Assessing the International Comovement of Equity Returns

The international comovement of equity returns has been viewed as reflecting either pervasive common shocks or local linkages between countries. This paper brings these perspectives together by assessing the comovement of equity returns in a dynamic model that allows for both common factors and spatial dependence, using quarterly data for 40 advanced and emerging countries over the past two decades, and including GDP growth, the real interest rate, and credit as fundamental variables. Estimation results employing a bias-corrected quasi-maximum likelihood approach provide strong indication that the cross-country dependence of equity returns results from both spatial effects and common shocks captured by a latent common factor -- weak and strong dependence, respectively. The factor exhibits a robust negative correlation with market measures of aggregate risk. Countries' exposure to the common factor rises with their extent of trade openness and the degree of rigidity of their exchange rate regime. Despite its simplicity, the empirical model fits the data well. All these results are robust to the use of alternative spatial weight matrices. The paper also shows that ignoring cross-country dependence leads to distorted parameter estimates and a marked deterioration of the explanatory power of the empirical model.


Policy Research Working Paper 8516
The international comovement of equity returns has been viewed as reflecting either pervasive common shocks or local linkages between countries. This paper brings these perspectives together by assessing the comovement of equity returns in a dynamic model that allows for both common factors and spatial dependence, using quarterly data for 40 advanced and emerging countries over the past two decades, and including GDP growth, the real interest rate, and credit as fundamental variables. Estimation results employing a bias-corrected quasi-maximum likelihood approach provide strong indication that the cross-country dependence of equity returns results from both spatial effects and common shocks captured by a latent common factor-weak and strong dependence, respectively. The factor exhibits a robust negative correlation with market measures of aggregate risk. Countries' exposure to the common factor rises with their extent of trade openness and the degree of rigidity of their exchange rate regime. Despite its simplicity, the empirical model fits the data well. All these results are robust to the use of alternative spatial weight matrices. The paper also shows that ignoring cross-country dependence leads to distorted parameter estimates and a marked deterioration of the explanatory power of the empirical model. This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/research. The authors may be contacted at gabate@worldbank.org and lserven@worldbank.org.

Introduction
The international synchronization of asset prices has attracted increasing attention in recent years. From the macroeconomic perspective, it represents a mechanism for the cross-country propagation of shocks, a matter of concern especially after the global financial crisis. From the finance perspective, it detracts from the presumed benefits of international portfolio diversification. Asset price synchronization has been on a rising trend, especially in the case of equity markets. Christoffersen, Errunza, Jacobs, and Langlois (2012) find that international dependence between stock markets has trended significantly upward since the 1970s, for both advanced and emerging countries. Jorda, Schularick, Taylor and Ward (2017), using data for 17 advanced countries spanning 150 years, find a marked long-run rise in the degree of international comovement of equity prices, above and beyond the comovement of the prices of other assets such as housing and debt. Using the same data, Bekaert and Mehl (2017) similarly find an increase in the conditional betas averaged across countries, in both value-weighted and unweighted terms. In turn, Cotter, Gabriel and Roll (2018) conclude that common global factors account for an increasing share of the variance of equity, bond and real estate returns across developed and emerging markets.
Much of the empirical literature attributes the international dependence of asset prices to pervasive shocks affecting a multitude of countries. A number of recent contributions view the shocks as reflecting a global financial cycle driven primarily by shifts in international investors' risk aversion -as well as monetary policy in center countries, notably the U.S. (see e.g., Rey (2013), Miranda-Agrippino and Rey (2018), Bruno and Shin (2015), Xu (2017)). 1 Empirically, the shocks driving asset prices across the world are typically modeled as latent common factors or, alternatively, summarized by a handful of variables capturing global financial conditions. This literature goes on to assess the determinants of countries' exposure to the global financial cycle, with particular attention to the insulating role of flexible exchange rates; see for example Rey (2013) and Bekaert and Mehl (2017).
Another strand of empirical literature has attributed the international comovement of asset prices to real and financial linkages between countries. 2 Bayoumi, Fazio and MacDonald (2007) show that bilateral geographic distance can account for much of the observed variation in the pairwise correlation of equity price indices across a sample of emerging markets. 3 Forbes and Chinn (2004) assess how trade and financial linkages between five center countries and 40 other countries shape the response of the latter countries' equity market returns to those of the former, concluding that bilateral trade plays the biggest role.
Didier, Love and Martinez-Peria (2012) examine the 1 Similarly, Forbes and Warnock (2012) stress the role of global investors' risk aversion for the international comovement of capital flows, while Barrot and Servén (2018) and Cerutti, Claessens and Rose (2017) evaluate the quantitative relevance of the global financial cycle for the observed patterns of flows.
2 In a similar vein, several papers have highlighted the role of bilateral linkages in the international propagation of sovereign and banking crises. Bolton and Jeanne (2011) stress banks' holdings of foreign sovereign debt as a key mechanism in the propagation of the Eurozone crisis. Hale, Kapan and Minoiu (2016) likewise document how interbank loans help transmit distress across national banking systems. 3 Chong, Wong, and Zhang (2011) likewise relate the pairwise correlations across a sample of advanced and emerging countries to bilateral distance plus other gravity variables, and find that distance has a robust negative effect. comovement of market returns of a large number of countries with U.S. returns during the global financial crisis, and find that bilateral financial linkages were the primary drivers. Asgharian, Hess and Liu (2013) use spatial econometrics techniques to explore the role of a variety of bilateral linkages (geographic, macroeconomic and financial) for the commonality of the returns of 41 stock markets. Bilateral trade emerges as the most important link. 4 While both of these literatures are concerned with the cross-country dependence of equity (and other asset) prices, methodologically they take very different views. The first literature stresses shocks with worldwide reach. In contrast, the second literature emphasizes interdependence between particular countries. These two views roughly correspond to the distinction between strong and weak cross-sectional dependence, respectively. Strong dependence arises from common shocks. Weak dependence reflects local interactions. 5 Strong dependence is typically analyzed with factor models (as done, for example, by Miranda-Agrippino and Rey (2018), Xu (2017) or Cotter, Gabriel and Roll (2018) for asset prices), while weak dependence is typically analyzed with spatial models highlighting geographic or economic distance (as in, e.g., Asgharian, Hess and Liu (2013)).
So far, the empirical literature has taken into account one or the other form of dependence -but not both. However, identifying correctly the type of cross-sectional dependence at work can be quite important. For example, ignoring strong dependence when it is present may lead to inconsistent estimates if the omitted common factors are correlated with the regressors (Pesaran and Tosetti (2011)). Conversely, introducing common factors in the estimation when only weak dependence is at play may similarly yield inconsistent estimates (Onatski (2012)). In turn, the consequences of neglecting spatial dependence when it is present depend on its precise form. Ignoring spatial dependence in the error term will only cause loss of efficiency; however, ignoring spatial dependence in the dependent variable and/or the independent variables may produce biased and inconsistent estimates of the parameters of the remaining variables (LeSage and Pace (2009)).
In reality, however, the two forms of dependence are likely to be simultaneously present. Thus, from the viewpoint of empirical modeling, the two perspectives should be viewed as complementary, rather than mutually exclusive.
In this paper we bring both approaches together. We analyze the international comovement of aggregate equity returns in a sample of 40 advanced and emerging countries, using an encompassing empirical framework including both spatial effects and common factors. This allows us to assess the respective roles of strong and weak cross-sectional dependence, and to illustrate the consequences of unduly omitting either (or both) of them. To date, very few papers have employed a similarly flexible 4 Aside from these direct links, indirect linkages across asset markets in different countries may also arise from the presence of common investors. As they adjust their international portfolio holdings in response to shocks, common investors become a source of asset price comovement (see, e.g., Broner, Gelos and Reinhart (2006)). The empirical relevance of this propagation mechanism has received special attention in the case of international mutual funds (e.g., Raddatz and Schmukler (2012)).
5 Strong and weak cross-sectional dependence can be defined in different ways. One commonly-used definition bases the distinction between them on the rate at which the largest eigenvalue of the covariance matrix of the cross-section units rises with the number of units; see Bailey, Kapetanios and Pesaran (2015). methodological framework; the notable exception is Bailey, Holly and Pesaran (2016), who examine the patterns of house prices across U.S. metropolitan areas.
We assume that spatial interactions occur through equity returns, which seems a natural way to model the interconnectedness of investors' portfolios. However, this implies that the two-step estimation methods employed by Bailey, Holly and Pesaran (2016), which assume that the interaction occurs through the spatial error, are not applicable to our setting. Instead, we use a quasi-maximum likelihood estimation approach recently developed by Shi and Lee (2017) that permits joint consideration of common factors and spatial dependence in a dynamic framework. Because the factors and their loadings are treated as parameters, whose number grows with sample size, they create a nuisance parameter problem. To address it, we use the bias correction procedure proposed by Shi and Lee (2017).
In light of the earlier literature, we experiment with alternative specifications of the spatial weight matrix summarizing interactions between countries. We use a bilateral trade weight matrix to capture real linkages across countries, and a bilateral foreign investment weight matrix to assess their financial linkages. In addition, we also present results using a bilateral geographical distance weight matrix.
Because our sample contains both advanced and emerging countries, and the degree of development of financial markets -as well as the extent of financial and real integration in the global economy -differs between both groups, we also estimate the empirical model of equity returns on a sub-sample of 25 advanced countries. This allows us to assess differences across both groups in the role of the various drivers of equity prices, as well as the extent of cross-sectional dependence.
Consistent with theoretical expectations, we find that real equity returns are positively related to real GDP growth, and negatively related to changes in the real interest rate and -in the extended country sample only -real credit growth, which suggests that in emerging countries with less-developed financial markets interest rates do not suffice to capture the financial environment faced by investors. These results show little variation across the three alternative specifications of the spatial weight matrix.
We also find strong evidence of spatial effects, summarized by a positive contemporaneous spatial lag and a negative spatial-time lag. Both are statistically significant in virtually all specifications, implying that local interactions are important to understand the international comovement of equity returns. In addition, equity returns reflect a common factor that is strongly positively correlated with average returns, and strongly negatively correlated with measures of aggregate risk, in line with the results of, e.g., Rey (2013), Miranda-Agrippino and Rey (2018), andXu (2017).
Our results also speak to the determinants of countries' exposure to global shocks, an issue at the core of the policy debate. We find that the impact of the common factor on equity returns is bigger in countries with more open trade accounts and more rigid exchange rate regimes. The latter result implies that, notwithstanding their worldwide reach, the choice of exchange rate regime still matters for countries' exposure to global financial shocks -which echoes the recent findings of Bekaert and Mehl (2017) and Barrot and Servén (2018).
Finally, we shed light on the importance of properly taking into account cross-sectional dependence. Ignoring it, by omitting both common factors and spatial effects, leads to a gross overstatement of the procyclical behavior of equity returns. It also weakens dramatically the estimated model's empirical performance. Including the common factor, while still omitting spatial effects, greatly helps correct these problems, at the cost of remaining residual (weak) dependence. In turn, allowing for spatial effects, while omitting the common factor, also improves the model fit, but leads to overstated spatial effects -exaggerating the propagation of shocks through local linkages -and strong residual dependence. Overall, these results confirm the need to account for cross-sectional dependence, both strong and weak, in empirical modeling of equity returns across countries.
The rest of the paper is organized as follows. Section 2 lays out the factor-augmented dynamic spatial model of equity returns employed in the paper. Section 3 presents the data. Section 4 reports the results, and the final section provides conclusions.

Analytical framework
To study the comovement of equity returns, we use a dynamic model that allows for both common factors and spatial dependence. This section describes the model and summarizes our estimation approach.

A factor-augmented dynamic spatial model of equity returns
Let g it denote the real equity returns in country i = 1, . . . , n at time t = 1, . . . , T , and let g t = (g 1t , ..., g nt ) . Following Shi and Lee (2017), we assume that g t follows a spatial dynamic panel data (SDPD) model of the form: Thus, each country's real equity return is related to current real equity returns in (economically) neighboring countries, W g t , where W is an n × n spatial weight matrix; lagged equity returns in the own country, g t−1 , as well as in neighboring countries W g t−1 ; a set of observable explanatory variables X t ; a set of r time-varying unobserved factors f t common to all countries; and a stochastic disturbance V t . This general specification allows for both spatial dependence and unobserved common factors. Spatial dependence, embedded in the spatially-lagged dependent variable Wg t as well as its time-lagged value Wg t−1 , reflects the effects of current and lagged equity returns of nearby countries on the equity returns of a particular country. The extent of spatial dependence is measured by the contemporaneous spatial autoregressive parameter ρ and the space-time lag coefficient λ. 6 The relative contribution of each country to the overall spatial effect is measured by the spatial weight matrix W, which can be understood as providing a measure of economic proximity between countries.
In turn, the unobserved common factors f t capture systemic shocks that affect stock returns across all countries. The n×r matrix of factor loadings Ψ measures the (possibly heterogeneous) effect of the factors on each country's equity returns.
The X variables comprise a set of macroeconomic fundamentals of equity returns. Following the standard valuation model, which states that equity returns are given by the present discounted value of expected future cash flows, we include in X the rate of growth of real GDP, which has been found to account for much of the variation in expected cash flows (e.g., Fama (1990)), and (the first difference of) the real interest rate, which affects the risk premium and the discount rate employed to bring future cash flows to present value terms, and therefore has a negative effect on equity returns (Chen, Roll and Ross (1986), Jensen and Johnson (1995)). Moreover, because the empirical sample employed in the paper includes countries with relatively undeveloped financial markets, in which observed interest rates may not reflect accurately the cost of financing, we add among the X variables the rate of growth of the real stock of credit to the private sector, to better capture financial conditions. The SDPD in equation (1) nests various models as special cases. For example, in the absence of spatial dependence (ρ = 0 and λ = 0), equation (1) simplifies to a factor-augmented model relating equity returns to observable fundamentals plus latent common factors. 7 In these specifications, the spatial dependence between countries is parameterized by the n×n spatial weight matrix W. The matrix is assumed to be non-stochastic, with the properties (i) W ij ≥ 0 for i = j, and (ii) W ij = 0 for i = j. The first property indicates that the elements of W are non-negative known constants. The second property states that countries are not neighbors to themselves. In empirical applications the weight matrix W typically is row-normalized, such that n i =j W ij = 1, see Anselin (1988). Further, define S = (I − ρW ). Assuming that S is invertible, and letting A = S −1 (βI + λW ), equation (1) can be written as g t = Ag t−1 + S −1 (X t θ + Ψf t + V t ). Recurrent substitution yields With a row-normalized spatial weight matrix W , the sequence {A h } ∞ h=0 is summable in absolute values, and the initial condition g 0 becomes asymptotically irrelevant when T → ∞, provided the model's parameters lie in the region R s = {(ρ, β, λ) : β + (λ − ρ)ω min + 1 > 0, β + λ + ρ − 1 < 0, β + λ − ρ + 1 > 0, β + (ρ + λ)ω min − 1 < 0}, where ω min is the smallest characteristic root of the weight matrix W , see Shi and Lee (2017). 8 Equation (2) shows that shocks to the error term and the explanatory variables in a particular location are propagated to all other units within the spatial system in 7 In turn, the common factor framework in (1) encompasses individual and time period fixed effects as a particular case (Shi and Lee (2017)). To see this, consider the specification where ζ = (ζ 1 ζ 2 · · ζ) are individual effects, and ξ t are time effects with ι = (1 1 · · 1) , where Ψ n = ζ 1 ζ 2 · · ζ 1 1 · · 1 and F T = 1 1 · · 1 ξ 1 ξ 2 · · ξ . 8 The parameter estimates reported below satisfy these restrictions in all cases.
accordance with the structure of the weight matrix, so that their impact diminishes with (economic) distance.

Estimation approach
Estimation of the model (1) poses some special issues due to its simultaneous consideration of common factors and spatial effects. Both features are also present in the empirical specification employed by Bailey, Holly and Pesaran (2016), who use a two-stage approach to estimate their model: they estimate the common factors and the model's parameters at the first stage, and the spatial effects at the second stage.
In their case, however, the spatial effects accrue through the error term, while here they accrue through the dependent variable. This implies that the two-stage estimation approach is not applicable in our setting. The reason is that ignoring the spatial effects in the first-stage estimation, as done by Bailey, Holly and Pesaran (2016), would yield inconsistent estimates.
In settings more similar to ours, Kuersteiner and Prucha (2015) propose a GMMtype estimator, while Bai and Li (2015) develop a quasi-maximum likelihood (QML) estimator. Below we employ the QML estimation approach recently developed by Shi and Lee (2017). We provide a brief outline next, and refer the reader to Shi and Lee (2017) for the full details and Appendix B for a brief summary.
In equation (1), let Z t = (g t−1 , Wg t−1 , X t ). Define the parameters of the model as η = (δ , ρ) with δ = (β, λ, θ ) , σ 2 , Ψ and F T . Then the quasi-log likelihood function of the model in equation (1) is ( Dropping the constant term − 1 2 log2π − 1 2 logσ 2 , this expression can be rewritten as While here the number of common factors r is assumed to be known, for the estimation it is determined using information criteria, as will be discussed below.
Due to the presence of the factors and their loadings, the number of parameters in the model increases with the sample size. Focusing on η as the parameter of interest, and concentrating out the factors and their loadings applying principal component analysis, the concentrated log-likelihood is where The QML estimator is derived from the optimization problem in equation (5). The estimate of the factor loadings Ψ is computed from the eigenvectors associated with the first r largest eigenvalues of (S . The estimate of F T is obtained analogously by switching T and n. The QML estimator of the regression coefficients is consistent and asymptotically normal. However, it may be asymptotically biased owing to an incidental parameters problem, specifically due both to the presence of predetermined regressors (the lagged dependent variable) and to the interaction between the spatial effects and the factor loadings. To tackle this problem, Shi and Lee (2017) develop a bias correction that yields an asymptotically normal, properly-centered estimator. The estimations reported below using common factors employ the bias-corrected estimator.

Data
We assess the international comovement of equity returns using a balanced panel data set comprising 40 advanced and emerging countries over 1995:1 to 2016:3 -a total of 3,480 quarterly observations. Because advanced countries tend to exhibit higher real and financial openness, as well as larger and more liquid financial markets, than other countries, we consider separately a subsample of 25 advanced economies, comprising 2,175 observations. Table A1 in the appendix provides the complete list of countries.
Following Hirata, Kose, Otrok and Terrones (2013), we measure equity prices by the stock market price index for each country. Real equity returns are then given by the first difference of the log of equity prices deflated by the consumer price index. 9 We collect the equity price index data from the OECD Statistics Database, except in the cases of Argentina, Hong Kong and Peru, for which we draw the data from Investing.com. 10 Figure 1 depicts the cross-country average of the real equity returns. It displays very similar fluctuations over the two country samples, including a sharp decline around the fourth quarter of 2008, at the onset of the global crisis.
As already mentioned, we consider three covariates of equity returns: real GDP, short term interest rates and private credit. The GDP data comes from the OECD Statistics Database and the International Financial Statistics (IFS) of IMF and complemented with national statistics sources as well as the Federal Reserve Bank of St. Louis Economic Data (FRED). Credit is measured by domestic credit to the private sector, drawn from DataStream, complemented with FRED and the Bank for International Settlements (BIS). All the variables are measured in real terms, using the consumer price index as deflator.  Notes: Equity returns are measured by the first difference of the log of real equity price indices, GDP growth is the first difference of the log of real GDP, ∆ Real interest rate is the first difference of the real interest rate, and Real credit growth is the first difference of the log of the real credit stock. The sample period covers 1995:1-2016:3. Table 1 reports summary statistics of real equity returns, real GDP growth, the growth rate of the real credit stock, and the first difference of the real interest rate. Except for the latter variable, all exhibit higher means in the full sample than in the advanced-country sample. Likewise, all the variables show higher volatility in the former than in the latter sample, with the exception of the GDP growth rate. The large equity return growth outlier shown in the table corresponds to the collapse of Iceland's stock market at the time of the global financial crisis.
Descriptive statistics of real equity returns for the individual countries are reported in Table A3 in the appendix. The mean and standard deviation exhibit considerable variation across countries. The mean ranges from 6.5% in Turkey to -0.5% in Greece, while the standard deviation ranges from a low value of 5.1% in New Zealand to a high value of 18.9% in Iceland.
The spatial weight matrix that connects cross-sectional units is a key element in the empirical implementation of the model. We experiment with three alternative specifications. First, we measure the economic distance between each pair of countries by the magnitude of their bilateral trade, following the view that bilateral trade intensities capture economic interactions and shock spillovers across countries, so that countries that trade more are economically more connected, see e.g. Frankel and Rose (1998).
To construct the bilateral trade weight matrix, we use information on bilateral trade taken from the IMF Direction of Trade Statistics (DOT). For the Czech Republic, Hungary, Poland, Slovakia and Slovenia, the trade data is collected from the Centre d'Etudes Prospectives et d'Informations Internationales (CEPII) online database. Specifically, for a pair of countries i and j, i = j, entry i,j of the trade spatial weight matrix W is defined as where Exports ij denotes the exports from country i to country j, and Imports ji are the imports of country i from country j. Once W has been computed, it is re-scaled dividing each of its elements by the sum of its corresponding row, so that the rows of the rescaled matrix sum to unity. 11 The second specification of the spatial weight matrix is based on bilateral foreign direct investment (FDI) positions. 12 The underlying logic is that the higher is the share of outward and/or inward investments of country i from country j, the more economically interdependent countries i and j are, resulting in larger spillovers from one country to the other; see, for instance, Asgharian, Hess and Liu (2013) and Chinn and Forbes (2004).
We construct the bilateral FDI weight matrix in a similar fashion to the bilateral trade weight matrix, using data on bilateral FDI positions taken from the OECD International Direct Investment Statistics, and replacing imports and exports in the above definition of W ij with the stock of outward foreign direct investment from country i to country j and the stock of inward foreign direct investment from country j to country i, respectively.
Finally, the third specification of the weight matrix is based on pure geographical distance. In particular, following Ertur and Koch (2007), we use a weight matrix W D based on inverse squared distance. 13 The elements of W D are defined (before row normalization) as where d ij is the great-circle distance between the capital cities of countries i and j. 14 11 Such row standardization of the weight matrix facilitates the interpretation of the model coefficients, see Anselin (1988).
12 Ideally, we would want to use the bilateral positions of countries' international portfolios, rather than just their FDI positions. However, such information is not available for our sample coverage. 13 We also experimented with a matrix based on the negative exponential of squared distance. The results were very similar to those obtained with inverse squared distance, and to save space they are not reported.
14 The great-circle distance, the shortest distance between any two country capitals, is computed as: To explore the relation between the common factor(s) and the 'push' variables taken in the literature as indicators of global financial conditions (as in, e.g., Miranda-Agrippino and Rey (2018)), we collect data on several measures of investors' risk perceptions -the Chicago Board Options Exchange Market Volatility Index (VIX), the Bank of America Merrill Lynch high-yield spread, the Bank of America Below Options-Adjusted Spread, and Moody's corporate bond yield spread. In addition, we also use the U.S. Federal Funds real interest rate, as well as the real effective exchange rate of the U.S., taken from FRED, and the risk appetite index of Bekaert, Engstrom, and Xu (2017).
To assess the covariates of countries' factor loadings, we collect information on selected policy and structural indicators of capital account openness, financial depth, stock market capitalization and the exchange rate regime. Following Barrot and Servén (2018), capital account openness is measured by the Chinn-Ito index, and financial depth is measured by domestic credit to the private sector as a percentage of GDP. Stock market capitalization is measured as the total value of all listed shares in a stock market as a percentage of GDP. Following Ghosh, Ostry, Kapan and Qureshi (2015), the exchange rate regime is categorized into three groups, and each is assigned a numeric value so that higher values denote more flexible regimes -i.e., Peg = 1, Intermediate = 2, and Float = 3. Table A2 in the appendix presents the data description, sources and other details.

Empirical results
Before proceeding with the estimation of the model, we examine the cross-country comovement of equity returns in the data. The average pairwise correlation of real equity returns equals .51 for the full sample and .67 for the advanced-country sample. A total of 754 of the 780 distinct pairwise correlations in the full country sample are positive, and 652 of them are statistically significant at the 5 percent significance level. 15 In turn, all of the 300 distinct pairwise correlations in the advanced-country sample are positive, and all are significantly different from zero at the 5 percent significance level. Overall, these figures reveal strong international comovement of equity returns, particularly among advanced countries. 16 where radius is the Earth's radius, and lat and long are, respectively, latitude and longitude for country capitals i and j. The latitude and longitude coordinates for each of the country capitals in our sample were collected from the CEPII database. 15 We approximate the standard error of a correlation coefficient r by 1−r 2 T −1 . 16 This result is in line with a large number of studies (e.g., Bekaert and Mehl (2017), Jorda, Schularick, Taylor and Ward (2017), Bekaert, Harvey, Kiguel and Wang (2016), Rey (2013), and Bekaert, Hodrick, and Zhang (2009)) that find significant comovement of asset returns across countries. The heat map in Figure 2 illustrates the patterns of the pairwise correlation of equity returns for the advanced-country sample. 17 Darker colors indicate higher correlations. The correlations in the figure range from .22 (corresponding to the Korea-Portugal pair) to .94 (the France-Germany correlation), with a median of .69. Visual inspection reveals that the correlations are consistently very high among European countries, whose equity returns appear to be tightly linked, except for Iceland, Finland and Greece. In contrast, with the exception of Australia, in the Asia-Pacific countries shown at the bottom of the graph equity returns bear a much weaker relation with those of other countries.
While the pairwise correlations provide a strong hint of cross-sectional dependence in the data, a more formal assessment can be made using two suitable statistics. The first one is the cross-sectional dependence (CD) test statistic of Pesaran (2015), which is based on simple averages of pairwise correlation coefficients. Specifically, the statistic is given by N j=i+1r ij where ther ij are the estimated pairwise correlation coefficients. Under the null of weak cross-sectional dependence, CD d − → N (0, 1) for N → ∞ and large T ; see Pesaran (2015).
The second statistic is the exponent of cross-sectional dependence of Bailey, Kapetanios and Pesaran (2015), defined by the standard deviation of the 17 To save space, we do not report the heat map of the full sample.
cross-sectional average of the observations. Specifically, the exponent α is given by , wherex t is a simple cross-sectional average of observations x it , i = 1, 2, ...n; t = 1, 2, ..., T . The exponent takes a value between 0 and 1. A value of 1 indicates strong cross-sectional dependence, of the type usually captured with (strong) factor models. 18 Table 2 reports the Pesaran CD test statistic and the exponent of cross-sectional dependence of the dependent and independent variables of the model, for both the advanced-country and the full sample. The CD test statistic for equity returns is above 100 for both samples, overwhelmingly rejecting the null. For the other variables, the statistic also provides evidence against weak dependence, although to a more limited extent. In both country samples, the highest CD statistic, aside from that of equity prices, corresponds to GDP growth. Notes: Equity returns are measured by the first difference of the log of real equity price indices, GDP growth is the first difference of the log of real GDP, ∆ Real interest rate is the first difference of the real interest rate, and Real credit growth is the first difference of the log of the real credit stock. 'Exponent of CSD' is the exponent of cross-sectional dependence, and values in parenthesis are its 95% confidence bands. The sample period covers 1995:1-2016:3.
Table 2 also reports the exponent of cross-sectional dependence along with 95% confidence bands, for both the advanced and full country samples. For equity prices, the estimated value of α is 1 in the advanced-country sample and .98 in the full sample. In both cases, the 95% confidence region reaches well above 1, providing clear indication of the presence of strong common factors in the equity returns data.
The table also shows that the exponent of cross-sectional dependence of GDP growth is very close to 1 in both country samples, with the confidence region including unity in both cases. This agrees with the evidence that GDP growth around the world reflects a common factor (Kose, Otrok and Whiteman (2003)). Finally, for the interest rate and credit variables the estimated exponents of cross-sectional dependence are below .9, except in the case of credit in the full country sample. However, for neither of them does the 95% confidence region include 1.

Model estimation results
In order to estimate the factor-augmented dynamic spatial model (1), we first need to determine the number of unobserved common factors. To do so, we use information criteria. Following Choi and Jeong (2018), we compute the IC p2 , BIC and HQ criteria, setting the maximum number of factors to 3. 19 For both the full and the advanced-country samples, we perform this calculation using in turn each of the three spatial weight matrices considered. However, the results are invariant to the choice of weight matrix. For both country samples, the BIC and HQ criteria select one factor. In turn, the IC p2 criterion selects two factors in the full sample, and three in the advanced-country sample. We opt for employing one factor in all the estimations below. 20 We next turn to the main estimation results. Table 3 reports model estimates for both the full sample (top panel) and the advanced-country sample (bottom panel). The three columns of the table correspond to the three alternative specifications of the spatial weight matrix -bilateral trade, bilateral foreign direct investment, and bilateral inverse distance.
Consider first the full-sample results in the top panel of the table. Across all specifications, the coefficient of the lagged dependent variable is positive and statistically significant, indicating a significant degree of inertia in equity returns. In turn, both real GDP growth and the change in the real interest rate carry significant coefficients, positive and negative, respectively, in accordance with theoretical expectations. In addition, the coefficient on credit growth is also positive and significant, consistent with the view that the interest rate does not suffice to capture financial conditions in the sample countries, likely due to financial market frictions; hence credit growth provides independent information on the financial environment faced by equity investors. Moreover, the estimated coefficients of these variables are very similar across the three specifications of the spatial weight matrices.
As for the spatial effects, the parameter estimate of the contemporaneous spatial lag is consistently positive, while that of the space-time lag is negative; their magnitude is largest under the bilateral trade weight matrix. The positive contemporaneous spatial lag implies that higher equity returns in a given country tend to raise those of neighboring countries through bilateral financial linkages. In turn, the negative sign of the spacetime lag accords with the pattern commonly found in dynamic models with spatial effects. 21 The estimated spatial effects are significant in all cases except for the spacetime lag under the bilateral FDI weight matrix. The implication is that information 19 Setting the maximum number of factors to 5 instead yields very similar results. 20 Because the factors and loadings are identified only up to a sign change, in the estimation we set their respective signs so that the sample country with the largest equity market (the U.S.) has a positive loading. 21 Tao and Yu (2012) show that, in a spatial setting, models of intertemporal choice often yield the non-linear restriction λ = −βρ in equation (1). With positive autocorrelation of the dependent variable (β > 0), and positive contemporaneous spatial spillovers (ρ > 0), the implication is that the lagged spatial spillover effect λ should be negative, as found here. about local interactions is important for understanding the patterns of equity returns across countries and over time.
The estimated models do a good job at accounting for the cross-sectional dependence of equity returns highlighted in Table 2. For all three models, the CD test statistic shown at the bottom of Table 3 reveals little evidence of residual cross-sectional correlation. The exponent of cross-sectional dependence is in all cases under 0.5, likewise suggesting that the residuals exhibit no strong cross-sectional dependence.
Results using the advanced-country sample are shown in the bottom panel of Table  3. For the most part, the estimates follow the same sign and significance patterns of the full-country estimates. There are some differences, however. Credit growth now carries an insignificant coefficient. This result is consistent with the idea that among advanced countries financial frictions are less pervasive than in other countries, and therefore the interest rate suffices to summarize financial conditions. Also, the GDP growth parameter estimate is much larger than in the full sample, indicating that in advanced economies equity returns track growth fluctuations more closely than in the rest. Finally, the estimated spatial effects are consistently larger, in absolute value, than in the full sample, likely reflecting the deeper real and financial linkages among advanced countries relative to the rest. Still, the residuals seem to exhibit some traces of weak cross-sectional dependence, as the CD statistics border on statistical significance and the exponents of cross-sectional dependence lie around .7, well above the value of their full-sample counterparts.
Despite its simplicity, the model does a good job at tracking equity returns, as summarized by the R 2 shown in Table 3. 22 . The model accounts for close to 60 percent of the variation of the dependent variable in the full sample, and over 70 percent in the advanced-country sample, with the highest value corresponding in both cases to the bilateral trade specification of the weight matrix. Table 4 further reports the R 2 by country for all the specifications estimated in Table  3. They range from over .8 (even .9 in the advanced-country sample) for a handful of advanced countries -France, Germany, the Netherlands, Sweden and the U.S. -to less than .1 for Colombia and .2 for Indonesia. In the full sample, the median R 2 is just under .7, and only 8 countries exhibit values under 0.5, while in the advanced-country subsample the median exceeds .75 and only two countries show values consistently below .5.

Transmission of spatial impacts
The fundamental implication of the dynamic spatial model is that a shock in a particular country affects not only the equity returns of that country itself, but also the equity returns of neighboring countries within the spatial system. In other words, incorporating the spatial interaction effects helps understand the nature and magnitude of spillover effects across countries. To illustrate the spatial spillovers implied by the estimates of the model, consider equation (1) rewritten as: Then, recursive substitution shows that the effect h-periods ahead of a one-time shock to V t is ∂g t+h ∂Vt = [(I − ρW ) −1 (βI + λW )] h (I − ρW ) −1 . The short-run effect is just ∂gt ∂Vt = (I − ρW ) −1 . Hence the contemporaneous impact of a shock hitting a particular country (i.e., a shock to a particular element of V t ) diminishes with distance at a rate that depends on the elements of the weight matrix W and the spatial coefficient ρ. It also declines over time at a rate that depends on β, λ and ρ. The larger (in absolute value) these parameters, the larger the eigenvalues of the transition matrix [(I − ρW ) −1 (βI + λW )], and the more persistent the effects of the shock.
For illustration, Figure 3 reports the impact on selected countries of one-time shocks to equity returns in the U.S., the U.K., Germany, Argentina and Turkey. The graphs on the left side of the figure show the responses obtained with the full-sample estimates using the FDI spatial weight matrix, and the graphs on the right side show the responses when using instead the bilateral trade weight matrix. In each case, the graphs show the contemporaneous response to a unit shock to equity returns, and the dynamics over the subsequent three quarters. 23 Two general features are worth noting. First, convergence is quite fast -after just four quarters, the impacts arising from the FDI linkages have virtually vanished; with the trade linkages, they vanish even faster. The reason is that the eigenvalues of the transition matrix turn out to be fairly small in absolute value (under 0.30), thus implying little persistence. Second, convergence is monotonic with the FDI matrix, but oscillating with the trade matrix. This is due to the larger negative estimate of the space-time lag in the latter specification.
The short-run effects are, in some cases, fairly substantial. For example, a unit shock to U.S. returns raises Mexican returns by 0.35 percent under the FDI specification of the weight matrix, and by a whopping 2.5 percent under the bilateral trade specification. In the latter case, U.S. shocks also have a larger impact on equity returns in Brazil and Argentina. Likewise, the impact of shocks to German stock returns on other countries is much larger under the bilateral trade specification of the weight matrix than under the FDI specification. The reason for these differences is twofold. First, the estimate of the contemporaneous spatial effect is almost twice as large under the former specification than under the latter. Second, the bilateral trade links of the countries shown with the U.S. and Germany are larger than their bilateral FDI links.

The common factor and the global financial cycle
An important element of the empirical model is the unobserved common factor driving equity returns around the world. Figure 4 depicts the common factor obtained from the model estimates using the bilateral trade matrix (column 1 of Table 3) along with the cross-country average equity returns, for both the advanced-country and the full samples. In both cases the common factor tracks average equity returns very closely. The estimated common factors are very similar across all the specifications shown in Table 3. For the full country sample, their pairwise correlations exceed .98; for the advanced-country subsample they exceed .74. Rey (2013), Xu (2017) and Miranda-Agrippino and Rey (2018) likewise find a common factor behind asset prices worldwide, which they interpret as reflecting global investors' risk aversion -or the 'global financial cycle' (see also Barrot and Servén (2018) and Bekaert and Mehl (2017)). Consistent with this logic, Figure 5 plots the common factor estimated in column 1 of Table 3 and the VIX (shown using an inverted scale), commonly used as a summary measure of perceived aggregate risk. The figure depicts both the advanced-country and the full sample common factors. Both show a strong association with the VIX, as argued by the literature.
However, the world financial cycle may reflect other global forces besides risk -e.g., global interest rates, or the U.S. real exchange rate, owing to the dominant role of the U.S. dollar in financial transactions worldwide (see, e.g., Avdjiev et al (2017) and Bruno and Shin (2015)). To assess their contribution to the observed pattern of equity returns around the world, we examine the relationship between the estimated common factor and selected global variables. 24 In particular, we consider the VIX plus other measures of global risk -the Bank of America Merrill Lynch High Yield Spread (HYS), the Bank of America Below Options-Adjusted Spread (BOAS), Moody's Corporate Bond Yield Spread (MCOY), and the risk appetite and uncertainty indices of Bekaert, Engstrom, and Xu (2017). In addition, we consider the U.S. federal funds rate, expressed in real terms, as well as the first difference of the log of the U.S. real effective exchange rate, defined so that an increase represents an appreciation. Table 5 reports the correlation of the common factors from the two country samples with each of these global variables. The correlations are negative in all cases, except for the U.S. short-term real interest rate, which exhibits a positive correlation with both global factors. All the correlations are statistically significant, except for that of the BOAS spread with the advanced-country factor. The strong negative correlation with the risk variables accords with the results of Miranda-Agrippino and Rey (2018) and Xu (2017), while the negative correlation with the real exchange rate confirms those of Bruno and Shin (2015), who argue that U.S. dollar real appreciation itself represents a global risk factor.   Bekaert, Engstrom, and Xu (2017). ***, **, and * denote significance at the 1%, 5%, and 10% level respectively. Table 6 reports regressions of the full-sample common factor on these global variables. We present regressions of the common factor on the VIX; the VIX plus the U.S. shortterm real interest rate; these two variables plus the real exchange rate of the U.S.; and the full set of global variables considered. The regressions show that the VIX alone can account for over 40 percent of the variation of the common factor. Adding the U.S. short-term real interest rate and real exchange rate raises the explanatory power of the regressions above 50 percent. Finally, adding the other risk measures further raises the R 2 to 66 percent, although several of them carry insignificant coefficients due to their high degree of collinearity.

The exposure to global financial shocks
Overall, the results in Table 6 support the view that the common factor driving equity returns across the world can be interpreted as a summary representation of global financial conditions. This raises the question of what determines countries' exposure to them -or, in other words, the sensitivity of their asset prices to global shocks. This is a first-order policy question that has attracted increased attention following the global financial crisis. In our model, the sensitivity of each country's equity returns to the common shocks is given by its respective factor loading. The estimated loadings (shown in Table A4 in the Appendix) are very similar across the three specifications in Table 3: in the full country sample, their pairwise correlations exceed .98, while for the advanced-country sample the correlation exceeds .86. However, the loadings display considerable variation across countries. In the full sample they are all positive, except for three emerging countries (Colombia, Indonesia and Slovakia) that exhibit very small negative values. On average, they are also larger among advanced countries than among emerging countries.
The largest values belong to major emerging markets -Argentina, Turkey -and small open advanced economies -Iceland, Greece. In the advanced-country sample,Iceland consistently exhibits the largest loading, especially when using the trade-based matrix to estimate the model.
It seems plausible to expect the loadings to vary systematically with key features of countries' structural and policy framework -such as their degree of financial development and/or international financial integration. To verify this conjecture, we regress the fullsample factor loadings on selected policy and structural indicators. 25 Specifically, the variables we consider include capital account openness, trade openness, financial depth, stock market capitalization and the exchange rate regime. In particular, the latter variable has been at the center of the policy debate following the claim by Rey (2013) that the global financial cycle renders virtually irrelevant any distinctions between exchange rate regimes regarding their ability to shelter the economy from external financial disturbances. 26 As the factor loadings do not change over time, the regressions only make use of the cross-sectional variation, and therefore the explanatory variables are measured by their respective average over the entire 21-year time sample. Over this time span the explanatory variables have surely undergone major changes, and this should tend to obscure their relationship with the loadings. Hence the regressions probably understate the strength of that relationship, and should be viewed with some caution. The table reports regressions on the variables shown of the factor loadings from the full sample estimates in the first column of Table 3. An increase in the value of the exchange rate regime variable denotes a more flexible regime. Trade openness is measured in logs. T-statistics in brackets computed with heteroscedasticity-consistent standard errors. The regressions include a constant. The sample comprises 25 observations. Table 7 reports the regression results using the factor loadings as dependent variable. The univariate regressions in the first five columns show that the loadings increase significantly with financial and trade openness, the size of the stock market, and domestic financial depth. In contrast, they decrease with a more flexible exchange rate regime. The last column of Table 7 shows that when all variables are jointly considered they can account for over 90 percent of the variation of the factor loadings. However, because of strong collinearity among the variables, only trade openness and the exchange rate regime remain individually significant. One implication is that, in spite of the worldwide reach of the global financial cycle, the choice of exchange regime continues to matter for the exposure of domestic asset prices to international shocks.

Sensitivity analysis
Finally, we examine the sensitivity of our main results to alternative ways of modeling the cross-country dependence of equity returns. Our methodological setting employs both common factors and spatial effects, in contrast with the earlier literature that opts for one or the other. We next assess how this choice affects our results. For this purpose, we re-estimate the model omitting the common factor and the spatial effects -first jointly and then in turn. 27 The results are shown in Table 8. In the first column, cross-sectional dependence is ignored altogether, and common factors and spatial effects are both omitted -i.e., in terms of equation (1), we impose ρ = λ = 0 and Ψ = 0. In the second column, the model includes a common factor but no spatial effects (i.e., ρ = λ = 0). The last three columns allow for spatial effects but rule out common factors (i.e., Ψ = 0); each of these columns corresponds to one specification of the spatial weight matrix. The top panel of Table 8 reports the results obtained with the full sample, and the bottom panel reports those obtained with the advanced-country sample.
The first column of Table 8 shows that ignoring cross-sectional dependence leads to distorted parameter estimates and to a marked deterioration of the model's empirical performance relative to that achieved when both spatial effects and common factors are allowed for (shown in Table 3). The GDP growth parameter estimate doubles relative to that in Table 3; most of the other coefficient estimates exhibit large changes too. Also, in the advanced country sample the coefficient estimate of the real interest rate becomes insignificant. Moreover, in both samples the CD statistic and the exponent of crosssectional dependence show overwhelming evidence of (strong) residual dependence. In addition, the overall fit of the model is quite poor, as it accounts for less than 20 percent of the variation of the dependent variable. The second column of Table 8 adds a common factor but omits spatial effects. The parameter estimates are now much closer to those in Table 3, with the only exception of the real interest rate in the advanced-country sample, whose coefficient remains small (in absolute value) and insignificant. The CD statistic continues to show evidence of cross-sectional dependence, but its value falls sharply relative to that in the first column. In turn, the exponent of cross-sectional dependence also declines well below 1. Together, these statistics suggest that the common factor succeeds in taking account of the strongest dependence in the data, but still leaves considerable (weak) dependence in the residuals. Lastly, the fit of the model shows a considerable improvement relative to the preceding column, with the R 2 rising threefold. The last three columns of Table 8 report estimates including spatial effects, for each of the three versions of the spatial weight matrix we consider, but excluding the common factor. The spatial effects are strongly significant -in fact, both their magnitude and their significance appear substantially overstated relative to the results shown in Table 3. This is particularly the case for the spatial lag, whose parameter estimate is much larger than in Table 3; its massive t-statistic hints at residual dependence. The values of the other parameter estimates lie in most cases between those in the first column and those in the applicable column of Table 3. In turn, the cross-sectional dependence statistics show in general lower values than in the first column of Table 8, but higher than in the factor-only model of the second column. In particular, the exponent of cross-sectional dependence is fairly close to that in the first column, suggesting that the spatial effects do little to ameliorate the strong dependence in the data. Lastly, the overall fit of the model, as measured by R 2 , improves substantially relative to column I with the addition of the spatial variables. However, it is worse than that of the factor-only model.
Overall, comparison of Tables 3 and 8 shows that both the common factor and the spatial effects contribute to the model's empirical performance -they complement each other in their ability to account for cross-sectional dependence, and to track the variation of the dependent variable. Table A5 in the appendix reports the individual-country R 2 for each of the estimated model specifications in Table 8. On the whole, the pattern is the same as in Table 8: the fit is very poor (R 2 of .3 or lower) when cross-sectional dependence is ignored, and improves substantially when either the common factor or spatial dependence is taken into account -although in both cases the fit is unsurprisingly poorer than that obtained when both features are jointly included, as shown in Table 4.
Finally, Table 9 reports further robustness checks on the specification of the empirical model. To save space, only results using the bilateral trade weight matrix are reported; however, results with the other matrices are not very different. The first column adds to the baseline specification in column 1 of Table 3 a spatial error term. The result is a deterioration in the empirical performance of the model. In the full sample, the spatial error is insignificant, and the rest of the estimates show little change relative to the baseline. In the advanced-country sample, the parameter estimates of the fundamental variables all become insignificant, and the residuals show strong symptoms of dependence.
The second column employs two factors in the estimation, rather than the single factor used in the baseline specification following the verdict of the information criteria. The main consequence is that the estimates of the spatial effects become smaller than in the baseline, as more of the cross-sectional dependence is taken up by the factors. This is especially visible in the advanced-country subsample; also, the evidence of residual dependence weakens. The overall fit of the model improves relative to that in Table 3. The last column of Table 9 attempts to address the strong dependence of the returns by cross-sectional de-meaning of the data prior to estimation, instead of adding common factors in the regression specification. This can be viewed as equivalent to a factor model imposing the implicit restriction that the loadings be constant across countries. The parameter estimates on the fundamental variables show some changes relative to Table  3 -particularly in the case of the real interest rate in the advanced-country sample, whose coefficient declines by half. In addition, the spatial effects appear overstated, especially in the full sample, and the huge t-statistic on the spatial lag hints at misspecification.

Conclusion
Equity returns display strong international comovement, especially across advanced countries. Existing empirical literature has modeled it as reflecting either localized real and/or financial linkages across countries, or pervasive common shocks -i.e., weak and strong cross-sectional dependence, respectively. In this paper we have brought both perspectives together by assessing the comovement of equity returns in a setting that allows for both spatial dependence and latent common factors, using quarterly equity price data over the years 1995 to 2016 for 40 advanced and emerging countries.
In the paper's framework, real equity returns are driven by three observable variables proxying for growth of the present value of anticipated dividends: the growth rates of GDP and real credit, and the change in the real interest rate. These variables are augmented with latent common factors and spatial effects accruing through equity prices themselves. We estimate the model using a bias-corrected quasi-maximum likelihood procedure recently developed by Shi and Lee (2017), alternatively considering all 40 sample countries, or a subsample of 25 advanced economies. To capture the interactions among countries, we employ alternative spatial weight matrices based on bilateral trade, bilateral FDI stocks, and geographic distance. To determine the number of latent common factors driving equity returns, we use a variety of information criteria. On the whole, they indicate the presence of a single factor for both country samples considered.
Estimation results reveal that, in accordance with prior expectations, real equity returns are significantly positively related to real GDP growth, and negatively related to changes in the real interest rate. In the advanced-country subsample, real credit growth has no significant effect on equity returns. However, in the full sample, which includes countries with less-developed financial markets, credit growth carries a positive and strongly significant coefficient. In addition, equity returns display significant inertia in both samples.
The results show little variation across the three alternative specifications of the spatial weight matrix. They also provide strong evidence of spatial effects. The contemporaneous spatial lag coefficient is consistently positive and significant, while the spatial-time lag carries a negative coefficient that is significant in most specifications.
The significant spatial effects imply that local interactions are important to understand the international comovement of equity returns.
In turn, the estimated common factor is strongly positively correlated with real equity returns, and strongly negatively correlated with market indicators of aggregate risk. Overall, the common factor can be interpreted as summarizing the 'global financial cycle' stressed by Rey (2013) and Miranda-Agrippino and Rey (2018).
Our results also shed light on the determinants of countries' exposure to global shocks, an issue at the core of the policy debate. We find that the impact of the common factor on equity returns is bigger in countries that exhibit higher trade openness and whose exchange rate regimes exhibit less flexibility. The latter result in particular suggests that, notwithstanding the worldwide reach of the global financial cycle, the choice of exchange rate regime still matters for countries' exposure to global financial shockswhich echoes the recent findings of Bekaert and Mehl (2017) and Barrot and Servén (2018).
Despite its simplicity, the empirical model does a good job at explaining observed equity returns. In the baseline specification, it accounts for close to 60 percent of the variance of equity returns in the full country sample, and 70 percent in the advancedcountry subsample.
Our empirical exercises underscore the importance of properly taking into account cross-sectional dependence. Ignoring it, by omitting both common factors and spatial effects, causes major distortions in the parameter estimates, leading in particular to a gross overstatement of the procyclical behavior of equity returns. It also results in a abysmal deterioration of the model's explanatory power. Adding the common factor, while still omitting spatial effects, helps correct these problems, but leaves evidence of residual (weak) dependence. In turn, allowing for spatial effects, while omitting the common factor, also improves the fit considerably, but leads to overstated spatial effects and strong residual dependence. Overall, these results confirm the need to account for cross-sectional dependence, both strong and weak, in empirical modeling of equity returns across countries.  Bekaert, Engstrom, and Xu (2017)