Stock Return Predictability and Variance Risk Premia: Statistical Inference and International Evidence

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Stock Return Predictability and Variance Risk Premia: Statistical Inference and International Evidence Tim Bollerslev, James Marrone, Lai Xu, and Hao Zhou 2011-52 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Stock Return Predictability and Variance Risk Premia: Statistical Inference and International Evidence ∗ Tim Bollerslev, James Marrone, Lai Xu, and Hao Zhou † ‡ § ¶ This Version: December 2011 Abstract Recent empirical evidence suggests that the variance risk premium, or the difference betweenrisk-neutralandstatisticalexpectationsofthefuturereturnvariation,predicts aggregate stock market returns, with the predictability especially strong at the 2-4 monthhorizons. WeprovideextensiveMonteCarlosimulationevidencethatstatistical finite sample biases in the overlapping return regressions underlying these findings can not“explain”thisapparentpredictability. Furthercorroboratingtheexistingempirical evidence,weshowthatthepatternsinthepredictabilityacrossdifferentreturnhorizons estimated from country specific regressions for France, Germany, Japan, Switzerland and the U.K. are remarkably similar to the pattern previously documented for the U.S. Defining a “global” variance risk premium, we uncover even stronger predictability and almost identical cross-country patterns through the use of panel regressions that effectively restrict the compensation for world-wide variance risk to be the same across countries. Our findings are broadly consistent with the implications from a stylized two-countrygeneralequilibriummodelexplicitlyincorporatingtheeffectsofworld-wide time-varying economic uncertainty. JEL classification: C12, C22, G12, G13. Keywords: Variance risk premium; return predictability; over-lapping return regressions; international stock market returns; global variance risk. ∗Theanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyother members of the research staff or the Board of Governors. We would like to thank Qianqiu Liu, Andrew Patton, GeorgeTauchen, Guofu Zhou, and seminar participants at the 2011 China International Conference in Finance (CICF) in Wuhan, the 2011 NBER-NSF Time Series Conference at Michigan State University, the 2011 Inquire Europe Autumn Seminar, Notre Dame University, and the Duke Financial Econometrics Lunch Group for their helpful comments. We would also like to acknowledge the Best Paper Award from CICF. Bollerslev’s research was supported by a grant from the NSF to the NBER, and CREATES funded by the Danish National Research Foundation. †Department of Economics, Duke University, Durham, NC 27708, USA, and NBER and CREATES, boller@duke.edu, 919-660-1846. ‡Department of Economics, University of Chicago, Chicago, IL 60637, USA, jmar@uchicago.edu, 773- 702-9016. §Department of Economics, Duke University, Durham, NC 27708, USA, lai.xu@duke.edu, 919-257-0059. ¶Division of Research and Statistics, Federal Reserve Board, Mail Stop 91, Washington DC 20551 USA, hao.zhou@frb.gov, 202-452-3360.

Stock Return Predictability and Variance Risk Premia: Statistical Inference and International Evidence This Version: October 11, 2011 Abstract Recent empirical evidence suggests that the variance risk premium, or the difference between risk-neutral and statistical expectations of the future return variation, predicts aggregate stock market returns, with the predictability especially strong at the 2-4 month horizons. We provide extensive Monte Carlo simulation evidence that statistical finite sample biases in the overlapping return regressions underlying these findings can not “explain” this apparent predictability. Further corroborating the existing empirical evidence, we show that the patterns in the predictability across different return horizons estimated from country specific regressions for France, Germany, Japan, Switzerland and the U.K. are remarkably similar to the pattern previously documented for the U.S. Defining a “global” variance risk premium, we uncover even stronger predictability and almost identical cross-country patterns through the use of panel regressions that effectively restrict the compensation for world-wide variance risk to be the same across countries. Our findings are broadly consistent with the implications from a stylized two-country general equilibrium model explicitly incorporating the effects of world-wide time-varying economic uncertainty. JEL classification: C12, C22, G12, G13. Keywords: Variance risk premium; return predictability; over-lapping return regressions; international stock market returns; global variance risk.

1 Introduction A number of recent studies have suggested that aggregate U.S. stock market return is predictable over horizons ranging up to a few quarters based on the difference between optionsimplied and actual realized variation measures, or the so-called variance risk premium (see, e.g., Bollerslev, Tauchen, and Zhou, 2009; Drechsler and Yaron, 2011; Gabaix, 2011; Kelly, 2011; Zhou, 2010; Zhou and Zhu, 2009, among others). These findings of apparent predictability over relatively short quarterly horizons have potentially far reaching implications for many issues in asset pricing finance. They are also distinctly different from the longer-run multi-year return predictability patterns that have been studied extensively in the existing literature, in which the predictability is typically associated with more traditional valuation measures such as dividend yields, P/E ratios, or consumption-wealth ratios (see, e.g., Fama and French, 1988; Campbell and Shiller, 1988b; Lettau and Ludvigson, 2001, among others). Motivated by these observations, the main goal of the present paper is to further examine the robustness and the scope of these striking new empirical findings. Our investigations are essentially twofold. First, to assess the validity of the statistical inference procedures underlying the empirical findings, we report the results from an extensive Monte Carlo simulation exercise designed to closely mimic the dynamic dependencies inherent in daily returns and variance risk premia. Our results clearly suggest that statistical biases can not “explain” the documented return predictability patterns. At the same time, the results also suggest that the use of finer sampled observations, say daily as opposed to monthly data as employed in the above cited studies, provides limited additional power to detect the predictability inherent in the variance risk premium. Second, in a separate effort to expand on and corroborate the existing empirical evidence pertaining to monthly U.S. returns, we extend the same basic ideas and regressions to several other countries. In so doing, we also define a “global” variance risk premium. We show that this simple aggregate measure of world-wide economic uncertainty results in even stronger predictability for all of the countries in the sample. We also show that these new empirical findings are broadly consistent with the implications from a stylized two-country 1

general equilibrium model that explicitly incorporates the effect of time-varying economic uncertainty across countries. The finite sample properties of overlapping long-horizon return regressions have been studied extensively in the existing literature. Boudoukh, Richardson, and Whitelaw (2008), for instance, have recently shown that even in the absence of any increase in the true predictability, the values of the R2’s in regressions involving highly persistent predictor variables and overlapping returns will by construction increase roughly proportional to the return horizon and the length of the overlap.1 In line with the procedures adapted in the existing literature, we will focus on the Newey and West (1987) and Hodrick (1992) type t-statistics. Both of these are robust asymptotically to heteroskedasticity and serial correlation in the residuals from the estimated regressions. Our simulation design is based on an empirically realistic bivariate VAR-GARCH-DCC model for the joint daily return and variance risk premium dynamics. We find both of the t-statistics to be reasonably well behaved, albeit slightly over-sized under the null hypothesis of no predictability. We also find that the Newey-West based t-statistics result in marginally more powerful (size adjusted) tests under the alternative. Moreover, directly in line with the results in the existing literature on long-horizon return predictability, the quantiles in the finite sample distribution of the R2’s from the regressions are spuriously increasing with the return horizon under the null of no predictability.2 At the same time, the R2’s implied by the daily VAR-GARCH-DCC model exhibit a distinct hump shape in the degree of predictability that closely mimics the pattern actually observed in U.S. return regressions. Guided by the Monte Carlo simulations, we rely on the Newey-West based t-statistics and monthly return regressions to summarize our new international evidence. Due to data availability and the required liquidity of options markets, we restrict our attention to the six major financial markets of France, Germany, Japan, Switzerland, the U.K., and the U.S. Our empiricalresultshowsthatthecountryspecificregressionsbasedonregressingeachcountry’s 1Closelyrelatedissuespertainingtotheuseofpersistentpredictorvariableshavealsobeenstudiedby,e.g., Stambaugh (1999), Ferson, Sarkissian, and Simin (2003), Baker, Taliaferro, and Wurgler (2006), Campbell and Yogo (2006), Ang and Bekaert (2007), and Goyal and Welch (2008), among others. 2Although the variance risk premium is not especially persistent at the monthly frequency typically employed in the literature, its first order autocorrelation is still in excess of 0.97 at the daily level. 2

return on its own variance risk premium result in similar hump-shaped regression coefficients and R2’s for all of the six countries. However, the degree of predictability afforded by the country specific variance risk premia and the statistical significance of the results generally are not as strong as the previously reported results for the U.S. market. These results naturally point to the possibility of world-wide variance risk, as opposed to the country specific variance risk premia, being priced. To investigate this idea, we construct a “global” variance risk premium, defined as a simple market capitalization weighted average oftheindividualcountryvarianceriskpremia. Restrictingtheeffectonthis“global”variance risk premium to be the same across countries in a panel return regression results in much stronger findings for all of the countries, with a uniform peak in the degree of predictability at the four month horizons. Moreover, the degree of predictability afforded by this “global” variance risk premium easily exceeds that of the implied and realized variation measures when included in isolation. It also clearly dominates that of other traditional predictor variables that have been shown to work well over longer annual horizons, including the P/E ratio. While this new international evidence indirectly corroborates the previous findings based exclusively on U.S. data reported in the studies cited above, importantly the results also point to the existence of even stronger predictability through the use of alternative definitions of world-wide variance risk.3 Our new empirical findings are, of course, related to the large existing literature on international stock return predictability (see, e.g., Harvey, 1991; Bekaert and Hodrick, 1992; Campbell and Hamao, 1992; Ferson and Harvey, 1993, among others). However, the focus of this literature has traditionally been on longer-run multi-year return predictability. By contrast, our results pertaining to the “global” variance risk premium concern much shorterrun within year predictability, and are essentially “orthogonal” to the findings reported in 3These results are also indirectly in line with those reported in a few other recent studies pertaining to other markets. In particular, in concurrent independent work, Londono (2010) finds that the U.S. variance risk premium predicts several foreign stock market returns. In a slightly different context, Mueller, Vedolin, and Zhou (2011) argue that the U.S. variance risk premium predicts bond risk premia, beyond the predictability afforded by forward rates, while Buraschi, Trojani, and Vedolin (2010) and Zhou (2010) show that the variance risk premium also helps predict credit spreads, over and above the typical interest rate predictor variables. 3

the existing literature.4 At the same time, however, the new empirical results are generally in line with the calibrations from a simple theoretical two-country model that explicitly incorporates the equilibrium effects of time-varying economic uncertainty across countries. The rest of the paper is organized as follows. Section 2 presents our Monte Carlo based simulation evidence pertaining to the statistical inference procedures underlying the existing empirical findings. Section 3 discusses our new international evidence and the results for our “global” variance risk premia measure, along with our equilibrium model based calibrations. Section 4 concludes. 2 General Setup and Monte Carlo Simulations The key empirical findings reported in Bollerslev, Tauchen, and Zhou (2009) (BTZ2009, henceforth), and the subsequent studies cited above, are based on simple OLS regressions of the returns on the aggregate market portfolio over monthly and longer return horizons on a measure of the one-month variance risk premium. In particular, let r and VRP t,t+τ t denote the continuously compounded return from time t to time t+τ and the variance risk premium at time t, respectively. Defining the unit time interval to be one trading day, the multi-period return regressions in BTZ2009 may then be expressed as special cases of, h 1 r t+(j−1)s,t+js = a s (h) + b s (h)VRP t + u t,t+hs (1) h Xj=1 for s = 20 (monthly) and return horizon hs, where t = 1,s+1,2s+1,...,T hs refer to the − specific observations used in the regression. Of course, the use of finer sampling frequencies, say s = 5 (weekly) or s = 1 (daily), may give rise to more powerful inference, and we will investigate that below. Meanwhile, it is well known that in the context of overlapping return observations, the regression in (1) can result in spuriously large and highly misleading regression R2’s, say R2(h), as the horizon h increases; see, e.g., the discussion and many references in Campbell, s 4Other recent studies highlighting short-run international predictability include Rapach, Strauss, and Zhou (2010) based on lagged U.S. returns, Ang and Bekaert (2007) and Hjalmarsson (2010) based on shortterm interest rates, and Bakshi, Panayotov, and Skoulakis (2011) based on the Baltic Dry Index. 4

Lo, and MacKinlay (1997). Similarly, the standard errors for the OLS estimates designed to take account of the serial correlation in us based on the Bartlett kernel advocated by t+hs,t Newey and West (1987) (NW, henceforth), and the modification proposed by Hodrick (1992) (HD, henceforth), can also both result in t-statistics for testing hypotheses about a (h) and s b (h) that are poorly approximated by a standard normal distribution. Most of the existing s analyses pertaining to these and other related finite sample biases, however, have been calibrated to situations with a highly persistent predictor variable, as traditionally used in long-horizon return regressions. Even though the variance risk premium is fairly persistent at the daily frequency, it is much less so at the monthly level, and as such one might naturally expect the finite sample biases to be less severe in this situation.5 Our Monte Carlo simulations discussed in the next section confirm this conjecture in an empirically realistic setting designed to closely mimic the joint dependencies in actual daily returns and variance risk premia. 2.1 Simulation Design The model underlying our simulations is based on daily S&P500 composite index returns (obtained from CRSP). The corresponding daily observations on the variance risk premium are defined as VRP t = IV t RV t−20,t , where we rely on the square of the new VIX index − (obtained from the CBOE) to quantify the implied variation IV , and the summation of t current and previous 20 trading days daily realized variances (obtained from the Oxford- Man Institute’s Realized Volatility Library) together with the squared overnight returns to quantify the total realized variation over the previous month RV t−20,t .6 The span of the data runs from February 1, 1996 to December 31, 2007, for a total of 2,954 daily observations. After some experimentation, we arrived at the following bivariate VAR(1)-GARCH(1,1)- 5The first order autocorrelation coefficient for the monthly U.S. variance risk premium analyzed in the empirical section below equals 0.50, and it is even lower for all of the other countries included in our analysis. By comparison, the first order autocorrelations for monthly dividend yields and P/E ratios, and other variables typically employed in the long-horizon regression literature, are around 0.95-0.99. 6ThisdirectlymirrorsthedefinitionofthevarianceriskpremiumemployedinBTZ2009. Forwardlooking measuresofVRP t thatalignIV t withameasureoftheexpectedvolatilityE t (RV t,t+20)havealsobeenusedin theliterature. However, thisrequiresadditionalmodelingassumptionsforcalculatingE t (RV t,t+20),whereas the VRP used here has the obvious advantage of being directly observable at time t. t 5

DCC model (see Engle, 2002, for additional details on the DCC model) for the two daily time series, r t−1,t = 1.958e-5 0.009r t−2,t−1 +0.025VRP t−1 +ǫ t,r − (0.001) − (0.016) (0.010) VRP t = 3.759e-5+0.033r t−2,t−1 +0.972VRP t−1 +ǫ t,vrp (0.001) (0.017) (0.010) σ2 = 1.280e-6+0.071ǫ2 +0.920σ2 t,r t−1,r t−1,r (1.68e-6) (0.004) (0.008) σ2 = 2.038e-7+0.133ǫ2 +0.871σ2 t,vrp t−1,vrp t−1,vrp (7.59e-6) (0.004) (0.028) 0.997 0.754 Q t =  (0.036) − (0.040) +0.011η t−1 η t ′ −1 +0.979Q t−1 0.754 1.023 (0.002) (0.004) − (0.040) (0.060)    R = diag Q −1Q diag Q −1, t t t t { } { } where η t ≡ ( σ ǫt t , , r r , σ ǫt t , , v v r r p p ) ′ , and E t−1 (η t ) = 0 and E t−1 (η t η t ′ ) = R t by assumption. The specific parameter values refer to Quasi Maximum Likelihood Estimates (QMLE) obtained under the auxiliary assumption of conditional normality, with robust standard errors following Bollerslev and Wooldridge (1992) in parentheses. With the exception of the lagged daily returns, most of the dynamic coefficients are highly significant at conventional levels. The model implies a strong negative (on average) correlation between the innovations to the return and VRP equations. This, of course, is consistent with the well documented “leverage” effect; see, e.g., Bollerslev, Sizova, and Tauchen (2011) and the many references therein. At the same time, as is evident from the equation for Q , and the corresponding t plot in the top panel in Figure 1, the value of the conditional correlation clearly varies over time, reaching a low of close to -0.85 toward the end of the sample. The bottom three panels in Figure 1 indicate that the distribution of the estimated standardized residuals from the model (i.e., cη F −1η , where F F ′ =R ) are well behaved and centered at zero, with t ≡ t t t · t t variances closbe to ubnityb, albeit nobt nobrmablly distributed.7 All in all, however, the model provides a reasonably good fit to the joint dynamic dependencies inherent in the two daily 7The sample means for cη and cη equal -0.044 and 0.088, the standard deviations equal 0.999 and t,1 t,2 1.007, while the skewness and kurtosis equal -0.469 and 0.894, and 4.913 and 7.860, respectively. Further diagnostic checks also reveabl that whbile the residuals from the return equation appear close to serially uncorrelated, there is some evidence for neglected longer-run serial dependencies in the equation for the variance risk premium. 6

series. As such, we will use this relatively simple-to-implement model as our basic data generating process for the Monte Carlo simulations, and our analysis of the finite sample properties of the NW and HD t-statistics, and R2(h)’s from the overlapping return regressions in equas tion (1).8 Our simulated finite sample distributions will be based on a total of 2,000 bootstrapped replications from the model. We will look at sample frequencies of s = 1 (“daily”), s = 5 (“weekly”) and s = 20 (“monthly”), and return horizons hs ranging up to 240 “days,” or 12 “months.” The number of observations for each of the simulated samples is fixed at T = 2,954 “days” (or 598 “weeks,” or 149 “months”), corresponding to the length of the actual sample used in the estimation of the VAR-GARCH-DCC model above. We begin with a discussion of the size and power properties of the two t-statistics. 2.2 Size and Power Ourcharacterizationofthedistributionsunderthenullhypothesisofnoreturnpredictability is based on restricting the coefficients associated with r t−2,t−1 and VRP t−1 in the return equation to be identically equal to zero, leaving all of the other coefficients at their estimated values. Table 1 reports the resulting simulated the 95th percentiles of the tNW and tHD test statistics, along with the regression R2’s. Directly in line with the evidence in the existing literature, both of the t-statistics exhibit non-trivial size distortions relative to the nominal one-side 95-percent critical value of 1.645. Also, the distortions tend to increase with the returnhorizonh. Moreover, consistentwiththeresultsreportedinHodrick(1992), thebiases for the NW based standard error calculations generally exceed those for the HD standard errors, and markedly more so the longer the return horizon. To more directly illustrate the results, we plot in the three left panels in Figure 2 the simulated 95-percent critical values for tNW (dashed lines) and tHD (solid lines) for s = 1,5,20. We also include in the figure the t-statistics obtained by running these same regressions 8The bandwidth in the Bartlett kernel employed in our implementation of the NW standard errors is set to m = [h+4 ((T hs)/100)2/9], where [] refers to the integer value. We also experimented with the ∗ − · reverse regression technique suggested by Hodrick (1992) for testing bs(h) = 0. The results, available upon request, were very similar to the ones for the HD t-statistic reported below. 7

on the actual daily, weekly and monthly data over the February 1996 through December 2007 sample period used in calibrating the simulated model. As the figure shows, the actual tNW-statistics systematically exceeds the simulated critical values for return horizons in the range of 2 to 3 months. This is true regardless of whether the regressions are based on daily, weekly, or monthly data. Meanwhile, the tHD-statistics generally do not exceed the simulated critical values and accordingly do not support the idea of return predictability. In order to better understand this discrepancy in the conclusions drawn from the two tests, we report in Table 2 the power of the tests to detect predictability implied by the unrestricted VAR-GARCH-DCC model. To facilitate comparisons we only report the sizeadjusted power for a 5-percent test. Not surprisingly, the power of both tests decrease with the return horizon. At the same time, the power of the tNW test systematically exceed that of the tHD test for return horizons less than a year, and the differences appear most pronounced at the 2-4 month horizons. These differences are also evident in the three right panels in Figure 2, which plot the relevant power curves. Comparing the simulations across the three different panels in the table and the figure also point to fairly small loses in terms of power when decreasing the sampling frequency of the data used in the regressions from s = 1 (“daily”) to s = 5 (“weekly”) to s = 20 (“monthly”). Guidedbythesefindingswewill baseoursubsequentempirical investigationson themost commonly used monthly return regressions and NW-based standard errors, recognizing that the finite sample distributions of the tNW-statistics tend to be slightly upward biased under the null of no predictability. 2.3 R2 In addition to the t-statistics associated with the b (h) coefficients, the R2(h)’s from the s s return regressions are often used to assess the strength of the relationship and the effectiveness of the predictor variable across different horizons. Of course, as previously noted above, it is well known that the biases exhibited by the t-statistics in the context of longhorizon return regressions carry over to the R2(h)’s, and that these need to be carefully s 8

interpreted in the context of persistent predictor variables (see, e.g., the aforementioned study by Boudoukh, Richardson, and Whitelaw, 2008, for a recent analysis, along with the many references therein). The corresponding columns in Table 1 show that, while less dramatic than the biases over multi-year return horizons, the R2(h)’s may still be quite different from zero under the s null of no predictability in the present setting. In particular, the 95th percentiles are around 5-6 percent at the 2-4 months horizon for all of the three sampling frequencies s = 1,5,20. Furthertothiseffect,weshowinthetoppanelinFigure3selectquantilesinthesimulated distribution of the R2(h)’s that obtain in the absence of any predictability. Consistent with 1 the findings in the extant literature pertaining to monthly observations and longer return horizons, allofthequantilesincreasemonotonicallywiththereturnhorizon, andthisincrease is especially marked for the higher percentiles. Intuitively as the horizon increases, the overlapping return regressions become closer to a spurious type regression. In addition to the simulated quantiles, we also include in the same figure the R2(h)’s 1 obtained from the actual return regressions based on the same daily data used in estimating the VAR-GARCH-DCC model. Comparing the actual R2(h)’s to the simulated percentiles 1 again suggest that the degree of predictability is most significant at the intermediate 2-4 months horizon. This, of course, is directly in line with the inference based on the t-statistics discussed in the previous section, and the prior empirical evidence reported in BTZ2009. The hump-shaped pattern in the actual R2(h)’s also closely mimics the patterns in the 1 simulated quantiles for the estimated VAR-GARCH-DCC model depicted in the bottom panel in Figure 3. Interestingly, this striking similarity with an apparent peak in the degree of predictability at the intermediate 2-4 months horizon arises in spite of the fact that the simulated model involves only first-order dynamics in the equations that describe the daily conditional means. To help understand this result, consider the VAR(1) corresponding to the conditional 9

mean dependencies in the Monte Carlo simulation design, r t−1,t = a 1 +b 1 r t−2,t−1 +c 1 VRP t−1 +ǫ r,t , VRP t = a 2 +b 2 r t−2,t−1 +c 2 VRP t−1 +ǫ vrp,t . Following Campbell (2001), it is possible to show that the population regression coefficients and R2’s from the overlapping return regressions in (1) may be expressed as,9 b(h) = c 1 − ch 2 + b +c b 1 − ch 2 −1 Cov(r t−1,t ,VRP t ) 1 1 1 2 (cid:18) 1 c (cid:19) (cid:18) 1 c (cid:19) Var(VRP ) 2 2 t − − +b b(h 1)+c b [b(h 2)+c b(h 3)+ +ch−3b(1)] , 1 − 1 2 − 2 − ··· 2 b(h)2 Var(VRP ) R2(h) = t . h [h−1Var( h j=1 r t−1+j,t+j )] P Hence, the strength of the predictability over different horizons h is primarily determined by the interaction between the short-run predictability, or Cov(r t−1,t ,VRP t ) and c 1 , and the own persistence of the VRP predictor variable and c . t 2 To illustrate this, the solid lines in each of the four panels in Figure 4 show the R2(h)’s implied by the unrestricted VAR(1) coefficient estimates used in the simulations. Indirectly confirming the satisfactory fit of the model, the theoretically implied population R2(h)’s are generally close to the R2(h)’s actually estimated from the sample regressions depicted by the star-dashed line in the previous Figure 3. Meanwhile, marginally decreasing the value of each of the VAR(1) coefficients, b , c , b and c , by ten percent, results in quite different 1 1 2 2 R2(h)’s, as shown by the dashed lines in Figure 4. In particular, the decrease in c has by far 2 the largest effect. Moreover, the value of c , and the own persistence of VRP , is intimately 2 t linked to the location of the maximum in the hump shaped predictability pattern.10 Taken as a whole, our Monte Carlo simulations and the new regression results based on daily U.S. returns discussed above clearly support the variance risk premium as a powerful predictor at the 2-4 month horizons. At the same time, the overlapping nature of the return 9We have omitted the implicit dependence on the sampling frequency s = 1 for notational simplicity. Further details concerning these derivations are available upon request. 10Thesesameideasalsounderlietheeconomicmechanismsandrisk-returntrade-offsacrossdifferentreturn horizons analyzed within the stylized equilibrium model setting in Bollerslev, Sizova, and Tauchen (2011). 10

regressions tend to attenuate the strength of the predictability somewhat. Hence, in an effort to further corroborate the existing empirical evidence pertaining exclusively to the U.S. market and data prior to the 2008 financial crisis, we next turn to a discussion of our new empirical findings involving more recent data and several other countries. 3 International Evidence Motivated by the Monte Carlo simulation results in the previous section, we will rely exclusively on the common benchmark monthly sampling frequency, along with the traditional NW-based standard errors and tNW-statistics, keeping in mind the finite sample biases documented above. We will restrict our analysis to France, Germany, Japan, Switzerland, the U.K., and the U.S., all of which have highly liquid options markets and readily available model-free implied variances for their respective aggregate market indexes (see Siriopoulos and Fassas, 2009, for a recent summary of the model-free and parametric options implied volatility indexes available for different countries). We begin with a brief discussion of the relevant data. 3.1 Data and Summary Statistics Our monthly aggregate market index returns are based on daily data for the French CAC 40 (obtained from Euronext), the German DAX 30 (obtained from Deutsche B¨orse), the JapaneseNikkei225, theSwissSMI,andtheU.K.FTSE100(allobtainedfromDatastream), and the U.S. S&P 500 (obtained from Standard & Poor’s). We use the sum of the daily squared returns over a month to construct end-of-month realized variances RVi for each t of the countries. The corresponding end-of-month model-free implied volatilities (IVi)1/2 t for the S&P 500 (VIX) were obtained from the CBOE, the CAC (VCAC) from Euronext, the DAX (VDAX) from Deutsche B¨orse, while those for the FTSE (VFTSE) and the SMI (VSMI) were both obtained from Datastream. Our data for the Japanese volatility index (VXJ) is obtained directly from the Center for the Study of Finance and Insurance at Osaka University (see Nishina, Maghrebi, and Kim, 2006, for a more detailed discussion of the VXJ 11

index). The sample period for each of the series extends from January 2000 to December 2010,andassuchalsoallowsforanout-of-samplevalidationoftheexistingempiricalevidence for the U.S. based exclusively on data prior to the recent financial crisis.11 In accordance with the empirical analysis in the previous section, the variance risk premium for each of the individual countries is simply defined by VRPi IVi RVi . The t ≡ t − t−20,t resulting time series plots in Figure 5 clearly show the dramatic impact of the financial crisis, and the exceptionally large volatility risk premia observed in the Fall of 2008 for all of the countries. Interestingly, the premium for the DAX, and to a lesser extent the SMI, were almost as large and negative as in 2001-2002. The standard set of summary statistics reported in Table 3 also show a remarkable coherence in the distributions of the variance risk premia and monthly excess returns for each of the countries.12 In particular, looking at Panel A the average excess returns are all negative, ranging from a high of -2.15 for Switzerland to a low of -6.52 for France, reflective of the often-called “lost decade.” Of course, the corresponding standard deviations all point to considerable variations in the returns around their sample means. The variance risk premia are all positive on average, ranging from a low of 4.13 for France to a high of 13.26 for Japan on a percentage-squared monthly basis. “Selling” volatility has been highly profitable on average over the last decade. Meanwhile, consistent with the visual impressions from Figure 5, all of the premia are significantly negatively skewed and exhibit large excess kurtosis. Even though implied and realized variances are both strongly serially correlated for all of the countries, the variance risk premia are generally not very persistent with the maximum first order serial correlation for the S&P 500 just 0.50. Turning to Panels B and C, the sample cross-country correlations are all fairly high, and with the exceptions of those for the Nikkei, the correlations for the returns all exceed 0.75, while those for the variance risk premia are in excess of 0.70. The similarity in the summary statistics in Table 3 and the time series plots in Figure 11The beginning of the sample coincides with the back-dated initial date of the NYSE Euronext volatility indices. 12The riskfree rates used in the construction of the excess returns were obtained from the Federal Reserve Board and Eurocurrency via Datastream. The use of excess returns, as opposed to raw returns, has almost no effect on the results from the return predictability regressions reported below. 12

5 across the different countries, naturally suggests that the same predictive relationship between the multi-period U.S. returns and variance risk premium may hold true for the other countries as well. The results discussed in the next subsection corroborate this conjecture. 3.2 Country Specific Variance Risk Premia Regressions In parallel to the general multi-period return regression defined in equation (1), our monthly return regressions for each of the individual countries may be expressed as, h −1ri = ai(h) + bi(h)VRPi + ui , (2) t,t+h t t,t+h where ri and VRPi refer to the h = 1,2,...,12 month excess return and variance risk t,t+h t premium for country i, respectively.13 The resulting regressions results for each of the six countries are reported in Table 4. Theactualestimatesfor bi(h)and thecorrespondingtNW-statisticsobviously differsomewhatacrossthecountries. However, withtheexceptionofFranceandtheU.S., theestimated coefficients all show the same general pattern starting out fairly low and insignificant at the shortest one-month horizon, rising to their largest values at 3-5 months, and then gradually declining thereafter for longer return horizons. These similarities are also evident in Figure 6, which displays the regression coefficients for the variance risk premia along with the conventional 95-percent confidence bands based on two NW standard errors.14 These general patterns in the estimated values of bi(h) naturally translate into very similar patterns in the corresponding regression R2(h)’s. In particular, looking at the plots in Figure 7, all of the R2(h)’s exhibit an almost identical hump-shaped pattern with the degree of predictability maximized at the 4 months horizon. Of course, the actual values of the R2(h)’s again vary somewhat across the different indices, achieving a maximum of only 0.96 percent for the Nikkei 225 compared to 14.18 percent for the S&P 500.15 13We omit the s = 20 monthly subscript on the regressors and regression coefficients for notational simplicity. 14Of course, the results from the Monte Carlo simulations reported in Table 1 indicate that the two standard error bands are likely somewhat conservative and need to be interpreted accordingly. 15Interestingly, this value of R2(4) for the U.S. exceeds that obtained with monthly data through the end of 2007, reported in BTZ2009 and Drechsler and Yaron (2011), as well as the corresponding daily results discussed in Section 2 above. 13

Takenaswhole, theresultsinTable4andthesignificanceofthecountryspecificVRP’sas predictorvariableshelptounderscorethesignificanceoftheexistingresultsbasedexclusively on the U.S. data. The similarities in the patterns obtained across countries also suggest that even stronger results may be available by pooling the regressions and entertaining the notion of a common “global” variance risk premium. We explore these ideas next. 3.3 Global Variance Risk Premium and Panel Regressions Our definition of a “global” variance risk premium is based on a simple capitalization weighted average of the country specific variance risk premia, 6 VRPglobal wiVRPi , t ≡ t t Xi=1 where i = 1,2,...,6 refer to each of the six countries included in our analysis.16 The end-ofmonth market capitalizations used in defining the weights wi are obtained from Thomson t Reuters Institutional Brokers’ Estimate System (I/B/E/S) via Datastream. The plot of the weights in Figure 8 shows that the U.S. market accounts for more than sixty percent through most of the sample period, with Japan a distant second. This relatively large weight assigned to the U.S. market in our definition of the “global” VRP index is also evident from the aforementioned summary statistics in Table 3. The results for the regressions obtained by replacing the country specific VRPi’s in t equation (2) with the new VRPglobal index, t h −1ri = ai(h) + bi(h)VRPglobal + ui , (3) t,t+h t t,t+h are reported in Table 5, along with the corresponding tNW-statistics. Comparing the results totheonesforthecountryspecificregressionsinTable4, revealsevenstrongercommonalities and uniform patterns across countries. The “global” VRP index serves as a highly significant predictor variable for all of the different country indexes, with tNW-statistics in excess of 5.0 at the 4 months horizon. Meanwhile, increasing h, the VRPglobal predictor variable always t 16This parallels the idea used in Harvey (1991) in the estimation of the world price of covariance risk. 14

becomes insignificant over the longer 9 and 12 months return horizons reported in the last two columns of the table. These striking cross country similarities are also immediately evident from the plots of theestimatedregressioncoefficientsandthetwoNW-basedstandarderrorbandsinFigure9. Not only do the individual country estimates for the bi(h)’s look very similar, the confidence bandsalsobecomemuchtightercomparedtothecountryspecificregressionsdiscussedabove. Further along these lines, Figure 10 shows the general patterns in the predictability, as measured by theR2(h)’s, to bevery similarly shaped for thedifferentcountries, with uniform peaks at the 4 months return horizon.17 Going one step further, we next restrict the coefficients for the “global” variance risk premium to be the same across countries, h −1ri = a(h) + b(h)VRPglobal + ui , (4) t,t+h t t,t+h as a way to further enhance the efficiency of the estimates. The corresponding panel regression estimates for the b(h)’s together with the NW-based t-statistics are reported in Table 6 (for additional details on the calculations, see, e.g., Petersen, 2009).18 As the table clearly shows, the use of panel regressions do indeed result in more accurate estimates, and a highly significant tNW-statistics of 11.21 at the 4-months horizon. Similarly, the average panel regression R2(h)’s across the six countries gradually rise from around one percent at the one-month horizon to a large 7.46 percent for the four-month returns, tapering off to zero for the longer 9-12 month return horizons. These key empirical findings are succinctly summarized in Figure 11, which plots the panel regression estimates for b(h) based on the country specific and “global” VRP measures along with two NW standard error bands (top two panels), and the corresponding 17The large weight assigned to the U.S. in our construction of the “global” risk premium means that fairly similar results are obtained by replacing the new VRPglobal in the regressions in equation (3) with t VRPS&P500. Theseadditionalresultsareavailableuponrequest. Comparableempiricalresultsbasedonthe t U.S. variance risk premium have also recently been reported in concurrent independent work by Londono (2010), who ascribes the predictability to informational frictions along the lines of Rapach, Strauss, and Zhou (2010). 18Wealsoexperimentedwiththetwo-wayclusteranalysisinCameron, Gelbach, andMiller(2011), resulting in qualitatively very similar findings. 15

panel regression R2(h)’s (bottom two panels). The VRPglobal-based regressions (depicted in the right two panels) obviously result in sharper coefficient estimates and stronger average predictability across the six countries, compared to the individual country VRPi regression (depicted in the left two panels). The panel regression R2(h)’s, of course, mask important cross-country differences in the degree of predictability. We therefore also show in Figure 12 the country specific implied R2(h)’sobtainedbyevaluatingtheregressionsinequation(3)atthemorepreciselyestimated ˆ common aˆ(h) and b(h) obtain from the panel regressions in equation (4). Interestingly, comparing Figure 12 to the earlier Figure 10 for the individual country regressions, it is clear that the added precision afforded by restricting the ai(h) and bi(h) coefficients to be the same across countries sacrifices very little in terms of the implied predictability. To assess the robustness of these impressive empirical findings, the next panel in Table 6 reports the results obtained by including a capitalization weighted average of the country specific P/E ratios as an additional regressor. Consistent with the results for the U.S. market in isolation reported in BTZ2009, the “global” P/E ratio adds nothing to the predictability afforded by VRPglobal within the one-year horizons reported in the table, leaving all of the estimates for b(h) and the R2(h)’s the same to within the second decimal place. The predictability of the “global” variance risk premium is effectively orthogonal to that documented in the existing literature based on more traditional macro-finance variables, such as the P/E ratio, dividend yields, and consumption-wealth ratios, which are typically only significant over longer multi-year return horizons (see, e.g., the classic studies by Fama and French, 1988; Campbell and Shiller, 1988b; Lettau and Ludvigson, 2001).19 To further highlight the predictive gains afforded by the use of our “global” VRP as opposed to the own country VRP’s, the last two panels in Table 6 report the results obtained by including each individual country’s premium in a panel regression, h −1ri = a(h) + b(h)VRPi + ui . (5) t,t+h t t,t+h 19Further corroborating the results for the U.S. market in BTZ2009, we also found that including the implied“global”varianceortherealized“global”variancetogetherwiththe“global”varianceriskpremium resulted in mostly insignificant coefficient estimates. These additional results are available upon request. 16

While the results still point to overall efficiency gains from the panel regression setting relative to the individual country specific regressions in Table 4, the magnitude of the return predictability is obviously much lower than for VRPglobal. The “global” variance risk premium is clearly a much better predictor of the future returns than the individual country specific premia. Also, including the country specific P/E ratios in the same panel regression setting again results in insignificant coefficient estimates, while the tNW-statistics for the variance risk premia remain highly significant at the intermediate 2-6 month horizons. Tohelpbetterunderstandtheeconomicmechanismsunderlyingthesenewempiricalfindings, we next present a stylized two-country equilibrium model. This relatively simple model provides a possible rationale for why the estimated “global” VRP regression coefficients are fairly similar across countries, and why the R2(h)’s for the panel regressions depicted in Figure 11 are generally larger for the “global” VRP than for the “local” VRP’s, except for the U.S. 3.4 Global Variance Risk in Equilibrium Our two-country model is based on a direct extension of the “long-run risk” model in BTZ2009.20 Specifically, denoting the geometric growth rate of consumption in country i by gi log(Ci /Ci), we will assume that t+1 ≡ t+1 t gi = µ +σ z , (6) t+1 g gi,t gi,t+1 σ2 = α ϕ +ν σ2 +ϕ √q z , (7) gi,t+1 σ q,i σ gi,t q,i t σ,t+1 q = α +ν q +ϕ √q z , (8) t+1 q q t q t q,t+1 where µ denotes mean growth rate, assumed to be constant and the same for the two coung tries, σ2 refers to the conditional variance of consumption growth for each of the countries, gi,t and q represents time-varying volatility-of-volatility, or aggregate world-wide economic unt certainty. In parallel to existing “long-run risk” models, we will assume that z and σ,t+1 z are independent i.i.d. N(0,1) process, and jointly independent of the two consumption q,t+1 20Even though the model explicitly excludes predictability in consumption growth, following the terminology of Bansal and Yaron (2004), we will refer to the basic setup as a “long-run risk” model. 17

growth shocks, z . For simplicity, we will fix the dynamic variance parameters α and gi,t+1 σ ν to be the same across the two countries.21 The system is normalized by fixing ϕ at σ q,1 unity. The scaling of the mean and volatility parameters for the second country by ϕ , in q,2 turn ensures that the two variance processes move proportional to each other. To complete the specification, we assume that the conditional covariance between z and z is gi,t+1 gj,t+1 determined by the process cv = α +ν cv +ϕ √q z . (9) t+1,ij cv cv t,ij cv,ij t σ,t+1 This trivially implies time-varying conditional correlations, unless the parameters are identical across the covariance and two variance processes.22 We assume that the two international equity markets are fully integrated. We further assume the existence of a global representative agent with a claim on the world aggregate consumption, defined as the per capita weighted average consumption in each of the two countries, say Cglobal. Moreover, this agent is endowed with Epstein-Zin-Weil recursive preft erences of the form U t = [(1 − δ)(C t global) 1− θ γ +δ(E t [U t 1 + − 1 γ])θ 1 ]1− θ γ. (10) In the specific calibration reported on below, we follow Bansal and Yaron (2004) and BTZ2009 in fixing the discount rate at δ=0.997, the risk aversion parameter at γ=10, and the intertemporal elasticity of substitution at ϕ=1.5. The parameters for the consumption dynamics are calibrated to mimic the U.S. as country “1”, and the U.K. as country “2”. In particular, following BTZ2009 we fix the base parameters for the U.S. at µ =0.0015, ν =0.979, α =0.00782(1 ν ), ν =0.80, α =1.0*10eg σ σ σ q q − 6, and ϕ =0.001, respectively. For simplicity, we treat the weights used in the calculation of q “global” consumption as constant and equal to ω =0.855 and ω =0.145, corresponding US UK to the consumption shares at the end of the sample. 21BansalandShaliastovich(2010)employsasimilarassumptionforthevariancedynamicsintheirrelated two-country model. By contrast, the two-country model in Londono (2010) involves separate shocks for the “leader” and “follower” countries, but assumes that all of the parameters driving the σ ’s and the country gi,t specific q ’s are the same across the two countries. i,t 22The consumption data discussed below strongly supports the notion of time-varying covariances (and correlations). By contrast, the aforementioned two-country model in Bansal and Shaliastovich (2010) postulates constant cross-country conditional covariances. 18

The relevance of allowing for time-varying cross-country covariances is highlighted by Figure 13, which plots exponentially weighted moving average estimates of U.S. variances, and U.S.-U.K. covariances and correlations from 1951 to 2009.23 Although the U.S.-U.K. consumption covariances clearly changes through time, the process is not as persistent as the process for the U.S. variance depicted in the top panel. We consequently set ν =0.85 cv and ϕ = ϕ 1/2 ϕ 1/2 . Finally, we set the parameter ϕ 1/2 =(2.5)1/2 1.581 to reflect cv,US,UK q,UK q,US q,UK ≈ the generally higher variability of UK consumption growth. Turningtotheactualcalibrationresults, thetoptwopanelsinFigure14showtheimplied regressioncoefficientsforthe“local”(dashedlines)and“global”(solidlines)VRPregressions for each of the two countries, while the bottom two panels show the implied R2’s from the “local” (dashed lines) and “global” (solid lines) VRP panel regressions.24 The model-implied regressions in the figure generally match the qualitative features in the actual international return regressions quite well. First, theimpliedslopecoefficientsforVRPglobal inthetwoindividualcountryregressions tend to be close across all horizons. For instance, at the four-month horizon, the model implied slope coefficients equal 0.34 and 0.33 for the U.S. and U.K., respectively, both of which are well within two standard errors of their corresponding estimates reported in Table 5. Of course, these numbers are also very close to the estimate of 0.32 for the six-country panel regression in Table 6. Second, the exposure to the “local” VRP is systematically lower than the exposure to the “global” VRP for the smaller country in the model (U.K.), directly mirroring the empirical results. Conversely, for the larger country (U.S.), the “local” VRP gives rise to marginally higher slope coefficients than the “global” VRP within the model, again directly mirroring the actual empirical results. Specifically, focusing again on the four-month horizon, the slope 23Theexponentialweightedmovingaveragesdepictedinthefigurearebasedonannualrealtotalconsumption expenditures from the Penn World database, and a smoothing parameter equal to λ=1 (1 0.06)4. Specifically, for the U.S. variance σ2 =(1 λ)σ2 +λ(gUS µ )2, and the U.S.-U. − K. co − variance US,t+1 − US,t t − US cv t 2 + 4T 1= h ( e 1 c − al λ ib ) r c a v t t e + d λ m (g o t U de S l − al µ so US im )(g p t U lie K s − eq µ u U it K y ), p w re i m th iu t m he s c f o o r r re t l h a e tio U n .S d . d efi an n d ed U a . c K co . r o d f in 6 g . l 8 y 5 . percent and 6.02 percent, respectively, along wdith a world-wdide risk free rate of 0.96 percent. Additional technical details concerning the solution of the model, together with explicit formulas for the regression coefficients and R2’s depicted in the figure, are relegated to Appendix A. 19

coefficient implied by the model equals 0.17 for the U.K. compared to 0.15 for the actual U.K. regression. In comparison, the model implied slope coefficient for the U.S. equals 0.40, compared to 0.36 for the actual “local” U.S. regression. Third, looking at the R2’s from the corresponding panel regressions in the bottom two panels of Figure 14, both of the plots exhibit a hump shaped pattern with an apparent peak at the 2-4 month horizons. This overall shape closely matches that for the actual six-country panel regressions depicted in the bottom two panels in Figure 11. Of course, the values of the R2’s from the theoretical model are somewhat muted compared to the sixcountry panel regressions R2’s. Importantly, however, the model implied panel regression R2’s based on VRPglobal uniformly dominate the “local” VRP panel regression R2’s. Again, these theoretical implications directly mirror the empirical results for the six-country panel regressions in Figure 11. Intuitively, the “global” VRP effectively isolates the aggregate world-wide economic uncertainty that is being priced in both markets, in turn providing better overall predictions for the future returns than the “local” VRP’s.25 In a sum, while the qualitative implications form our stylized equilibrium model are generally in line with the international predictability patterns documented in the data, some of the quantitative implications from the model fall short in explaining the magnitude of the effects. However, we purposely kept the model relatively simple, involving only two independent volatility shocks. It is certainly possible that by extending the basic model setup to include additional sources of covariance, or correlation, risks, a full-fledged riskbased explanation for the new international evidence may be feasible. 4 Conclusion A number of recent studies have argued that the aggregate U.S. stock market return is predictableoverrelativelyshort2-4monthhorizonsbythedifferencebetweenoptionsimplied and actual realized variances, or the so-called variance risk premium. We provide extensive 25We also experimented with other calibrations and model specifications. In particular, restricting the c c o ov n a d r i i t a io n n c a e l t c o or b r e el p at r i o o p n or a t t io ρ n = al 0. t 1 o 8 t a h s e i U n . B S. an v s a a r l ia a n n c d e c S v h t a ,u li s a ,u s k to = v √ ic ϕ h q ( u 2 k 0 σ 1 g 2 0 us ) ρ , , re a s n u d lt fi i x n in d g ra t m he at i i m ca p ll l y ied low co e n r s R ta 2 n ’s t (less than 0.03 percent across all return horizons) for VRPglobal. 20

Monte Carlo simulation evidence that this newly documented predictability is not due to finite sample biases in the statistical inference procedures, and that the hump-shape in the degree of predictability with a maximum at the 2-4 month horizons is entirely consistent with the implications from an empirically realistic bivariate daily time series model for the returns and variance risk premia. Further corroborating the existing empirical evidence for the U.S. market, we show that the same basic predictive relationship between future returns and current variance risk premia holds true for a set of five other countries, although the magnitude of the predictability and the statistical significance of the own country variance risk premia tend to be somewhat muted relative to those for the U.S. Meanwhile, regressing the individual country returns on a capitalization weighted “global” variance risk premium, results in almost identical shapes in the degree of predictability across horizons and uniformly larger t-statistics for all of the countries in the sample. Further restricting the regression coefficients and the compensation for the “global” variance risk to be the same across countries, we find even stronger results and highly significant test statistics, with the degree of predictability maximized at the four month horizon. By contrast, the predictability documented in the existing literature based on more traditional macro-finance variables are generally only significant over longer multi-year return horizons. These new empirical findings naturally raise the question of why the “global” variance risk premium works so well as a predictor variable, and why the predictability is restricted to within-year horizons. Building on the equilibrium based model in Bollerslev, Tauchen, and Zhou (2009), we argue that the “global” variance risk premium may be seen as a proxy for world-wide aggregate economic uncertainty. We also show why this “global” variance risk premium may serve as a more effective predictor variable for future international equity returns than the own country’s individual variance risk premium. Alternatively, following the analysis in Bekaert, Engstrom, and Xing (2009), the variance risk premium may be interpreted as a measure of aggregate risk aversion in world financial markets, or a summary measure of disagreements in beliefs across international market participants, as discussed in Buraschi, Trojani, and Vedolin (2010). All of these competing 21

explanations are likely at work to some degree, and we leave it for future research to more clearly sort out the extent to which each of these competing explanations best accounts for thestronginternationalreturnpredictabilityembodiedinthe“global”varianceriskpremium documented here. A Two-Country Equilibrium Model Solution Following Epstein and Zin (1989), the logarithm of the world unique intertemporal marginal rate of substitution, m =log(M ), must satisfy, t+1 t+1 m = θlog(δ) θψ −1g +(θ 1)r , (A.1) t+1 t+1 t+1 − − where r refers to the time t to t + 1 logarithmic return on the “global” consumption t+1 asset, and g denotes the corresponding “global” consumption growth rate.26 Further, t+1 utilizing the standard Campbell and Shiller (1988a) log-linearization technique, the “world” and country specific returns may be expressed as, r = k +k w w +g , (A.2) t+1 0 1 t+1 t t+1 − ri = k +k wi wi +gi , (A.3) t+1 i,0 i,1 t+1 − t t+1 where w and wi denote the logarithmic price-consumption ratios for the “world” and the t t two individual countries, respectively.27 Following the standard approach in the “long-run risk” literature, we proceed by conjecturing solutions to w and wi of the form,28 t t w = A + A σ2 +A q +A cv , (A.4) t+1 0 σj gj,t+1 q t+1 cv,ij t+1,ij X 26For notational simplicity, here and throughout the Appendix, we omit the “global” superscript on the relevant variables. 27Forthecalibrationexercisediscussedinthemaintextwesetk1 =k US,1 =k UK,1=0.9. Theconstantsk0 and k i,0 only enter the expressions for A0 and A i,0 below, which are not actually needed for the calculations of the regression coefficients, R2’s, and equity premia. 28In the following, unless explicitly noted, all of the summations are over the two countries, running from j =1 to 2. 22

wi = A + A σ2 +A q +A cv . (A.5) t+1 i,0 i,σj gj,t+1 i,q t+1 i,cv,ij t+1,ij X Combining the equations for ri and wi above, with equation (6) for gi in the main t+1 t+1 t+1 text, the equilibrium return for country i may alternatively be expressed as, 2 ri = c + A σ2 +A q +A cv +√q k [A z +A ϕ z ]+σ z , t+1 i,r ri,gl gl,t ri,q t ri,cv,ij t,ij t i,1 i,ϕ σ,t+1 i,q q q,t+1 gi,t gi,t+1 X l=1 (A.6) where c = log(δ) + ϕ −1µ , A = A (k ν 1), A = A (k ν 1), A = i,r − g ri,gj i,σj i,1 σ − ri,q i,q i,1 q − ri,cv,ij A (k ν 1), and A = A ϕ +A ϕ . i,cv,ij i,1 cv − i,ϕ i,σj qj i,cv,ij cv,ij Next, utilizing the standardPno-arbitrage condition E t (exp(r t+1 +m t+1 )) = 1, the parameters for the “world” in equation (A.4) may be solved as,29 log(δ)+(1 ϕ −1)µ +k +k [ A α ϕ +A α +A α ] A = − g 0 1 σj σ q,j q q cv,ij cv , 0 1 Pk 1 − (γ 1)2ω ω i j A = − , cv,ij θ(1 k ν ) 1 cv − (γ 1)2ω2 A = − j , σj 2θ(1 k ν ) 1 σ − A = θ −1ϕ −2k −2((1 k ν ) [(1 k ν )2 θ2k4ϕ2( A ϕ +A ϕ )2]1/2). q q 1 − 1 q − − 1 q − 1 q σj qj cv,ij cv,ij X Similarly, the parameters for the individual countries in equation (A.5) may be solved as,30 log(δ)+(1 ϕ −1)µ +k +k [ A α ϕ +A α +A α ] A = − g i,0 i,1 i,σj σ q,j i,q q i,cv,ij cv,ij , i,0 1Pk i,1 − (2γ 1)ω ω γω i j j A = A + − − , i,cv,ij cv,ij (1 k ν ) 1 cv − A = (1 θ)A 1 − k 1 ν σ + γ2ω j 2+I i=j ( − 2γω j +1) , i,σj − σj1 k ν 2(1 k ν ) i,1 σ i,1 σ − − k (1 k ν ) 1 i,1 q A = (1 θ)A + − i,q k − q ϕ2k2 i,1 q i,1 k ϕ −2k −2((1 k ν )2 θ2k2 ϕ2 k2( A ϕ +A ϕ )2 2(θ 1)A (1 1 ) − q i,1 − i,1 q − i,1 q{ 1 σj qj cv,ij cv,ij − − q − k X i,1 2 + [0.5( A ϕ +A ϕ )2k2 +(0.5 θ)k2( A +A ϕ )2 θ2 i,σj qj i,cv,ij cv,ij i,1 − 1 σj cv,ij cv,ij X X +(θ 1)k k ( A ϕ +A ϕ )( A ϕ +A ϕ ] )1/2. − 1 i,1 σj qj cv,ij cv,ij i,σj qj i,cv,ij cv,ij } X X 29Note, the aforementioned restrictions that γ > 1 and ϕ > 1, readily imply that the impact coefficient associated with the volatility and correlation state variables are negative; i.e. A < 0, A < 0, and cv,ij σj A <0. q 30Intuitively,thelargerthecovarianceforthe“small”country,themoreriskythecountry. Also,ingeneral, the more volatile the consumption of country i, the less risky is country j. 23

Going one step further and building on the derivations in BTZ2009, the two country specific VRP’s may be approximated as, VRPi (θ 1)k A q , (A.7) t ≈ − 1 vrpi,q t where A = k2 A ϕ2(A2 +A2 ϕ2)+A ϕ . vrpi,q i,1 q q i,ϕ i,q q ϕ q,i Based on these expressions, it is now possible to derive the slope coefficients from regressing country i’s return on country j’s VRP, 1−νh A q β (h) = ri,q1−νq , (A.8) i,j h(θ 1)k A − 1 vrpj,q as well as the slope coefficient from regressing country i’s return on the global VRP, 1−νh A q β (h) = ri,q1−νq . (A.9) i h(θ 1)k A 1 vrp,q − The final expressions for the “global” and “local” panel regressions discussed in the main text may be derived analogously. In particular, it is possible to show that 1 2 2(θ 1)2k2A2 Var(q ) R2 (h) = β (h) − 1 vrp,q t , (A.10) global (cid:18)2 j (cid:19) Var( h rj ) X m=1 t+m P P and −2 2 Var(VRPj) R2 (h) = Var2(VRPj) β (h)Var2(VRPj) t , local (cid:16)X t (cid:17) (cid:16)X j,j t (cid:17) PVar( h m=1 r t j +m ) P P (A.11) where Var(VRP j )=(θ 1)2k2A2 Var(q ), t − 1 vrpj,q t h h−1 Var( ri ) = hVar(ri )+2 (h s)(A +B ), t+m t+1 − i,s i,s mX=1 Xs=1 ϕ2 ϕ Var(ri ) = A2 Var(q )+(A2 +A2 q,j +2A A q,j )Var(σ2 )+A2 Var(cv ) t+1 ri,q t ri,gi ri,gj ϕ2 ri,gi ri,gj ϕ gi,t ri,cv,ij t,ij q,i q,i 2 +2 A A Cov(σ2 ,cv )+k2 [(A2 +A2 ϕ2)E(q )]+E(σ2 ), ri,gl ri,cv,ij gl,t t,ij i,1 i,ϕ i,q q t gi,t Xl=1 24

ϕ2 ϕ A = A2 νsVar(q )+νs(A2 +A2 q,j +2A A q,j )Var(σ2 ) i,s ri,q q t σ ri,gi ri,gj ϕ2 ri,gi ri,gj ϕ gi,t q,i q,i 2 +A2 νs Var(cv )+ A A (νs +νs)Cov(σ2 ,cv ), ri,cv,ij cv t,ij ri,gl ri,cv,ij cv σ gl,t t,ij Xl=1 2 B = k E(q )(νs−1 A A ϕ +νs−1A A ϕ2+νs−1A A ϕ ), i,s i,1 t σ ri,gl i,ϕ ql q ri,q i,q q cv ri,cv,ij i,ϕ cv,ij Xl=1 and the model-implied moments entering the above expressions are given by, α α ϕ α ω ϕ α E(q ) = q , E(σ2 ) = σ q,i , E(σ2 ) = σ i q,i , E(cv ) = cv , t 1 ν gi,t 1 ν g,t 1P ν t,ij 1 ν q σ σ cv − − − − ϕ2E(q ) ϕ2 E(q ) ϕ2 E(q ) Var(q ) = q t , Var(σ2 ) = qi t , Var(cv ) = cv,ij t , t 1 ν2 gi,t 1 ν2 t,ij 1 ν2 − q − σ − cv ϕ ϕ E(q ) ϕ ϕ E(q ) Cov(σ2 ,σ2 ) = q,i q,j t , Cov(σ2 ,cv ) = ql cv,ij t . gi,t gj,t 1 ν2 gl,t t,ij 1 ν ν − σ − σ cv,ij 25

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Table 1 Simulated Size and R2 The table reports the simulated 95-percentiles in the finite sample distributions of tNW and tHD for testing the hypothesis that b (h) = 0 based on the return predictability regression in equation (1), along with the s adjusted R2 from the regression. The data are generated from the VAR-GARCH-DCC model discussed in the main text, restricting the coefficients in the conditional mean equation for the returns to be equal to zero. The “daily” return regressions are based on 2,954 observations, while the “weekly” and “monthly” regressions involve 598 and 149 observations, respectively. All of the simulations are based on a total of 2,000 replications. h tNW tHD adj.R2 h tNW tHD adj.R2 h tNW tHD adj.R2 Daily Weekly Monthly 20 2.445 2.182 3.036 4 2.434 2.212 2.971 1 2.260 2.276 3.017 40 2.602 2.112 4.804 8 2.600 2.078 4.800 2 2.520 2.187 4.837 60 2.815 2.085 5.756 12 2.805 2.016 5.790 3 2.788 2.083 5.774 80 2.969 2.107 6.324 16 2.967 2.031 6.345 4 2.941 2.106 6.315 120 3.054 2.152 7.639 24 3.095 2.098 7.615 6 3.220 2.124 7.502 180 3.289 2.171 8.594 36 3.322 2.113 8.482 9 3.314 2.163 8.192 240 3.400 2.229 8.951 48 3.476 2.146 8.682 12 3.509 2.186 8.679 Table 2 Simulated Power The table reports the simulated power of the size-adjusted 5-percent tNW and tHD statistics for testing the null hypothesis of no predictability and b (h) = 0 in the return regression in equation (1). The data are s generatedfromtheVAR-GARCH-DCCmodeldiscussedinthemaintext. The“daily”returnregressionsare basedon2,954observations,whilethe“weekly”and“monthly”regressionsinvolve598and149observations, respectively. All of the simulations are based on a total of 2,000 replications. h pwNW pwHD h pwNW pwHD h pwNW pwHD Daily Weekly Monthly 20 0.919 0.873 4 0.910 0.852 1 0.886 0.807 40 0.888 0.808 8 0.877 0.799 2 0.845 0.762 60 0.804 0.725 12 0.793 0.729 3 0.768 0.711 80 0.710 0.630 16 0.709 0.639 4 0.686 0.627 120 0.567 0.474 24 0.553 0.481 6 0.507 0.497 180 0.375 0.352 36 0.371 0.352 9 0.368 0.350 240 0.288 0.283 48 0.285 0.289 12 0.277 0.302 30

Table 3 Summary Statistics Themonthlyexcessreturnsareinannualizedpercentageform. Thevarianceriskpremiaareinmonthlypercentage-squaredform. The“global”index of variance risk premium are defined in the main text. The sample period extends from January 2000 to December 2010. Panel A: Excess Returns and Variance Risk Premia CAC40 DAX30 FTSE100 Nikkei225 SMI S&P500 GlobalIndex rt −rf,t VRPt rt −rf,t VRPt rt −rf,t VRPt rt −rf,t VRPt rt −rf,t VRPt rt −rf,t VRPt VRPt Mean -6.52 4.13 -2.76 6.01 -4.82 8.16 -6.12 13.26 -2.15 5.36 -3.70 7.69 8.15 Std. Dev 67.72 41.45 82.53 32.74 52.87 32.63 73.08 35.26 52.04 29.03 57.82 34.08 32.14 Skewness -0.62 -4.97 -0.85 -2.90 -0.68 -5.35 -0.79 -5.23 -0.70 -4.21 -0.67 -5.06 -5.62 Kurtosis 3.74 43.09 5.45 16.67 3.56 47.38 4.67 52.11 3.57 31.83 3.93 39.41 49.01 AR(1) 0.11 0.27 0.08 0.07 0.07 0.34 0.15 0.18 0.27 0.12 0.16 0.50 0.46 Panel B: Correlations for Excess Returns CAC40 DAX FTSE100 Nikkei225 SMI S&P500 CAC40 1.00 0.94 0.89 0.59 0.84 0.85 DAX 1.00 0.83 0.56 0.81 0.82 FTSE100 1.00 0.62 0.80 0.87 Nikkei225 1.00 0.56 0.64 SMI 1.00 0.77 S&P500 1.00 Panel C: Correlations for Variance Risk Premia CAC40 DAX FTSE100 Nikkei225 SMI S&P500 Global CAC40 1.00 0.83 0.89 0.72 0.83 0.85 0.90 DAX 1.00 0.78 0.59 0.88 0.71 0.77 FTSE100 1.00 0.78 0.86 0.90 0.94 Nikkei225 1.00 0.69 0.72 0.81 SMI 1.00 0.74 0.81 S&P500 1.00 0.99 Global 1.00 31

Table 4 Country Specific Regressions The results are based on the monthly regression in equation (2). tNW-statistics are reported in parentheses. The sample period extends from January 2000 to December 2010. Index Monthly Horizon 1 2 3 4 5 6 9 12 Constant -7.41 -8.07 -8.13 -8.30 -8.35 -8.30 -8.52 -8.39 (-1.04) (-1.16) (-1.18) (-1.20) (-1.19) (-1.18) (-1.14) (-1.10) CAC 40 VRPi 0.22 0.22 0.22 0.21 0.16 0.11 0.06 0.04 t (2.71) (3.59) (3.30) (3.79) (3.86) (3.01) (1.37) (0.88) Adj. R2 0.99 2.61 3.89 4.88 3.19 1.36 -0.03 -0.32 Constant -2.72 -4.58 -4.78 -5.28 -5.29 -4.76 -4.63 -4.80 (-0.30) (-0.54) (-0.58) (-0.66) (-0.66) (-0.58) (-0.54) (-0.56) DAX 30 VRPi -0.01 0.19 0.19 0.24 0.22 0.13 0.07 0.10 t (-0.06) (1.10) (1.51) (2.04) (2.58) (1.55) (1.44) (2.25) Adj. R2 -0.77 0.34 0.76 2.40 2.42 0.39 -0.30 0.52 Constant -4.67 -5.50 -6.28 -6.56 -6.51 -6.38 -6.08 -5.86 (-0.82) (-0.99) (-1.19) (-1.25) (-1.23) (-1.20) (-1.12) (-1.09) FTSE 100 VRPi -0.02 0.05 0.13 0.15 0.13 0.10 0.02 -0.01 t (-0.20) (0.91) (3.02) (3.86) (4.14) (2.22) (0.59) (-0.29) Adj. R2 -0.76 -0.61 0.93 2.55 1.84 0.94 -0.72 -0.82 Constant -6.00 -6.75 -8.63 -8.79 -7.97 -7.57 -6.10 -5.67 (-0.80) (-0.90) (-1.15) (-1.20) (-1.10) (-1.04) (-0.79) (-0.73) Nikkei 225 VRPi -0.01 0.03 0.14 0.16 0.10 0.08 0.00 0.01 t (-0.09) (0.27) (1.46) (1.80) (1.22) (0.91) (0.04) (0.36) Adj. R2 -0.77 -0.75 0.29 0.95 0.11 -0.19 -0.83 -0.81 Constant -2.39 -3.07 -3.24 -3.81 -3.91 -3.87 -3.77 -3.76 (-0.36) (-0.47) (-0.51) (-0.62) (-0.64) (-0.63) (-0.59) (-0.57) SMI VRPi 0.04 0.15 0.15 0.24 0.22 0.18 0.10 0.09 t (0.37) (1.76) (1.49) (2.37) (3.24) (2.88) (2.43) (3.42) Adj. R2 -0.71 0.40 0.73 3.55 3.53 2.59 0.71 0.78 Constant -6.93 -6.88 -7.16 -7.09 -6.65 -6.08 -5.31 -5.02 (-1.30) (-1.29) (-1.38) (-1.34) (-1.23) (-1.11) (-0.95) (-0.94) S&P 500 VRPi 0.42 0.40 0.39 0.36 0.28 0.18 0.04 0.00 t (5.11) (5.29) (8.43) (8.80) (6.52) (3.83) (0.90) (0.13) Adj. R2 5.40 8.72 13.13 14.18 9.40 4.06 -0.54 -0.84 32

Table 5 “Global” Variance Risk Premium Regressions The results are based on the monthly regression in equation (3). tNW-statistics are reported in parentheses. The sample period extends from January 2000 to December 2010. Index Monthly Horizon 1 2 3 4 5 6 9 12 Constant -8.51 -10.00 -9.96 -10.09 -9.79 -9.16 -8.58 -8.28 (-1.15) (-1.39) (-1.43) (-1.44) (-1.39) (-1.29) (-1.15) (-1.09) CAC 40 VRPglobal 0.24 0.35 0.33 0.33 0.26 0.16 0.04 0.01 t (2.44) (4.52) (6.21) (8.21) (6.95) (3.58) (0.96) (0.22) Adj. R2 0.58 4.22 5.99 7.43 5.27 1.87 -0.59 -0.83 Constant -4.72 -6.44 -6.66 -6.96 -6.55 -5.55 -4.56 -4.39 (-0.54) (-0.75) (-0.81) (-0.85) (-0.79) (-0.67) (-0.53) (-0.51) DAX 30 VRPglobal 0.24 0.37 0.38 0.39 0.33 0.20 0.05 0.02 t (2.43) (2.61) (6.26) (7.90) (4.07) (2.32) (0.72) (0.31) Adj. R2 0.10 3.20 5.10 7.21 5.85 1.91 -0.63 -0.82 Constant -5.87 -6.52 -7.00 -7.03 -6.89 -6.58 -6.08 -5.83 (-1.09) (-1.23) (-1.38) (-1.38) (-1.33) (-1.26) (-1.13) (-1.09) FTSE 100 VRPglobal 0.13 0.18 0.22 0.22 0.18 0.13 0.02 -0.01 t (1.78) (1.85) (3.44) (5.07) (3.79) (2.21) (0.56) (-0.38) Adj. R2 -0.16 1.40 4.23 5.55 4.30 2.02 -0.70 -0.78 Constant -7.31 -7.70 -8.72 -8.66 -8.08 -7.49 -6.00 -5.45 (-0.94) (-1.03) (-1.19) (-1.20) (-1.12) (-1.04) (-0.79) (-0.71) Nikkei 225 VRPglobal 0.15 0.16 0.24 0.25 0.19 0.12 -0.01 -0.01 t (1.13) (2.51) (4.44) (5.05) (3.15) (1.67) (-0.17) (-0.19) Adj. R2 -0.36 0.11 1.97 2.81 1.77 0.47 -0.81 -0.84 Constant -3.58 -4.53 -4.76 -5.08 -4.89 -4.50 -3.65 -3.45 (-0.55) (-0.71) (-0.77) (-0.82) (-0.79) (-0.73) (-0.57) (-0.53) SMI VRPglobal 0.17 0.28 0.29 0.32 0.27 0.20 0.05 0.02 t (1.51) (5.78) (5.32) (7.70) (8.39) (5.62) (1.52) (0.59) Adj. R2 0.40 4.10 6.10 8.92 7.54 4.47 -0.34 -0.76 Constant -7.03 -6.95 -7.41 -7.40 -6.88 -6.29 -5.49 -5.20 (-1.32) (-1.32) (-1.46) (-1.43) (-1.30) (-1.17) (-1.00) (-0.98) S&P 500 VRPglobal 0.41 0.38 0.40 0.38 0.29 0.20 0.06 0.03 t (4.98) (5.01) (6.93) (7.88) (5.70) (3.40) (1.24) (0.68) Adj. R2 4.43 7.05 12.04 13.69 9.04 4.31 -0.16 -0.62 33

Table 6 Panel Regressions The results are based on on the monthly “global” and county-specific panel regressions in equations (4) and (5), respectively. NW-based t-statistics are reported in parentheses. The sample period extends from January 2000 to December 2010. “Global” Regressors Horizon (mos) 1 2 3 4 5 6 9 12 Constant -6.17 -7.02 -7.42 -7.54 -7.18 -6.60 -5.73 -5.44 (-2.15) (-2.51) (-2.74) (-2.78) (-2.64) (-2.42) (-2.12) (-2.07) VRPglobal 0.22 0.29 0.31 0.31 0.25 0.17 0.04 0.01 t (4.66) (6.60) (9.70) (11.21) (9.43) (6.04) (1.41) (0.41) Adj. R2 1.08 3.43 5.85 7.46 5.72 2.81 0.04 -0.12 Constant -14.90 -10.73 -2.25 4.05 4.41 2.03 -8.12 -13.83 (-0.63) (-0.47) (-0.11) (0.20) (0.23) (0.11) (-0.46) (-0.83) VRPglobal 0.22 0.28 0.32 0.32 0.26 0.18 0.03 0.00 t (4.45) (5.34) (9.34) (10.81) (8.17) (5.28) (1.18) (0.03) log(P /E )global 2.70 1.14 -1.60 -3.57 -3.57 -2.66 0.73 2.59 t t (0.37) (0.16) (-0.25) (-0.59) (-0.62) (-0.47) (0.14) (0.53) Adj. R2 0.99 3.32 5.75 7.51 5.80 2.81 -0.08 -0.06 Country-Specific Regressors Horizon (mos) 1 2 3 4 5 6 9 12 Constant -5.23 -6.00 -6.43 -6.68 -6.50 -6.18 -5.79 -5.64 (-1.80) (-2.12) (-2.35) (-2.47) (-2.39) (-2.27) (-2.13) (-2.12) VRPi 0.12 0.18 0.20 0.22 0.18 0.12 0.05 0.04 t (1.85) (3.31) (4.86) (5.95) (5.82) (4.42) (2.00) (1.68) Adj. R2 0.27 1.40 2.82 4.30 3.27 1.73 0.22 0.19 Constant 11.57 11.54 11.83 11.92 10.79 9.20 5.48 2.19 (0.96) (0.96) (1.00) (0.99) (0.89) (0.75) (0.45) (0.19) VRPi 0.13 0.19 0.22 0.24 0.20 0.14 0.06 0.05 t (2.07) (3.52) (5.33) (6.38) (6.09) (4.59) (2.30) (1.94) log(Pi/Ei) -5.55 -5.80 -6.03 -6.13 -5.70 -5.06 -3.70 -2.57 t t (-1.46) (-1.50) (-1.59) (-1.59) (-1.47) (-1.29) (-0.96) (-0.69) Adj. R2 0.51 1.99 3.79 5.59 4.58 2.90 1.04 0.63 34

Implied Correlations −0.5 −0.6 −0.7 −0.8 −0.9 96 97 98 99 00 01 02 03 04 05 06 07 08 10 8 6 4 2 0 −2 −4 −6 −8 −10 −10 −8 −6 −4 −2 0 2 4 6 8 10 cη 1 ηc 2 Standardized Errors Standardized Error, cη 1 400 200 0 −10 −5 0 5 Standardized Error, cη 2 400 200 0 −5 0 5 10 Figure 1 Estimated VAR-GARCH-DCC Model The first panel plots the daily conditional correlations between the returns and the variance risk premium implied by the estimated VAR(1)-GARCH(1,1)-DCC model described in the main text. The lower left and right two panels provide a scatterplot and histograms, respectively, for the standardized residuals from the estimated model, cη . The daily sample used in estimating the model spans the period from February 1, t 1996 to December 31, 2007, for a total of 2,954 daily observations. b 35

Daily Daily 5 100 4 80 3 60 2 40 1 20 0 1 20 40 60 80 120 180 240 1 20 40 60 80 120 180 240 Weekly Weekly 5 100 4 80 3 60 2 40 1 20 0 1 4 8 12 16 20 24 28 32 36 40 44 48 52 1 4 8 12 16 20 24 28 32 36 40 44 48 52 Monthly Monthly 5 100 4 80 3 60 2 40 1 20 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Figure 2 Simulated Size and Power The upper left panel reports the 95-percentiles in the finite-sample distributions of the tNW (dash line) and tHD (solid line) based on simulated “daily” data from the restricted VAR-GARCH-DCC model under the null of no predictability. The dashed and solid star lines refer to the corresponding t-statistics for actual daily U.S. S&P 500 returns spanning February 1, 1996 to December 31, 2007. The middle and bottom two left panels give the results for the simulated “weekly” and “monthly” data, together with the results based ontheactualweeklyandmonthlyS&P500returns. Therightthreepanelsgivesimulated“daily,” “weekly” and“monthly”percentagepowerbasedontheunrestrictedVAR-GARCH-DCCmodelandthesize-adjusted 5-percent tNW (dashed line) and tHD (solid line) statistics. 36

2 R Under Null 9 R2j 8 Qt95 7 Qt10 6 Qt25 5 Qt50 4 Qt75 Qt90 3 2 1 0 −1 1 10 20 40 60 80 120 180 240 2 R Under Alternative 40 R2j 35 Qt10 30 Qt25 Qt50 25 Qt75 20 Qt90 Mode 15 10 5 0 1 10 20 40 60 80 120 180 240 Figure 3 Simulated R2 The top panel in the figure plots the quantiles in the finite-sample distribution of the R2 from the return regression in equation (1) and simulated “daily” date from the restricted VAR-GARCH-DCC model under the null of no predictability. The star dashed line refer to the corresponding R2’s in actual daily U.S. S&P 500 returns spanning February 1, 1996 to December 31, 2007. The bottom panel reports the quantiles in the simulated finite-sample distribution based on the unrestricted VAR-GARCH-DCC model. 37

2 2 Implied R and b Implied R and b 1 2 5 5 4 4 3 3 2 2 1 1 0 0 1 20 60 100 140 180 220 250 1 20 60 100 140 180 220 250 2 2 Implied R and c Implied R and c 1 2 5 5 4 4 3 3 2 2 1 1 0 0 1 20 60 100 140 180 220 250 1 20 60 100 140 180 220 250 Figure 4 Implied R2 The solid lines in each of the four panels show the R2(h)’s implied by the formula in Section 2.3 in the main text and the estimated unrestricted VAR-GARCH-DCC model. The dashed lines in each of the four panels show the implied R2(h)’s for a 10-percent decrease in the values of the b1, b2, c1, and c2 VAR coefficients, respectively. 38

CAC 40 Nikkei 225 200 200 100 100 0 0 −100 −100 −200 −200 −300 −300 −400 −400 DAX SMI 100 100 50 50 0 0 −50 −50 −100 −100 −150 −150 −200 −200 −250 FTSE 100 S&P 500 100 100 0 0 −100 −100 −200 −200 −300 −300 2000 2002 2004 2006 2008 2010 2000 2002 2004 2006 2008 2010 Figure 5 Variance Risk Premia ThefigureshowsthemonthlyvarianceriskpremiaVRPi forFrance(CAC40),Japan(Nikkei225),Germany t (DAX 30), Switzerland (SMI 20), the U.K. (FTSE 100), and the U.S. (S&P 500). The risk premia are constructed by subtracting the actual realized variation from the model-free options implied variation. The sample period spans January 2000 to December 2010. 39

CAC 40 Nikkei 225 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0 0.1 −0.1 0 −0.2 −0.1 −0.3 DAX SMI 0.6 0.5 0.4 0.4 0.3 0.2 0.2 0.1 0 0 −0.2 −0.1 −0.4 −0.2 FTSE 100 S&P 500 0.3 0.6 0.5 0.2 0.4 0.1 0.3 0.2 0 0.1 −0.1 0 −0.2 −0.1 2 4 6 8 10 12 2 4 6 8 10 12 Horizon (months) Horizon (months) Figure 6 Country Specific Regression Coefficients The figure shows the estimated regression coefficients for VRPi for each of the country specific return t regressions reported in Table 4, together with two NW-based standard error bands. The regressions are based on monthly data from January 2000 to December 2010. 40

CAC 40 Nikkei 225 5 1 4 0.5 3 2 0 1 −0.5 0 −1 −1 DAX SMI 2.5 4 2 3 1.5 2 1 0.5 1 0 0 −0.5 −1 −1 FTSE 100 S&P 500 3 15 2 10 1 5 0 0 −1 −5 2 4 6 8 10 12 Horizon (months) Horizon (months) Figure 7 Country Specific Regression R2’s The figure shows the adjusted R2(h)’s for the country specific return regressions reported in Table 4. The regressions are based on monthly data from January 2000 to December 2010. 41

1 0.9 0.8 0.7 CAC 40 0.6 DAX 30 FTSE 100 0.5 Nikkei 225 SMI 20 0.4 S&P 500 0.3 0.2 0.1 0 2000 2002 2004 2006 2008 2010 Figure 8 Market Capitalization ThefigureshowstherelativemarketcapitalizationbyaggregateindexforFrance(CAC40),Germany(DAX 30), the U.K. (FTSE 100), Japan (Nikkei 225), Switzerland (SMI 20), and the U.S. (S&P 500). 42

CAC 40 Nikkei 225 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −0.1 −0.1 −0.2 DAX SMI 0.8 0.6 0.5 0.6 0.4 0.4 0.3 0.2 0.2 0.1 0 0 −0.2 −0.1 FTSE 100 S&P 500 0.4 0.6 0.5 0.3 0.4 0.2 0.3 0.2 0.1 0.1 0 0 −0.1 −0.1 2 4 6 8 10 12 2 4 6 8 10 12 Horizon (months) Horizon (months) Figure 9 “Global” VRP Regression Coefficients The figure shows the coefficient estimates for VRPglobal from the return regressions reported in Table 5, t togetherwithtwoNW-basedstandarderrorbands. TheregressionsarebasedonmonthlydatafromJanuary 2000 to December 2010. 43

CAC 40 Nikkei 225 8 3 6 2 4 1 2 0 0 −2 −1 DAX SMI 8 10 8 6 6 4 4 2 2 0 0 −2 −2 FTSE 100 S&P 500 6 15 5 10 4 3 5 2 1 0 0 −1 −5 2 4 6 8 10 12 2 4 6 8 10 12 Horizon (months) Horizon (months) Figure 10 “Global” VRP Regression R2’s ThefigureshowstheadjustedR2(h)’sfromregressingtheindividualcountryreturnsonVRPglobal reported t in Table 5. The regressions are based on monthly data from January 2000 to December 2010. 44

Country−Specific Regressors Global Regressors 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 −0.05 −0.05 −0.1 −0.1 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 −1 −1 2 4 6 8 10 12 2 4 6 8 10 12 Horizon (months) Horizon (months) Figure 11 Panel Regression Coefficients and R2’s The top two panels show the estimated panel regression coefficients from regressing the returns on the individual country variance risk premia VRPi and the “global” variance risk premium VRPglobal, respectively, t t reported in Table 6, together with two NW-based standard error bands. The bottom two panels show the R2(h)’sfromthesame two panel regressions. Theregressionsarebasedonmonthly data fromJanuary 2000 through December 2010. 45

CAC 40 Nikkei 225 8 3 6 2 4 1 2 0 0 −1 −2 −4 −2 DAX SMI 8 10 8 6 6 4 4 2 2 0 0 −2 −2 FTSE 100 S&P 500 5 15 4 10 3 2 5 1 0 0 −1 −2 −5 2 4 6 8 10 12 2 4 6 8 10 12 Horizon (months) Horizon (months) Figure 12 “Global” VRP Panel Regression R2’s The figure shows the adjusted R2(h)’s implied by the VRPglobal panel regressions reported in the top panel t in Table 5. The regressions are based on monthly data from January 2000 to December 2010. 46

U.S. variance 7 6 5 4 3 2 1 0 U.S.−U.K. covariance 4 3 2 1 0 −1 U.S.−U.K. correlatoin 1 0.8 0.6 0.4 0.2 0 −0.2 1951 1960 1970 1980 1990 2000 2009 Figure 13 Consumption Growth Variances, Covariances, and Correlations The figure shows exponentially weighted moving average estimates for U.S. consumption growth variances, andcovariancesandcorrelationswithU.K.consumptiongrowth. Theestimatesarebasedonannualtotalreal consumption expenditures from 1951 to 2009, and a exponential smoothing parameter of λ=1 (1 0.06)4. − − The variances and covariances are both scaled by a factor of 104. 47

U.S. U.K. 0.5 0.5 0.45 0.45 0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 0 3 3 2 2 1 1 2 4 6 8 10 12 2 4 6 8 10 12 Horizon (months) Horizon (months) Figure 14 Equilibrium VRP Regression Coefficients and R2’s The figure shows the implications from the calibrated stylized two-country general equilibrium model. The upper two panels plot the slope coefficients in the country specific regressions for each of the two countries based on the “local” VRP’s (dashed lines) and “global” VRP (solid lines). The lower two panels show the impliedpanelregressionR2’sbasedonthe“local”(dashedline)and“global”(solidline)VRPs,respectively. 48