Uncertainty and Economic Activity: Identification Through Cross-country Correlations 9

Transcription

1 Uncertainty and Economic Activity: Identification Through Cross-country Correlations 9 Ambrogio Cesa-Bianchi M. Hashem Pesaran Alessandro Rebucci March 2, 217 Abstract Uncertainty behaves countercyclically, but interpreting this correlation in structural terms is difficult because the direction of causation may run both ways. In this paper, we take a multicountry 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 factors by assuming different patterns of correlation across countries for volatility and growth innovations 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 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 growth variance. Keywords: Business Cycle, Common Factors, Financial Cycle, Growth, Identification, Uncertainty, Volatility. JEL Codes: E44, F44, G15. 9 Preliminary draft. The views expressed in this paper are solely those of the authors and should not be taken to represent those of the Bank of England. Bank of England and CfM. ambrogio.cesa-bianchi@bankofengland.co.uk. University of Southern California and Trinity College, Cambridge. pesaran@dornsife.usc.edu. Johns Hopkins University and NBER. arebucci@jhu.edu. 1

2 1 Introduction During the global financial crisis, the world economy experienced a sharp and synchronized contraction 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 28, 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 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 9 percent of world GDP. The start of the sample period in this chart is 199:Q1-211:Q2 and is constrained by the availability of the VIX index. The sample period in our application starts in 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 (29)) and the household s side (Kimball (199), Leduc and 1 The ensuing literature is now voluminous. Here we focus on the studies more directly related to our paper. See Bloom (214) for a recent survey of newer and older contributions. 2 See, for instance, Baker and Bloom (213), Carriere-Swallow and Cespedes (213), Nakamura, Sergeyev, and Steinsson (217). 2

3 Liu (212), Fernandez-Villaverde et al. (211)). 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, 26, Fostel and Geanakoplos, 212, Bachmann and Moscarini, 211, Tian, 212, Decker et al., 214). Yet, other models stress interaction effects with financial frictions via an increase in the risk premium (e.g., Christiano et al., 214, Gilchrist et al., 213, Arellano et al., 212). 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 9 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- 3 Theoretically, the impact of uncertainty on activity could also be positive. For example Mirman (1971) shows that, 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 capital is 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

4 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., 211). This is equivalent to assume that volatility innovations share at least one more strong common 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 Correlation Argentina Australia Austria Belgium Brazil Canada China Chile Finland France Germany India Indonesia Italy Japan Korea Malaysia Mexico Netherlands Norway New Zealand Peru Philippines South Africa Singapore Spain Sweden Switzerland Thailand Turkey United Kingdom United States Note. Average cross-country pairwise correlation of volatility (light bars) and GDP growth (dark bars). The volatility measures are computed as in (42). The dotted lines correspond to the average pairwise correlations across countries, at.48 and.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.48 and.17, respectively. As we shall see in the paper, we find comparable differences by using 4 The approach is reminiscent of the identification through hetroskedasticity of Rigobon (23). The key difference is that it cannot be applied to a single country in isolation from the rest of the world economy. 4

5 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 (215) 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 (213)) 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. (21, 23) and Barndorff-Nielsen and Shephard (22, 24), 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 5 See Tables 2 and 3, and Figures C.3 and C.4. 6 For example, Lewis and Liu (217) 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

6 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 volatility 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 country 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 quantitatively 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. (215), Clark et al. (216), Giglio et al. (216). 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. (215) we document that it is the non-fundamental common 6

7 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 (213) estimate a battery of small open economy VARs for 2 advanced and 2 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. (214), Gilchrist et al. (213) and Arellano et al. (212). Using an unbalanced panel of 6 countries, Baker and Bloom (213) 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. (212) 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 (29), 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

8 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 contemporaneous correlation between volatility and growth, for a large collection of N countries: v it = λ i f t + u it, (1) y it = γ i f t + ε it, (2) for i = 1, 2,..., N. Here, v it denotes a volatility measure and y it 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 f t has zero mean and finite variance and is normalized to one for simplicity. The innovations u it and ε it 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 f t, therefore, the correlation between u it and ε it 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 f t 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, f t 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). 7 As we shall see below, the main results of the analysis carry through to the fully dynamic model that we use in our application. 8

9 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 u it and ε it, the common factor f t, and the factor loadings λ i and γ i 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 f t, unless we exclude f t from one of the tow equations. In fact, the covariance matrix of v it and y it is given by: Θ i = λ2 i + σ 2 u,i λ i γ i γ 2 i + σ2 ε,i (3) This provides three independent restrictions, but the parameters of the model are four, (λ i, γ i ), and (σ 2 u,i, σ2 ε,i ). Their identification can be achieved only by imposing at least one additional exclusion restriction, e.g. either λ i = or γ i =. 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 u it and ε it. 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 ( v ω,t ) and GDP growth ( ȳ ω,t ) 8 The model (1)-(2) is observationally equivalent to a standard structural vector autoregression model. Thus, it is subject to the same identification requirements. See appendix for a proof. 9

10 as follows: v ω,t = ȳ ω,t = N ẘ i v it, (4) i=1 N w i y it, (5) i=1 where ẘ i and w i 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 f t, the loadings (λ i and γ i ), the the weights (ẘ i and w i ), and the innovations (u it and ε it ): Assumption 1 (Loadings) The factor loadings λ i and γ i, are independently and identically distributed across i, independent of the common factors f t, for all i and t, with means λ and γ. Furthermore they satisfy the following conditions: N 1 λ = as N. N i=1 N λ 2 i = O(1) and N 1 γ 2 i = O(1), (6) N ẘ i λ i and γ = i=1 i=1 N w i γ i, (7) i=1 Assumption 2 (Weights) Let w = (w 1, w 2,..., w N ) and ẘ = (ẘ 1, ẘ 2,..., ẘ N ) be N 1 vectors of non-stochastic weights with N i=1 w i = 1 and N i=1 ẘi = 1. We assume that growth weights w are granular, i.e.: w = O p (N 1), w i w = O p(n 1 2 ) i, (8) while volatility weights ẘ 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 ε t = (ε 1t, ε 2t,..., ε Nt ) and u t = (u 1t, u 2t,..., u Nt ) be Σ εε = V ar (ε t ) and Σ uu = V ar (u t ), respectively. They satisfy: ϱ max (Σ u ) = O(N), (9) ϱ max (Σ ε ) = O(1). (1) 1

11 where ϱ 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 (22)). It ensures that f t is a strong (or pervasive) factor for both volatility and activity (see Chudik et al., 211), and it can be estimated using principal components or the cross-section averages 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 198s, 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 crosscountry correlation means that, asymptotically, as N becomes large, the average pairwise correlations 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 Σ u 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, f t can be identified (up to a constant) by ȳ ω,t = N i=1 w i y it. 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: v ω,t = λf t + ū ω,t, (11) ȳ ω,t = γf t + ε ω,t, (12) 9 Similarly different patterns of cross country correlations have been documented also for consumption growth and and equity returns. See, for instance, Tesar (1995) and Lewis and Liu (215). 11

12 where ε ω,t = ẘ ε t and ū ω,t = w u t. Because of Assumption 3: V ar ( ε ω,t ) = w Σ ε w ( w w ) λ max (Σ ε ), (13) and: V ar ( ε ω,t ) O ( w w ) = O ( N 1), (14) and hence: ε ω,t = O p (N 1/2). (15) Using this in (12), since γ under Assumption 1, we have: f t = γ 1 ȳ ω,t + O p (N 1/2), (16) which allows us to recover f t form ȳ ω,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, f t cannot be recovered from v ω,t. In fact, since λ max (Σ u ) = O(N), it follows that V ar (ū ω,t ) = w Σ u w will generally not converge to zero. 1 Similarly, under Assumptions 2 and 3, only the principal components or cross-section averages of output growth can be used to identify f t (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 f t. 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 u it 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: 1 The difference in the way in which the volatility and GDP growth innovations are correlated across countries also ensures that we do not end up with a perfect relationship between v t and y t when N tends to infinity. 12

13 Assumption 4 The residual cross-section correlation of u it, conditional on f t, can also be decomposed in a strong factor (g t ) and a weak component (η it ), namely u it = θ i g t + η it, (17) where η it is a volatility innovation that is now cross-sectionally weakly correlated, like ε it. Hence, letting the variance-covariance matrix of the N 1 vector η t = (η 1t, η 2t,..., η Nt ) be Σ ηη = V ar (η t ), we also have ϱ max (Σ ηη ) = O(1), (18) and: N 1 N i=1 θ 2 i = 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 u it is given by (17), the model becomes: v it = λ i f t + θ i g t + η it, (2) y it = γ i f t + ε it. (21) Moreover, conditional on f t, for N large enough, g t is given by the following linear combination of v ω,t and ȳ ω,t for N large enough: g t = θ 1 v ω,t λ θγ ȳ ω,t + O p ( N 1/2). (22) Proof. To see this, substitute (16) and (17) into (11) and apply the same logic as before to η it. 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 equations on the second strong factor g t, 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 13

14 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 g t. Our identifications restrictions are consistent, for instance, with the international disaster risk model of Lewis and Liu (217), 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, g t can be interpreted as the common jump component of the model. More generally, we could interpret g t 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 g t 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 (214)). In this case, g t 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 f t and especially g t directly. Denote with f t and g t an estimate of f t and g t. An estimate of f t can be simply obtained from (16). If we normalize γ = θ = 1, which is innocuous, f t can be approximated by world growth as f t = ȳ ω,t. Given (22), g t can then be approximated by the residuals of a regression of world volatility v ω,t on world growth ȳ ω,t, namely g t = v ω,t ˆβ ȳ ω,t. Once observable factors are obtained, one can easily estimate their country-specific impact on volatilities and growth and the 11 See, for instance, a global preference shock of the kind studied by Albuquerque, Eichenbaum, Luo, and Rebelo (215) would induce such a representation. 14

15 associated innovations as follows: v it = λ i ft + θ i g t + η it, (23) y it = γ i ft + ε it. (24) Note here that, f t and g t are an orthonormal transformation of f t and g t. Therefore, the residuals of (2)-(21) and (23)-(24) will be exactly the same. We also note that f t and g t 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 y it, the correlation is assumed to vanish asymptotically. In the case of v it, 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 (215)). 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

16 results above. In particular, we want to show (i) how to identify the factors f t and g t 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: v it = a iv + φ i,11 v i,t 1 + φ i,12 y i,t 1 + λ i f t + θ i g t + η it, (25) y it = a iy + φ i,21 v i,t 1 + φ i,22 y i,t 1 + γ i f t + ε it, (26) on which we already imposed assumptions 3 and 4. That is, in vector format: z it = a i + Φ i z i,t 1 + Γ i f t + ξ it, for t = 1, 2,..., T, (27) where z it = (v it, y it ) and: a i = a iv a iy, Φ i = φ i,11 φ i,12 φ i,21 φ i,22, Γ i = λ i θ i γ i, f t = f t g t, ξ it = η it ε it. (28) The matrix Γ i of contemporaneous factor loadings is triangular because of our identification assumptions 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 ξ it are serially uncorrelated (over t), and cross-sectionally weakly correlated (over i), with zero mean and a positive definite covariance matrix, Ω i, for i = 1, 2,..., N. Assumption 6 (Common factors) The 2 1 vector of unobserved common factors, f t = (f t, g t ), is covariance stationary with absolute summable autocovariances, and is distributed independently of the country-specific shocks, ξ it for all i, t and t. Fourth order moments of f lt, for l = 1, 2, are also bounded. 16

17 Assumption 7 (Factor Loadings) The factor loadings λ i, θ i, and γ i (i.e., the non-zero elements of Γ i ) are independently and identically distributed across i, and independent of the common factors f t, for all i and t, with means λ, θ, and γ. Furthermore, they have bounded second moments and: Γ = E (Γ i ) = λ θ γ, is invertible (namely γθ ). Assumption 8 (Coefficients) The constants a i are bounded, Φ i and Γ i are independently distributed for all i, the support of ϱ (Φ i ) lies strictly inside the unit circle, for i = 1, 2,..., N, and the inverse of the polynomial Λ (L) = l= Λ ll l, where Λ l = E ( Φ l i) exists and has exponentially decaying coefficients. We can then prove the following proposition: Proposition 2 Under Assumptions 5-8, the unobserved common factors, f t and g t in (27) can be approximated by: f t = γ 1 ȳ ω,t + ) c 1,l z ω,t l + O p (N 1 2, (29) l=1 g t = θ 1 v ω,t λ θγ ȳ ω,t + c 2,l z ω,t l + O p (N 1 2 l=1 ), (3) where: v ω,t = N ẘ i v it, ȳ ω,t = i=1 N w i y it, z ω,t = ( v ω,t, ȳ ω,t ), (31) i=1 and c 1,l and c 2,l are 1 2 vectors of coefficients. Proof. See Appendix A.2. We note here that c 1,l and c 2,l are the first and second rows of the 2 2 matrix C l as defined in the Appendix. As Pesaran and Chudik (214) and Chudik and Pesaran (215) showed, if slope heterogeneity is not extreme (i.e., if these matrices of coefficients do not differ too much across countries) and C l decays exponentially in l, the infinite order distributed lag functions in z ω,t 17

18 in the above derivations can be truncated. In practice, Pesaran and Chudik (214) and Chudik and Pesaran (215) recommend a lag length l 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, f t and g t 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: f t = ȳ ω,t + p c 1,l z ω,t l + O p (N 1/2), (32) l=1 g t = v ω,t λ ȳ ω,t + p ) c 2,l z ω,t l + O p (N 1 2. (33) l=1 Substituting these approximations in (27) we obtain the cross-section augmented VAR models: v it = a iv + φ i,11 v i,t 1 + φ i,12 y i,t 1 + θ i v ω,t + β i ȳ ω,t + p ) ψ il z ω,t l + η it + O p (N 1 2 (34), p ) y it = a iy + φ i,21 v i,t 1 + φ i,22 y i,t 1 + γ i ȳ ω,t + γ i c 1,l z ω,t l + ε it + O p (N 1 2 where β i = λ i θ i λ and ψ il = (λ i c 1,l + θ i c 2,l ). Note here that only ȳ ω,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 θ i, β i, and γ i. But these coefficients might be difficult to interpret due to the possible non-zero correlation between ȳ ω,t and v ω,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 z ω,t are filtered out. To this end we stack (32) and (33) by T (abstracting from the order O p ( N 1/2 ) terms) and 12 One could use the orthogonalized components of ȳ ω,t and v ω,t using the Cholesky factor. However, such a procedure could invalidate the triangular form of Γ i applied to the true underlying factors, f t and g t. Also focusing on the orthogonalized components of ȳ ω,t and v ω,t, ignores the contributions of z ω,t l for l = 1, 2,..., p to the estimation of f t and g t. l=1 l=1 (35) 18

19 get: f = ȳ ω + Z ω C 1, (36) g = v ω λ ȳ ω + Z ω C 2, (37) where ȳ ω = ( ȳ ω,1, ȳ ω,2,..., ȳ ω,t ), v ω = ( v ω,1, v ω,2,..., v ω,t ), and Z ω = (τ T, z ω, 1, z ω, 1,..., z ω, p ). Inclusion of τ T in Z ω 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 residuals of the the following OLS regressions: ȳ ω = Z ω Ĉ 1 + f, (38) v ω = ˆλ f + Z ω Ĉ 2 + 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: p v it = a iv + φ i,11 v i,t 1 + φ i,12 y i,t 1 + β i,11 ft + β i.12 g t + ψ v,il z ω,t l + η it, (4) y it = a iy + φ i,21 v i,t 1 + φ i,22 y i,t 1 + β i,21 ft + l=1 p ψ y,il z ω,t l + ε it. (41) Notice here that by construction the estimates of η it and ε it based on the above regressions should be numerically identical to those obtained using the non-orthogonalized factors ȳ ω and v ω. It is also important to note that since f t and g t are based on residuals from regressions of f t and g t on the p th order lagged values, τ T, z ω,t 1, z ω,t 2,..., z ω,t p, then it also follows that f t and g t have zero means (in-sample) and for a sufficiently large value of p, they are also serially uncorrelated. Therefore, ft and g t can be viewed as global innovations (or shocks) to the underlying factors, f t l=1 19

20 and g t. In Appendix A.4 we show how to compute impulse responses and variance decompositions to shocks f t and g t as well as η it and ε it. 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 (η it and ε it ) 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 f t and g t. 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. (21, 23), Barndorff-Nielsen and Shephard (22, 24)). 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 measurement 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 markets 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 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 2

21 VIX Index, which is an average of the daily option price implied volatility for the S&P5 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. (214) 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 (29), Fernandez-Villaverde et al. (211) 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. (21), Bloom et al. (27) and Gilchrist et al. (213) use dispersion measures of firm-level stock market returns; Bloom et al. (212) 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 summarizing 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 instance, Zarnowitz and Lambros (1987), Popescu and Smets (21), and Bachmann et al. (213). Finally, model based measures, such as those Jurado et al. (215) and Ludvigson et al. (215) 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

22 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 P κit (τ). We then compute the realized volatility for quarter t, asset of type κ, and country i as: D t v κit = D 1 t (r κit (τ) r κit ) 2 (42) τ=1 where r κit (τ) = ln P κit (τ) and r κit = Dt 1 Dt τ=1 r κit(τ) is the average daily price changes over the quarter t, and D t is the number of trading days in quarter t. The scaling factor D t 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, D t = 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 1-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 P κit (τ) in the above expression replaced by P κit (τ)/p it, where P it 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 211 (depending on data availability). Figure 3 plots the U.S. realized volatility measure we constructed with the VIX index (as plotted in Figure 1). 13 We measure P it by the consumer price index (CP I it) 14 Cesa-Bianchi et al. (214) construct them for exchange rates, bond prices, and commodity prices and report their unconditional properties. 22

23 The chart shows that the two measures co-move very closely. Figure 3 Quarterly U.S. Equity Realized Volatility And The Vix Index 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). The VIX Index is the quarterly average of the daily Chicago Board Options Exchange Market Volatility Index from Bloomberg. The sample period is 199:Q1-211:Q 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

24 5.1 Persistence A battery of summary statistics on our volatility and real GDP series support our model specification 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.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 =, 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 conclusions. 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. Therefore, it looses some information one way or the other. In our data set, there are 16 countries with 24