 firm-quarter observations of which 380are used as lagged “pre-sample” observations so that an effective sample size of 949 firm-quarter observations remains. For 932 of these observations we also observe span.

59

Appendix D Definition of manufacturing industriesTable 15 presents the definition of the 14 industries we use. They are based on the original24 two-digit manufacturing industries, which are defined by the WZ08 (WZ stands for theGerman Wirtschaftszweig) code of the German Statistical Office. Since not all these industryhave a large number of observations, we aggregate, for the purposes of this paper, some upinto 14 manufacturing industries. The column with the number of observations by industryrefers to our baseline sample of 2,762 observations.

Table 15: Definition of industries

Industry Industry WZ08 Industry WZ08 name No. of obs.1 10, 11, 12 Food products; Beverages; Tobacco products 1842 13, 14, 15 Textiles; Wearing apparel;

Leather and related products66

3 16, 17, 31 Wood, products of wood and cork except fur-niture, articles of straw and plaiting materials;Paper and paper products; Furniture

286

4 18 Printing and reproduction of recorded media 1915 19, 20, 21 Coke and refined petroleum products;

Chemicals and chemical support;Basic pharmaceutical products and pharmaceu-tical preparations

262

6 22 Rubber and plastic products 2287 23 Other non-metallic mineral products 1338 24 Basic metals 1209 25 Fabricated metal products, except machinery

and equipment324

10 26 Computer, electronic and optical products 10211 27 Electrical equipment 20112 28 Machinery and equipment n.e.c. 44513 29, 30 Motor vehicles, trailers and semi-trailers;

Other transport equipment116

14 32, 33 Other manufacturing; Repair and installation ofmachinery and equipment

104

All 2,762

60

Appendix E Detailed summary statisticsIn this appendix, we report summary statistics for the answers to the questions in our surveymodule. Table 16 pools all firm-quarter observations and reports mean, standard deviation,and key quantiles for this pooled sample. The numbers here reflect variation both in the timeseries and in the cross section of firms. For Table 17, we compute, for each individual firm, thetime series mean and standard deviations. The panel reports mean, standard deviation andquantiles of the cross sectional distributions of firm-level statistics. The number of observa-tions for (functions of) forecast errors naturally drops because, in order to compute firm-levelforecast errors, we need to observe the expected sales growth rate and the realized sales growthrate of a firm in two consecutive quarters; for details see Appendix C. In addition, we presentin this appendix the same two summary statistics tables split by our three firm characteristics:firm size (Tables 18 to 25), trend (Tables 26 to 31), and turbulence (Tables 32 to 39).

61

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nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r40

01.

747.

87-6

.67

-1.9

21.

695.

310

.29

Mea

nby

Firm

:For

ecas

tsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r39

92.

616.

57-3

.75

-.62.

255

9M

ean

byFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

400

-4.4

87.

4-1

3-8

.31

-3.3

5-.1

52.

72M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

400

7.87

8.02

.81

3.41

6.67

10.6

116

.67

Mea

nby

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

400

12.3

47.

355.

097.

1510

.57

15.5

522

.33

Mea

nby

Firm

:For

ecas

ter

ror

389

-.23

10.4

8-1

0-3

.57

03.

58

Mea

nby

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el38

9.6

9.74

-8.3

3-3

.22

03.

89.

67M

ean

byFi

rm:F

orec

ast

erro

rfr

omiid

mod

el38

9.5

57.

67-6

.8-2

.35

.22.

638.

67M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r38

99.

449.

552.

254

711

.67

17.5

7

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r39

711

.42

9.22

3.44

5.48

8.59

13.7

23.4

9St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

396

7.36

7.1

1.6

2.89

5.24

9.72

14.9

2St

d.D

ev.b

yFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

397

8.09

7.13

2.12

3.18

6.28

10.8

16.3

3St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

397

8.13

7.78

2.16

3.43

5.87

10.3

115

.52

Std.

Dev

.by

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

397

5.85

5.05

2.04

2.87

4.67

7.5

10.6

1St

d.D

ev.b

yFi

rm:F

orec

ast

erro

r33

810

.17

9.79

2.29

4.04

7.46

12.7

320

.21

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el33

812

.37

13.3

82.

074.

048.

0914

.57

29.6

9St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omiid

mod

el33

87

7.19

.71

2.3

4.51

9.9

16.4

2St

d.D

ev.b

yFi

rm:A

bsol

ute

fore

cast

erro

r33

86.

446.

281.

412.

514.

338.

0813

.63

Not

es:

P10

toP9

0de

note

the

corr

espo

ndin

gpe

rcen

tile

sof

the

dist

ribu

tion

.Th

esu

mm

ary

stat

isti

csfo

ral

lva

riab

les

inth

eup

per

pane

lar

eba

sed

onth

eba

selin

esa

mpl

esp

anw

ith

400

firm

s,w

ith

two

exce

ptio

ns:

first

,one

firm

did

not

answ

erqu

esti

on2.

b,w

hich

isw

hyw

ear

ele

ftw

ith

399

firm

-qua

rter

obse

rvat

ions

wit

ha

fore

cast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter.

Seco

nd,t

hesu

mm

ary

stat

isti

csfo

rfo

reca

ster

ror

vari

able

sar

eba

sed

onth

eba

selin

esa

mpl

efo

reca

stw

ith

afo

reca

ster

ror,

lead

ing

to38

9fir

ms

(see

Tabl

e14

inA

ppen

dix

C).

For

com

pari

son,

we

also

com

pute

the

fore

cast

erro

rfr

oma

rand

omw

alk

mod

elan

dfr

oman

iidpr

oces

s,re

spec

tive

ly,b

ased

onth

esa

me

sam

ple

asfo

rth

efir

ms’

fore

cast

erro

rs.A

sfo

rth

esu

mm

ary

stat

isti

csin

the

low

erpa

nel,

not

allfi

rms

have

asu

ffici

ent

num

ber

toco

mpu

tea

stan

dard

devi

atio

n.Th

isex

plai

nsth

edi

ffer

ence

innu

mbe

rof

firm

sbe

twee

nth

eup

per

and

low

erpa

nel.

63

Tabl

e18

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omti

nyfir

ms,

pool

ed

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r13

1-2

.72

18.0

6-3

0-1

00

1015

Abs

olut

eva

lue

ofsa

les

grow

thra

tein

the

prev

ious

quar

ter

131

13.0

712

.71

03

1020

30Fo

reca

stsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r13

0-1

.25

14.8

7-2

0-1

00

515

Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r13

1-1

2.37

18.4

7-5

0-2

0-1

00

5Be

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r13

17.

4817

.27

-10

05

1530

Span

betw

een

wor

stan

dbe

stca

sefo

reca

st13

119

.85

16.3

94

1015

3040

Fore

cast

erro

r86

-2.7

18.5

-30

-10

05

15Fo

reca

ster

ror

from

rand

omw

alk

mod

el86

-.44

21.5

4-2

0-5

05

17Fo

reca

ster

ror

from

iidpr

oces

s86

113

.92

-16.

67-6

.67

2.31

7.5

14.6

7A

bsol

ute

fore

cast

erro

r86

12.6

513

.70

310

1535

Tabl

e19

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omti

nyfir

ms,

byfir

m

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r19

-2.6

10.7

3-2

0-1

0.5

-1.3

67.

149.

71M

ean

byFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

191.

867.

1-9

-3.5

74.

436.

9210

.63

Mea

nby

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r19

-8.2

28.

61-2

6-1

3.83

-5.8

3-2

.29

2.5

Mea

nby

Firm

:Bes

tca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r19

9.94

8.1

.58

6.67

1015

.38

18.7

5M

ean

byFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st19

18.1

67.

719

13.5

17.8

621

.67

32.5

Mea

nby

Firm

:For

ecas

ter

ror

19-3

.53

11.8

-15

-12

-1.7

73.

58

Mea

nby

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el19

-.78

8.19

-15

-5.5

-12.

612

.73

Mea

nby

Firm

:For

ecas

ter

ror

from

iidpr

oces

s19

1.34

9.22

-11.

67-4

.51.

57.

2712

.5M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r19

12.5

9.95

2.5

6.67

8.67

17.5

33.6

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r19

16.7

410

.54

4.54

8.76

13.5

922

.36

36.0

6St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

1912

.02

10.1

43.

314.

2110

.49

15.3

37.4

7St

d.D

ev.b

yFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

1912

.86

9.02

3.36

512

.22

17.9

625

.03

Std.

Dev

.by

Firm

:Bes

tca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r19

13.5

312

.43

2.48

5.73

10.3

114

.56

42.4

3St

d.D

ev.b

yFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st19

9.77

6.31

4.28

5.48

8.23

12.2

16.2

6St

d.D

ev.b

yFi

rm:F

orec

ast

erro

r18

12.8

910

.13

3.21

3.56

10.3

718

.85

28.2

8St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

1816

.42

21.9

72.

714.

7310

.78

15.2

835

.23

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

iidpr

oces

s18

11.4

211

.03

2.89

4.51

7.07

13.6

624

.75

Std.

Dev

.by

Firm

:Abs

olut

efo

reca

ster

ror

188.

677.

222.

083.

214.

9713

.78

19.6

8

64

Tabl

e20

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omsm

allfi

rms,

pool

ed

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r64

62.

416

.55

-15

-73

1020

Abs

olut

eva

lue

ofsa

les

grow

thra

tein

the

prev

ious

quar

ter

646

11.4

412

.18

15

1015

22Fo

reca

stsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r61

93.

710

.9-5

03

815

Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r64

6-4

.53

11.5

3-2

0-1

0-2

05

Best

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

646

9.72

12.4

40

310

1520

Span

betw

een

wor

stan

dbe

stca

sefo

reca

st64

614

.25

10.2

74

710

2030

Fore

cast

erro

r80

1-.1

512

.97

-12

-50

511

Fore

cast

erro

rfr

omra

ndom

wal

km

odel

801

.56

17.2

2-1

3-5

05

15Fo

reca

ster

ror

from

iidpr

oces

s80

1.4

310

.29

-8.4

4-3

.2.3

33.

739.

78A

bsol

ute

fore

cast

erro

r80

18.

1910

.05

02

510

20

Tabl

e21

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omsm

allfi

rms,

byfir

m

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r10

82.

549

-7-3

.06

1.35

6.53

12.5

Mea

nby

Firm

:For

ecas

tsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

83.

537.

64-4

.87

-.48

36.

812

.67

Mea

nby

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

8-5

.01

8.39

-15.

83-9

.33

-3.0

2-.3

54.

2M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

108

9.88

9.5

1.2

4.08

7.57

13.3

120

Mea

nby

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

108

14.8

98.

716.

678.

0912

.519

.45

27.5

Mea

nby

Firm

:For

ecas

ter

ror

104

1.33

14.4

4-1

3.25

-5.2

8.6

35.

513

Mea

nby

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el10

4.2

9.93

-10.

18-4

.50

510

Mea

nby

Firm

:For

ecas

ter

ror

from

iidpr

oces

s10

4.0

28.

38-9

.5-2

.78

.39

39.

17M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r10

411

.66

12.4

52.

755

8.75

1520

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r10

812

.64

103.

976.

79.

5415

.18

24.6

1St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

107

7.19

6.45

1.15

2.74

5.79

9.69

14.4

8St

d.D

ev.b

yFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

108

8.01

6.22

1.89

3.13

6.7

11.0

618

.22

Std.

Dev

.by

Firm

:Bes

tca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

88.

156.

722.

513.

686.

1310

.42

15.3

2St

d.D

ev.b

yFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st10

86.

714.

672.

33.

395.

758.

4913

.93

Std.

Dev

.by

Firm

:For

ecas

ter

ror

8012

.33

10.8

32.

865.

879.

9215

.48

22.1

8St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

8012

.78

11.5

72.

14.

89.

0316

.76

26.5

6St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

806.

596.

330

2.09

4.94

8.87

16.5

4St

d.D

ev.b

yFi

rm:A

bsol

ute

fore

cast

erro

r80

7.85

6.8

1.41

3.18

6.43

9.73

18.5

8

65

Tabl

e22

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omm

ediu

m-s

ized

firm

s,po

oled

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r1,

296

1.71

14.2

9-1

4-5

39

15A

bsol

ute

valu

eof

sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r1,

296

9.83

10.5

11

37

1220

Fore

cast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

1,28

02.

1310

.44

-90

25

10W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

1,29

6-4

.611

.79

-20

-10

-20

5Be

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r1,

296

6.92

11.4

30

25

1020

Span

betw

een

wor

stan

dbe

stca

sefo

reca

st1,

296

11.5

38.

744

510

1520

Fore

cast

erro

r80

1-.1

512

.97

-12

-50

511

Fore

cast

erro

rfr

omra

ndom

wal

km

odel

801

.56

17.2

2-1

3-5

05

15Fo

reca

ster

ror

from

iidpr

oces

s80

1.4

310

.29

-8.4

4-3

.2.3

33.

739.

78A

bsol

ute

fore

cast

erro

r80

18.

1910

.05

02

510

20

Tabl

e23

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omm

ediu

m-s

ized

firm

s,by

firm

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r18

01.

857.

73-6

.06

-1.1

82

5.53

10.9

2M

ean

byFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

179

2.67

6.47

-3.0

7-.5

72.

275.

48.

25M

ean

byFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

180

-4.1

37.

33-1

2.58

-8-3

.33

02.

76M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

180

7.63

7.64

.94

3.41

710

.48

16.6

7M

ean

byFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st18

011

.76

6.33

5.33

7.15

10.6

714

.59

20.2

1M

ean

byFi

rm:F

orec

ast

erro

r17

6-.9

29.

47-1

0-3

.45

-.37

2.5

7.5

Mea

nby

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el17

61.

3910

.62

-7.2

9-2

.48

.17

3.62

9.56

Mea

nby

Firm

:For

ecas

ter

ror

from

iidpr

oces

s17

61.

117.

84-5

.14

-1.9

4.3

52.

999.

5M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r17

69.

078.

842

46.

2711

.29

18

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r17

811

.26

9.32

3.4

5.38

8.18

13.5

723

.08

Std.

Dev

.by

Firm

:For

ecas

tsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r17

87.

387.

681.

832.

895.

139.

0215

.14

Std.

Dev

.by

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r17

88.

258.

022.

193

6.13

10.8

15.6

2St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

178

8.12

8.23

1.95

3.46

5.73

10.2

616

.43

Std.

Dev

.by

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

178

5.51

5.27

22.

744.

467.

079.

67St

d.D

ev.b

yFi

rm:F

orec

ast

erro

r16

19.

679.

792.

53.

977.

0711

.77

18.6

5St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

161

12.3

213

.92.

123.

547.

5313

.76

31.1

2St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

161

6.82

7.17

12.

234.

048.

7615

.5St

d.D

ev.b

yFi

rm:A

bsol

ute

fore

cast

erro

r16

16.

136.

11.

262.

354.

247.

7812

.02

66

Tabl

e24

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omla

rge

firm

s,po

oled

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r68

91.

912

.6-1

0-4

37

12A

bsol

ute

valu

eof

sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r68

98.

249.

721

35

1015

Fore

cast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

681

1.69

9.52

-8-1

25

10W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

689

-3.7

99.

91-1

5-1

0-2

05

Best

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

689

5.95

11.0

7-2

15

1015

Span

betw

een

wor

stan

dbe

stca

sefo

reca

st68

99.

748.

773

58

1020

Fore

cast

erro

r42

7.0

212

.33

-10

-40

510

Fore

cast

erro

rfr

omra

ndom

wal

km

odel

427

.07

16.4

2-1

1-5

05

12Fo

reca

ster

ror

from

iidpr

oces

s42

7.1

410

.22

-9.3

1-3

.21

-.21

37.

13A

bsol

ute

fore

cast

erro

r42

76.

9710

.16

02

59

15

Tabl

e25

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omla

rge

firm

s,by

firm

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r93

1.47

5.57

-5.2

5-1

1.5

4.75

8M

ean

byFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

931.

555.

09-4

-.51.

433.

66.

67M

ean

byFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

93-3

.77

5.72

-10.

71-7

.22

-3.4

3-.3

71.

67M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

935.

566.

050

3.09

4.8

7.75

11.2

9M

ean

byFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st93

9.33

5.64

3.67

5.38

811

.516

Mea

nby

Firm

:For

ecas

ter

ror

90.0

15.

2-5

-2.6

.33

3.14

5.33

Mea

nby

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el90

-.17

7.85

-9-3

.08

-.52.

887.

31M

ean

byFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

90-.0

95.

99-6

.05

-2.8

3-.3

21.

633.

62M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r90

6.96

5.46

2.13

3.4

5.5

9.5

13.3

8

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r92

9.21

6.99

3.22

4.91

7.42

10.8

316

.5St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

926.

565.

471.

692.

884.

429.

1913

.58

Std.

Dev

.by

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r92

6.9

5.3

1.81

3.14

5.66

9.04

12.0

6St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

927.

016.

372.

042.

924.

759.

3213

.61

Std.

Dev

.by

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

924.

674.

191.

692.

373.

665.

587.

81St

d.D

ev.b

yFi

rm:F

orec

ast

erro

r79

8.37

8.13

2.02

3.31

5.55

11.6

17.6

8St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

7911

.11

11.4

31.

413.

547.

7813

.44

27.3

7St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

796.

796.

78.7

12.

34.

0410

.05

16.4

4St

d.D

ev.b

yFi

rm:A

bsol

ute

fore

cast

erro

r79

5.14

5.57

1.39

2.14

3.54

5.8

11.5

5

67

Tabl

e26

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

om’b

ad’t

rend

firm

s,po

oled

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r66

4-5

.92

16.9

7-2

1-1

5-5

210

Abs

olut

eva

lue

ofsa

les

grow

thra

tein

the

prev

ious

quar

ter

664

12.7

212

.71

510

1525

Fore

cast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

648

-.36

13.7

5-1

5-5

05

10W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

664

-9.8

915

.79

-25

-20

-10

05

Best

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

664

6.53

15.6

1-1

00

510

20Sp

anbe

twee

nw

orst

and

best

case

fore

cast

664

16.4

212

.92

510

1520

30Fo

reca

ster

ror

386

-4.2

817

.58

-22

-10

-35

14Fo

reca

ster

ror

from

rand

omw

alk

mod

el38

64.

2822

.01

-15

-52

1029

Fore

cast

erro

rfr

omiid

proc

ess

386

4.95

13.1

9-8

.15

-.93.

6410

.92

19.3

3A

bsol

ute

fore

cast

erro

r38

612

.11

13.4

30

48

1525

Tabl

e27

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

om’b

ad’t

rend

firm

s,by

firm

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r10

0-6

.11

7.99

-13.

56-9

.39

-5.2

9-2

.55

-1.5

2M

ean

byFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

99.2

99.

03-8

.6-3

.57

-.83

3.33

10.6

7M

ean

byFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

100

-9.1

10.0

8-1

9.36

-14.

02-9

.09

-3.5

8.7

1M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

100

7.16

10.0

4-1

.18

16.

1411

.15

18.7

5M

ean

byFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st10

016

.26

8.18

7.67

10.4

814

.51

2028

.64

Mea

nby

Firm

:For

ecas

ter

ror

97-5

.49

11.2

-17.

5-1

1.75

-3.5

.33

5.5

Mea

nby

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el97

5.27

12.9

2-7

.50

39.

5620

Mea

nby

Firm

:For

ecas

ter

ror

from

iidpr

oces

s97

5.83

8.95

-3.5

.71

2.63

1117

.4M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r97

12.6

9.5

46

1015

.71

25

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r98

14.6

210

.49

5.42

8.01

10.8

118

.78

29.8

1St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

9710

.19

9.53

2.52

4.58

7.45

12.3

219

.94

Std.

Dev

.by

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r98

11.5

79.

563.

565.

779.

5714

.05

22.4

5St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

9811

.18

10.8

62.

884.

87.

7513

.23

22.4

5St

d.D

ev.b

yFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st98

8.15

7.68

2.89

3.93

6.41

9.12

13.9

3St

d.D

ev.b

yFi

rm:F

orec

ast

erro

r81

13.0

211

.15

4.04

6.08

10.1

616

.89

22.2

3St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

8116

.25

17.2

53.

545

9.57

18.1

941

.12

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

iidpr

oces

s81

9.76

8.69

2.12

3.54

7.07

13.1

423

.76

Std.

Dev

.by

Firm

:Abs

olut

efo

reca

ster

ror

818.

617.

552.

523.

546.

4111

.31

17.2

1

68

Tabl

e28

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

om’n

orm

al’t

rend

firm

s,po

oled

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r1,

472

1.61

10.1

1-1

0-4

26

11A

bsol

ute

valu

eof

sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r1,

472

7.28

7.19

13

510

15Fo

reca

stsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r1,

447

1.81

7.93

-50

25

10W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

1,47

2-4

.05

8.8

-15

-7.5

-20

3Be

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r1,

472

6.13

8.94

02

510

15Sp

anbe

twee

nw

orst

and

best

case

fore

cast

1,47

210

.18

7.83

35

915

20Fo

reca

ster

ror

940

-.03

10.4

3-1

0-5

05

10Fo

reca

ster

ror

from

rand

omw

alk

mod

el94

00

12.5

5-1

0-5

05

12Fo

reca

ster

ror

from

iidpr

oces

s94

0.2

8.2

-6.7

1-2

.06

.29

3.06

7.28

Abs

olut

efo

reca

ster

ror

940

6.73

7.96

02

59

15

Tabl

e29

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

om’n

orm

al’t

rend

firm

s,by

firm

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r20

11.

533.

45-1

.25

01.

63.

144.

88M

ean

byFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

201

24.

51-2

.50

1.7

3.8

5.63

Mea

nby

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r20

1-3

.93

4.98

-10

-7-3

.3-.7

51

Mea

nby

Firm

:Bes

tca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r20

16.

396.

531.

673.

185.

18.

1311

.5M

ean

byFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st20

110

.32

6.46

4.27

5.8

8.71

12.6

718

.64

Mea

nby

Firm

:For

ecas

ter

ror

197

-.22

5.42

-5-2

.75

-.41.

676

Mea

nby

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el19

7.1

25.

99-5

.83

-20

2.5

7.5

Mea

nby

Firm

:For

ecas

ter

ror

from

iidpr

oces

s19

7.1

94.

65-4

.11

-1.3

9.2

32.

24.

7M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r19

76.

855.

641.

713.

25.

438.

7113

.75

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r20

08.

866.

322.

94.

897.

5410

.92

16.1

5St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

200

5.8

4.78

1.31

2.49

4.39

7.35

12.3

6St

d.D

ev.b

yFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

200

6.48

4.96

1.59

2.58

5.23

8.76

12.4

Std.

Dev

.by

Firm

:Bes

tca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r20

06.

194.

871.

922.

675.

067.

811

.65

Std.

Dev

.by

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

200

4.54

2.94

1.64

2.33

3.93

5.76

8.73

Std.

Dev

.by

Firm

:For

ecas

ter

ror

178

8.15

6.94

2.08

3.54

6.5

10.6

115

.67

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el17

89.

358.

631.

413.

467.

0712

.518

.87

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

iidpr

oces

s17

85.

65.

67.7

12.

063.

826.

4513

.67

Std.

Dev

.by

Firm

:Abs

olut

efo

reca

ster

ror

178

4.95

4.23

1.15

2.12

3.6

6.99

10.5

4

69

Tabl

e30

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

om’g

ood’

tren

dfir

ms,

pool

ed

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r62

610

.03

16.5

9-5

310

1525

Abs

olut

eva

lue

ofsa

les

grow

thra

tein

the

prev

ious

quar

ter

626

13.3

414

.06

25

1015

29Fo

reca

stsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r61

55.

8911

.41

-52

510

15W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

626

-.95

11.2

8-1

1-5

05

10Be

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r62

611

.15

13.0

60

510

1525

Span

betw

een

wor

stan

dbe

stca

sefo

reca

st62

612

.19.

054

610

1520

Fore

cast

erro

r33

83.

9116

.49

-10

-22

919

Fore

cast

erro

rfr

omra

ndom

wal

km

odel

338

-2.7

920

.84

-17

-7-2

213

Fore

cast

erro

rfr

omiid

proc

ess

338

-3.7

12.3

-14.

57-7

-3.1

8-.1

76.

33A

bsol

ute

fore

cast

erro

r33

810

.21

13.5

10

25

1224

Tabl

e31

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

om’g

ood’

tren

dfir

ms,

byfir

m

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r99

10.1

5.46

5.2

6.57

8.83

12.3

317

.86

Mea

nby

Firm

:For

ecas

tsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r99

6.15

5.78

03.

145.

48.

2513

.57

Mea

nby

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r99

-.92

5.9

-8-3

.57

-.62.

16.

17M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

9911

.58

7.37

4.5

7.4

1014

.36

23.5

7M

ean

byFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st99

12.5

6.55

6.2

7.67

10.7

16.2

522

.5M

ean

byFi

rm:F

orec

ast

erro

r95

5.12

14.3

7-5

13.

578

14.5

Mea

nby

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el95

-3.1

610

.42

-15.

5-6

.5-3

16.

67M

ean

byFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

95-4

.09

8.1

-13.

83-6

.83

-2.7

-.79

3.33

Mea

nby

Firm

:Abs

olut

efo

reca

ster

ror

9511

.58

13.7

43

4.33

7.75

1425

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r99

13.4

411

.21

3.43

5.59

9.62

18.2

331

.13

Std.

Dev

.by

Firm

:For

ecas

tsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r99

7.76

7.39

1.6

35.

329.

7916

.26

Std.

Dev

.by

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r99

7.9

6.89

2.24

2.99

5.49

10.8

19.1

2St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

999.

027.

872.

513.

615.

9211

.51

20St

d.D

ev.b

yFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st99

6.21

4.25

2.38

2.95

58.

2112

.04

Std.

Dev

.by

Firm

:For

ecas

ter

ror

7911

.812

.53

1.41

3.78

7.59

17.9

725

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el79

15.1

915

.95

2.12

4.24

9.19

17.6

844

.55

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

iidpr

oces

s79

7.34

7.77

.71

2.52

3.69

10.6

18.4

8St

d.D

ev.b

yFi

rm:A

bsol

ute

fore

cast

erro

r79

7.57

7.76

1.15

2.28

4.68

10.4

918

.63

70

Tabl

e32

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omlo

wtu

rbul

ence

firm

s,po

oled

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r74

82.

835.

73-5

03

510

Abs

olut

eva

lue

ofsa

les

grow

thra

tein

the

prev

ious

quar

ter

748

4.99

3.99

02

57

10Fo

reca

stsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r73

92.

324.

36-2

02

57

Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r74

8-1

.87

5.3

-10

-50

14

Best

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

748

5.36

5.09

02

58

10Sp

anbe

twee

nw

orst

and

best

case

fore

cast

748

7.23

4.97

34

610

14Fo

reca

ster

ror

453

.49

4.64

-5-2

03

6Fo

reca

ster

ror

from

rand

omw

alk

mod

el45

3-.4

94.

7-5

-30

25

Fore

cast

erro

rfr

omiid

proc

ess

453

-.47

3.68

-4.9

2-2

.18

-.29

1.4

3.78

Abs

olut

efo

reca

ster

ror

453

3.51

3.07

01

35

8

Tabl

e33

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omlo

wtu

rbul

ence

firm

s,by

firm

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r10

23.

024.

81-2

.38

2.79

5.11

9.71

Mea

nby

Firm

:For

ecas

tsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

12.

573.

27-1

.1.8

32.

274

6.5

Mea

nby

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

2-1

.84

4.16

-6.8

9-3

.87

-1.2

7.6

32.

83M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

102

5.61

3.64

1.88

3.15

4.95

7.57

10.8

3M

ean

byFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st10

27.

463.

73.

674.

86.

769

12M

ean

byFi

rm:F

orec

ast

erro

r10

0.5

33.

32-3

.27

-1.0

8.3

52.

133.

8M

ean

byFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

100

-.36

3.23

-3.9

5-2

.12

-.63

1.86

3.42

Mea

nby

Firm

:For

ecas

ter

ror

from

iidpr

oces

s10

0-.5

82.

86-3

.86

-2.0

5-.3

81.

192.

32M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r10

03.

62.

271

23.

54.

665.

93

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r10

13.

931.

312.

272.

884.

024.

975.

42St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

101

2.78

1.79

1.1

1.6

2.54

3.45

4.57

Std.

Dev

.by

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

13.

412.

081.

252.

12.

874.

495.

96St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

101

3.48

2.1

1.64

2.17

2.97

4.21

5.97

Std.

Dev

.by

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

101

3.24

1.85

1.46

2.07

2.58

45.

48St

d.D

ev.b

yFi

rm:F

orec

ast

erro

r90

3.54

2.12

.71

2.02

3.42

5.13

6.46

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

rand

omw

alk

mod

el90

3.66

2.41

.71

23.

184.

956.

74St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

902.

51.

780

1.1

2.31

3.54

5.06

Std.

Dev

.by

Firm

:Abs

olut

efo

reca

ster

ror

902.

211.

44.5

51.

142.

133.

084.

03

71

Tabl

e34

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omm

ediu

mlo

wtu

rbul

ence

firm

s,po

oled

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r68

9.4

68.

72-1

0-5

15

10A

bsol

ute

valu

eof

sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r68

97.

025.

181

35

1014

Fore

cast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

672

1.36

7.19

-6-1

25

10W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

689

-4.2

98.

36-1

5-1

0-3

05

Best

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

689

5.39

7.81

-21

510

15Sp

anbe

twee

nw

orst

and

best

case

fore

cast

689

9.68

6.66

45

1010

19Fo

reca

ster

ror

403

-.53

8.49

-10

-50

510

Fore

cast

erro

rfr

omra

ndom

wal

km

odel

403

1.16

10.2

8-9

-40

613

Fore

cast

erro

rfr

omiid

proc

ess

403

.66

7.11

-5.3

-2.2

.33

3.5

8.36

Abs

olut

efo

reca

ster

ror

403

6.23

5.79

02

59

13

Tabl

e35

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omm

ediu

mlo

wtu

rbul

ence

firm

s,by

firm

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r98

.72

5.92

-5.8

6-1

.43

.29

37.

08M

ean

byFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

981.

724.

87-4

.14

-.73

1.77

4.4

6.25

Mea

nby

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r98

-3.9

85.

76-1

1.33

-6.6

7-3

.43

-.17

1.6

Mea

nby

Firm

:Bes

tca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r98

5.76

5.65

02.

575.

678.

1311

.2M

ean

byFi

rm:S

pan

betw

een

wor

stan

dbe

stca

sefo

reca

st98

9.74

4.99

5.4

6.86

8.71

11.5

15.5

Mea

nby

Firm

:For

ecas

ter

ror

96-.4

36.

61-5

.67

-3.0

50

3.17

6M

ean

byFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

961.

458.

29-5

-1.6

4.3

94

8M

ean

byFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

96.8

16.

28-4

.63

-1.5

4.5

72.

375.

62M

ean

byFi

rm:A

bsol

ute

fore

cast

erro

r96

6.72

4.64

2.83

4.37

5.81

8.05

11.3

3

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r98

7.31

1.25

6.03

6.47

7.51

8.11

8.62

Std.

Dev

.by

Firm

:For

ecas

tsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r98

5.36

3.44

1.47

3.01

4.85

7.09

9.01

Std.

Dev

.by

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r98

5.91

3.51

2.26

3.41

5.43

7.5

11.3

4St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

985.

923.

62.

53.

245.

227.

511

.06

Std.

Dev

.by

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

984.

493.

332.

082.

693.

515.

387.

58St

d.D

ev.b

yFi

rm:F

orec

ast

erro

r82

7.25

3.79

3.54

4.36

7.02

8.66

11.1

5St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

828.

025.

212.

54.

437.

4410

.41

13.6

5St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

824.

84

1.41

2.12

3.71

6.45

8.76

Std.

Dev

.by

Firm

:Abs

olut

efo

reca

ster

ror

824.

672.

661.

412.

834.

296.

217.

35

72

Tabl

e36

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omm

ediu

mhi

ghtu

rbul

ence

firm

s,po

oled

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r68

11.

1611

.66

-15

-10

210

15A

bsol

ute

valu

eof

sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r68

19.

516.

832

510

1520

Fore

cast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

661

1.34

9.52

-10

-22

510

Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r68

1-6

.59

10.9

8-2

0-1

0-5

05

Best

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

681

7.12

9.85

02

510

20Sp

anbe

twee

nw

orst

and

best

case

fore

cast

681

13.7

18.

755

810

2025

Fore

cast

erro

r40

9-.5

112

.42

-15

-90

515

Fore

cast

erro

rfr

omra

ndom

wal

km

odel

409

.41

14.0

5-1

5-5

08

17Fo

reca

ster

ror

from

iidpr

oces

s40

9.7

79.

99-8

.82

-3.1

81

5.2

10.9

2A

bsol

ute

fore

cast

erro

r40

99.

278.

270

37

1420

Tabl

e37

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omm

ediu

mhi

ghtu

rbul

ence

firm

s,by

firm

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r10

01.

155.

84-6

.07

-2.5

51.

284.

757.

38M

ean

byFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

100

1.57

5.46

-5.0

4-1

.31

1.44

4.67

7.81

Mea

nby

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

0-6

.14

6.5

-13.

79-9

.77

-5.6

9-1

.44

.7M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

100

7.34

6.09

1.13

3.27

7.11

10.4

813

.79

Mea

nby

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

100

13.4

86.

257

8.94

12.5

617

21.7

5M

ean

byFi

rm:F

orec

ast

erro

r96

-.75

7.8

-10

-5-1

4.07

8M

ean

byFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

961

9.05

-8-4

.57

05.

7510

Mea

nby

Firm

:For

ecas

ter

ror

from

iidpr

oces

s96

.86

6.57

-4.9

4-2

.35

.26

3.33

7.27

Mea

nby

Firm

:Abs

olut

efo

reca

ster

ror

969.

514.

494.

336.

568.

7812

.45

16

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r10

010

.88

2.27

8.81

9.71

10.8

212

.25

13.2

6St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

997.

454.

442.

654.

436.

669.

9812

.76

Std.

Dev

.by

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

08.

54.

922.

55.

148.

0611

.37

13.5

1St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

100

7.61

4.44

2.94

4.9

6.77

10.1

612

.42

Std.

Dev

.by

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

100

6.14

2.63

2.76

4.24

5.82

8.15

9.65

Std.

Dev

.by

Firm

:For

ecas

ter

ror

8110

.97

5.48

3.42

7.07

10.5

914

.52

18.3

8St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

8111

.76.

984.

637.

0710

.15

15.3

618

.12

Std.

Dev

.by

Firm

:For

ecas

ter

ror

from

iidpr

oces

s81

7.13

5.47

.96

3.54

5.45

11.2

113

.44

Std.

Dev

.by

Firm

:Abs

olut

efo

reca

ster

ror

817.

054.

312.

363.

56.

438.

9212

.73

73

Tabl

e38

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omhi

ghtu

rbul

ence

firm

s,po

oled

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Sale

sgr

owth

rate

inth

epr

evio

usqu

arte

r64

42.

325

.67

-30

-15

315

.530

Abs

olut

eva

lue

ofsa

les

grow

thra

tein

the

prev

ious

quar

ter

644

19.3

717

39

1525

40Fo

reca

stsa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r63

83.

917

.48

-15

-55

1020

Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r64

4-6

.65

18.6

6-3

0-1

5-5

310

Best

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

644

12.0

519

.73

-10

010

2030

Span

betw

een

wor

stan

dbe

stca

sefo

reca

st64

418

.713

.51

610

1525

35Fo

reca

ster

ror

399

-.423

.67

-27

-11

010

25Fo

reca

ster

ror

from

rand

omw

alk

mod

el39

9.7

529

.94

-31

-14

015

40Fo

reca

ster

ror

from

iidpr

oces

s39

91.

217

.86

-20.

31-9

.83

1121

Abs

olut

efo

reca

ster

ror

399

16.4

517

15

1022

40

Tabl

e39

:Sum

mar

yst

atis

tics

ofsu

rvey

answ

ers

and

deri

ved

vari

able

sfr

omhi

ghtu

rbul

ence

firm

s,by

firm

Var

iabl

eO

bsM

ean

Std.

Dev

.P1

0P2

5P5

0P7

5P9

0

Tim

ese

ries

mea

nby

firm

Mea

nby

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r10

02.

0212

.43

-10.

95-5

.62

2.61

8.14

16.8

5M

ean

byFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

100

4.54

10.2

1-6

.04

-1.2

13.

678.

8817

.08

Mea

nby

Firm

:Wor

stca

sesa

les

grow

thra

tefo

rth

ecu

rren

tqu

arte

r10

0-5

.99

10.7

7-1

6.02

-12.

71-7

.46

06.

45M

ean

byFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

100

12.7

511

.97

.65

5.14

11.6

717

.75

25.5

Mea

nby

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

100

18.7

48.

139.

9213

16.7

22.7

531

.25

Mea

nby

Firm

:For

ecas

ter

ror

97-.3

18.1

2-2

0-8

.25

-.37

6.75

15M

ean

byFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

97.3

714

.88

-15

-7.5

-.87

7.13

15.3

3M

ean

byFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

971.

1612

.08

-13.

83-6

.19

.58.

6717

.33

Mea

nby

Firm

:Abs

olut

efo

reca

ster

ror

9718

.09

14.2

66.

59.

6714

.44

20.7

135

Tim

ese

ries

stan

dard

devi

atio

nby

firm

Std.

Dev

.by

Firm

:Sal

esgr

owth

rate

inth

epr

evio

usqu

arte

r98

23.8

210

.36

14.2

516

.44

20.8

527

.87

40.4

1St

d.D

ev.b

yFi

rm:F

orec

ast

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

9814

9.98

4.09

6.76

11.6

817

.02

27.5

4St

d.D

ev.b

yFi

rm:W

orst

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

9814

.68

9.71

4.35

8.16

12.2

419

.12

26.0

8St

d.D

ev.b

yFi

rm:B

est

case

sale

sgr

owth

rate

for

the

curr

ent

quar

ter

9815

.66

11.1

55.

018.

7312

.58

20.4

129

.05

Std.

Dev

.by

Firm

:Spa

nbe

twee

nw

orst

and

best

case

fore

cast

989.

67.

683.

764.

927.

9811

.73

16.2

6St

d.D

ev.b

yFi

rm:F

orec

ast

erro

r85

19.2

414

.07

6.08

9.71

15.4

222

.82

39.5

9St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omra

ndom

wal

km

odel

8526

.41

18.4

36.

6911

.55

23.5

337

.48

53.4

1St

d.D

ev.b

yFi

rm:F

orec

ast

erro

rfr

omiid

proc

ess

8513

.77

9.33

3.35

7.07

11.8

118

.38

26.3

7St

d.D

ev.b

yFi

rm:A

bsol

ute

fore

cast

erro

r85

12.0

48.

83.

466.

0610

.36

17.0

223

.02

74

Appendix F Additional regressions for the cross sectionTa

ble

40:R

egre

ssio

nsof

tim

ese

ries

aver

ages

ofab

solu

tefo

reca

ster

rors

,for

ecas

terr

ors,

and

unco

n-di

tion

alvo

lati

litie

son

firm

char

acte

rist

ics,

byfir

m(1

)(2

)(3

)(4

)(5

)(6

)(7

)(8

)(9

)(1

0)D

epen

dent

vari

able

:av

g.ab

s.FE

avg.

abs.

FEav

g.ab

s.FE

avg.

FEav

g.FE

avg.

FEvo

lati

lity

vola

tilit

yvo

lati

lity

vola

tilit

y

Dum

my

smal

lfirm

s-0

.841

4.86

6-2

.552

0.23

2(2

.546

)(3

.003

)(2

.336

)(1

.504

)D

umm

ym

ediu

m-s

ized

firm

s-3

.437

2.61

7-4

.062∗

0.48

4(2

.331

)(2

.743

)(2

.218

)(1

.427

)D

umm

yla

rge

firm

s-5

.543∗∗

3.54

5-6

.313∗∗∗

0.49

0(2

.306

)(2

.704

)(2

.226

)(1

.399

)D

umm

y’b

ad’s

ales

grow

thtr

end

5.74

6∗∗∗

-5.2

71∗∗∗

5.96

4∗∗∗

1.72

7∗∗

(1.0

44)

(1.2

00)

(1.2

22)

(0.7

32)

Dum

my

’goo

d’sa

les

grow

thtr

end

4.73

2∗∗∗

5.34

3∗∗∗

4.43

4∗∗∗

1.59

3∗∗

(1.4

64)

(1.5

22)

(1.1

91)

(0.6

36)

Dum

my

med

ium

low

turb

ulen

ce3.

123∗∗∗

-0.9

563.

347∗∗∗

3.26

1∗∗∗

(0.5

25)

(0.7

52)

(0.1

51)

(0.2

05)

Dum

my

med

ium

high

turb

ulen

ce5.

908∗∗∗

-1.2

737.

081∗∗∗

6.81

9∗∗∗

(0.5

12)

(0.8

62)

(0.1

82)

(0.2

50)

Dum

my

high

turb

ulen

ce14

.49∗∗∗

-0.8

2720

.25∗∗∗

19.7

8∗∗∗

(1.4

66)

(1.8

69)

(1.0

71)

(1.0

05)

Con

stan

t12

.50∗∗∗

6.85

2∗∗∗

3.60

0∗∗∗

-3.5

34-0

.219

0.52

715

.55∗∗∗

8.97

6∗∗∗

3.91

2∗∗∗

2.89

5∗∗

(2.2

33)

(0.4

03)

(0.2

27)

(2.6

48)

(0.3

87)

(0.3

32)

(2.1

02)

(0.4

33)

(0.1

24)

(1.3

93)

No.

ofob

serv

atio

ns38

938

938

938

938

938

940

040

040

040

0N

o.of

firm

s38

938

938

938

938

938

940

040

040

040

0N

o.of

para

met

ers

(exc

l.in

terc

ept)

32

33

23

32

38

R-s

quar

ed0.

036

0.07

70.

320.

013

0.13

0.00

200.

029

0.08

00.

670.

68

Not

es:

avg.

abs.

FEde

note

sth

eti

me-

seri

esav

erag

eof

the

firm

-lev

elab

solu

tefo

reca

ster

ror,

avg.

FEis

the

tim

e-se

ries

aver

age

ofth

efir

m-l

evel

fore

cast

erro

r,an

dvo

lati

lity

deno

tes

the

unco

ndit

iona

lvo

lati

lity,

that

is,

the

tim

e-se

ries

stan

dard

devi

atio

nof

firm

-lev

elsa

les

grow

th.R

esul

tsfr

omO

LSre

gres

sion

s.St

anda

rder

rors

inpa

rent

hese

s,cl

uste

red

byfir

m;*

p<

0.10

,**

p<

0.05

,***

p<

0.01

.

75

Appendix G Uncertainty and change by firm characteristicsIn this appendix, we show that the V-shaped relationship between sales growth and subjectiveuncertainty, first shown in Figure 3 in Section 3.1 holds separately, and in a quantitativelysimilar manner, for all firm-level subgroups: the four firm size groups, the three growth trendgroups, and the four turbulence groups. To be specific, the solid lines in the following figuresrepresent nonparametric regression lines. They are the predictions from a kernel-weightedlocal polynomial regression of degree zero with an Epanechnikov kernel where the bandwidthwas selected based on the rule of thumb suggested by Fan and Gijbels (1996). The dashed linesdepict the predicted values from a piecewise linear regression of subjective uncertainty on pastsales growth, with a break at zero.

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. tiny firmsLinear full sample Linear tiny firms

Tiny firms

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. small firmsLinear full sample Linear small firms

Small firms

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. medium sized firmsLinear full sample Linear medium sized firms

Medium sized firms

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. large firmsLinear full sample Linear large firms

Large firms

Figure 10: Relationship between subjective uncertainty (span) and sales growth in the previousquarter for four different firm size groups: tiny, small, medium, and large firms (full sample= blue and thin lines, subsample = red and bold lines).

76

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. trend 'bad'Linear full sample Linear trend 'bad'

Trend 'bad'

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. trend 'normal'Linear full sample Linear trend 'normal'

Trend 'normal'

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. trend 'good'Linear full sample Linear trend 'good'

Trend 'good'

Figure 11: Relationship between subjective uncertainty (span) and sales growth in the previousquarter for three different firm trend growth groups: ‘bad’, ‘normal’, and ‘good’ trend growth(full sample = blue and thin lines, subsample = red and bold lines).

77

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. turbulence lowLinear full sample Linear turbulence low

Turbulence low

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. turbulence med. lowLinear full sample Linear turbulence med. low

Turbulence medium low

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. turbulence med. highLinear full sample Linear turbulence med. high

Turbulence medium high

1020

3040

5060

Subj

ectiv

e un

certa

inty

(spa

n)

-100 -50 0 50 100Sales growth rate in the previous quarter

Nonparam. full sample Nonparam. turbulence highLinear full sample Linear turbulence high

Turbulence high

Figure 12: Relationship between subjective uncertainty (span) and sales growth in the previousquarter for four different firm volatility groups: low, medium low, medium high, and highvolatility (full sample = blue and thin lines, subsample = red and bold lines).

78

Appendix H Modelling conditional volatilityIn this appendix, we provide details on how we estimate the dynamics of “statistical” uncer-tainty experienced by firms, that is, the two preliminary steps summarized in Section 7. Thefirst step, documented in Appendix H.1, is to calculate two sets of forecast errors: one basedon the subjective forecast made by firms, corrected for firm-specific bias, and one based ona statistical model forecast. The second step, documented in Appendix H.2, is to estimatedynamic models of conditional volatility for both the bias-adjusted subjective forecast errorsand for the statistical forecast errors.

H.1 Bias-adjusted subjective and statistical forecast errors

In order to clean observed forecast errors of firm-specific bias, we estimate regressions ofsurvey-provided forecast errors on fixed characteristics and use the resulting residuals as ourcleaned errors. In terms of the representation (5) from Section 3.2, this removes the part of thebias b(si

t, xi) that depends on fixed characteristics xi. We do this in order to focus on beliefdynamics, that is, we are interested in the response of span to temporary surprises experiencedby firms, not surprises firms routinely experience because they make biased forecasts.

Specifically, we regress the survey-provided forecast errors on our previously defined size,trend, and turbulence dummies as well as their pairwise interactions. To prevent overfitting,we apply the LASSO estimator (Tibshirani, 1996) to select a subset of relevant regressors.31

We choose the LASSO tuning parameter τ by minimizing Mallows’s Cp statistic as suggestedby Efron et al. (2004). The LASSO then selects 11 predictors. In particular, denote the highand low growth dummies by gd1 and gd3, the medium low, medium high and high volatilitydummies by vd2, vd3, and vd4, and the small, medium and large size dummies by sd2, sd3, andsd4, respectively. The selected predictors are then gd1, gd3, as well as the interactions sd2 · gd1,sd3 · gd1, sd3 · gd3, sd2 · vd2, sd2 · vd3, sd3 · vd4, sd4 · vd4, gd1 · vd4, gd3 · vd4.

To obtain a bias-adjusted forecast error, we compute the OLS residuals of a regression ofthe forecast error on these predictors. In doing so, we follow the recommendations in Belloniand Chernozhukov (2013) who argue that the LASSO should select the relevant regressors andOLS should estimate the regression coefficients (see also Lehrer and Xie, 2017, for a relatedapplication). We note that the distributional properties of these bias-adjusted forecast errorsare very similar to those of the raw forecast error directly from the survey data. In addition,the result, documented in Section 6, that the V-shaped nexus between previous-quarter salesgrowth and subjective uncertainty is robust to including forecast errors, and also the resultthat previous-quarter sales growth drive out forecast errors, both hold when we use thesebias-adjusted forecast errors.

To provide an alternative benchmark for firm-level forecast errors, we construct a set of

31The LASSO is a standard shrinkage estimator popular in “big data” analysis (Varian, 2014) as it recovers thecorrect (sparse) model with high probability (Hastie et al., 2017). By requiring that the L1 norm of the coefficientvector does not exceed a certain threshold, say, τ, the LASSO restricts many coefficients to zero and thus helps tobalance the bias-variance tradeoff. This is why the LASSO and related estimators are widely applied in data-richenvironments (Bai and Ng, 2008; Manzan, 2015; Elliott and Timmermann, 2016).

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statistical forecast errors by using statistical forecasting models an econometrician would pre-sumably consider. In particular, we regress sales growth on its own lag as well as size, growth,and turbulence dummies. We allow for an asymmetric response to past growth, as in ourspan regressions. We also note that, since trend and turbulence are defined using sample mo-ments, they are, strictly speaking, not part of the information set of a firm. At the same time,firms have longer samples than we have that speak to their trend growth and volatility. Ourassumption here is that trend and turbulence reflect medium-term prospects known to firms.

The regression coefficients of these various forecasting models are reported in Table 41. Thespecifications in columns (1)-(3) allow for an asymmetric effect of past sales growth, whereasthe specifications in columns (4)-(6) restrict this effect to be symmetric. The forecast errorsfrom all six specifications are highly correlated, with correlation coefficients at 0.93 or above.Since the model selection criteria AIC and BIC favor specification (1), we report, in whatfollows, the results based on that specification.

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Table 41: Regressions of sales growth on past sales growth and firm characteristicsDependent variable:sales growth in quarter t (1) (2) (3) (4) (5) (6)

Negative sales growth in quarter t− 1 -0.269∗ -0.141 -0.130(0.141) (0.145) (0.128)

Positive sales growth in quarter t− 1 -0.0357 0.0392 0.0495(0.0907) (0.0978) (0.0856)

Sales growth in quarter t− 1 -0.132∗∗∗ -0.0365 -0.0247(0.0414) (0.0455) (0.0462)

Dummy small firms 5.314∗∗∗ 7.798∗∗∗ 5.298∗∗∗ 7.776∗∗∗

(1.815) (2.255) (1.747) (2.192)Dummy medium-sized firms 4.996∗∗∗ 7.408∗∗∗ 5.082∗∗∗ 7.432∗∗∗

(1.669) (2.136) (1.607) (2.073)Dummy large firms 5.000∗∗∗ 7.228∗∗∗ 5.013∗∗∗ 7.159∗∗∗

(1.689) (2.127) (1.632) (2.057)Dummy ’bad’ sales growth trend -7.143∗∗∗ -6.726∗∗∗

(0.967) (0.936)Dummy ’good’ sales growth trend 8.005∗∗∗ 8.253∗∗∗

(1.095) (1.112)Dummy medium low turbulence -1.717∗∗∗ -2.068∗∗ -1.434∗∗∗ -1.827∗

(0.572) (0.959) (0.500) (0.929)Dummy medium high turbulence 0.427 -0.882 0.973 -0.411

(0.833) (1.117) (0.642) (0.968)Dummy high turbulence -0.143 -0.274 1.343 0.954

(1.561) (1.691) (1.149) (1.384)Intercept -3.243∗∗ -5.059∗∗ 1.164 -2.881∗ -4.663∗∗ 2.015∗∗∗

(1.610) (2.092) (0.811) (1.534) (2.011) (0.439)

No. of obs. 1,329 1,329 1,329 1,329 1,329 1,329No. of firms 292 292 292 292 292 292R-squared 0.12 0.021 0.0047 0.12 0.018 0.001AIC 10696.7 10837.8 10848.1 10702.2 10839.9 10851.5BIC 10753.8 10884.5 10863.7 10754.1 10881.5 10861.9

Notes: results from OLS regressions. They are based on the baseline sample forecast as defined in Table 14 inAppendix C which includes all quarter-firm observations for which a forecast is available but not necessarily aspan, that is, 1,664 observations. In addition, because we want to compute forecast errors with these statisticalforecasts and use them in dynamic models, we have to eliminate isolated forecast errors that have no forecasterror surrounding them and thus end up with 1,329 observations. Standard errors in parentheses, clustered byfirm. * p < 0.10, ** p < 0.05, *** p < 0.01.

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H.2 Estimating conditional volatility

We now proceed to estimate conditional volatility models on both types of forecast errors:biased-corrected subjective forecast errors and statistical forecast errors. We have operational-ized subjective uncertainty with span, the difference between best and worst case scenarios.A natural “objective” counterpart would be the length of a forecast interval constructed bythe econometrician, for example, the difference between an upper and lower quantile of theconditional distribution of forecast errors. In the broad class of distributions which belong tothe location-scale family that forecast interval length is simply a multiple of the distribution’sstandard deviation. We, therefore, choose the conditional standard deviation of forecast er-rors as our measure of “objective” uncertainty. We select and estimate a volatility model thatoptimally describes the data as indicated by information criteria.

Let eit+1 be the (bias-adjusted) forecast error of firm i in the quarter beginning in t, and

denote its conditional standard deviation by σit , which is the econometrician’s implementation

of σ(si

t, xi) from equation (5). Our choice of functional form mirrors our analysis of subjectiveuncertainty: we write σi

t as a function of past growth, past forecast errors and fixed firmcharacteristics, summarized in a vector zi

t. We thus use a restricted version of the powerGARCH model (Ding et al., 1993; Ding and Granger, 1996; Karanasos and Kim, 2006). Whereasthe unrestricted power GARCH model conditions (σi

t)p on past information, I i

t , where p is apower coefficient to be estimated, we impose the restriction p = 1 to model the conditionalstandard deviation.

Our conditional volatility model then has the general form:32

eit+1 = σi

t εit+1, εi

t+1|I it ∼ N(0, 1) (8)

with a conditional standard deviation equation:

σit = exp(β0 + β′1zi

t) + α1(|eit|+ γei

t) + α2σit−1. (9)

The conditional volatility equation (9) allows, in some specifications, for an asymmetriceffect of the past absolute forecast error measured by the coefficient γ, because asymmetrywas found to be relevant in explaining subjective uncertainty.33 The conditional volatilityequation also contains two types of explanatory variables through an exponential link functionwhich ensures that effects are always positive. The first type consists of size, trend, andturbulence dummies which are essentially time-invariant and thus control for different levelsof conditional volatility for subgroups of firms. Our analysis in the main text indicated thatthese dummies are sufficient to capture the bulk of time-invariant heterogeneity in subjective

32In the estimation, the mean equation (8) includes an intercept, µ, to account for a nonzero sample mean thatarises because we apply the bias adjustment of the forecast error to all 1,329 observations but have to estimatethe volatility model on an effective sample of those 949 observations for which a lag is available.

33The empirical unconditional distribution of the bias-adjusted subjective forecast errors is essentially symmet-ric with a sample skewness of 0.1. A test of the null hypothesis that the population skewness is zero cannot berejected (p-value of 0.2). Taking both reasons on balance, we, therefore, experiment with both symmetric andasymmetric specifications.

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uncertainty. The second type includes positive and negative sales growth in the previousquarter which we found to be highly relevant to explain the dynamics of subjective uncertainty.

To find a reliable parsimonious specification, we estimate several restricted versions of(8)-(9) by maximum likelihood. Specification (1) adds no additional control variables (β1 =0), specifications (2)-(4) allow, respectively, only for size, trend, and turbulence dummies,specification (5) allows for only positive and negative sales growth rate in the previous quarter,and (6) adds all variables together. All specifications are estimated either assuming symmetriceffects of past forecast errors (γ = 0) or allowing for asymmetry (γ unrestricted).

To select among these specifications, we use two information criteria, AIC and BIC, whichare commonly used in applied work with GARCH models (Nelson, 1991; Zivot, 2009). In fi-nite samples, the BIC typically favors overly sparse models, while the AIC picks models witha more generous number of parameters; see Efron et al. (2004, pp. 230-235) for a general dis-cussion and Lütkepohl (2005) for asymptotic and simulation evidence in a time series context.Hence, the models chosen by AIC and BIC may be thought of giving upper and lower boundsin terms of richness of parametrization.

Table 42: Model selection criteria for different specifications of the conditional volatility model

Symmetric (γ = 0) Asymmetric (γ 6= 0)

Specification k AIC BIC k AIC BIC

(1) no controls 4 7,256.39 7,275.81 5 7,253.13 7,277.41(2) only size dummies 7 7,249.95 7,283.94 8 7,248.10 7,286.94(3) only growth trend dummies 6 7,173.30 7,202.43 7 7,174.46 7,208.45(4) only turbulence dummies 7 6,901.40 6,935.38 8 6,903.38 6,942.22(5) only sales growth rate 6 7,106.90 7,136.03 7 7,104.63 7,138.62(6) all controls 14 6,871.59 6,939.57 15 6,870.96 6,943.79

Notes: k denotes the number of parameters. All specifications are estimated by maximum likelihood using949 observations and 380 pre-sample observations on which we condition as explained in Appendix C. Modelselection is for the bias-corrected subjective forecast errors.

The selection results for the bias-corrected subjective forecast errors reported in Table 42suggest that the inclusion of turbulence dummies, whether by themselves in specification (4)or jointly with the other control variables in specification (6), is essential for model fit: all otherspecifications generate much larger information criteria. Deciding between specifications (4)and (6) is less obvious. In both the symmetric and the asymmetric case, the AIC favors theinclusion of all controls while the BIC picks the turbulence dummies alone. However, thedifferences in terms of AIC are large (29.81 and 32.42) while the differences in terms of BICare small (4.19 and 1.57). Given that the BIC tends to select overly parsimonious models andbased on the classification of Kass and Raftery (1995) that only BIC differences of more thansix are “strong”, we prefer, on balance, specification (6).

Since neither information criterion gives us clear guidance whether to prefer the symmetricor the asymmetric specification, we report, in Table 43, the coefficient estimates for both.It turns out that the asymmetry parameter γ is not statistically different from zero while

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Table 43: Conditional volatility equation (9) estimated by maximum likelihoodDependent variable: Firms’ forecast errors Statistical forecast errors

(1) (2) (3) (4)

Mean equation

Intercept (µ) 0.298 0.291 0.171 0.122(0.252) (0.251) (0.270) (0.281)

Volatility equation: baseline parameters

Lagged absolute FE (α1) 0.0852∗ 0.102∗ 0.00830 0.00918(0.0511) (0.0538) (0.0682) (0.0579)

Lagged volatility (α2) 0.235∗∗∗ 0.215∗∗∗ 0.236∗∗ 0.229∗∗∗

(0.0874) (0.0799) (0.0921) (0.0824)Asymmetry (γ) 0 0.478 0 3.359

(.) (0.317) (.) (21.85)

Volatility equation: parameters of predetermined regressors

Dummy medium low volatility 0.504∗∗∗ 0.506∗∗∗ 0.518∗∗∗ 0.517∗∗∗

(0.0867) (0.0886) (0.0725) (0.0708)Dummy medium high volatility 0.794∗∗∗ 0.797∗∗∗ 0.873∗∗∗ 0.879∗∗∗

(0.0873) (0.0874) (0.0684) (0.0674)Dummy high volatility 1.336∗∗∗ 1.322∗∗∗ 1.519∗∗∗ 1.520∗∗∗

(0.110) (0.111) (0.101) (0.0964)Negative sales growth in t− 1 -0.0286∗∗∗ -0.0309∗∗∗ -0.0259∗∗∗ -0.0278∗∗∗

(0.00910) (0.00879) (0.00792) (0.00801)Positive sales growth in t− 1 0.0131∗∗ 0.0107∗ 0.00710 0.00468

(0.00515) (0.00576) (0.00522) (0.00605)Dummy small firms -0.139 -0.138 -0.142 -0.127

(0.118) (0.119) (0.0867) (0.0884)Dummy medium-sized firms -0.223∗∗ -0.230∗∗ -0.182∗∗ -0.170∗∗

(0.105) (0.105) (0.0744) (0.0766)Dummy large firms -0.248∗∗ -0.269∗∗ -0.170∗∗ -0.156∗

(0.111) (0.111) (0.0822) (0.0871)Dummy ’bad’ sales growth trend 0.194∗∗ 0.167∗ -0.0281 -0.0456

(0.0882) (0.0858) (0.0652) (0.0673)Dummy ’good’ sales growth trend 0.199∗∗ 0.209∗∗ 0.0648 0.0818

(0.0948) (0.0958) (0.0815) (0.0857)Intercept (β0) 1.101∗∗∗ 1.125∗∗∗ 1.204∗∗∗ 1.199∗∗∗

(0.180) (0.167) (0.151) (0.146)

Number of observations 949 949 949 949Number of firms 292 292 292 292

Notes: All specifications are estimated by maximum likelihood using 949 observations and 380 pre-sample ob-servations on which we condition as explained in Appendix C. Standard errors in parentheses, clustered by firm.* p < 0.10, ** p < 0.05, *** p < 0.01.

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the estimates of the other coefficients are largely unaffected by restricting it to zero. Wethus conclude that the symmetric specification (6) is a sufficient description of the conditionalvolatility process that drives the data.

For the statistical forecast errors we fit the same symmetric and asymmetric volatility mod-els as for the firms’ subjective forecast errors, see columns (3) and (4) of Table 43. Again, theasymmetry parameter is not significantly different from zero, and restricting it to zero leavesthe other parameter estimates essentially unchanged. Therefore, we take again the symmetricspecification as a sufficient description of the conditional volatility process that characterizesthe statistical forecast errors.

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