Uncertainty and Economic Activity:

Identification Through Cross-country Correlations9

Ambrogio Cesa-Bianchi † M. Hashem Pesaran ‡ Alessandro Rebucci §

March 20, 2017

Abstract

Uncertainty behaves countercyclically, but interpreting this correlation in structural terms isdifficult because the direction of causation may run both ways. In this paper, we take a multi-country approach to identification and model the interaction between uncertainty and economicactivity without restricting the direction of economic causation. We assume that uncertaintyand activity are driven by two common factors, a fundamental and non-fundamental factor,as well as country-specific shocks. We measure uncertainty and activity with realized equitymarket volatility and GDP growth, respectively. We identify and estimate these two factors byassuming different patterns of correlation across countries for volatility and growth innovationsthat are in accordance with stylized facts of the data. We then estimate the impact of shocksto these factors on country-specific volatility and growth as well as their importance relativeto each other, and to country-specific shocks. We find that the fundamental factor accountsfor most of the unconditional association between volatility and growth, but explains only asmall fraction of the variance of country-specific volatility. We also find that country-specificvolatility shocks are not important for growth, but shocks to the common non-fundamentalfactor do explain some fraction of the growth variance.

Keywords: Business Cycle, Common Factors, Financial Cycle, Growth, Identification, Uncer-tainty, Volatility.JEL Codes: E44, F44, G15.

9Preliminary draft. The views expressed in this paper are solely those of the authors and should not be taken torepresent those of the Bank of England.†Bank of England and CfM. Email: ambrogio.cesa-bianchi@bankofengland.co.uk.‡University of Southern California and Trinity College, Cambridge. Email: pesaran@dornsife.usc.edu.§Johns Hopkins University and NBER. Email: arebucci@jhu.edu.

1

1 Introduction

During the global financial crisis, the world economy experienced a sharp and synchronized contrac-

tion in economic activity and an exceptional increase in volatility. Indeed, after the VIX Index (a

commonly used measure of market volatility) spiked during the second half of 2008, world growth

collapsed (Figure 1). Since then, in part as a response to this dramatic event, there has been

a renewed and strong interest in the relationship between uncertainty (defined and measured in

various ways) and economic activity over the business cycle.1

Figure 1 Quarterly World Gdp Growth And VIX Index

1990 1993 1996 1999 2002 2005 2008-2

-1

0

1

2

0

15

30

45

60

World GDP (percent, left ax.) VIX (Index, right ax.)

Note. World GDP growth is a PPP-GDP weighted average of the quarter on quarter GDP growth(in percent) of 32 advanced and developing economies–the same used in our empirical application–covering more than 90 percent of world GDP. The start of the sample period in this chart is1990:Q1-2011:Q2 and is constrained by the availability of the VIX index. The sample period in ourapplication starts in 1979. See the Appendix for data sources.

It is now well established that empirical measures of uncertainty behave countercyclically in the

United States and in most other countries around the world.2 But interpreting this correlation in

economic or structural terms is challenging because the direction of causation can run in both ways.

From a theoretical standpoint, uncertainty can cause economy activity to slow and contract through

a variety of mechanisms on both the firm’s (see for instance Bernanke (1983), Dixit and Pindyck

(1994) and, more recently, Bloom (2009)) and the household’s side (Kimball (1990), Leduc and

1The ensuing literature is now voluminous. Here we focus on the studies more directly related to our paper. SeeBloom (2014) for a recent survey of newer and older contributions.

2See, for instance, Baker and Bloom (2013), Carriere-Swallow and Cespedes (2013), Nakamura, Sergeyev, andSteinsson (2017).

2

Liu (2012), Fernandez-Villaverde et al. (2011)).3 But it is also possible that uncertainty responds

to fluctuations in economic activity. In fact, examples of how spikes in uncertainty may be the

result of adverse economic conditions rather than being a driving force of economic downturns are

numerous (see Van Nieuwerburgh and Veldkamp, 2006, Fostel and Geanakoplos, 2012, Bachmann

and Moscarini, 2011, Tian, 2012, Decker et al., 2014). Yet, other models stress interaction effects

with financial frictions via an increase in the risk premium (e.g., Christiano et al., 2014, Gilchrist

et al., 2013, Arellano et al., 2012).

In this paper, we take a multi-country approach to identification and model the interaction

between uncertainty and economic activity without restricting the direction of economic causation.

We assume that uncertainty and activity are driven by two common factors, a fundamental and

non-fundamental factor, as well as country-specific shocks. We measure uncertainty and activity

with realized equity market volatility and GDP growth, respectively. We identify and estimate these

two common factors by assuming different patterns of correlation across countries for volatility and

growth that are in accordance with stylized facts of the data. We then estimate the impact of shocks

to these factors on country-specific volatility and growth as well as their importance relative to each

other, and to country-specific shocks. We find that the fundamental factor accounts for most of

the unconditional association between volatility and growth, but explains only a small fraction of

the forecast error variance of country-specific volatility. We also find that country-specific volatility

shocks are not important for growth, but shocks to the common non-fundamental factor do explain

some fraction of the forecast error variance of GDP growth.

We consider a collection of countries representing more than 90 percent of the world economy.

For each country, we focus on GDP growth and realized equity market volatility. We start by

assuming that all growth and volatility series can share at least one common fundamental factor.

This is consistent with a simple, multi-country CAPM in which world growth affect the price of

country equity claims. To achieve identification of the fundamental factor, we then assume that

growth innovations are much less correlated across countries than volatility innovations. Techni-

3Theoretically, the impact of uncertainty on activity could also be positive. For example Mirman (1971) showsthat, if there is a precautionary motive for savings, then higher volatility should lead to higher investments. Oi (1961),Hartman (1976) and Abel (1983) show that if labor can be freely adjusted, the marginal revenue product of capitalis convex in price; in this case, uncertainty may increase the level of the capital stock and, therefore, investment.However, these theories are not consistent with the counterciclycal nature of uncertainty measures.

3

cally, we assume weak cross-sectional dependence of country-specific innovations to GDP growth

rates, and strong cross-country correlation for volatility innovations (in the sense of Chudik et al.,

2011). This is equivalent to assume that volatility innovations share at least one more strong com-

mon factor than GDP growth innovations. Under the auxiliary and testable assumption that the

volatility series share only one additional strong common factor (which we label ‘non-fundamental’),

country-specific GDP growth does not load on the second factor, and the loading matrix becomes

triangular. That is, volatility series must load contemporaneously on both the fundamental and the

non-fundamental factor, while growth series load only on the fundamental factor. The crucial point

to note here is that the identification strategy applies to a large enough collection of countries, as

opposed to a single country considered in isolation. The approach is simple and transparent, and

can be applied in other contexts with more factors and more variables.4

Figure 2 Cross-country Correlation Of Volatility and GDP Growth

Argen

tina

Austra

lia

Austri

a

Belgium

Brazil

Can

ada

China

Chile

Finland

Franc

e

Ger

man

yIn

dia

Indo

nesiaIta

ly

Japa

n

Korea

Malay

sia

Mex

ico

Net

herla

nds

Nor

way

New

Zea

land

Peru

Philip

pine

s

South

Afri

ca

Singa

pore

Spain

Swed

en

Switz

erland

Thaila

nd

Turke

y

Unite

d Kin

gdom

Unite

d Sta

tes

0

0.2

0.4

0.6

Co

rre

latio

n

Note. Average cross-country pairwise correlation of volatility (light bars) and GDP growth (darkbars). The volatility measures are computed as in (42). The dotted lines correspond to the averagepairwise correlations across countries, at 0.48 and 0.17 for volatility and GDP growth, respectively.

Our key identification assumption is in accordance with patterns of cross-country correlation of

the raw data and the country specific shocks that we document in the paper. For instance, Figure

2 plots the average cross-country pairwise correlation of volatility and GDP growth series across

countries. The average for the volatility is more than twice the average for the growth series, at

0.48 and 0.17, respectively. As we shall see in the paper, we find comparable differences by using

4The approach is reminiscent of the identification through hetroskedasticity of Rigobon (2003). The key differenceis that it cannot be applied to a single country in isolation from the rest of the world economy.

4

principal component analysis. And even more striking differences between the pairwise correlation

of country specific volatility and growth innovations.5 Note also that these patterns are similar

to those documented by Tesar (1995), Lewis and Liu (2015) for equity return and consumption

growth correlations.

As the paper shows, the first common factor can be approximated by world growth, up to a

normalization constant. Thus, we can interpret it as a common fundamental driver of equity price

volatility, country growth and equity returns, consistent with basic variants of the international

CAPM. The interpretation of the second factor common only to volatility is more open. It is

well known that the international CAPM has a number of counter factual predictions. While one

could write down an two-factor asset pricing model that is consistent with the properties of the

data and the identification assumptions that we made, the approach in this paper is to start from

the properties of the data without taking a stance on the specific model that might be generating

them.6

Consistent with evidence that we report, we assume that volatility shares at least one more

strong factor than growth. This second factor, not shared by growth, could be interpreted as a

non-fundamental global driver specific to financial markets, such as ‘exuberance’, ‘bubbles’, ‘panics’,

etc. Alternatively, one could also think about the United States as “small and granular ” in global

production and consumption because relatively close to trade in goods and services, but “large and

dominant ” in global finance (in the sense that all volatility series must load onto it as discussed

by Chudik and Pesaran (2013)) because of its very large and open capital markets. Nonetheless,

the paper is ultimately agnostic on the deeper economic interpretation of the second factor that we

identify.

To measure economic uncertainty, we build on the contributions of Andersen et al. (2001, 2003)

and Barndorff-Nielsen and Shephard (2002, 2004), and we compute realized equity price volatility

for a given quarter by using daily stock returns on 32 advanced and emerging economies. In the

paper, we consider and discuss several other measures of volatility and argue that they are either

not suitable for the purpose of our analysis or not readily available for a large collection of countries

5See Tables 2 and 3, and Figures C.3 and C.4.6For example, Lewis and Liu (2017) show that a disaster risk model with both country-specific and common

disaster risk can generate patterns of equity return and consumption growth close to the data.

5

like the one we need in our analysis. We measure activity by GDP growth.

The paper reports three main findings. First, and most importantly, while unconditional volatil-

ity behaves countercyclically for most of the 32 countries in our sample, conditional on the identified

fundamental common factor, the contemporaneous correlation between country-specific volatility

and growth shocks is negligible. This result suggests that most of the unconditional correlation

between volatility and growth can be accounted for by shocks to the fundamental common factor.

Combined with the fact that, conditional on the two factors identified, country specific volatility

and growth shocks are weakly correlated across countries, this result also makes the identification

of country specific shocks trivial as it implies a near-diagonal variance-covariance matrix.

Second, we show that the second non-fundamental factor has much larger impact on coun-

try growth than country-specific volatility shocks. However, it can explain only a small fraction

of the growth variance in the United states as well as in the typical economy represented by a

PPP-GDP weighted average of all countries in the our sample. This implies that country specific

volatility shocks are largely diversified away in the world economy, but common non-fundamental

disturbances can potentially inflict severe damage as we saw during the global financial crisis.

Third and finally, we find that the endogenous component of equity market volatility is quanti-

tatively very small. Country volatilities are largely driven by shocks to the second non-fundamental

factor and by country-specific volatility shocks. Thus, we estimate a negligible share of volatility

variance explained by the fundamental common factor or by country-specific growth shocks. This

means that the endogenous component of volatility is small.

The paper is related two at least two different strands of the literature. The first strands

acknowledges that volatility may be endogenous and could be driven by business or financial cycles.

So, some of its components might be endogenous to macroeconomic dynamics, while others might

be exogenous—See for instance Ludvigson et al. (2015), Clark et al. (2016), Giglio et al. (2016).

A key difference relative to these contributions, is that our fundamental common factor, which is

the main source of endogeneity, is common to both macroeconomic and financial variables (GDP

growth and equity market volatility), as opposed to be common only macroeconomic variables. As

we argue in the paper, this is a crucial property of our fundamental factor. Consistent with the

empirical results of Ludvigson et al. (2015) we document that it is the non-fundamental common

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component of volatility that is most harmful for growth and has some weight in growth variance.

We reach this important conclusions with identification assumptions we believe are simple and

empirically verifiable in principle.

A second strand of the literature assumes that uncertainty is exogenous and explicitly focuses

on the international dimension of the relation between uncertainty and activity, with findings

consistent with those for the United States. For instance, Carriere-Swallow and Cespedes (2013)

estimate a battery of small open economy VARs for 20 advanced and 20 emerging market economies

in which the VIX index is assumed to be determined exogenously. Their results show that emerging

market economies suffer deeper and more prolonged impacts from uncertainty shocks, and that a

substantial portion of such larger impact can be explained by the presence of credit constraints in

the case of emerging market economies, which is in accordance with the recent work of Christiano

et al. (2014), Gilchrist et al. (2013) and Arellano et al. (2012). Using an unbalanced panel of 60

countries, Baker and Bloom (2013) provide evidence of the counter-cyclicality of different proxies

for uncertainty, such as stock market volatility, sovereign bond yields volatility, exchange rate

volatility and GDP forecast disagreement. Finally, Hirata et al. (2012) use a factor-augmented

VAR (FAVAR), with factors computed based on data for 18 advanced economies and a recursive

identification scheme in which the volatility variable is ordered first in the VAR. They find that, in

response to an uncertainty (volatility) shock, GDP falls and then rebounds consistent with Bloom

(2009), although the impact is smaller. While these papers share with this one an international

focus, the distinctive feature of our empirical analysis is to consider a whole collection of countries

interacting with each other, as opposed to a collection of countries taken in isolation. Indeed, our

identification strategy exploits the contemporaneous correlation (or lack thereof) between country

volatility and growth innovations to achieve identification of both common an country-specific

components and shows that volatility is largely exogenous as opposed to assuming it from the

outset of the analysis.

The rest of the paper is organized as follows. Section 2 discusses the model that we propose

and the identification of the two factors. Section 3 extends the analysis to a fully dynamic and

heterogeneous model. Section 4 gives the details of how we construct our proxy measure of volatility.

Section 5 reports key stylized facts of the data, including the cross country correlation structure

7

that is crucial for identification purposes. Section 6 reports all the empirical results. Section

7 concludes. Proof of our derivations, details on the data sources, and selected country-specific

results are in Appendix.

2 A Simple Static Factor Model

In this section, we present a static version of the model that we use in the empirical analysis and

discuss its identification and interpretation.7 Omitting any dynamics and deterministic components

for exposition purposes, we posit the following common factor representation for the contempora-

neous correlation between volatility and growth, for a large collection of N countries:

vit = λift + uit, (1)

∆yit = γift + εit, (2)

for i = 1, 2, ..., N . Here, vit denotes a volatility measure and ∆yit is an economic activity indicator

for country i (say, real GDP growth like in our application, simply called “output” or “growth”

for brevity in the paper). The factor ft has zero mean and finite variance and is normalized to

one for simplicity. The innovations uit and εit are serially uncorrelated with zero means and finite

variances, but can be correlated to each other both within and between countries. Conditional on

the common factors ft, therefore, the correlation between uit and εit captures any contemporaneous

causal relation between volatility and growth, on which we do not impose any restriction.

This representation is quite general and can be motivated by both theory and empirical evidence.

From a theoretical perspective, one can think of ft as a fundamental factor, such as world technology,

which affects all countries GDP growth rates and country equity price index volatilities at the same

time. For example, in a general equilibrium version of the international CAPM with complete

asset markets, ft is world growth and it affects all country growth rates and return volatilities

contemporaneously. While this frictionless benchmark cannot account for many stylized facts of

country equity returns, it provides a useful starting point to consider more shocks and frictions

(See for instance Obstfeld and Rogoff, Chapter 5 for a discussion).

7As we shall see below, the main results of the analysis carry through to the fully dynamic model that we use inour application.

8

From an empirical perspective, it is well documented and we will show below for our sample that

GDP growth and volatility share a high and negative contemporaneous correlation at the country

level, for all countries we consider, like in the case of the United States. Moreover volatility and

growth are also correlated across countries to a different degrees.

2.1 Identifying the Fundamental Factor through Cross-Country Correlation

In order to identify the innovations uit and εit, the common factor ft, and the factor loadings λi

and γi of (1)-(2) we need to impose some restrictions. The model (1)-(2) applies to all N countries.

Consider the generic country i, for instance the United States. If we consider the United States in

isolation from the rest of the world, we cannot identify the parameters of the model above, even

assuming that we know the factor ft, unless we exclude ft from one of the tow equations. In fact,

the covariance matrix of vit and ∆yit is given by:

Θi =

λ2i + σ2

u,i λiγi

− γ2i + σ2

ε,i

(3)

This provides three independent restrictions, but the parameters of the model are four, (λi, γi), and

(σ2u,i, σ

2ε,i). Their identification can be achieved only by imposing at least one additional exclusion

restriction, e.g. either λi = 0 or γi = 0.8

The main idea of this paper is to achieve identification of all model parameters by placing

restrictions on the whole collection of countries N that exploit the different patterns of correlation

across countries of uit and εit. The identification strategy that we propose is general and can

be applied to any panel of time series with the same properties of cross-section correlation that

we assume. The model could have more factors provided that we consider more cross sections of

variables. However, what is crucial is to look at the whole collection of N units, as opposed to one

unit at a time, taken in isolation.

To illustrate the strategy, let’s define global (or world) volatility (vω,t) and GDP growth (∆yω,t)

8The model (1)-(2) is observationally equivalent to a standard structural vector autoregression model. Thus, it issubject to the same identification requirements. See appendix for a proof.

9

as follows:

vω,t =N∑i=1

wivit, (4)

∆yω,t =

N∑i=1

wi∆yit, (5)

where wi and wi denote two sets of aggregation weights. These weights can be the same or differ

for each variable. To achieve identification, we then make the following assumptions on the factor

ft, the loadings (λi and γi), the the weights (wi and wi), and the innovations (uit and εit):

Assumption 1 (Loadings) The factor loadings λi and γi, are independently and identically dis-

tributed across i, independent of the common factors ft, for all i and t, with means λ and γ.

Furthermore they satisfy the following conditions:

N−1N∑i=1

λ2i = O(1) and N−1

N∑i=1

γ2i = O(1), (6)

λ =

N∑i=1

wiλi 6= 0 and γ =

N∑i=1

wiγi 6= 0, (7)

as N →∞.

Assumption 2 (Weights) Let w = (w1, w2, ..., wN )′ and w = (w1, w2, ..., wN )′ be N × 1 vectors of

non-stochastic weights with∑N

i=1wi = 1 and∑N

i=1 wi = 1. We assume that growth weights w are

granular, i.e.:

||w|| = Op(N−1),

wi||w||

= Op(N− 1

2 ) ∀i, (8)

while volatility weights w are left unrestricted, so they may or may not be granular.

Assumption 3 (Cross-section correlations) Let the variance-covariance matrices of the N × 1

vectors εt = (ε1t, ε2t, ..., εNt)′ and ut = (u1t, u2t, ..., uNt)

′ be Σεε = V ar (εt) and Σuu = V ar (ut),

respectively. They satisfy:

%max (Σu) = O(N), (9)

%max (Σε) = O(1). (10)

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where %max(·) is the largest eigenvalue associated with the each of the two covariance matrices.

Assumption 1 is standard in the factor literature (see, for instance, Assumption B in Bai and

Ng (2002)). It ensures that ft is a strong (or pervasive) factor for both volatility and activity (see

Chudik et al., 2011), and it can be estimated using principal components or the cross-section aver-

ages of country-specific observations. Assumption 2 requires that individual countries’ contribution

to world growth is of order 1/N . In contrast, it leaves the volatility weights unrestricted, implying

that some country could have higher weight in global volatility. This is consistent with the notion

that, since the 1980s, world growth became a progressively more diversified process, while financial

markets continued to be dominated by the largest and most developed countries like the United

States.

Assumption 3 is crucial. It says that the volatility innovations are strongly correlated across

countries, while GDP growth innovations are weakly correlated across countries. Weak cross-

country correlation means that, asymptotically, as N becomes large, the average pairwise corre-

lations across countries of output growth innovations tends to zero, since the largest eigenvalue

of their variance-covariance matrix is bounded in N . On the other hand, strong cross-sectional

correlation means that the largest eigenvalue of Σu grows with the size of the cross-section N . As

we shall see, empirically, this key assumption is in accordance with the properties of both the data

we use and the residual that we obtain from the model estimation.9

Proposition 1 Under the assumptions made, for N large enough, ft can be identified (up to a

constant) by yω,t =∑N

i=1wi∆yit.

Proof. Consider the collection of countries (1)-(2) for i = 1, 2, ..., N . Under Assumption 1, and

using the definitions in (4)-(5), we have the following model for the global variables:

vω,t = λft + uω,t, (11)

∆yω,t = γft + εω,t, (12)

9Similarly different patterns of cross country correlations have been documented also for consumption growth andand equity returns. See, for instance, Tesar (1995) and Lewis and Liu (2015).

11

where εω,t = w′εt and uω,t = w′ut. Because of Assumption 3:

V ar (εω,t) = w′Σεw ≤(w′w

)λmax (Σε) , (13)

and:

V ar (εω,t) ≤ O(w′w

)= O

(N−1

), (14)

and hence:

εω,t = Op

(N−1/2

). (15)

Using this in (12), since γ 6= 0 under Assumption 1, we have:

ft = γ−1∆yω,t +Op

(N−1/2

), (16)

which allows us to recover ft form ∆yω,t up to the scalar 1/γ.

It is important to observe here that, if, as we assume in accordance to the data, the volatility

innovations are strongly correlated across countries, ft cannot be recovered from vω,t. In fact,

since λmax (Σu) = O(N), it follows that V ar (uω,t) = w′Σuw will generally not converge to zero.10

Similarly, under Assumptions 2 and 3, only the principal components or cross-section averages

of output growth can be used to identify ft (up to an affine transformation). Therefore, pooling

observations on all volatility variables (or all volatility and growth variables) and then extracting

their principal components would not identify ft.

2.2 The Non-Fundamental Factor

The identification assumptions made imply additional restrictions on the data that we will exploit

in our application. To develop them, notice that, because of Assumption 3, the cross-section of

volatility innovations uit must share at least one additional strong factor that is not shared by the

panel of growth innovations. It could share more than one additional strong component, but it

must share at least one more. So, for simplicity, and again without loss of generality, we can make

the following auxiliary assumption:

10The difference in the way in which the volatility and GDP growth innovations are correlated across countries alsoensures that we do not end up with a perfect relationship between vt and ∆yt when N tends to infinity.

12

Assumption 4 The residual cross-section correlation of uit, conditional on ft, can also be decom-

posed in a strong factor (gt) and a weak component (ηit), namely

uit = θigt + ηit, (17)

where ηit is a volatility innovation that is now cross-sectionally weakly correlated, like εit. Hence,

letting the variance-covariance matrix of the N×1 vector ηt = (η1t, η2t, ..., ηNt)′ be Σηη = V ar (ηt),

we also have

%max (Σηη) = O(1), (18)

and:

N−1N∑i=1

θ2i = O(1). (19)

Given the auxiliary Assumption 4, the second strong factor that must be shared by the volatility

series can also be identified from the data.

Corollary 1 If uit is given by (17), the model becomes:

vit = λift + θigt + ηit, (20)

∆yit = γift + εit. (21)

Moreover, conditional on ft, for N large enough, gt is given by the following linear combination of

vω,t and ∆yω,t for N large enough:

gt = θ−1vω,t −λ

θγ∆yω,t +Op

(N−1/2

). (22)

Proof. To see this, substitute (16) and (17) into (11) and apply the same logic as before to ηit.

It is now evident that the assumed, different pattern of correlation across countries of volatility

and growth innovations implicitly provides a restriction on the factor loadings of the growth equa-

tions on the second strong factor gt, and yields country models that are lower triangular in the

factor loading matrix. The peculiarity of our approach is that the crucial restriction is not imposed

at the level of individual country, but on the cross-section correlation of the whole collection of

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countries. Indeed, it is easy to show that, even assuming such a triangular factor loading matrix,

with a single country, we could not identify the factor and the other model parameters in model

(1)-(2).

A second common factor in volatility and growth can easily justified by asset pricing models

with preference shocks.11 Our identification strategy by cross country correlations, however, implies

that growth does not load contemporaneously on gt. Our identifications restrictions are consistent,

for instance, with the international disaster risk model of Lewis and Liu (2017), who show that a

model with time-varying probability of disaster and both world and country-specific disaster risk

generates patters of consumption growth and equity return correlations across countries consistent

with the data and the assumptions we made. In this framework, gt can be interpreted as the

common “jump” component of the model. More generally, we could interpret gt as ‘exuberance’,

‘bubbles’, ‘panics’, consistent with models in which non-fundamental determinants are included

to improve empirical asset pricing performance. A third way to to interpret gt is to think about

the United States as “granular” in global production and consumption, but “dominant ” in global

financial markets (in the sense that all volatility series must load onto it as discussed by Pesaran

and Chudik (2014)). In this case, gt would simply be the volatility of the dominant market in the

world.

2.3 Model Estimation When the Factors Are Not Observed

We now want to estimate the factors, their impacts on the country-specific volatilities and GDP

growth rates, and the innovations taking into account that we cannot observe ft and especially gt

directly. Denote with ft and gt an estimate of ft and gt. An estimate of ft can be simply obtained

from (16). If we normalize γ = θ = 1, which is innocuous, ft can be approximated by world

growth as ft = ∆yω,t. Given (22), gt can then be approximated by the residuals of a regression of

world volatility vω,t on world growth ∆yω,t, namely gt = vω,t− β∆yω,t. Once observable factors are

obtained, one can easily estimate their country-specific impact on volatilities and growth and the

11See, for instance, a global preference shock of the kind studied by Albuquerque, Eichenbaum, Luo, and Rebelo(2015) would induce such a representation.

14

associated innovations as follows:

vit = λift + θigt + ηit, (23)

∆yit = γift + εit. (24)

Note here that,ft and gt are an orthonormal transformation of ft and gt. Therefore, the residuals

of (20)-(21) and (23)-(24) will be exactly the same. We also note that ft and gt are are orthogonal

to each other by construction.

In summary, the key point of our identification strategy is that the nature of the cross-country

correlation of the two series of innovations must differ. In the case of ∆yit, the correlation is

assumed to vanish asymptotically. In the case of vit, the innovations are assumed to share some

residual correlation across countries even if the cross section is large (or, in the limit, if N goes

to infinity). Moreover, under these assumptions the matrix of contemporaneous factor loadings is

recursive, and observable and orthonormal factors that are orthogonal to each other can be easily

estimated from the data simply by means of OLS. Importantly, as we shall see below, this crucial

assumption is in accordance with the stylized facts that characterize our data as well as existing

evidence on the cross country correlations of consumption growth and equity return data (see Tesar

(1995); and Lewis and Liu (2015)). As we will show below, in fact, volatility series correlate across

countries much more strongly than and GDP growth series.

The identification strategy that we propose is general and can be applied to any panel of time

series with the same properties of cross-section correlation that we assume. The model could have

more factors provided that we consider more cross sections of variables. However, what is crucial

is to focus at the whole collection of N units, as opposed to one unit at a time, taken in isolation.

3 A Dynamic Heterogeneous Two-factor Model

Empirically, it is important to allow for unrestricted dynamic interaction between volatility and

growth to capture any led or lagged interaction. We therefore now embeds our identification

strategy in a fully dynamic, heterogeneous model. As we shall see, adding a dynamics that differ

across countries, while requiring additional assumptions and derivations, does not alter the main

15

results above. In particular, we want to show (i) how to identify the factors ft and gt up to an

orthonormal transformation; (ii) how to estimate the factors and the innovations from the data

assuming they are not observable; (iii) and how to estimate the dynamic impact and the relative

importance of the factors on country-specific volatility and growth series.

Consider the following version of our model, with dynamics restricted to the first order for

simplicity, without loss of any generality:

vit = aiv + φi,11vi,t−1 + φi,12∆yi,t−1 + λift + θigt + ηit, (25)

∆yit = aiy + φi,21vi,t−1 + φi,22∆yi,t−1 + γift + εit, (26)

on which we already imposed assumptions 3 and 4. That is, in vector format:

zit = ai + Φizi,t−1 + Γift + ξit, for t = 1, 2, ..., T, (27)

where zit = (vit,∆yit)′ and:

ai =

aiv

aiy

, Φi =

φi,11 φi,12

φi,21 φi,22

, Γi =

λi θi

γi 0

, ft =

ft

gt

, ξit =

ηit

εit

. (28)

The matrix Γi of contemporaneous factor loadings is triangular because of our identification as-

sumptions on the cross-section correlation of the residuals as we discussed in the previous section.

Because of the dynamic nature of the model, other assumptions are modified as follows:

Assumption 5 (Innovations) The country-specific shocks ξit are serially uncorrelated (over t),

and cross-sectionally weakly correlated (over i), with zero mean and a positive definite covariance

matrix, Ωi, for i = 1, 2, ..., N .

Assumption 6 (Common factors) The 2× 1 vector of unobserved common factors, ft = (ft, gt)′,

is covariance stationary with absolute summable autocovariances, and is distributed independently

of the country-specific shocks, ξit′ for all i, t and t′. Fourth order moments of f`t, for ` = 1, 2, are

also bounded.

16

Assumption 7 (Factor Loadings) The factor loadings λi, θi, and γi (i.e., the non-zero elements of

Γi) are independently and identically distributed across i, and independent of the common factors

ft, for all i and t, with means λ, θ, and γ. Furthermore, they have bounded second moments and:

Γ = E (Γi) =

λ θ

γ 0

,

is invertible (namely γθ 6= 0).

Assumption 8 (Coefficients) The constants ai are bounded, Φi and Γi are independently dis-

tributed for all i, the support of % (Φi) lies strictly inside the unit circle, for i = 1, 2, ..., N , and

the inverse of the polynomial Λ (L) =∑∞

`=0 Λ`L`, where Λ` = E

(Φ`i

)exists and has exponentially

decaying coefficients.

We can then prove the following proposition:

Proposition 2 Under Assumptions 5-8, the unobserved common factors, ft and gt in (27) can be

approximated by:

ft = γ−1∆yω,t +

∞∑`=1

c′1,`zω,t−` +Op

(N−

12

), (29)

gt = θ−1vω,t −λ

θγ∆yω,t +

∞∑`=1

c′2,`zω,t−` +Op

(N−

12

), (30)

where:

vω,t =

N∑i=1

wivit, ∆yω,t =

N∑i=1

wi∆yit, zω,t = (vω,t,∆yω,t) , (31)

and c′1,` and c′2,` are 1× 2 vectors of coefficients.

Proof. See Appendix A.2.

We note here that c′1,` and c′2,` are the first and second rows of the 2× 2 matrix C` as defined

in the Appendix. As Pesaran and Chudik (2014) and Chudik and Pesaran (2015) showed, if slope

heterogeneity is not extreme (i.e., if these matrices of coefficients do not differ too much across

countries) and C` decays exponentially in `, the infinite order distributed lag functions in zω,t

17

in the above derivations can be truncated. In practice, Pesaran and Chudik (2014) and Chudik

and Pesaran (2015) recommend a lag length ` equal to T 1/3 where T is the number of sample

observations. For our sample period this means 5 lags.

As we noted earlier, ft and gt are identified up to a 2×2 rotation matrix. To proceed, we impose

again the normalization restriction that γ = θ = 1. Normalization restrictions are innocuous and

do not affect the final estimating equation that identifies the idiosyncratic shocks (which are also

of interest). After truncating the infinite distributed lag functions with the formula above we have:

ft = ∆yω,t +

p∑`=1

c′1,`zω,t−` +Op

(N−1/2

), (32)

gt = vω,t − λ∆yω,t +

p∑`=1

c′2,`zω,t−` +Op

(N−

12

). (33)

Substituting these approximations in (27) we obtain the cross-section augmented VAR models:

vit = aiv + φi,11vi,t−1 + φi,12∆yi,t−1 + θivω,t + βi∆yω,t +

p∑`=1

ψ′i`zω,t−` + ηit +Op

(N−

12

),(34)

∆yit = aiy + φi,21vi,t−1 + φi,22∆yi,t−1 + γi∆yω,t + γi

p∑`=1

c′1,`zω,t−` + εit +Op

(N−

12

)(35)

where βi = λi− θiλ and ψi` = (λic1,` + θic2,`). Note here that only ∆yω,t is included in the output

growth equation.

The above model can now be estimated consistently by least squares so long as N and T are

sufficiently large. This would yield estimated residuals and the factor loadings θi, βi, and γi. But

these coefficients might be difficult to interpret due to the possible non-zero correlation between

∆yω,t and vω,t.12 A second complication is that the factors depend on lagged variables. It is

therefore useful to consider an orthonormal transformation that yields orthogonal factors once the

effects of past values of zω,t are filtered out.

To this end we stack (32) and (33) by T (abstracting from the order Op(N−1/2

)terms) and

12One could use the orthogonalized components of ∆yω,t and vω,t using the Cholesky factor. However, such aprocedure could invalidate the triangular form of Γi applied to the true underlying factors, ft and gt. Also focusingon the orthogonalized components of ∆yω,t and vω,t, ignores the contributions of zω,t−` for ` = 1, 2, ..., p to theestimation of ft and gt.

18

get:

f = ∆yω + ZωC1, (36)

g = vω − λ∆yω + ZωC2, (37)

where ∆yω = (∆yω,1,∆yω,2, ...,∆yω,T )′, vω = (vω,1, vω,2, ..., vω,T )′, and Zω = (τT , zω,−1, zω,−1, ..., zω,−p).

Inclusion of τT in Zω ensures that the filtered factors have mean zeros. We can now establish the

following proposition.

Proposition 3 The orthogonalized filtered factors f and g can be recovered from the data as resid-

uals of the the following OLS regressions:

∆yω = ZωC1 + f , (38)

vω = λf + ZωC2 + g. (39)

Proof. See Appendix A.3.

Given the orthogonal factors, substituting them in (34)-(35), we can compute their impact and

relative importance based on the following regressions:

vit = aiv + φi,11vi,t−1 + φi,12∆yi,t−1 + βi,11ft + βi.12gt +

p∑`=1

ψ′v,i`zω,t−` + ηit, (40)

∆yit = aiy + φi,21vi,t−1 + φi,22∆yi,t−1 + βi,21ft +

p∑`=1

ψ′∆y,i`zω,t−` + εit. (41)

Notice here that by construction the estimates of ηit and εit based on the above regressions should

be numerically identical to those obtained using the non-orthogonalized factors ∆yω and vω. It is

also important to note that since ft and gt are based on residuals from regressions of ft and gt on

the pth order lagged values, τT , zω,t−1, zω,t−2, ...., zω,t−p, then it also follows that ft and gt have

zero means (in-sample) and for a sufficiently large value of p, they are also serially uncorrelated.

Therefore, ft and gt can be viewed as global innovations (or shocks) to the underlying factors, ft

19

and gt. In Appendix A.4 we show how to compute impulse responses and variance decompositions

to shocks ft and gt as well as ηit and εit.

In summary, we saw that generalizing the assumptions made before, we can estimate consistently

the country-specific impact and relative importance of the two factors, as well as the innovations

(ηit and εit) even if the model is dynamic and heterogeneous. Next we want to apply this framework

to study the relation between volatility and growth, conditional on the factors the estimated factors

ft and gt. But before doing that we need to discuss how we measure volatility in a multi-country

setting.

4 Measurement

As a proxy for uncertainty, in our application, we use asset price volatility. Asset price volatility

has been used extensively in the theoretical and empirical finance literature and implicitly assumes

that uncertainty and risk can be characterized in terms of probability distributions. It therefore

abstracts from the Knightian notion of uncertainty, which refers to the idea that some types of

risks can not as such be characterized.

Specifically, we use a measure of realized volatility based on the summation of intra-period,

higher-frequency stock price squared returns (see, for example, Andersen et al. (2001, 2003),

Barndorff-Nielsen and Shephard (2002, 2004)). The idea of realized volatility can be easily adapted

for use in macro-econometric models by summing squares of daily returns within a given quarter

to construct a quarterly measure of market volatility. This approach can be extended to include

intra-daily return observations when available, but this could contaminate the measures with mea-

surement errors due to market micro-structure and jumps in intra-daily returns. In addition, like

implied volatility measures that we discuss below, intra-daily returns are not available for all mar-

kets that we want to consider and, when available, tend to cover a relatively short time period as

compared to our data period that begins in 1979.

In the finance literature, the focus of the volatility measurement has now shifted to market-based

implied volatility measures obtained from option prices, but these measures are not yet available

for long time periods for a meaningful number of countries. This explains the popularity of the US

20

VIX Index, which is an average of the daily option price implied volatility for the S&P500 index

also as a proxy for global uncertainty (see Figure 1). However, a key input for the implementation

of our identification strategy is the availability of country-specific measures for a large number of

countries. So, implied volatility is not suitable for our purposes.

If we consider a panel of country-specific equity prices (such as, for example, firm level or

sectorial equity prices) a different measure of volatility can be computed as the cross-sectional

dispersion of asset prices. Computing country-specific indices of cross-sectional dispersion for the

32 countries in our sample, however, would require a large amount of data —which in many cases

would not be available on a long historical sample. Moreover, Cesa-Bianchi et al. (2014) show that

realized volatility and cross-sectional dispersion are closely related. So, in our application, we will

focus on realized volatilities.

Realized volatility and cross-sectional dispersion encompass most measures of uncertainty and

risk proposed in the literature that would be suitable to implement our identification strategy.

Schwert (1989b), Ramey and Ramey (1995), Bloom (2009), Fernandez-Villaverde et al. (2011) use

aggregate time series volatility (i.e., summary measures of dispersion over time of output growth,

stock market returns, or interest rates); Leahy and Whited (1996), Campbell et al. (2001), Bloom

et al. (2007) and Gilchrist et al. (2013) use dispersion measures of firm-level stock market returns;

Bloom et al. (2012) use cross-sectional dispersion of plant, firm, and industry profits, stocks, or

total factor productivity.

The literature has also used uncertainty measures based on expectation dispersion: while sum-

marizing the range of disagreement among individual forecasters at a point in time, these measures

do not give information about the uncertainty surrounding the individual’s forecast. See, for in-

stance, Zarnowitz and Lambros (1987), Popescu and Smets (2010), and Bachmann et al. (2013).

Finally, model based measures, such as those Jurado et al. (2015) and Ludvigson et al. (2015)

could be in principle computed for all countries in our sample, but this would likely encounter data

constraints.

The rest of this section provides a precise definition of the realized volatility measure that we

use in the empirical analysis and also briefly describes the data set we assembled to compute them.

We then report some stylized facts on the cross-section comovement of volatility and GDP growth

21

that are in accordance with the identification assumptions we make.

4.1 Volatility

To construct a quarterly measures of realized volatility for many countries we begin with the daily

price of an asset of type κ, in country i, measured on close of day τ in quarter t, denoting it by

Pκit(τ). We then compute the realized volatility for quarter t, asset of type κ, and country i as:

vκit =

√√√√D−1t

Dt∑τ=1

(rκit(τ)− rκit)2 (42)

where rκit(τ) = ∆ lnPκit(τ) and rκit = D−1t

∑Dtτ=1 rκit(τ) is the average daily price changes over the

quarter t, and Dt is the number of trading days in quarter t. The scaling factor√Dt allows us to

express the realized volatility measures at quarterly rates: in this way we obtain realized volatility

measures that are consistent with the remaining macroeconomic time series that we shall use in our

empirical analysis (which are at quarterly frequency, too). For most time periods, Dt = 3×22 = 66,

which is larger than the number of data points typically used in the construction of daily realized

market volatility in finance. In the case of intra-day observations prices are usually sampled at

10-minutes interval which yields around 48 intra-daily returns in an 8 hour-long trading day.

The realized volatility measures can also be computed for real asset prices, with Pκit(τ) in the

above expression replaced by Pκit(τ)/Pit, where Pit is the general price level in country i for quarter

t, but they yield very similar results and in our application they are not reported.13 While the

measure above can be constructed for several asset classes, in this paper, we shall focus on equity

prices. So hereafter we drop the subscript κ.14

The sources of the data and their sampling information are reported in Appendix B. To construct

quarterly measures of country-specific realized volatility, we first collect daily prices of stock market

equity indices for 32 advanced and emerging economies. For each equity index, the data set spans

up to 8479 daily observations from 1979 to 2011 (depending on data availability). Figure 3 plots

the U.S. realized volatility measure we constructed with the VIX index (as plotted in Figure 1).

13We measure Pit by the consumer price index (CPIit)14Cesa-Bianchi et al. (2014) construct them for exchange rates, bond prices, and commodity prices and report their

unconditional properties.

22

The chart shows that the two measures co-move very closely.

Figure 3 Quarterly U.S. Equity Realized Volatility And The Vix Index

1990 1994 1998 2002 20060

0.1

0.2

0.3

0.4

0

15

30

45

60

RV US Equity (left ax.) VIX Index (right ax.)

Note. RV US Equity is the U.S. realized volatility measure we constructed in equation (42). TheVIX Index is the quarterly average of the daily Chicago Board Options Exchange Market VolatilityIndex from Bloomberg. The sample period is 1990:Q1-2011:Q2.

4.2 Economic Activity

Defining and measuring a country’s level of economic activity over the business cycle, in principle, is

also open to debate. Consistent with the theoretical framework presented above, in our application,

we shall use real GDP growth (calculated as the first difference of the log-level). We find similar

results when we detrend real GDP with a deterministic quadratic trend or the HP filter (results

not reported, but available from the authors on request).

5 Stylized Facts

In this section we describe key unconditional properties of the data that are relevant for our model

specification and identification. We focus first on persistence of volatility and real GDP to support

of model specification. We then look at the pattern of cross-country correlations, which is crucial

for identification. Finally, we look at country correlation between volatility and growth.

23

5.1 Persistence

A battery of summary statistics on our volatility and real GDP series support our model specifica-

tion in terms of log-difference of real GDP and level of the volatility variables.

It is well known that GDP levels are non stationary. This is also true also in our sample. As

Table C.1 in the Appendix shows, the persistence of the log-level of GDP is very high (on average

around 1). Moreover, the null of a unit root is not rejected by a standard ADF test for any of the

32 countries in our sample.

Differently, the levels of volatility, even though persistent, tend to be mean reverting. Table C.2

in the Appendix shows that the first order auto-correlation coefficient is high, on average about 0.6.

However, standard ADF tests reject the null hypothesis that the volatility variables have a unit

root. We also run a test for fractional integration, for comparison with the finance literature, and

find that volatility is stationary. We conclude from this evidence that volatility is best modeled in

levels.

5.2 Cross-country Correlations of Volatility and Growth

The cross-country correlations of our data are crucial for our identification strategy. In order

to gauge to what extent our volatility and output variables co-move across countries we use two

techniques: standard principal component analysis and pair-wise correlation analysis. The average

bilateral (or pairwise) correlation in a panel of series over i = 0, 1, ..., N measures the degree of

comovement of country i with all others in the collection N . The overall average for all countries

N , therefore, is a summary measure of the cross-country correlation in the whole collection N . For

robustness and comparability, we also use standard principal component (PC) analysis.

In the case of a balanced panel, both approaches can be used and should lead to similar con-

clusions. But in the case of unbalanced panels, like ours, the average pairwise correlation has the

advantage that it can be computed using the longest sample period available for each bilateral pair.

Principal component analysis, instead, must be applied to the shortest sample period available

common to all series, or to the maximum number of countries with the full sample period. There-

fore, it looses some information one way or the other. In our data set, there are 16 countries with

24

volatility series covering the full sample period 1979:Q1-2011:Q2. So we apply principal component

analysis to these 16 series. Instead, the panel of GDP series is balanced.

The first principal component on these 16 volatility series explains 65 percent of the total

variation in the first difference of vit. In contrast, the first principal component in the panel of

32 GDP growth series (first difference of the log-levels) accounts for only 18 percent of the total

variation.15

Table 1 Average Pairwise Correlations of Volatility and GDP

Panel A: Volatility (Level)

Pair. Corr. Pair. Corr. Pair. Corr.

Argentina 0.21 India 0.36 Philippines 0.43Australia 0.54 Indonesia 0.37 South Africa 0.48Austria 0.44 Italy 0.47 Singapore 0.58Belgium 0.53 Japan 0.52 Spain 0.55Brazil 0.15 Korea 0.46 Sweden 0.58Canada 0.55 Malaysia 0.39 Switzerland 0.56China 0.61 Mexico 0.51 Thailand 0.46Chile 0.42 Netherlands 0.54 Turkey 0.35Finland 0.33 Norway 0.55 United Kingdom 0.57France 0.56 New Zealand 0.49 United States 0.59Germany 0.60 Peru 0.44

Panel B: GDP (Log-Difference)

Pair. Corr. Pair. Corr. Pair. Corr.

Argentina 0.05 India -0.02 Philippines 0.05Australia 0.15 Indonesia 0.11 South Africa 0.18Austria 0.19 Italy 0.31 Singapore 0.20Belgium 0.27 Japan 0.20 Spain 0.22Brazil 0.17 Korea 0.11 Sweden 0.21Canada 0.23 Malaysia 0.21 Switzerland 0.23China 0.07 Mexico 0.17 Thailand 0.19Chile 0.15 Netherlands 0.22 Turkey 0.13Finland 0.23 Norway 0.13 United Kingdom 0.19France 0.28 New Zealand 0.15 United States 0.23Germany 0.24 Peru 0.04

Note. Panel (A) refers to the volatility measures as computed in (42). Panel (B) refers to GDP in

log-differences. The average pairwise correlations across countries are 0.48 and 0.17 for volatility and

GDP, respectively. Same statistics as in Figure 2.

The pairwise correlation analysis yields similar results. The average pairwise correlations for

all countries are reported in Table 1. Lets focus first on the overall averages for the two variables:

15Note that the first component in the panel of GDP log-levels explains 97 percent of the total variation, but it ismost likely driven by the unit root or near unit root behavior of this variable.

25

the average pairwise correlation of the country-specific realized volatility measures is 0.48. In

contrast, the average pairwise correlation of real GDP series is only 0.17 for the first differences of

the log-level. As we can see from Table 1, the pairwise correlations of volatility and growth have

a very similar pattern across countries, thus providing evidence that are in accordance with our

identification assumption.

In sum, pairwise correlation analysis and principal component analysis lead to similar conclu-

sions, despite the different information content due to the unbalanced nature of the volatility panel.

They suggest that the cross-country correlation of realized volatility is much larger than that of

GDP growth, fully consistent with our identification assumptions.

5.3 Country Correlations Between Volatility and Growth

The countercyclical behavior of the U.S. stock market volatility is a well known stylized fact.16

Does this stylized fact hold in our large sample of countries? Figure 4 plots the country-specific

contemporaneous correlation for all countries in the panel. It reveals that, for most countries, there

is a strong negative and statistically significant association between volatility and GDP growth.

On average, this correlation is about −0.2, ranging from a maximum of −0.46 in the case of

Argentina to a minimum of −0.004 for Australia. As the confidence intervals illustrates most of

these associations are also statistically significant.

In Figure 5, we also look at the typical dynamic pattern of correlation in the cross-section at

lags and leads. Here, the darker dots representing the median and the shaded areas the 10-90

percentile range of the distributions. For each country i = 1, ..., N , led and lagged correlations

between growth and volatility are computed as follows:

ri(±n) = COR(∆yit, vi,t±n) n = 0, 1, ..., 4, (43)

where ∆yit is quarterly growth rate of real GDP in country i, vi,t±n is the realized volatility in

country i, and n denotes leads and lags. The lead/lag pattern found confirms the merit to work

16See, for example, Schwert (1989a) and Schwert (1989b). On the volatility of firm-level stock returns see Campbellet al. (2001), Bloom et al. (2007) and Gilchrist et al. (2013); on the volatility of plant, firm, industry and aggregateoutput and productivity see Bloom et al. (2012) and Bachmann and Bayer (2013); on the behavior of expectations’disagreement see Popescu and Smets (2010) and Bachmann et al. (2013).

26

Figure 4 Country Correlation between Volatility and Growth Data

Argen

tina

Thaila

nd

Brazil

Indo

nesia

Korea

Swed

en

Japa

n

Unite

d Sta

tes

Switz

erland

Malay

sia

Philip

pine

s

Unite

d Kin

gdom

New

Zea

land

Mex

ico

Net

herla

nds

Turke

y

China

Italy

Spain

Ger

man

y

Can

ada

Belgium

South

Afri

ca

Franc

e

Singa

poreIn

dia

Nor

way

Austri

a

Finland

Chile

Austra

liaPer

u

-0.6

-0.4

-0.2

0

0.2

0.4C

orr

ela

tio

n

Note. The correlations are computed over the period 1979:Q1–2011:Q2 (subject to data availability asdetailed in the Appendix). The dots represent the country-specific contemporaneous correlations, andthe lines represent 95% confidence intervals.

with a dynamic framework with at least one lag and suggests that volatility may lead changes in

growth.

Figure 5 Lead/lag Country Correlation between Volatility andGrowth Data

-4 -3 -2 -1 0 +1 +2 +3 +4

-0.4

-0.2

0

0.2

Co

rre

latio

n

Note. The correlations are computed over the period 1979:Q1–2011:Q2 (subject to data availabilityas detailed in the Appendix). The dots represent the median of the distribution across countries atthe given lag or lead; the shaded areas describe the cross-country 10-90 percentile range.

6 Results

In this section we report the estimation results. First, we estimate the factors (ft and gt) using (38)

and (39), and we look at their time series properties. Armed with an estimate of the factors ft and

gt, we then estimate the model (40)-(41) with OLS to recover, for each country, the country-specific

27

volatility (ηit) and growth innovations (εit). Next, with these innovations, we compute conditional

country and cross-country correlations. Finally we will compute impulse responses and variance

decompositions to both the factors (ft and gt) and the country-specific shocks (ηit) and (εit).

6.1 Estimated Factors

The factors ft and gt are recovered from the OLS estimation of (38) and (39). Figure 6 plots them.

As they are standardized, they have zero mean and unit variance. By construction, they are also

serially uncorrelated and orthogonal to each other. The figure shows that the largest decline in the

fundamental factor was after the 1979 second oil shocks and after the 2008 Lehman’s collapse. The

largest increases in the non-fundamental factor were in correspondence to the 1987 stock market

crash and again at the time of the 2008 Lehman’s collapse. These peaks and through correspond

well known global recessions and two episodes of exceptional increase in volatility.

6.2 Cross-country Correlations of Country-specific Shocks

Next, we explore the cross-country correlation of the residuals from the estimation of (40)-(41) to

assess plausibility of our identification assumptions. As we did not impose direct restrictions on the

cross-country correlation structure of these residuals, their properties are informative as to whether

our identification assumptions might be inconsistent with the results obtained.

Table 2 reports the average pairwise correlation of the volatility innovations (ηit) and GDP

growth innovations (εit) across countries. These are the same statistics as in Table 1. Panel (A)

reports the pairwise correlations of the volatility innovations. Panel (B) reports those of the growth

innovations.

The results are in strong accordance with Assumption 3. They show that, if we condition on

both factors, f and g, both sets of pairwise correlations are negligible, with overall averages below

0.05.

For illustrative purposes, in Table 3 below, we also report the same statistics when we condition

only on ft, like in model (1)-(2), rather than on both ft and gt like in model (20)-(21); that is,

the average pairwise correlation between the volatility innovations (uit) and the growth innovations

(εit), rather than the pairwise correlation between ηit and εit.

28

Figure 6 Estimated Fundamental (f) And Non-fundamental (g) Factors

(A) Fundamental Factor f

1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009

-4

-2

0

2

(B) Non-fundamental Factor g

1979 1982 1985 1988 1991 1994 1997 2000 2003 2006 2009

-2

0

2

4

Note. The factors (f and g) are computed using (38) and 39) with using 1 lag of zit. Both factors(f and g) are standardized. The dotted lines represent one-standard deviations bands around themean.

Table 3 shows that, if we condition only on f , the volatility innovations display a pairwise

correlation comparable to that of the raw data, of about 0.4 − 0.5. In contrast, the pairwise

correlations of the growth innovations are negligible, below 0.1 in absolute value for all countries

regardless whether we condition on both factors or only the fundamental one. In the case of the

United States, for instance, the pairwise correlation of volatility innovations remains more than 0.5

conditional on ft alone. In contrast, the pairwise correlation of the GDP growth residuals drops

to 0.07 if we condition on both factors. By comparison the pairwise correlation of the growth

innovations of is −0.06.

In other words, Table 3 illustrates that, after conditioning on ft alone —which is common to

both GDP growth and volatility series— not much commonality is left in the growth innovations;

but the volatility innovations still share a large common component. We regard this evidence as

29

Table 2 Average Pairwise Correlations of Volatility and GDP Growth In-novations (Conditional on both f and g)

Panel A: Volatility Innovations

Pair. Corr. Pair. Corr. Pair. Corr.

Argentina 0.01 India -0.09 Philippines 0.05Australia 0.12 Indonesia 0.01 South Africa 0.08Austria 0.07 Italy 0.07 Singapore 0.14Belgium 0.11 Japan -0.03 Spain 0.10Brazil -0.19 Korea 0.02 Sweden 0.14Canada 0.06 Malaysia 0.05 Switzerland 0.14China -0.21 Mexico 0.09 Thailand 0.00Chile 0.04 Netherlands 0.10 Turkey 0.07Finland 0.03 Norway 0.16 United Kingdom 0.11France 0.13 New Zealand 0.10 United States 0.08Germany 0.11 Peru 0.04

Panel B: Growth Innovations

Pair. Corr. Pair. Corr. Pair. Corr.

Argentina -0.03 India -0.08 Philippines 0.04Australia -0.01 Indonesia -0.01 South Africa 0.05Austria 0.05 Italy 0.09 Singapore 0.04Belgium 0.09 Japan -0.03 Spain 0.06Brazil 0.02 Korea 0.06 Sweden 0.09Canada 0.02 Malaysia 0.03 Switzerland 0.04China -0.19 Mexico 0.05 Thailand 0.09Chile 0.01 Netherlands 0.06 Turkey 0.00Finland 0.06 Norway 0.00 United Kingdom 0.02France 0.08 New Zealand 0.06 United States -0.06Germany 0.06 Peru 0.05

Note. Volatility innovations (η) are the residuals of equation (40). GDP innovations (ε) are the residuals

of equation (41). The average pairwise correlation across countries is 0.05 and 0.03 for the volatility

innovations and the GDP innovations respectively. Same statistics as in Figure C.3.

strongly supportive of our identification Assumption 3.

6.3 Country Correlations Between Volatility and GDP Growth Innovations

We saw earlier in Figure 4 that the raw volatility and growth data (vit and ∆yit, respectively)

display a strong (and statistically significant) negative correlation at the country level. How much

of that association is accounted for by our common factors ft and gt? To answer this question, and

similarly to what we have done above, we now compute the correlations between the volatility and

growth innovations (ηit and εit) conditional on both ft and gt, or conditional only on ft.

Figure 7 reports the results. It plots the contemporaneous correlation together with a 95

30

Table 3 Average Pairwise Correlations of Volatility and GDP Growth In-novations (Conditional on f only)

Panel A: Volatility Innovations

Pair. Corr. Pair. Corr. Pair. Corr.

Argentina 0.26 India 0.29 Philippines 0.31Australia 0.54 Indonesia 0.30 South Africa 0.51Austria 0.44 Italy 0.44 Singapore 0.53Belgium 0.50 Japan 0.48 Spain 0.54Brazil 0.06 Korea 0.41 Sweden 0.55Canada 0.57 Malaysia 0.38 Switzerland 0.54China 0.49 Mexico 0.50 Thailand 0.41Chile 0.42 Netherlands 0.54 Turkey 0.35Finland 0.23 Norway 0.56 United Kingdom 0.56France 0.55 New Zealand 0.43 United States 0.57Germany 0.55 Peru 0.52

Panel B: Growth Innovations

Pair. Corr. Pair. Corr. Pair. Corr.

Argentina -0.03 India -0.08 Philippines 0.04Australia -0.01 Indonesia -0.01 South Africa 0.05Austria 0.05 Italy 0.09 Singapore 0.04Belgium 0.09 Japan -0.03 Spain 0.06Brazil 0.02 Korea 0.06 Sweden 0.09Canada 0.02 Malaysia 0.03 Switzerland 0.04China -0.19 Mexico 0.05 Thailand 0.09Chile 0.01 Netherlands 0.06 Turkey 0.00Finland 0.06 Norway 0.00 United Kingdom 0.02France 0.08 New Zealand 0.06 United States -0.06Germany 0.06 Peru 0.05

Note. Volatility innovations (u) are the residuals of a version of equation (40), where we control only for

f (but not for g). GDP innovations (ε) are the residuals of equation (41) as before. The average pairwise

correlation across countries is 0.45 and 0.03 for the volatility innovations and the GDP innovations

respectively. Same statistics as in Figure C.4.

percent confidence interval between ηit and εit from (40)-(41) for all countries in our sample. The

conditional correlation between volatility and growth innovations falls dramatically conditional on

ft and gt, and it is much weaker than in the raw data. As we can see, the confidence interval now

includes zero for all but 5 countries, which are all highly volatile emerging markets. The common

factors therefore absorb a large portion of the association between volatility and growth at quarterly

frequency. For the United States, for instance, this correlation drops to −0.06 and is statistically

not significant. Interestingly, we essentially the same results even if we do not model the second

strong factor gt explicitly.17

17Results not reported but available from the authors on request.

31

Figure 7 Country Correlation between Volatility and Growth Innova-tions

Argen

tina

China

Brazil

Malay

sia

Indo

nesia

Japa

n

Swed

en

Thaila

nd

Switz

erland

New

Zea

land

South

Afri

caIn

dia

Ger

man

y

Korea

Net

herla

nds

Austri

a

Can

ada

Mex

ico

Nor

way

Philip

pine

s

Unite

d Kin

gdom

Singa

pore

Italy

Turke

y

Belgium

Spain

Unite

d Sta

tes

Austra

lia

Franc

e

Finland

ChilePer

u

-0.6

-0.4

-0.2

0

0.2

0.4C

orr

ela

tion

Note. Volatility innovations (η) are the residuals of equation (40) GDP innovations (ε) are theresiduals of equation (41). The dots represent the country-specific contemporaneous correlations,and the lines represent 95% confidence intervals.

Overall, these results suggest that country-specific volatility and growth series share a significant

common component at quarterly frequency, and that conditioning on our fundamental factor ft

absorbs most of it. But note that this evidence does not suggests that time-varying changes in

volatility are driven mostly by shocks to the fundamental factor ft. That is, while shocks to the

fundamental factor ft can account for most of the contemporaneous correlation between volatility

and growth, it does not not necessarily accounts for the time variation in volatility. In fact, as we

will see below, shocks to ft will turn out to explain a very small share of the volatility variance

over time.

6.4 Impulse Responses

We compute impulse responses to shocks to the orthogonal factors ft and gt following the derivations

in the Appendix A.4. We consider a one-standard deviation shock to each of the two factors and

Figure 8 reports the results. The figure plots both country-specific responses (thin lines) and a

weighted average with PPP GDP weights (thick line).

Panel (A) plots the response of volatility to a positive, one-standard-deviation shock to the com-

mon fundamental factor ft. It shows that volatility falls in many (but not all) countries when ft

rises. This reflects an endogenous response of volatility to fundamental changes in the world econ-

32

omy and it is consistent with volatility increasing when growth tanked during the global financial

crisis in 2008.

Panel (B) plots the growth response to the same shock. It shows that growth loads positively

in almost all countries on the fundamental factor ft, with effects lasting up to 4-5 quarters. This

is consistent with existing evidence on the international business cycle, which stresses the role of a

world factor, along with regional and country factors in driving the business cycle (e.g., Kose et al.

(2003)).

Panel (C) and (D) report the response of volatility and growth to a positive, one-standard-

deviation shock to the non-fundamental factor gt. Country-specific volatility increases on impact

in all countries in response to this shock, and then gradually declines. Growth declines only with a

time lag, with some differences in the intensity of the response across countries. Notice here that,

the delayed response of growth to this shocks follows from our identification assumptions through

cross-country correlations, but it is not imposed explicitly on country models like in a Cholesky

decomposition of the variance-covariance matrix ordering growth first. These responses suggest

that a positive shock to this factor is a “global bad news”. Note finally that both volatility and

growth appear to respond more persistently to an innovation in gt than in ft, possibly reflecting a

bigger role for learning and other information frictions in its transmission.

6.4.1 Comparing Common and Country-specific Shocks

For the case of the United States, we now compare the response to shocks to global factors with

responses to the country-specific shocks ηit and εit assuming that the latter is orthogonal to all

other country-specific shocks in the model. The details of the computations are spelled out in

Appendix A.4.

This strong assumption is justified by the evidence reported earlier in Table 2 and Figure 7

that most of the contemporaneous correlations between volatility and growth innovations, within

or between countries, is absorbed by the common factors ft and gt. For most countries, in fact,

these conditional correlations were negligible. Thus, as a first approximation, we assume that the

variance covariance matrix of the country-specific innovations, for the whole collection of countries

N , is diagonal. We then check that we obtain the same results by allowing for non-zero correlations.

33

Figure 8 Country Volatility and Growth Impulse Responses to CommonShocks

5 10 15 20

-2

-1

0

1

(A) vi to a shock to f

5 10 15 20

0

0.2

0.4

0.6

0.8

(B) ∆yi to a shock to f

5 10 15 20

0

1

2

3

4

(C) vi to a shock to g

5 10 15 20

-0.8

-0.6

-0.4

-0.2

0

(D) ∆yi to a shock to g

Note. One-Standard Deviation sock to ft and gt. Thin lines are individual countries responses.The solid line is the PPP-GDP Weighted average. Impulse responses computed as in the AppendixA.4.

This is done by compute the same impulse responses using the generalized method of Pesaran and

Shin (1998), which allows for such non-zero correlations.

Figure 9 plots the response of US volatility and growth data to the common fundamental factor

(ft) and compares it to country-specific growth εit assuming that, as we justified above, the latter is

orthogonal to all other country-specific shocks in the model. The comparison illustrates that while

the common and country-specific shocks have a similar impact on growth, the common fundamental

shock has a much deeper and more persistent effect on volatility than the country-specific shock to

US growth.

Figure 10 reports the same comparison for a shock to the non-fundamental factor (gt) and

a country-specific shock to US volatility. This second comparison illustrates that while the non-

fundamental common factor has a similar effect on volatility than a country-specific shock, it has

a much deeper and more persistent impact on growth.

34

Figure 9 US Impulse Responses to Fundamental Shock

5 10 15 20

-0.4

-0.2

0

US Volatility (vi)

5 10 15 20

0

0.2

0.4

US Real GDP (∆yi)

εUS

f

Note. One-standard-deviation shocks. The thin line is the response to a country-specific US growthshock. The thick line is the response to a global fundamental shock (also reported in Figure 8).Impulse responses computed as in the Appendix A.4.

Figure 10 US Impulse Responses to Non-fundamental Shock

5 10 15 20

1

2

3

US Volatility (vi)

5 10 15 20

-0.1

-0.05

0

US Real GDP (∆yi)

ηUS

g

Note. One-standard-deviation shocks. The thin line is the response to a country-specific USvolatility shock. The thick line is the response to a global non-fundamental shock (also reported inFigure 8). Impulse responses computed as in the Appendix A.4.

6.5 Variance Decompositions

While impulse responses illustrate the transmission mechanism of a given shock identified in the

model, variance decompositions speak to the importance of such a shock for the time-variation

of the endogenous variables at different time horizons relative to the other model shocks. For all

countries, we decompose the total variance of volatility and growth and attribute it to shocks to

the orthogonal factors ft and gt as well as all country-specific shocks in the model εit and ηit.18

We analyze first the forecast error variance decomposition of the United States and then the one of

the ‘typical country’ represented by a PPP-GDP weighted average of the country-specific variance

18See Appendix A.4 for the details on the derivations. Like in the case of the response functions, the results belowwere computed assuming that the variance-covariance matrix of the model is diagonal. We then find essentially thesame results computing these decompositions using generalized variance decompositions of Pesaran and Shin (1998),which takes any residual contemporaneous correlation into account.

35

decompositions.

6.5.1 United States

Figure 11 reports the results for the United States, for both volatility and growth. The figure

shows that US volatility is driven largely by shocks to the non-fundamental global factor (more

than 50 percent). The country-specific volatility shock also explains a significant portion of the

total variance (about 20 percent). Shocks to the fundamental common factor explain a very small

variance share (less than 10 percent). All other country-specific shocks have essentially no role in

explaining the variance of US volatility.

Figure 11 US Variance Decompositions

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9 10 11 12

US Volatility (vit)

Othergfεη

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9 10 11 12

US Real GDP (Δyit)

Othergfεη

Note. Country Variance Decompositions as reported in Figure B.1. For both growth and volatilitythe figure reports the variance decompositions with respect to all the shocks in the model. Other isan aggregate of all other country-specific shocks other than the US shock. Variance decompositionscomputed as in Appendix A.4.

This evidence has important implications. First, it suggests that the component of US volatility

that is endogenous to common fundamental forces in the world economy (or the US economy itself)

is very small. Second, it also suggests that, while global non-fundamental shocks have a much

larger impact that US specific volatility shocks when they occur, they account for a smaller share

of the volatility variance.

The figure also shows that US growth variance is driven mostly by country-specific growth

shocks (about 50 percent in the long-run) and shocks to the global fundamental factor. The

global fundamental factor, in particular, explains about 30 percent of the growth variance, in line

36

with results in the international business cycle literature (see, for instance, Kose et al. (2003)).

Its importance for growth variance increases slowly over time and stabilizes after 1.5 − 2 years.

US volatility shocks and all other country-specific shocks have a negligible importance for growth

variance according to these results.

6.5.2 Typical Economy

We find similar results for the typical country represented by a PPP-GDP weighted average of

the country-specific variance decompositions (Figure 12). Relative to the US, we see a similar

60 − 40 percent split for the volatility variance between the common non-fundamental factor and

the country-specific volatility shock. The share of variance explained by the common fundamental

factor is similarly small, below 10 percent. In the case of the typical economy, we find an even

larger share of growth variance explained by the country-specific growth shock than in the US case.

We note, however, a large heterogeneity that is documented in Appendix, where we report results

country by country in Figures C.1 and C.2.

Figure 12 Typical Country Variance Decomposition

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9 10 11 12

Volatility (vit), average

Othergfεη

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9 10 11 12

Real GDP (Δyit), average

Othergfεη

Note. PPP-GDP Weighted Average of Country Variance Decompositions. Country Variance De-compositions reported in Figure B.1. For both growth and volatility the figure reports the weightedaverage of the variance decompositions with respect to all the shocks in the model. Other is anaggregate of all other country-specific shocks other than the own country. Variance decompositionscomputed as in Appendix A.4.

This evidence is consistent with the notion that the endogenous component of volatility is

relatively small. Moreover, even though the country-specific factor explains a large portion of the

volatility variance, it seems that volatility has a secondary role in explaining growth variance. This

37

means that, a global bad news can potentially inflict big damage to the world economy, but most

of the world economy variance is not driven by such shocks.

7 Conclusions

The global financial crisis spurred renewed academic interest on quantifying the association between

uncertainty on macroeconomic dynamics. In this paper, we study the interrelation between realized

equity price volatility and GDP growth without taking a stance on the direction of causation.

To do so we assume volatility and growth are driven by two common factors. By taking a

multi-country approach, as opposed to studying a single economy in isolation, we can identify and

estimate these two factors, as well as country-specific volatility and growth shocks. Identification

exploits different patterns of correlation across countries of country specific volatility and GDP

growth shocks that we find in the data.

We find that shocks to the common fundamental factor accounts for most of the unconditional

correlation between volatility and growth at the country level. However, the fundamental factor and

the country-specific growth shock can explain only a small fraction of country volatility variance.

This implies that the endogenous component of volatility is small, or that volatility is largely driven

by shocks to the non-fundamental common factor and its own country-specific dynamics. Finally,

we also find that even though shocks to the non-fundamental factor have a much larger impact on

growth than country-specific volatility shocks, in most countries (including the United States), the

non-fundamental factor explains less than 10 percent of growth variance in the long run.

Overall, we interpret this evidence as suggesting that the financial and the business cycle share

a small common component, and the financial cycle is driven mostly by its own dynamics that

affects the business cycle only when the shocks are global in nature.

38

References

Abel, A. B. (1983): “Optimal Investment under Uncertainty,” American Economic Review, 73,228–33.

Andersen, T. G., T. Bollerslev, F. X. Diebold, and P. Labys (2001): “The Distribution ofRealized Exchange Rate Volatility,” Journal of the American Statistical Association, 96, 42–55.

——— (2003): “Modeling and Forecasting Realized Volatility,” Econometrica, 71, 579–625.

Arellano, C., Y. Bai, and P. Kehoe (2012): “Financial Markets and Fluctuations in Uncer-tainty,” Unpublished manuscript.

Bachmann, R. and C. Bayer (2013): “’Wait-and-See’ business cycles?” Journal of MonetaryEconomics, 60, 704–719.

Bachmann, R., S. Elstner, and E. R. Sims (2013): “Uncertainty and Economic Activity:Evidence from Business Survey Data,” American Economic Journal: Macroeconomics, 5, 217–249.

Bachmann, R. and G. Moscarini (2011): “Business Cycles and Endogenous Uncertainty,” 2011Meeting Papers 36, Society for Economic Dynamics.

Bai, J. and S. Ng (2002): “Determining the Number of Factors in Approximate Factor Models,”Econometrica, 70, 191–221.

Baker, S. R. and N. Bloom (2013): “Does Uncertainty Reduce Growth? Using Disasters asNatural Experiments,” NBER Working Papers 19475, National Bureau of Economic Research,Inc.

Barndorff-Nielsen, O. E. and N. Shephard (2002): “Estimating quadratic variation usingrealized variance,” Journal of Applied Econometrics, 17, 457–477.

——— (2004): “Econometric Analysis of Realized Covariation: High Frequency Based Covariance,Regression, and Correlation in Financial Economics,” Econometrica, 72, 885–925.

Bernanke, B. S. (1983): “Irreversibility, Uncertainty, and Cyclical Investment,” The QuarterlyJournal of Economics, 98, 85–106.

Bloom, N. (2009): “The Impact of Uncertainty Shocks,” Econometrica, 77, 623–685.

——— (2014): “Fluctuations in Uncertainty,” Journal of Economic Perspectives, 28, 153–176.

Bloom, N., S. Bond, and J. V. Reenen (2007): “Uncertainty and Investment Dynamics,”Review of Economic Studies, 74, 391–415.

Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. J. Terry (2012):“Really Uncertain Business Cycles,” NBER Working Papers 18245, National Bureau of EconomicResearch, Inc.

Campbell, J. Y., M. Lettau, B. G. Malkiel, and Y. Xu (2001): “Have Individual StocksBecome More Volatile? An Empirical Exploration of Idiosyncratic Risk,” Journal of Finance,56, 1–43.

39

Carriere-Swallow, Y. and L. F. Cespedes (2013): “The impact of uncertainty shocks inemerging economies,” Forthcoming Journal of international Economics.

Cesa-Bianchi, A., M. H. Pesaran, and A. Rebucci (2014): “Uncertainty and EconomicActivity: A Global Perspective,” Tech. rep.

Christiano, L., R. Motto, and M. Rostagno (2014): “Risk Shocks,” American EconomicReview, 104, 27–65.

Chudik, A. and M. H. Pesaran (2013): “Econometric Analysis of High Dimensional VARsFeaturing a Dominant Unit,” Econometric Reviews, 32, 592–649.

——— (2015): “Common correlated effects estimation of heterogeneous dynamic panel data modelswith weakly exogenous regressors,” Journal of Econometrics, 188, 393–420.

Chudik, A., M. H. Pesaran, and E. Tosetti (2011): “Weak and strong cross???section de-pendence and estimation of large panels,” Econometrics Journal, 14, C45–C90.

Clark, T. E., A. Carriero, and M. Massimiliano (2016): “Measuring Uncertainty and ItsImpact on the Economy,” Working Paper 1622, Federal Reserve Bank of Cleveland.

Decker, R., P. D’Erasmo, and H. Moscoso Boedo (2014): “Market Exposure and Endoge-nous Firm Volatility over the Business Cycle,” Unpublished manuscript.

Dixit, A. K. and R. S. Pindyck (1994): Investment under Uncertainty, Princeton UniversityPress.

Fernandez-Villaverde, J., P. Guerron-Quintana, J. F. Rubio-Ramirez, and M. Uribe(2011): “Risk Matters: The Real Effects of Volatility Shocks,” American Economic Review, 101,2530–61.

Fostel, A. and J. Geanakoplos (2012): “Why Does Bad News Increase Volatility and DecreaseLeverage?” Journal of Economic Theory, 147, 501–525.

Giglio, S., I. Dew-Becker, and D. Berger (2016): “Contractionary Volatility or VolatileContractions?” Tech. rep.

Gilchrist, S., J. Sim, and E. Zakrajsek (2013): “Uncertainty, Financial Frictions, and Irre-versible Investment,” Unpublished manuscript.

Hartman, R. (1976): “Factor Demand with Output Price Uncertainty,” American EconomicReview, 66, 675–81.

Hirata, H., M. A. Kose, C. Otrok, and M. E. Terrones (2012): “Global House Price Fluctu-ations: Synchronization and Determinants,” in NBER International Seminar on Macroeconomics2012, National Bureau of Economic Research, Inc, NBER Chapters.

Jurado, K., S. C. Ludvigson, and S. Ng (2015): “Measuring Uncertainty,” American Eco-nomic Review, 105, 1177–1216.

Kimball, M. S. (1990): “Precautionary Saving in the Small and in the Large,” Econometrica, 58,53–73.

40

Kose, M. A., C. Otrok, and C. H. Whiteman (2003): “International Business Cycles: World,Region, and Country-Specific Factors,” American Economic Review, 93, 1216–1239.

Leahy, J. V. and T. M. Whited (1996): “The Effect of Uncertainty on Investment: SomeStylized Facts,” Journal of Money, Credit and Banking, 28, 64–83.

Leduc, S. and Z. Liu (2012): “Uncertainty shocks are aggregate demand shocks,” Tech. rep.

Lewis, K. K. and E. X. Liu (2015): “Evaluating international consumption risk sharing gains:An asset return view,” Journal of Monetary Economics, 71, 84–98.

——— (2017): “Disaster Risk and Asset Returns: An International Perspective,” NBER WorkingPapers 23065, National Bureau of Economic Research, Inc.

Ludvigson, S. C., S. Ma, and S. Ng (2015): “Uncertainty and Business Cycles: ExogenousImpulse or Endogenous Response?” NBER Working Papers 21803, National Bureau of EconomicResearch, Inc.

Mirman, L. J. (1971): “Uncertainty and Optimal Consumption Decisions,” Econometrica, 39,179–85.

Oi, W. (1961): “The desirability of price stability under perfect competition,” Econometrica, 29(1),58–64.

Pesaran, H. H. and Y. Shin (1998): “Generalized impulse response analysis in linear multivariatemodels,” Economics Letters, 58, 17–29.

Pesaran, M. H. and A. Chudik (2014): “Aggregation in large dynamic panels,” Journal ofEconometrics, 178, 273–285.

Popescu, A. and F. R. Smets (2010): “Uncertainty, Risk-taking, and the Business Cycle inGermany,” CESifo Economic Studies, 56, 596–626.

Ramey, G. and V. A. Ramey (1995): “Cross-Country Evidence on the Link between Volatilityand Growth,” American Economic Review, 85, 1138–51.

Rigobon, R. (2003): “Identification Through Heteroskedasticity,” The Review of Economics andStatistics, 85, 777–792.

Schwert, G. W. (1989a): “Business cycles, financial crises, and stock volatility,” Carnegie-Rochester Conference Series on Public Policy, 31, 83–125.

——— (1989b): “Why Does Stock Market Volatility Change over Time?” Journal of Finance, 44,1115–53.

Tesar, L. L. (1995): “Evaluating the gains from international risk sharing,” Carnegie-RochesterConference Series on Public Policy, 42, 95–143.

Tian, C. (2012): “Riskiness Choice and Endogenous Productivity Dispersion over the BusinessCycle,” PIER Working Paper Archive 12-025, Penn Institute for Economic Research, Departmentof Economics, University of Pennsylvania.

Van Nieuwerburgh, S. and L. Veldkamp (2006): “Learning asymmetries in real businesscycles,” Journal of Monetary Economics, 53, 753–772.

41

Zarnowitz, V. and L. A. Lambros (1987): “Consensus and Uncertainty in Economic Predic-tion,” Journal of Political Economy, 95, 591–621.

42

A Proofs

A.1 SVAR Equivalence

In this section, we show that model (1)-(2) is observationally equivalent to a standard structuralvector autoregression model. Thus, it is subject to the same identification requirements.

To see this, consider the following alternative specification of the relation between volatility andgrowth:

vit = αi∆yit + ζit, (A.1)

∆yit = βivit + υit. (A.2)

where ζit and υit are assumed to be orthogonal. Now, rewrite this model in a more compact formas: [

1 −αi−βi 1

] [vit

∆yit

]=

[ζitυit

], (A.3)

where the covariance matrix between vit and ∆yit is given by:

Ωi =1

|1− αiβi|

[σ2ζ,i + α2

iσ2υ,i αiσ

2υ,i + βσ2

ζ,i

− σ2ζ,i + β2σ2

υ,i

](A.4)

as long as αiβi 6= 1. The parameters of interest are: (αi, βi) and (σ2ζ,i, σ

2υ,i). It is evident that, if

αi and βi are both free parameters, the model is not identified.

Identification of this model is typically achieved by assuming a specific direction of causalitybetween the observable variables, e.g. by assuming that activity causes volatility or vice-versa,contemporaneously. This can be accomplished in a variety of ways: with timing or exclusionrestrictions, i.e. by assuming that one variable cannot respond contemporaneously to shocks tothe other (e.g., volatility shocks are faster than real activity shocks or that volatility is simplyexogenous); with long run restrictions, i.e. by assuming that volatility shocks need to vanish inthe long run, while activity shocks do not; with sign restrictions based on economic priors on thesign and possibly magnitude of the impact effect. Alternatively, identification can be achievedby choosing a suitable instrumental variable for one of the two shocks (like a deeper measure ofuncertainty such as natural disasters or other measures of tail risk). One way or the other, toachieve identification of this relation for any individual country we need to impose one ”exclusion”restriction at the level of individual countries: something akin to αi = 0 or βi = 0.

In contrast, our strategy relies on a collection of countries as opposed to an individual countryin isolation, and we refrain from imposing the missing restriction at the country level. So, in themodel above, we are going to make assumptions on the linkages among N countries in terms ofvolatility and growth innovations, as opposed to assumptions on each of them taken individually.In particular, we will exploit the fact that uit and εit have a different structure of contemporaneouscorrelation across countries as documented in Figure 2 and Table 2.

A.2 Unobservable Factors with Heterogeneous Dynamics

In this section we prove that, even if the model is dynamic and heterogeneous, our factors canbe recovered from a distributed lag of the world averages of country-specific volatility and GDP

43

growth rates. Specifically, given:

zit = ai + Φizi,t−1 + Γift + ξit, (A.5)

where zit = (vit,∆yit)′ and

ai =

(aivaiy

), Φi =

(φi,11 φi,12

φi,21 φi,22

), Γi =

(λi θiγi 0

), ft =

(ftgt

), ξit =

(ηitεit

). (A.6)

we want to prove that:

ft = γ−1∆yω,t +∞∑`=1

c′1,`zω,t−` +Op

(N−1/2

), (A.7)

gt = θ−1vω,t −λ

θγ∆yω,t +

∞∑`=1

c′2,`zω,t−` +Op

(N−1/2

), (A.8)

where

vω,t =N∑i=1

wivit, ∆yω,t =N∑i=1

wi∆yit, zω,t = (vω,t,∆yω,t) .

and c′1,` and c′2,` are 1× 2 vectors of coefficients.

Proof. Under Assumption 8, assuming that zit, i = 1, 2, ..., N started a long time ago, we have:

zit = µi +∞∑`=0

Φ`iΓift−` + ζit, (A.9)

where:

µi = (I2 −Φi)−1ai and ζit =

∞∑`=0

Φ`iξi,t−`. (A.10)

Pre-multiplying both sides of (A.9) by (wi) and summing over i yields:

zωt = µω +

∞∑`=0

A`,N ft−` + ζωt (A.11)

where:zωt =

∑Ni=1wizit,

µω =∑N

i=1wiµi,

A`,N =∑N

i=1wiΦ`iΓi,

ζωt =∑N

i=1wiζit.

(A.12)

Under Assumption 5, ζit are cross-sectionally weakly correlated, given that the wi are granular.Then using results in Pesaran and Chudik (2014), it readily follows that:

ζωt = Op

√√√√ N∑

i=1

w2i

= Op

(N−

12

), for each t. (A.13)

44

Also, under Assumptions 7 and 8:

A`,N − E (A`,N ) =N∑i=1

wi

[Φ`iΓi − E

(Φ`iΓi

)]= Op

√√√√ N∑

i=1

w2i

= Op

(N−

12

). (A.14)

By using this in (A.11), and noting that under 7 and 8 we find that:

E(Φ`iΓi

)= E

(Φ`i

)E (Γi) = Λ`Γ,

we also have:

zωt = µω +

∞∑`=0

Λ`Γft−` +Op

(N−

12

)= µω +

( ∞∑`=0

Λ`L`

)Γft +Op

(N−

12

)= µω + Λ (L) Γf t +Op

(N−1/2

).

But under Assumptions 7 and 8, Γ and Λ (L) are both invertible and:

ft = Γ−1Λ−1 (L) (zωt − µω) +Op

(N−

12

),

where:

Γ−1 =

(0 γ−1

θ−1 − λθγ

),

Λ−1 (L) = I2 + B1L+ B2L2 + ....

(noticing that Λ0 = I2). Hence,

ft = Γ−1 (zωt − µω) +(C1 + C2L+ C3L

2 + ....)

(zω,t−1 − µω) +Op

(N−

12

),

where Ci = Γ−1Bi, for i = 1, 2, ..., N . But given the lower triangular form of Γ−1:

ft = γ−1∆yω,t +

∞∑`=1

c′1,`zω,t−` +Op

(N−

12

), (A.15)

gt = θ−1vω,t −(λ

θγ

)∆yω,t +

∞∑`=1

c′2,`zω,t−` +Op

(N−

12

), (A.16)

where vω,t, ∆yω,t, zω,t are defined as above.

A.3 Observable and Orthogonal Factors

In this section we show how to extract orthogonal factors from the data that are an orthonormaltransformation of the unobservable ones to assess their impact and relative importance on country

45

volatility and growth. Let’s start from the truncated approximation of the unobservable factors:

f = ∆yω + ZωC1, (A.17)

g = vω − λ∆yω + ZωC2, (A.18)

where ∆yω = (∆yω,1,∆yω,2, ...,∆yω,T )′, vω = (vω,1, vω,2, ..., vω,T )′, Zω = (τT , zω,−1, zω,−2, ..., zω,−p),and zω,−i = (∆yω, vω,1).19 We want to prove that the orthogonalized filtered factors f and g canbe recovered as residuals from the following OLS regressions:

∆yω = ZωC1 + f (A.19)

vω = λf + ZωC2 + g (A.20)

Proof. Let MZω= IT − Zω

(Z′ωZω

)−1Z′ω, and note that:

MZωf = MZω

∆yω (A.21)

MZωg = MZω

vω − λMZω∆yω (A.22)

since MZωZωC1 = 0. We set the first normalized factor, that we label f , to MZω

f = MZω∆yω;

and set the second factor, that we label g, as the a linear combination of MZωf and MZω

g such

that f ′g = 0. This can be achieved selecting λ so that:

f ′g = ∆y′ωMZω

(MZω

vω − λMZω∆yω

)= 0. (A.23)

The value of λ that solves this equation is given by:

λ =∆y′ωMZω

vω

∆y′ωMZω∆yω

. (A.24)

Note here that λ is the OLS estimate the coefficient of a regression of MZωvω on MZω

∆yω. Hence,the orthogonalized factors are

f = MZω∆yω, (A.25)

g = MZωvω − λMZω

∆yω. (A.26)

In practice, this implies that f can be recovered as the residual from an OLS regression of ∆yω onzω,t−` with ` = 1, ..., p:

∆yω = ZωC1 + f (A.27)

While g can be recovered as the residual from an OLS regression of vω on f and zω,t−` with` = 1, ..., p:

vω = λf + ZωC2 + g (A.28)

19The inclusion of τT in Zω ensures that the filtered factors have zero meanmean zeros.

46

A.4 Computing Impulse Responses and Variance Decompositions

In this section, we derive the impulse responses and variance decompositions reported in Section6. By using (40) and (41), and abstracting from the constant to simplify notation, we have:

zit = Φizi,t−1 +

p∑`=1

ψi,`zω,t−` + Bift + ξit, for i = 1, 2, ..., N, (A.29)

Bi =

(βi,11 βi,12

βi,21 0

), ft =

(ftgt

). (A.30)

Note here that zω,t =∑N

i=1wi∆zit = Wzt, where zt = (z′1t, z′2t, ..., z

′Nt)′, and W is a 2×2N matrix

of weights. Stacking over i we obtain:

zt = Φzt−1 +

p∑`=1

ψ`Wzt−` + Bf t + ξt, (A.31)

where ξt = (ξ′1t, ξ′2t, ..., ξ

′Nt)′ and:

Φ =

Φ1 0 · · · 00 Φ2 · · · 0...

... · · ·...

0 0 · · · ΦN

, ψ` =

ψ1,`

ψ2,`...

ψN,`

, B =

B1

B2...

BN

. (A.32)

The high-dimensional VAR in (A.31) can now be used to compute impulse responses to differentshocks, global or idiosyncratic. If p = 1 then:

zt = (Φ +ψ`W) zt−1 + Bf t + ξt, (A.33)

which is a factor-augmented VAR(1) in 2N variables. Define now Ψ = Φ +ψ`W. Then:

zt = (I−ΨL)−1(Bf t + ξt), (A.34)

or:zt = (I−ΨL)−1Bf t + (I−ΨL)−1ξt = (I−ΨL)−1Bf t + et, (A.35)

where et is orthogonal to ft′ for all t and t′. Then, noting that:

∞∑`=0

A`L` = (I + ΨL + Ψ2L2 + ...)B =(I−ΨL)−1B, (A.36)

we have:

zt =∞∑`=0

A`ft−` +∞∑`=0

C`ξt−`, for ` = 0, 1, 2, ..., (A.37)

where A` = Ψ`B, and C` = Ψ`.

47

A.4.1 Responses to Common and Country-specific Shocks

Let ei be a selection vector such that e′izt picks the ith element of zt. Also let sf = (1, 0)′ and

sg = (0, 1)′, the vectors that select ft and gt from ft; namely,

s′f ft ≡ ft, s′g ft ≡ gt. (A.38)

Recall now that ft and gt have zero mean, unit variance and are orthogonal to each other. Thenthe impulse response to a positive unit shock to ft or gt are given by:

IRi,f ,n = e′iAnsf and IRi,g,n = e′iAnsg for n = 0, 1, 2, ..., (A.39)

To derive impulse response functions for country-specific shocks, i.e., the jth element of ξt), weneed to make assumptions about the correlation between volatility and growth innovations withineach country and across countries. Since the elements of ξt are weakly correlated across countries,by construction, they have some, but limited correlation across countries (See Tables 2 and 3).We also documented that, conditional on the common factors f and g, the country correlation ofvolatility ad growth innovations are statistically insignificant for all except four countries.

As a first approximation, therefore, we will assume that the variance-covariance matrix of theVAR residuals in (A.33) is diagonal. Under this assumption, the impulse response function of apositive, unit shock to the jth element of ξt on the the ith element of zt is given by:

IRi,ξj ,n =√ωjje

′iCnej , (A.40)

where ωjj is the (estimate) of the variance of the jth country-specific shock and ej is a selectionvector such that e′jzt picks the jth element of zt.

We then compare these responses to the generalized impulse responses of Pesaran and Shin(1998). The latter are given by given by:

GIRi,ξj ,n =e′iCnΩej√

ωjj, (A.41)

where Ω = (ωij) is the estimate of the covariance of ξt. They take any non-zero correlation inthe variance-covariance matrix into account, but depend on the order in which the variables areincluded in the system. They coincide with responses to orthogonalized innovations only if thevariance-covariance matrix is diagonal. In other words, if the shocks were orthogonal, then wewould have GIRi,ξj ,n = IRi,ξj ,n.

A.4.2 Variance Decompositions

Under the assumption that the variance-covariance matrix of all country specific shocks is diagonal,the relative importance of shocks to country volatility and growth for all countries (ξjt, for j =

1, 2, ...., 2N) and shocks to our two common factors that ft and gt, is easily characterized.

Let V Di,f ,n and V Di,g,n be the share of the n-step ahead forecast error variance of the ith

variable in zt that is accounted for by ft and gt, respectively, and V Di,j the variance share of a

48

generic country specific shock, then:

V Di,f ,n =

n∑=0

(e′iA`sf )2

n∑=0

e′iA`A′`ei +

n∑=0

e′iC`ΩC′`ei

, n = 1, 2, ....,H, (A.42)

V Di,g,n =

n∑=0

(e′iA`sg)2

n∑=0

e′iA`A′`ei +

n∑=0

e′iC`ΩC′`ei

, n = 1, 2, ....,H, (A.43)

V Di,j =

ω−1jj

n∑=0

(e′iC`Ωej

)2

n∑=0

e′iA`A′`ei +

n∑=0

e′iC`ΩC′`ei

, j = 1, 2, ...., 2N, n = 1, 2, ....,H; (A.44)

where V Di,f ,n +V Di,g,n +∑2N

j=1 V Di,j = 1 if Ω = diag(ω11, ω22, ..., ω2N,2N ). Like in the case of theimpulse responses, we can then check whether generalized variance decompositions that allow fora full variance-covariance matrix give the same or different results.

A.4.3 Extension to Higher Order VARs

Consider the VAR(p) model:

zt = Ψ1zt−1 + Ψ2zt−2 + · · ·+ Ψpzt−p + Bf t + ξt, (A.45)

zt is a 2N × 1 vector and Ψi are 2N × 2N matrices of coefficients. The impulse responses andforecast error variance decompositions can be computed using the above formulae after re-writingthe model (A.45) in the form of an infinite order moving-average representation, namely:

zt =

∞∑`=0

A`ft−` +

∞∑`=0

C`ξt−`, for ` = 0, 1, 2, ..., (A.46)

where the matrices A` and C` are determined by the recursive relations:

Aj = Φ1Aj−1 + Φ2Aj−2 + · · ·+ ΦpAj−p, j = 1, 2, . . . , (A.47)

Cj = Φ1Aj−1 + Φ2Aj−2 + · · ·+ ΦpAj−p, j = 1, 2, . . . , (A.48)

with

A0 = Bm, and Aj = 0, for j < 0,

C0 = Im, and Cj = 0, for j < 0.

B Data and Summary Statistics

Real GDP and CPI data come from standard sources (See the following web page and posted dataappendix for details: https://sites.google.com/site/gvarmodelling/.

For equity prices we use the MSCI Index in local currency. The data source for the dailyequity price indexes is Bloomberg. The countries included in the sample are the following: Ar-

49

gentina, Australia, Austria, Belgium, Brazil, Canada, Chile, China, Finland, France, Germany,India, Indonesia, Italy, Japan, Korea, Malaysia, Mexico, Netherlands, Norway, New Zealand, Peru,Philippines, Saudi Arabia, South Africa, Singapore, Spain, Sweden, Switzerland, Thailand, Turkey,United Kingdom, and United States (1979-2011).

The list of Bloomberg tickers is the following: MSELTAG, MSDLAS, MSDLAT, MSDLBE,MSELTBR, MSDLCA, MSELTCF, MSELTCH, MSDLFI, MSDLFR, MSDLGR, MSELTIA, MSELT-INF, MSDLIT, MSDLJN, MSELTKO, MXMY, MSELTMXF , MSDLNE, MSDLNO, MSDLNZ,MSELTPR, MSELTPHF, MSELTSA, MGCLSA, MSDLSG, MSDLSP, MSDLSW, MSDLSZ, MSELT-THF, MSELTTK, MSDLUK, MSDLUS.

C Additional Results

In this appendix, we report selected country-by-country results, including summary statistics ofreal GDP and volatility series that support our choice for the specification of the model. Table C.1reports the summary statistics of the log-levels of real GDP for each country. Table C.2 reportsthe same set of summary statistics for the volatility series.

We also report country-specific forecast error variance decompositions for both the volatilityand GDP series in Figures C.1 and C.2.

50

Table

C.1

SummaryStatisticsRorRealGDP

(Log-L

evels)

AR

GA

UST

LIA

AU

ST

RIA

BE

LB

RA

CA

NC

HIN

AC

HL

FIN

FR

AN

CE

GE

RM

Obs

129

129

129

129

129

129

129

129

129

129

129

Mea

n4.5

04.4

24.4

54.4

74.5

14.4

34.2

04.3

14.4

64.4

74.4

7M

edia

n4.4

74.3

94.4

64.4

74.5

14.4

04.1

94.3

94.4

24.4

74.5

1M

ax

5.1

24.9

34.7

94.7

65.0

04.8

15.8

95.0

74.8

44.7

44.7

3M

in4.1

63.9

04.1

04.1

44.1

04.0

22.7

03.5

74.0

64.1

74.1

8St.

Dev

.0.2

50.3

10.2

20.1

90.2

40.2

50.9

10.4

80.2

20.1

80.1

8A

uto

Corr

.1.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

0Skew

.0.7

00.0

3-0

.08

-0.0

80.2

60.0

00.1

3-0

.07

0.1

1-0

.09

-0.3

7K

urt

.2.5

61.7

21.6

71.6

72.1

91.7

01.9

41.5

51.8

11.7

11.7

2A

DF

0.9

2-0

.16

-0.5

1-0

.62

0.9

0-0

.87

0.9

10.0

6-0

.61

-1.3

0-1

.02

AD

F(p

)0.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

0C

V0.0

50.0

70.0

50.0

40.0

50.0

60.2

20.1

10.0

50.0

40.0

4

IND

IAIN

DN

SIT

ALY

JA

PA

NK

OR

MA

LM

EX

NE

TH

NO

RN

ZL

DP

ER

Obs

129

129

129

129

129

129

129

129

129

129

129

Mea

n4.3

44.4

24.4

84.5

04.2

34.2

34.4

24.4

24.3

34.7

74.5

5M

edia

n4.3

14.5

24.5

14.5

74.3

84.3

84.3

84.4

24.3

44.7

54.4

9M

ax

5.3

95.1

74.6

94.7

25.0

45.0

84.8

14.7

54.7

05.1

15.2

2M

in3.4

43.5

84.1

84.0

93.1

63.2

63.9

94.0

73.8

24.4

44.1

1St.

Dev

.0.5

60.4

40.1

50.1

80.5

90.5

60.2

40.2

20.2

70.2

10.2

8A

uto

Corr

.1.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

00.9

9Skew

.0.1

9-0

.19

-0.4

2-0

.86

-0.4

0-0

.19

0.1

3-0

.11

-0.2

00.2

30.7

8K

urt

.1.9

21.8

91.8

62.3

31.8

71.6

31.6

21.5

81.6

71.6

92.6

4A

DF

2.8

2-0

.52

-2.2

2-2

.89

-2.8

2-1

.04

-0.2

0-0

.93

-1.9

8-0

.37

1.2

0A

DF

(p)

0.0

00.0

00.0

00.1

00.1

00.0

00.0

00.0

00.0

00.0

00.0

0C

V0.1

30.1

00.0

30.0

40.1

40.1

30.0

60.0

50.0

60.0

40.0

6

PH

LP

SA

FR

CSA

RB

IASIN

GSP

AIN

SW

ESW

ITZ

TH

AI

TU

RK

UK

US

Obs

129

129

129

129

129

129

129

129

129

129

129

Mea

n4.4

94.5

44.5

54.1

84.4

44.4

84.5

24.3

34.3

94.4

64.4

0M

edia

n4.4

24.4

74.4

84.2

84.4

24.4

44.5

04.5

24.4

04.4

44.4

0M

ax

5.1

14.9

85.2

05.1

84.8

34.8

44.7

95.0

55.0

74.8

04.7

7M

in4.0

74.2

04.1

33.0

64.0

44.1

54.2

33.3

63.7

34.0

93.9

5St.

Dev

.0.3

00.2

20.2

80.6

20.2

60.2

10.1

50.5

30.3

80.2

30.2

7A

uto

Corr

.1.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

01.0

0Skew

.0.4

80.6

30.6

5-0

.17

-0.0

50.1

7-0

.01

-0.4

4-0

.04

-0.0

7-0

.18

Kurt

.1.9

82.1

52.5

51.7

61.7

11.8

12.0

31.8

21.9

31.7

11.7

1A

DF

0.8

81.4

33.2

0-0

.93

-1.2

3-0

.09

-0.0

6-1

.75

-0.3

9-1

.35

-1.8

4A

DF

(p)

0.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

00.0

0C

V0.0

70.0

50.0

60.1

50.0

60.0

50.0

30.1

20.0

90.0

50.0

6

Note.

Sum

mary

stati

stic

sof

the

log-l

evel

of

real

GD

P.

51

Table

C.2

SummaryStatisticsforCountryVolatility(L

evels)

AR

GA

UST

LIA

AU

ST

RIA

BE

LB

RA

CA

NC

HIN

AC

HL

FIN

FR

AN

CE

GE

RM

Obs

94

129

129

129

86

129

74

94

94

129

129

Mea

n20.8

27.7

16.8

86.8

524.5

67.3

214.6

48.4

114.7

18.7

29.1

3M

edia

n16.1

46.9

75.5

45.5

815.7

56.1

612.9

27.3

011.1

67.3

68.0

9M

ax

99.7

234.2

231.0

224.3

5183.1

933.9

041.5

323.8

077.1

828.0

133.5

9M

in7.5

83.3

51.4

62.7

56.5

63.2

06.6

34.2

14.1

13.9

53.7

2St.

Dev

.14.1

13.6

74.9

23.8

026.4

24.2

06.7

03.4

69.9

23.9

03.7

2A

uto

Corr

.0.5

70.3

40.7

50.6

20.8

10.6

20.5

80.3

90.5

40.4

80.4

5Skew

.2.8

93.9

92.0

22.1

33.8

42.8

61.3

81.8

23.0

32.0

42.6

5K

urt

.14.1

625.9

98.2

98.2

720.0

015.4

05.4

97.4

218.0

68.6

316.2

4A

DF

-3.2

1-3

.60

-2.9

2-3

.51

-3.5

7-3

.37

-2.7

2-3

.78

-2.6

1-3

.45

-3.8

3A

DF

(p)

0.0

50.0

10.0

50.0

10.0

10.0

50.1

00.0

10.1

00.0

50.0

1C

V0.6

80.4

80.7

10.5

61.0

80.5

70.4

60.4

10.6

70.4

50.4

1

IND

IAIN

DN

SIT

ALY

JA

PA

NK

OR

MA

LM

EX

NE

TH

NO

RN

ZL

DP

ER

Obs

97

94

129

129

129

125

94

129

125

94

74

Mea

n12.1

213.7

19.9

79.0

011.7

69.3

611.5

97.9

010.6

88.6

213.4

4M

edia

n10.4

011.5

29.1

78.6

210.0

07.9

110.8

26.8

99.6

47.6

012.5

7M

ax

29.3

556.9

727.9

634.2

634.0

347.1

029.9

029.8

139.3

323.6

844.9

5M

in5.4

14.1

13.6

12.5

52.9

72.6

45.1

13.4

94.7

43.7

66.1

4St.

Dev

.5.1

78.2

04.4

84.2

45.5

96.1

84.5

54.2

74.9

33.5

86.1

4A

uto

Corr

.0.4

90.4

10.5

50.4

90.6

90.4

40.4

70.5

90.4

70.4

90.5

7Skew

.1.1

62.3

71.5

42.1

11.4

33.0

91.6

82.3

22.9

61.5

52.1

0K

urt

.3.7

511.0

96.1

712.1

55.4

615.9

97.0

29.9

015.6

66.2

710.7

0A

DF

-3.1

6-2

.38

-3.8

9-3

.93

-2.4

5-3

.70

-3.0

7-3

.27

-3.7

0-2

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Note.

Sum

mary

stati

stic

sof

the

level

of

vola

tility

.

52

Figure

C.1

VolatilityForecast

ErrorVarianceDecompositions

10

20

0

50

10

0

VO

L -

AR

G

10

20

0

50

10

0

VO

L -

AU

ST

LIA

10

20

0

50

10

0

VO

L -

AU

ST

RIA

10

20

0

50

10

0

VO

L -

BE

L

10

20

0

50

10

0

VO

L -

BR

A

10

20

0

50

10

0

VO

L -

CA

N

10

20

0

50

10

0

VO

L -

CH

INA

10

20

0

50

10

0

VO

L -

CH

L

10

20

0

50

10

0

VO

L -

FIN

10

20

0

50

10

0

VO

L -

FR

AN

CE

10

20

0

50

10

0

VO

L -

GE

RM

10

20

0

50

10

0

VO

L -

IN

DIA

10

20

0

50

10

0

VO

L -

IN

DN

S

10

20

0

50

10

0

VO

L -

IT

AL

Y

10

20

0

50

10

0

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JA

PA

N

10

20

0

50

10

0

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L -

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R

10

20

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10

0

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L -

MA

L

10

20

0

50

10

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X

10

20

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0

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TH

10

20

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10

0

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R

10

20

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50

10

0

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L -

NZ

LD

10

20

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10

20

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PH

LP

10

20

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SA

FR

C

10

20

0

50

10

0

VO

L -

SIN

G

10

20

0

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10

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L -

SP

AIN

10

20

0

50

10

0

VO

L -

SW

E

10

20

0

50

10

0

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L -

SW

ITZ

10

20

0

50

10

0

VO

L -

TH

AI

10

20

0

50

10

0

VO

L -

TU

RK

10

20

0

50

10

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L -

UK

10

20

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50

10

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L -

US

ηi

εi

fg

Other

Note.Other

repre

sent

the

vari

ance

share

of

oth

erco

untr

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ecifi

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ock

s.

53

Figure

C.2

GDP

GrowthForecast

ErrorVarianceDecompositions

10

20

0

50

10

0

GD

P -

AR

G

10

20

0

50

10

0

GD

P -

AU

ST

LIA

10

20

0

50

10

0

GD

P -

AU

ST

RIA

10

20

0

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10

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BE

L

10

20

0

50

10

0

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P -

BR

A

10

20

0

50

10

0

GD

P -

CA

N

10

20

0

50

10

0

GD

P -

CH

INA

10

20

0

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10

0

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P -

CH

L

10

20

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10

0

GD

P -

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10

20

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50

10

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P -

FR

AN

CE

10

20

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10

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RM

10

20

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DIA

10

20

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P -

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10

20

0

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10

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P -

IT

AL

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10

20

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PA

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10

20

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10

20

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50

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10

20

0

50

10

0

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P -

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10

20

0

50

10

0

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P -

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TH

10

20

0

50

10

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P -

NO

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10

20

0

50

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NZ

LD

10

20

0

50

10

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PE

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10

20

0

50

10

0

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P -

PH

LP

10

20

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10

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FR

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10

20

0

50

10

0

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P -

SIN

G

10

20

0

50

10

0

GD

P -

SP

AIN

10

20

0

50

10

0

GD

P -

SW

E

10

20

0

50

10

0

GD

P -

SW

ITZ

10

20

0

50

10

0

GD

P -

TH

AI

10

20

0

50

10

0

GD

P -

TU

RK

10

20

0

50

10

0

GD

P -

UK

10

20

0

50

10

0

GD

P -

US

ηi

εi

fg

Other

Note.Other

repre

sent

the

vari

ance

share

of

oth

erco

untr

ysp

ecifi

csh

ock

s.

54

Figure C.3 Cross-country Correlation of Volatility and GDP GrowthInnovations Conditional on f and g

Argen

tina

Austra

lia

Austri

a

Belgium

Brazil

Can

ada

China

Chile

Finland

Franc

e

Ger

man

yIn

dia

Indo

nesiaIta

ly

Japa

n

Korea

Malay

sia

Mex

ico

Net

herla

nds

Nor

way

New

Zea

land

Peru

Philip

pine

s

South

Afri

ca

Singa

pore

Spain

Swed

en

Switz

erland

Thaila

nd

Turke

y

Unite

d Kin

gdom

Unite

d Sta

tes

0

0.2

0.4

0.6C

orr

ela

tio

n

Note. Average cross-country pairwise correlation of volatility (light bars) and GDP growth (darkbars) innovations conditional on f and g. The volatility measures are computed as in (42). Thedotted lines corresponds to the average pairwise correlations across countries, at 0.05 and 0.03 forvolatility and GDP growth, respectively. Same statistics as in Table 2

Figure C.4 Cross-country Correlation of Volatility and GDP GrowthInnovations Conditional only on f

Argen

tina

Austra

lia

Austri

a

Belgium

Brazil

Can

ada

China

Chile

Finland

Franc

e

Ger

man

yIn

dia

Indo

nesiaIta

ly

Japa

n

Korea

Malay

sia

Mex

ico

Net

herla

nds

Nor

way

New

Zea

land

Peru

Philip

pine

s

South

Afri

ca

Singa

pore

Spain

Swed

en

Switz

erland

Thaila

nd

Turke

y

Unite

d Kin

gdom

Unite

d Sta

tes

0

0.2

0.4

0.6

Co

rre

latio

n

Note. Average cross-country pairwise correlation of volatility (light bars) and GDP growth (darkbars) innovations conditional on f only. The volatility measures are computed as in (42). Thedotted lines corresponds to the average pairwise correlations across countries, at 0.45 and 0.03 forvolatility and GDP growth, respectively. Same statistics as in Table 3

55