International Journal of Forecasting 29 (2013) 469–478

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International Journal of Forecasting

journal homepage: www.elsevier.com/locate/ijforecast

Are macroeconomic variables useful for forecasting thedistribution of U.S. inflation?Sebastiano Manzan a, Dawit Zerom b,∗

a Department of Economics & Finance, Baruch College, CUNY, United Statesb California State University at Fullerton, United States

a r t i c l e i n f o

Keywords:DistributionEconomic indicatorsInflationPredictabilityQuantiles

a b s t r a c t

Much of the inflation forecasting literature examines the ability of macroeconomicindicators to predict the mean inflation accurately. For the period after 1984, the existingempirical evidence largely suggests that the likelihood of predicting inflation accuratelyusing macroeconomic indicators is no better than a random walk model. We expandthe scope of inflation predictability by exploring whether macroeconomic indicators areuseful in predicting the distribution of inflation. We consider six commonly-used macroindicators and core/non-core versions of the Consumer Price Index (CPI) and the PersonalConsumption Expenditure (PCE) deflator as measures of inflation. Based on monthly dataand for the forecast period after 1984, we find that some of the macro indicators, suchas the unemployment rate, housing starts and the term spread, provide significant out-of-sample predictability for the distribution of core inflation. An analysis of the quantiles of thepredictive distribution reveals interesting patterns which would otherwise be ignored byexisting inflation forecasting approaches that rely only on forecasting the mean. We alsoillustrate the importance of inflation distribution forecasting in evaluating some eventswhich are of policy interest by focusing on predicting the likelihood of deflation.© 2013 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

s. P

1. Introduction

Forecasting the behavior of inflation plays a central rolein the conduct of monetary policy, due to the lagged im-pact of the central bank’s actions on economic activity.Thus, accuracy is important when predicting the effects ofthe many shocks that hit the economy on the future dy-namics of inflation. The standard approach for forecastinginflation has been the Phillips curve (PC) model, which,in its expectation-augmented version, assumes a trade-off between unexpected inflation and unemployment, or,more generally, indicators of real economic activity. De-spite its long-lasting success, recent empirical evidence onthe effectiveness of the PC model has been far from unani-mous. Stock and Watson (1999) provide a detailed study

∗ Corresponding author.E-mail address: dzerom@fullerton.edu (D. Zerom).

0169-2070/$ – see front matter© 2013 International Institute of Forecasterhttp://dx.doi.org/10.1016/j.ijforecast.2013.01.005

of the out-of-sample forecast accuracy of the PC by us-ing an extensive set of macroeconomic variables. Using theforecast evaluation period January 1970–September 1996,their conclusion is that PC models have better forecastingperformances (compared to univariate time seriesmodels)when using the unemployment rate as well as other lead-ing indicators of economic activity (e.g., the output gap andcapacity utilization). They also find that combining infor-mation or models might provide better results than sim-ply relying on a few indicators. However, Atkenson andOhanian (2001) provide empirical evidence suggesting theopposite, albeit for a different forecast evaluation period,January 1984–November 1999, where they report thatPC models are no better than the naïve model, which as-sumes that the inflation expected over the next 12monthswill be equal to the inflation experienced over the pre-vious 12 months. Fisher, Liu, and Zhou (2002) conduct asystematic comparison of the forecasting accuracy at theone-year horizon in different sub-periods, and find that the

ublished by Elsevier B.V. All rights reserved.

470 S. Manzan, D. Zerom / International Journal of Forecasting 29 (2013) 469–478

PC forecasts only outperformed the naïve forecasts in the1977–1984 window. There is evidence that the decliningpredictive power of macroeconomic indicators is typicalalso of other variables (see D’Agostino, Giannone, & Surico,2006; and Rossi & Sekhposyan, 2010). For a comprehensivesurvey and a discussion of the outstanding issues in infla-tion forecasting, see Stock and Watson (2008).

In this paper, departing from the existing focus onconditional mean forecasting, we explore whether leadingindicators of economic activity are useful in predicting thedistribution of future inflation. Despite the extensive bodyof literature on inflation forecasting which is available,little or no attention has been paid to the examinationof whether indicators of economic activity carry usefulinformation about the dynamics of higher moments,beyond the mean. For example, having some idea of theconditional second-order moment of future inflation canbe vital to an assessment of the risk to inflation stabilityas a result of macroeconomic shocks. Greenspan (2004)discusses this issue in the following terms: ‘‘Given ourinevitably incomplete knowledge about key structuralaspects of an ever-changing economy and the sometimesasymmetric costs or benefits of particular outcomes, acentral bank needs to consider not only the most likelyfuture path for the economy, but also the distribution ofpossible outcomes about that path. The decision-makersthen need to reach a judgment about the probabilities,costs, and benefits of the various possible outcomes underalternative choices for policy’’ (p. 37). While the averagefuture inflation may signal the direction of the economy,it cannot help policy-makers to evaluate the risks ofdeviations from the most likely path, or the cost to theeconomy of such deviations. In a recent paper, Kilianand Manganelli (2008) introduced a model in which themonetary policy maker is viewed as a risk manager whois trying to balance the risks to inflation and outputstability. In this framework, if the preferences of the policymaker are assumed to be quadratic and symmetric, thenthe only relevant moment (of the inflation and outputdistributions) is the conditional mean. However, theyprovide evidence of a departure of the preferences fromsuch a benchmark. All of the elements above point to thesuggestion that forecasting the distribution of inflationrepresents a relevant tool in the conduct of monetarypolicy. In fact, formany years the Bank of England has beenpublishing the so-called ‘‘fan charts’’, which represent theBank’s subjective forecasts about the future distribution ofinflation.

We use quantile regression to incorporate macroeco-nomic variables into the prediction of the conditionaldistribution of future inflation. The approach considersseveral conditional quantiles of future inflation, and by do-ing so, offers more flexibility (than, for example, the con-ventional PC models) in capturing the possible roles ofmacroeconomic indicators in predicting the different partsof the inflation distribution. For instance, one may be ableto investigate whether some periods of low or high infla-tion are driven by some macroeconomic indicators. Suchinformation surely cannot be delivered by PC-type mod-els that deal only with predicting the average inflation.We find strong empirical evidence of predictability for U.S.

core monthly inflation using indicators of economic activ-ity, particularly the unemployment rate, housing starts andthe term spread. Importantly, the empirical findings applyto a forecast evaluation period that is intentionally chosento be post-1984, when the existing literature shows thatthe macroeconomic indicators were not relevant for pre-dicting future average inflation (see Stock&Watson, 2007).We attribute this result to the ability of our approach toaccount for the varying predictive effects of economic in-dicators on core inflation at different quantiles of its distri-bution. For example, when considering the unemploymentrate, we find that it adds no predictive value to the averagecore inflation (consistent with the prevailing empirical ev-idence), while providing statistically significant improve-ments to the left tail (or lower quantiles) of the inflationforecast distribution. Reflecting this finding,whenweuse aforecast density that incorporates the unemployment rateto predict the likelihood of deflation (out of sample), theapproach appropriately assigns a probability close to zeroto the likelihood of deflation for the period 1996–2007.This clearly illustrates the way in which a macroeconomicvariable may offer weak or no predictability of inflation inthe conditional mean sense, but can still help improve theforecast density of inflation.

A possible explanation for these findings is that the pub-licmight form inflation expectations that aremore respon-sive to macroeconomic news when inflation is in the tailsof the distribution. This can happen if themonetary policy-maker has a target (either implicit or explicit) which is lo-cated near the center of the distribution. When inflationfluctuates around the center of the distribution, expecta-tions are well-anchored to the target, in the sense that thepublic believes that the central banks will be able to main-tain price stability. However, when inflation deviates fromthe target significantly, the publicmight distrust the abilityofmonetary policy to bring inflation back to the target, andthus may form expectations that rely more on macroeco-nomic variables which represent more reliable indicatorsof future inflation.

The number of papers that deal with density forecastsof inflation is quite limited, and much of the existing liter-ature focuses largely on evaluating the absolute accuracyof inflation density forecasts with respect to dynamicaland distributional mis-specifications. For example, Clark(2011) considers a Bayesian VAR model with stochasticvolatility, and finds that accounting for the time variationin the volatility is essential to the production of accuratedensity forecasts (in an absolute sense) of U.S. inflation,especially for the post-1985 period. A similar conclusionwas also reached by Jore, Mitchell, and Vahey (2010) us-ing AR and VAR models that allow for structural breaks.On the other hand, our approach is a model selection one,where we examine the ability of macroeconomic indica-tors to improve density forecasts of U.S. inflation. We dothis by comparing the relative accuracy of the density fore-casts (using scoring rules defined from density and quan-tile forecasts) implied by a quantile regression approachthat incorporates a macroeconomic indicator with that ofa benchmark model that does not include the macroeco-nomic indicator. Amisano and Giacomini (2007) also con-sider a model selection approach, but focus on the influ-ence of nonlinearity on the accuracy of U.S. inflation den-sity forecasts. In particular, they compare the relative accu-racy of the out-of-sample density forecasts implied by the

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linear model with that of Markov switching Phillips curvemodels.

The rest of the paper is organized as follows. In Sec-tion 2 we discuss econometric approaches, including ourproposal, to the estimation of the conditional forecast dis-tribution of future inflation. Section 3 outlines a test of pre-dictive accuracy that is used to evaluate the conditionaldistributionmodels discussed in Section 2. In Section 4, wepresent the empirical findings of the paper, with discus-sion. Finally, Section 5 concludes.

2. Econometric methodology

We denote the annualized inflation over an h-monthperiod by Y h

t = (1200/h)[log Pt − log Pt−h], and the one-month annualized inflation by Yt = 1200[log Pt−log Pt−1],where Pt is the level of the price index in month t . Also,let Xt be some indicator of real economic activity, suchas the unemployment rate. A baseline specification whichis often used in forecasting inflation is the Phillips curve(PC) model. The PC model (see Stock & Watson, 1999)postulates that changes in the h-month inflation, Y h

t+h,depend on recent changes in the one-month inflation, aswell as past and present values of a candidate economicindicator,

Y ht+h − Yt = µh

0 + βh(L)∆ Yt + γ h(L)Xt + Upct+h, (1)

whereµh0 is a constant,β

h(L) and γ h(L) are lag polynomialswritten in terms of the lag operator L, and Upc

t+h is theerror term. Note that in the above specification we followStock and Watson (2007) and assume that the inflationrate Yt has a unit root, although other papers take theopposite view and consider inflation to be stationary (e.g.,Ang, Bekaert, & Wei, 2006).

In evaluating the forecasting performance of the PCmodel, it is often compared with two univariate models:the autoregressive (AR) model and the naïve (randomwalk) model. Although simple, these two time seriesmodels are very competitive benchmarks. The AR modelis a special case of the PC model where no data on presentand past values of Xt are included, i.e.,

Y ht+h − Yt = µh

0 + βh(L)∆ Yt + Uart+h, (2)

where Uart+h is the error term. The naïve model (see

Atkenson & Ohanian, 2001) specifies that the expectedinflation over the next h months is equal to inflation overthe previous hmonths, i.e.,

Y ht+h = Y h

t + Uaot+h, (3)

where Uaot+h is the error term. In the rest of the paper, we

will refer to the naïve model as the AO model.

2.1. Proposed approach

Note that in the PC model, the economic indicatoris assumed to affect the inflation rate only through the

conditional mean, which implies that its error Upct+h should

be independent of the past and present values of Xt(and lags of ∆Yt ). In other words, any effect of themacroeconomic variable on the conditional distribution offuture inflation (Y h

t+h) will be limited to the conditionalmean. Thus, the PC approach ignores the possibility ofeconomic indicators carrying useful information aboutthe dynamics of higher moments, and hence helpingto improve the accuracy of the forecast distribution ofinflation. To circumvent this limitation of PC models,we focus instead on the forecast distribution of Y h

t+h,conditional on the available information set at time t .

Because the existing empirical evidence (for the periodafter 1984) largely suggests that the likelihood of predict-ing inflation accurately using economic indicators is nobetter than a random walk model, we constrain the con-ditional mean of the forecast distribution of Y h

t+h to followthe AO model (see Eq. (3)). Let the past and present val-ues of a particular economic indicator variable, X , be de-noted by the vector Xt = (Xt , . . . , Xt−q+1). The use of theAO model implies that Xt does not carry any relevant in-formation for predicting the mean future inflation. How-ever, unlike the AO model, we allow Xt to have an effecton the higher-order moments of Y h

t+h, and, more generally,on the conditional distribution of Y h

t+h, which is not per-mitted in the PC model. Under the proposed set-up, if wefind (relative) predictability in the conditional distributionof Y h

t+h using Xt , but no (relative) predictability by the PCmodel compared to the benchmark AOmodel, we can thenconclude that the predictability is occurring at parts of thedistribution beyond the conditional mean.

The proposed approach is denoted by AO-U|X , wherethe conditional mean of Y h

t+h follows the AO model and Xtis used tomodel the conditional quantile of the AO forecasterror, Uao

t+h, which can be considered to be stationary.We should note that the direct nature of our forecastsintroduces overlapping observations in Uao

t+h which canbe assumed to follow a MA(h − 1) process. This serialcorrelation may impact the accuracy of the estimatedconditional quantiles,1 and we therefore decided to carryout the conditional quantile step using the residuals(denoted by Uao

t+h) from a fitted MA(h − 1) model on Uaot+h.

Denote the α ∈ (0, 1) conditional quantiles of Uaot+h and

Uaot+h conditional on the information set available at time

t by Qt+h|t(α) and Qt+h|t(α), respectively. We estimateQt+h|t(α) using the quantile regressionmodel (see Koenker& Bassett, 1978),

Qt+h|t(α) = δ0,α +

qk=1

δk,αXt−k+1. (4)

Although the local effect of Xt−k+1 on the α-quantile isassumed to be linear, the model is very flexible, because

1 Since the quantiles are affected by the variance, the MA(h − 1)structure has the potential to bias the quantile forecasts, especially forlarge values of h. To the best of our knowledge, there are no methodscurrently available for dealing with MA errors in a quantile regressionframework.

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each slope coefficient δk,α is allowed to differ acrossquantiles. This is a useful property, since it providesguidance as to where in the distribution of Y h

t+h theindicator Xt has a significant effect.

Finally, we construct the desired conditional distribu-tion (or density) of Y h

t+h from the conditional quantile esti-mates Qt+h|t(α), where the latter is obtained by taking the

product of ˆQ t+h|t(α) in Eq. (4) and the standard deviationof the fittedMAmodel (see above). In our implementation,the sequence α is chosen to be between 0.05 and 0.95 in0.01 intervals. Further, to guarantee monotonicity of theconditional quantile estimates, and hence validity of theimplied conditional distribution, we apply the proposal ofChernozhukov, Fernandez-Val, andGalichon (2010),whichconsists of rearranging the (possibly crossing) Qt+h|t(α)into a monotonic rearranged quantile curve.

3. Measuring the relative predictability

Suppose that we have a pair of models, k and j,where we consider model k as the benchmark. We areinterested in measuring the relative accuracy of modelj when h-step-ahead out-of-sample density or quantileforecasts are available.2 Our benchmark is the AO model(see Eq. (3)).3 The accuracy of the benchmark modelis compared separately with those of two alternativemodels which incorporate the effects of macroeconomicindicators. For the latter, we consider the PC model(see Eq. (1)) and the proposed AO-U|X model. Weadopt a rolling window scheme when generating out-of-sample density or quantile forecasts from the variousmodels. The first forecast is for January 1985, and themodels are estimated on the windows 1959:1–1984:1and 1959:1–1984:7, for h = 12 and 6, respectively.It should be noted that the forecast evaluation periodis chosen deliberately to be post-1984, where currentresearch shows that macroeconomic indicators do notadd much to the predictability of the mean inflation.The next forecast is for February 1985, and the length ofthe estimation window is kept constant by dropping one

2 An alternative approach consists of evaluating the absolute accuracyof a model’s forecasts rather than the accuracy relative to a benchmark.Berkowitz (2001) provided an early paper in the literature proposingseveral tests based on the Probability Integral Transform (PIT), anda recent application to macroeconomics is provided by Mitchell andWallis (2011). In results not reported in this paper, we evaluated theforecasts using absolute criteria, specifically the Anderson–Darling testfor uniformity of the PIT, an independence test of the transformed PIT,and the efficiency test of the transformed PIT. Overall, the results indicatethat the absolute predictability tests might be helpful in indicating themisspecification of the various forecastingmodels. However, they are notvery useful in selecting the most appropriate model in situations wheremost forecasting models show signs of misspecification, as is the case inthe current study. We have therefore decided to discuss and report onlythe relative forecast evaluation criteria.3 To ensure consistency, we estimate the forecast densities and

quantile forecasts in a similar fashion to that of the AO-U|X model, exceptthat the quantile regression in Eq. (4) is based on regressing the errorsUaot+h on a constant only (and hence, their respective error distributions

are assumed to be independent of economic indicators). The PC model istreated similarly.

observation at the beginning of the sample. For example,the estimation window is updated to (1959:2–1984:2)for h = 12. We continue doing this until forecasts forDecember 2007 have been computed.4 In this manner, wecollect a total of n = 276 density or quantile forecasts foreach model.

We use scoring rules to compare the pairs of densityor quantile forecasts. A scoring rule is a loss functionwhose arguments are the density/quantile forecast andthe realization of the future observation. In this paper,we consider two such score functions: the LogarithmicScore (LS) function and theWeightedQuantile Score (WQS)function. Let f jt+h|t(·) denote the forecast density of Y h

t+h,conditional on the information set available at time tobtained from model j. The LS function is defined asLSjt+h|t = log f jt+h|t(Y

ht+h). On the other hand, the WQS

is defined from quantile forecasts, and hence allows theforecaster to compare the performances of competingmodels on specific areas of the forecast distribution. Forexample, central banks may be interested in the likelihoodof ‘‘unattractive’’ economic outcomes, such as deflationor elevated inflation, and the proper evaluation of suchoutcomes requires forecast densities that are particularlyaccurate in the tails. The weighted quantile score is anatural choice when themore relevant areas of the densityare specified. Let Q j

t+h|t(α) denote the α-quantile forecastof Y h

t+h, conditional on the information set available attime t , obtained from model j. Instead, the WQS function(Gneiting & Raftery, 2007; Gneiting & Ranjan, 2011) isgiven byWQSjt+h|t =

10 QSjt+h|t(α)ω(α)dα,5 whereQSjt+h|t

(α) = [Q jt+h|t(α) − Y h

t+h][1(Yht+h ≤ Q j

t+h|t(α)) − α] is thequantile loss and 1(·) denotes the indicator function. Fourweight functions are of interest: (1) ω(α) = 1,6 whichis a uniform weight and provides an overall evaluation ofthe forecast distribution; (2) ω(α) = α(1 − α), whichconcentrates the weight in the middle of the distribution;(3) ω(α) = (1 − α)2, which assigns more weight tothe left tail of the distribution; and (4) ω(α) = α2,which puts more weight on the right tail. The latter twoweights help us examine possible asymmetries in thepredictive power of an economic indicator X . In order to

4 For the PC model, we select the lag order for ∆Yt using the AkaikeInformation Criterion (AIC), recursively for each rolling window sample.For both the PC model and the proposed AO-U|X model, we find thatincluding lags of Xt does not provide forecast improvements over the nolag case, and therefore all of the results in the next section are based onconditioning on current values of economic indicators, i.e., Xt .5 In our implementation, we replace this with a discrete version191

91i=1 ω(αi)QS

jt+h|t (αi), where αi ∈ [0.05, 0.95] in 0.01 intervals.

6 This version of WQS can be viewed as a direct alternative to the LS,although the test results from these scores will not necessarily lead tosame conclusions (see for example Tables 1 and 2, where WQS-unif andLS sometimes reach different conclusions). These differences are due totheways in which the scores are defined. The LS is obtained by evaluatingthe logarithm of the forecast density at the realized value, and hence LS islarge (small) when the realized value lies in the high (low) density region.On the other hand, the QWS-unif is evaluated based on the distances (viaquantile loss) of the quantile forecasts from the realized value. In thissense, the LS seems to be a more appealing metric when the focus is onassessing the whole forecast density.

S. Manzan, D. Zerom / International Journal of Forecasting 29 (2013) 469–478 473

put the contribution of the paper in the context of theexisting empirical evidence on U.S. inflation forecasting,we also evaluate the models in terms of their accuracyfor predicting the conditional mean. We denote the scorefunction by SS and define it as the square of the forecasterror, i.e., SSjt+h|t = (ϵ

jt+h)

2. For example, when j is the AOmodel, the forecast error is simply Uao

t+h.Now, given two competing models, k and j, where

model k is considered the benchmark (AO) and modelj is the alternative model (PC or AO-U|X), model j issaid to be relatively more accurate in terms of a partic-ular metric S ∈ {LS,WQS-unif,WQS-center,WQS-left,WQS-right, SS} if its average score

Sjn =

1n

t

S jt+h|t (5)

is higher than that of model k, Skn. Note that in our rolling

setting, the average score in Eq. (5) is computed based ont ∈ (1984:7–2007:6) for h = 6 and t ∈ (1984:1–2006:12)for h = 12. We consider the approach proposed byAmisano and Giacomini (2007) and Giacomini and White(2006) for evaluating the hypothesis of equal forecastaccuracy based on the test statistics

tn(S) =√nSkn − S

jn

σn, (6)

where σn is a heteroscedasticity and autocorrelationconsistent (HAC) standard error estimator of the differencein scores Skt+h|t − S jt+h|t . Assuming suitable regularityconditions, the statistic tn(S) is asymptotically standardnormal under the null hypothesis of vanishing expectedscore differentials. In the case of rejection, model j ispreferred if tn(S) is negative, and model k is preferred iftn(S) is positive.

4. U.S. inflation forecasts

We use four measures of the monthly price index(Pt ): the Consumer Price Index for all items (CPI), theCPI excluding food and energy (core-CPI), the PersonalConsumption Expenditure deflator (PCE), and the PCEexcluding food and energy (core-PCE). We follow therecent inflation forecasting literature (see Ang et al.,2006; and Stock & Watson, 1999) and include six of theindicators of economic activity that are often considered aspredictors of inflation, namely the civilian unemploymentrate (UNEM), the index of industrial production (IP), realpersonal consumption expenditure (INC), employees onnon-farm payrolls (WORK), housing starts (HS), and theterm spread (SPREAD), defined as the yield on the 5-yearTreasury bond minus the 3-month Treasury bill. Thus, wehave X ∈ {UNEM, IP, WORK, HS, INC, SPREAD}. All of thedata (on Pt and Xt ) were gathered from the Federal ReserveBank of Saint Louis database FRED, and the sample periodspans the period from January 1959 until December 2007.7Some of the leading indicators (IP, INC, and WORK) arenot stationary, and we therefore consider these variables

7 The macroeconomic series consist of revised data available at theJanuary 2008 vintage, due to the lack of a comprehensive real-timedataset at the monthly frequency.

in gap form, where the long-run trend is modeled using aHodrick and Prescott (1997) filter (HP) with a parameterequal to 14,400 (as is typical for monthly data).8 The trendis estimated using only information which is available atthe time the forecast is made.

4.1. Summary of the tn(S) test results

The results for h = 6 and h = 12 are given in Ta-bles 1 and 2, respectively. For both values of h, the PCmodel is either significantly worse than or no better thanthe AO model in terms of the conditional mean predictiveaccuracy for all values of X and all inflation measures. Infact, for h = 12, all of the tn(SS) values are positive, andthe majority are also significant. This evidence confirmsearlier results in the literature (e.g., Atkenson & Ohanian,2001) relating to the difficulty of outperforming the ran-dom walk benchmark based on Phillips-curve models inthe post-1984 period. The tests of the relative accuracy ofdensity/quantile forecasts of the PC models against the AOnaïve model lead, in most cases, to the conclusion reachedabove for the comparison of the mean forecasts. This is notsurprising, since themodels differ only in their conditionalmean specification, and therefore, evaluating their meanor distribution forecasts is very likely to lead to similar out-comes. It should be noted that no tn(SS) values are reportedfor AO-U|X , as its conditional mean forecast is the same asthat of the AO model by construction.

When using the LS statistics, the comparison of thedensity forecasts of the AO-U|X models with the AObenchmark results in several strong rejections (negativetn(S) values which are in bold) for the core measures ofinflation, but only in a few cases for PCE and CPI at bothh = 6 and h = 12. When using WQS-unif, the resultsare broadly similar to those for LS, although statisticallysignificant rejections occur for a smaller number of cases.The WQS-left (see tn(WQS-left)) shows that most of thebenefit from including the macroeconomic indicators ismore apparent on the left tail of the forecast distributionof inflation. Although these improvements are limitedmainly to core-inflation, the UNEM, HS and SPREADindicators also show improvements over the AO model(see tn(WQS-center)) for predicting the middle part of theforecast distribution of inflation. For example, consideringcore-PCE (for both values of h), the SPREAD indicator leadsto a forecast improvement over the AOmodel at all parts ofthe forecast distribution. Note that, even for the headlineinflation measures, most of the economic indicators leadto better quantile forecasts (where tn(S) is negative),although the improvements are not statistically significant.

4.2. Is there episodic predictability?

The relative forecast performance results reported inTables 1 and 2 are based on test statistics that average(aggregate) performances over a period of 23 years

8 We also considered a quadratic trend, as did Ang et al. (2006), withresults very similar to those for the HP filter. In order to conserve space,we decided to report only the results of the HP filter.

474 S. Manzan, D. Zerom / International Journal of Forecasting 29 (2013) 469–478

Table 1h = 6.

X Method Core PCE Core CPISS LS WQS-

unifWQS-center

WQS-left

WQS-right

SS LS WQS-unif

WQS-center

WQS-left

WQS-right

UNEM PC 3.271 3.124 3.032 2.852 0.832 3.963 1.599 5.049 3.401 2.59 1.799 3.701AO-U|X −4.321 −1.933 −1.283 −3.415 −0.476 −3.639 −2.230 −1.663 −2.623 −0.442

IP PC 2.929 3.734 3.411 3.171 3.088 3.600 1.711 5.253 2.994 2.696 3.571 2.686AO-U|X −2.738 −1.380 −1.051 −2.665 −0.355 −2.619 −0.618 −0.319 −2.327 −0.053

INC PC 3.649 3.828 3.722 3.495 3.302 3.545 1.760 5.256 3.379 2.68 4.744 2.265AO-U|X −3.309 −1.317 −0.838 −2.850 −1.191 −3.241 −1.220 −0.771 −1.588 −1.202

WORK PC 2.879 3.546 3.264 3.095 3.091 3.033 1.384 3.683 2.156 1.933 2.800 1.984AO-U|X −2.640 −1.153 −0.8131 −2.350 −0.621 −2.897 −0.682 −0.415 −2.582 −0.114

HS PC 2.115 2.629 2.151 1.869 1.529 2.922 2.398 4.532 3.211 2.701 2.312 2.007AO-U|X −5.357 −3.199 −2.738 −4.167 −1.577 −2.467 −1.122 −0.961 −0.532 −1.529

SPREAD PC 2.930 3.396 3.049 2.759 3.059 2.416 1.272 6.132 4.000 2.821 4.731 1.814AO-U|X −6.054 −2.680 −2.029 −3.744 −1.933 −2.001 −1.520 −1.184 −2.551 −1.381

PCE CPIAR 2.061 2.731 2.304 1.986 1.618 3.079 −0.360 −0.470 −0.123 −0.234 −0.492 1.046

UNEM PC 1.883 2.254 1.588 1.255 0.862 2.907 0.115 0.061 0.192 0.116 −1.058 1.898AO-U|X 1.006 −0.639 −0.606 −0.717 −0.286 1.427 −0.949 −1.102 −0.522 0.071

IP PC 2.118 2.348 2.275 2.091 0.617 3.207 0.432 0.228 0.500 0.546 −0.609 1.893AO-U|X 0.753 −0.611 −0.448 −1.743 0.683 1.690 −0.671 −0.543 −2.830 1.492

INC PC 1.477 1.719 1.491 1.455 −0.209 2.429 0.008 0.197 −0.074 0.055 −1.388 1.112AO-U|X 0.043 −0.734 −0.533 −2.552 0.495 1.106 −1.188 −1.049 −2.945 0.662

WORK PC 2.127 2.432 2.375 2.228 0.915 2.934 0.683 0.113 0.699 0.753 −0.493 1.978AO-U|X 0.671 −0.451 −0.291 −2.114 0.891 1.658 −0.585 −0.516 −3.032 1.800

HS PC 1.269 1.622 1.009 0.802 0.897 1.300 0.613 0.382 0.397 0.421 0.339 0.072AO-U|X −1.622 −1.578 −1.541 −0.989 −1.388 0.630 −1.214 −1.233 −0.460 −1.206

SPREAD PC 2.043 2.694 2.163 1.792 1.861 2.751 −0.389 −0.471 −0.182 −0.322 −0.322 0.818AO-U|X −0.102 −1.622 −1.371 −1.727 −1.227 1.198 −1.560 −1.652 −0.990 −0.929

Negative (positive) values of the test statistic tn(S) in Eq. (6) indicate that AO-U|X is more (less) accurate than the AO benchmark model, and significantvalues at the 5% level are shown in italics. The SS score function evaluates themean forecasts, the LS evaluates the density forecasts, and theWQS evaluatesthe distribution forecasts, focusing on the complete range of the variable (WQS-unif), the center of the distribution (WQS-center), and the left and righttails (WQS-left and WQS-right).

(1985:1–2007:12). It is possible that the tests may haveoverlooked performance differences between models thatonly occur at certain times during the forecast period. Here,we investigate the possibility of episodic predictability,in the sense that economic indicators might providemore accurate forecasts only during certain parts of theforecasting period. To do so, we conduct a fluctuationtest that evaluates the (relative) performances of twocompeting forecast models and signals whether one of theforecasts is significantlymore accurate at any point in time(given appropriate critical values). Using similar notationsas in Section 3, and for a window size of m, we define theaverage score for model j at time t and for h = 12 as

Sjt =

1m

t+m/2i=t−m/2

S ji+h|i, t ∈ (1990 : 1–2002 : 12).

In this application, we set m equal to 120, such that theaverage score for each t is based on a two-sided rollingwindow of 10 years, and consider the yearly horizon h,which seems more relevant from a policy perspective.We follow Giacomini and Rossi (2010), and test the equalforecast performance of models k and j based on the teststatistics

Ft(S) =√m

Skt − S

jt

σt, (7)

where σt is a heteroscedasticity and autocorrelationconsistent (HAC) standard error estimator of the differencein scores calculated on the m-month window centeredat t . The asymptotic distribution of Ft(S) under the nullhypothesis of equal performance is non-standard. Criticalvalues for various values of µ = m/n and two significancelevels are provided by Giacomini and Rossi (2010). In ourcase, we have n = 276 and m = 120, resulting inµ ≈ 0.43. Rejections occur at the 5% (10%) level against atwo-sided alternative when maxt |Ft(S)| > 2.890 (2.626),while the one-sided critical value is ±2.624 (±2.334).Note that the variation over time in Ft(S) also containsvaluable information. Model j is more accurate at time tif Ft(S) crosses the lower bound at time t , and model k ismore accurate if Ft(S) crosses the upper bound. To savespace, we consider only the score function S = LS, whichcompares the density forecasts of the AO-U|X model (j)against those of the AO model (k).9 The results for CPI andPCE are given in Fig. 2, and those for core-CPI and core-PCEare shown in Fig. 1.

9 We also considered the fluctuation test that is based on theWQS-unifscore, and reached similar conclusions overall to those for the one basedon the LS function.

S. Manzan, D. Zerom / International Journal of Forecasting 29 (2013) 469–478 475

Table 2h = 12.

X Method Core PCE Core CPISS LS WQS-

unifWQS-center

WQS-left

WQS-right

SS LS WQS-unif

WQS-center

WQS-left

WQS-right

UNEM PC 3.318 3.386 2.731 2.438 0.895 3.330 1.613 5.742 3.790 2.852 2.054 3.722AO-U|X −3.571 −2.201 −1.658 −3.005 −0.924 −2.880 −2.415 −1.946 −2.294 −1.639

IP PC 3.016 4.432 3.356 3.043 3.174 5.272 1.642 5.403 2.975 2.754 3.112 3.086AO-U|X −2.815 −1.827 −1.367 −2.875 −1.549 −0.973 −1.010 −0.751 −1.867 −0.516

INC PC 4.207 4.609 3.664 3.328 3.134 4.942 2.127 5.490 3.404 2.944 4.021 2.656AO-U|X −2.850 −1.334 −0.851 −2.479 −1.727 −2.283 −1.421 −0.974 −1.976 −1.213

WORK PC 2.556 4.091 2.969 2.716 3.066 4.029 1.279 3.958 2.162 1.928 2.696 2.370AO-U|X −2.961 −1.919 −1.523 −2.908 −1.547 −1.173 −1.161 −0.944 −1.829 −0.493

HS PC 2.409 3.258 2.287 1.896 1.581 2.520 2.737 3.954 2.990 2.657 1.969 1.735AO-U|X −5.264 −3.701 −3.206 −3.007 −2.110 −2.592 −1.653 −1.385 −1.410 −1.536

SPREAD PC 2.838 3.931 2.777 2.283 2.741 2.943 0.959 6.244 4.397 3.213 5.032 3.710AO-U|X −3.847 −2.672 −2.057 −2.960 −1.911 −2.045 −1.159 −0.749 −2.102 −1.261

PCE CPIAR 2.045 2.360 1.771 1.455 2.523 2.112 1.524 1.248 1.116 0.699 1.920 2.120

UNEM PC 2.143 2.175 1.644 1.375 1.442 2.519 1.878 1.386 1.243 0.846 0.962 2.577AO-U|X −0.531 −1.106 −0.972 −1.927 −0.862 0.506 −1.418 −1.444 −0.819 −0.743

IP PC 2.334 2.479 2.071 1.846 2.272 2.425 1.331 1.317 1.127 0.891 0.900 2.395AO-U|X −1.032 −1.076 −0.952 −2.338 −0.416 0.411 −1.549 −1.457 −2.365 −0.647

INC PC 2.127 2.207 1.797 1.695 1.728 1.940 1.104 1.361 0.847 0.681 0.577 1.698AO-U|X −0.724 −0.704 −0.552 −2.729 −0.033 0.410 −1.125 −1.018 −2.888 −0.064

WORK PC 2.319 2.556 2.195 1.956 2.393 2.593 1.496 1.224 1.215 0.945 1.075 2.543AO-U|X −0.892 −1.151 −0.997 −2.389 −0.629 0.368 −1.971 −1.851 −2.084 −1.108

HS PC 2.008 1.988 1.377 1.206 1.905 0.888 1.730 2.208 1.243 1.099 1.649 0.661AO-U|X −2.937 −2.019 −1.827 −2.166 −2.583 −0.237 −1.571 −1.534 −0.841 −2.493

SPREAD PC 2.323 2.360 1.876 1.617 2.485 2.151 1.502 1.403 1.284 0.974 1.952 2.169AO-U|X −1.276 −0.799 −0.518 −1.512 −1.690 0.797 0.006 0.029 0.190 −0.882

Negative (positive) values of the test statistic tn(S) in Eq. (6) indicate that AO-U|X is more (less) accurate than the AO benchmarkmodel, and the significantvalues at the 5% level are shown in italics. The SS score function evaluates themean forecasts, the LS evaluates the density forecasts, and theWQS evaluatesthe distribution forecasts, focusing on the complete range of the variable (WQS-unif), the center of the distribution (WQS-center), and the left and righttails (WQS-left and WQS-right).

Focusing on Fig. 2a, which corresponds to CPI, observethat there is evidence of predictability by most economicindicators (Ft(LS) takes a negative value) in the period1997–2002, although none of them are statistically signif-icant at conventional levels. Note that for INC GAP and HS,Ft(LS) is negative (although not significant) during most ofthe period 1989–2002. For PCE (see Fig. 2b), the pattern ofthe Ft(LS) is similar to that of CPI, but with a better pre-dictability (compared to the AOmodel) after 1997 formosteconomic indicators. The best performance is achieved byHS, for which Ft(LS) is negative and significant at the 10%level in the periods 1990–1992 and mid-1995–1997.

Fig. 1a gives the time path of Ft(LS) for core-CPI.The fluctuation test for core-CPI shows a much betterperformance of the AO-U|X compared to the case of CPIinflation, especially before 1997. Comparing Fig. 1a withFig. 2a, the most striking difference is observed for UNEMand HS, where these economic indicators become relevantin forecasting core-CPI, but are irrelevant for forecastingCPI. Notice from Fig. 1a that the Ft(LS) values for bothUNEM and HS are always negative throughout the forecastperiod. Fig. 1b shows the time path of Ft(LS) for core-PCE. This measure of inflation is the one with the greatestevidence of predictability, in the sense that all of theeconomic indicators contribute to the outperformance of

the benchmark throughout the forecast period (Ft(LS) isnegative). Notice that HS provides a significant degree ofpredictability throughout the forecast period.

In summary, there is ample evidence of predictabilityfor all versions of inflation, although the evidence isstronger for the core versions of inflation, i.e., core-CPI andcore-PCE. In particular, the core-PCE measure of inflationshows the most significant evidence of predictability,especially when using UNEM and HS as predictors. Thefluctuation test is a useful tool in uncovering relevantpredictability information that is overlooked by thetn(LS) test. Relying only on the tn(LS) test gives theimpression that all of the economic indicators consideredare uninformative in predicting the CPI and PCE versionsof inflation. However, the more detailed predictabilitypicture provided by the fluctuation test shows that manyof the indicators are indeed relevant, although the betterperformances seem to be overwhelmed by periods of poorperformance, which, on average, results in statisticallyinsignificant tn(LS) values.

4.3. Distribution forecasts of inflation

In this paper we ask the question, are macroeconomicvariables useful in forecasting the distribution (beyond

476 S. Manzan, D. Zerom / International Journal of Forecasting 29 (2013) 469–478

a

b

Fig. 1. Test statistic (Ft (LS)) values, where model j = AO-U|X (X =

UNEM, IP GAP, INC GAP, HS, SPREAD) is compared against the benchmarkk = AO model. The dashed horizontal lines represent the 5% (±2.890)and 10% (±2.626) critical values. The vertical lines indicate the NBERrecession dates.

the mean) of U.S. inflation in the post-1984 period?Several relative accuracy tests of density/quantile forecastsconfirm that some economic indicators provide significantlevels of predictability of the distribution of inflation fora large part of the post-1984 period. Furthermore, mostof the evidence of predictability occurs at the left tail ofthe inflation forecast distribution. In order to offer moreinsights into these results, we examine selected quantiles(α = 5%, 50% and 95%) of the forecast distribution ofinflation by focusing on core-PCE, which shows the mostpredictability.We summarize the results for theUNEMandHS indicators and focus on h = 12.

Unemployment rate (UNEM)

In the top portion of Fig. 3, we display the quantile(at 5%, 50% and 95%) forecasts of core-PCE for the period1985:1–2007:12, based on the benchmark AO model(shown with broken lines) and the AO-U|X (shown withsolid lines), where X = UNEM. In the lower panel, wepresent the time series plot of UNEM. Note that it is shiftedforward by 12 months so that it becomes aligned with thetarget date. For example, the UNEM value in January 1985actually refers to that of January 1984, which represents

a

b

Fig. 2. Test statistic (Ft (LS)) values, where model j = AO-U|X (X =

UNEM, IP GAP, INC GAP, HS, SPREAD) is compared against the benchmarkk = AO model. The dashed horizontal lines represent the 5% (±2.890)and 10% (±2.626) critical values. The vertical lines indicate the NBERrecession dates.

the value of UNEM used to produce the distributionforecasts for the target date (January 1985).

Until about 1995, no noticeable differences are ob-served in the quantile forecasts of the two models. How-ever, after 1995, the forecast distribution from the AO-U|Xmodel shifts upward away from the AO model, and thelargest gap between the quantile forecasts occurs at the 5%quantile level. Although it is not shown in the plot, largegaps also occur at other tail quantile levels. This observa-tion may be attributed to the upward pressure on inflationderived from the persistent decrease in the unemploymentrate in the late 1990s, as can be observed from the time se-ries plot of UNEM (see the lower portion of Fig. 3). In thelate 1990s and early 2000, unemployment was at histori-cally low levels, approaching 4%. Consistent with the exis-tence of a trade-off between unemployment and inflationrates, the low unemployment rate shifted the whole fore-cast distribution of inflation to higher levels (compared tothe AO model distribution forecasts), with a much largershift at the lower tail quantile of the distribution. In addi-tion, there appears to be a tendency for the forecast distri-butions of core PCE to gradually become narrower towardthe end of the sample. This characteristic is common to

S. Manzan, D. Zerom / International Journal of Forecasting 29 (2013) 469–478 477

Fig. 3. (Top plot) The (red) solid lines denote the quantile forecasts fromthe j = AO-U|X model (X = UNEM) for core-PCE at the 5%, 50%, and95% quantile levels, the (blue) dashed lines denote the same quantilelevels for the k = AO model, and the circles show the realized core-PCEinflation values. (Bottom plot) The unemployment series (shifted forwardby h = 12 months). The vertical lines indicate the NBER recession dates.(For the interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

both forecasting methods, and can be attributed to the de-cline in inflation volatility captured by the rolling windowestimation (see Clark, 2011, and Jore et al., 2010, for sim-ilar findings). By comparing the AO and AO-U|X models,both estimated on a rolling window, we can account forthe post-1984 decline in inflation volatility, butwe are alsoable to show that macroeconomic variables introduce anadditional level of time-variation in the quantile forecasts,which, as the previous section has shown, delivers moreaccurate forecasts than those from the AO model.

Housing starts (HS)

Fig. 4 displays quantile (5%, 50% and 95%) forecastsof core-PCE for the period 1985:1–2007:12 from the AO(shown with broken lines) and AO-U|X (shown with solidlines) models, where X = HS. Before 1995, the quantileforecasts (at both the 50% and 5% levels) of the AO-U|Xtrack those of the AO model closely. On the other hand,the two quantile forecasts at the 95% level appear todiffer appreciably. Although they are not shown here, thequantile forecasts at other right tail quantile levels behavesimilarly. This observation may be due to the strongerdownward pressure on high levels of inflation resultingfrom the slow and steady decrease in housing starts (fromthe mid-1980s to the beginning of 1990), as seen in thetime plot of HS (see the lower portion of Fig. 4).

Between 1995 and 1998, the two models generatequantile forecasts that are closely comparable. After 1998,the quantile forecasts start to diverge again, with thelargest gap between the quantile forecasts being at the5% quantile level. Notice from the time plot of HS (shownin the lower portion of Fig. 4) that the slow and steadyincrease in housing starts (from 2001 to 2007) exerts anupward pressure on the whole distribution of inflation,

Fig. 4. (Top plot) The (red) solid lines denote the quantile forecasts fromthe j = AO-U|X model (X = HS) for core-PCE at the 5%, 50%, and 95%quantile levels, the (blue) dashed lines denote the same quantile levels forthe k = AO model, and the circles show the realized core-PCE inflationvalues. (Bottom plot) The housing starts series (shifted forward by h = 12months). The vertical lines indicate the NBER recession dates. (For theinterpretation of the references to colour in this figure legend, the readeris referred to the web version of this article.)

with the pressure seeming to be more pronounced at thelower tail quantile.

4.4. Probability of deflation

In the beginning of 1998, a debate started on thepossibility of the U.S. economy entering a period ofdeflation, and this was also the topic of a speech givenby the Federal Reserve Board Chairman on January 3rd,1998 (see Greenspan, 1998). Using this historical fact asa motivation, we present a complementary (to relativedensity/quantile forecast evaluation tests) approach tothe assessment of forecast distributions of inflation byfocusing on the accuracy of the methods in predicting thelikelihood of deflation (negative inflation). We provide abrief illustration using core-PCE and the unemploymentrate, with a focus on h = 12. In Fig. 5, we display theprobability of deflation for the period 1996:7–2007:12, aspredicted by the benchmark AO model and the AO-U|Xmodel, where X = UNEM.

From the figure, it can be observed that a forecasterwho was using the AO model would have predictedthe probability of deflation to be larger than 5% forseveralmonths over the years, with the highest probabilityreaching 18.2% for the June 1999 target date. In contrast,for the same June 1999, the AO-U|X model predictedonly a 1.18% chance of deflation. Overall, the use of theunemployment rate leads to a likelihood of deflationwhichis close to zero throughout the period 1996:7–2007:12,and thus, it appears to provide more accurate predictionsof the event.

5. Conclusion

Forecasting the behavior of inflation plays a centralrole in economic policy-making, due to the inherently

478 S. Manzan, D. Zerom / International Journal of Forecasting 29 (2013) 469–478

Fig. 5. (Top plot) The estimated probability of deflation (defined as anegative inflation rate) for core-PCE from the k = AO model and j =

AO-U|X (X = UNEM) for the period 1996:7–2007:12. (Bottom plot) Thecore PCE series at the target date.

forward-looking nature of economic decisions. Typically,inflation forecasting focuses on modeling the conditionalmean or the most likely outcome. While they are relevant,relying only on the dynamics of the conditional meanleaves out other interesting aspects of the inflation process,such as the dynamics which occur at higher moments ofthe inflation distribution. For example, the central bankmay be interested in evaluating ‘‘unattractive’’ outcomesfor the economy, such as deflation or high inflation, andmodels that rely only on the conditional mean will notprovide the tools required for performing such evaluations.An accurate characterization of the complete distributionof future inflation, beyond the conditional mean, isneeded.

This paper examines whether indicators of economicactivity carry relevant information about the dynamicsof higher moments of inflation, and hence help toimprove the accuracy of density forecasts. Our findingsindicate that, for the core inflation measures in particular,conditioning the dynamics of the inflation distribution onthe leading indicators provides more accurate forecastsrelative to the random walk model. This is due to therelevance of the activity indicators in forecasting thequantile effects that take place far away from the centerof the inflation distribution. Overall, our results revealthat economic variables are more useful indicators of thedynamics of the lower tails of the inflation distribution.This finding can be of particular interest to policy makerswhen evaluating the likelihood of certain events, such aswhether inflation will be above or below a certain level inthe future.

Acknowledgments

We are grateful to Barbara Rossi and seminar partic-ipants at Baruch College, Rutgers University, SNDE 2009Symposium and the 2009 NBER Summer Institute, as well

as to three anonymous referees and the Editor, for helpfulcomments that significantly improved the paper.

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SebastianoManzan is assistant professor of Economics at Baruch College,City University of New York. His research interests are in appliedeconometrics and forecasting. His recent results have appeared in suchjournals as the Journal of Money, Credit and Banking, International Journalof Forecasting, and Econometric Reviews.

Dawit Zerom is Professor of Business Statistics at California StateUniversity—Fullerton. His research interests are in applied statistics andforecasting. His work has been published in such journals as the Reviewof Economic Studies, Management Science, and International Journal ofForecasting.