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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 933, June 2008 --- Screen Reader Version*

Predicting Global Stock Returns^*

Erik Hjalmarsson^†

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

I test for stock return predictability in the largest and most comprehensive data set analyzed so far, using four common forecasting variables: the dividend- and earnings-price ratios, the short interest rate, and the term spread. The data contain over 20,000 monthly observations from 40 international markets, including 24 developed and 16 emerging economies. In addition, I develop new methods for predictive regressions with panel data. Inference based on the standard fixed effects estimator is shown to suffer from severe size distortions in the typical stock return regression, and an alternative robust estimator is proposed. The empirical results indicate that the short interest rate and the term spread are fairly robust predictors of stock returns in developed markets. In contrast, no strong or consistent evidence of predictability is found when considering the earnings- and dividend-price ratios as predictors.

Keywords: Cross-sectional dependence, panel data, pooled regression, predictive regression, stock return predictability

JEL classification: C22, C23, G12, G15

1  Introduction

Our empirical knowledge regarding the predictability of stock returns by variables such as the dividend-price ratio has been subject to constant updating over time. Early work by Fama and French (1988, 1989) and Campbell and Shiller (1988) concluded that there is generally strong evidence of predictability. Recent studies that use more robust econometric methods, such as Campbell and Yogo (2006) and Lewellen (2004), still find evidence of predictability, but their results are much less conclusive than the earlier studies.

Despite the mixed evidence and uncertainty regarding stock return predictability, there have been surprisingly few attempts at furthering our understanding by using data other than that of the U.S. stock-market. Since the predictable component of stock returns must be small, if indeed one does exist, there seems to be little chance of reaching a decisive conclusion using U.S. data alone, which effectively provides only one time-series at the market level. There has, of course, been some analysis of predictability in international stock returns, but many of the results are based on relatively small data sets and non-robust econometric methods.¹ In addition, most international results are based only on individual time-series regressions and very little analysis has been conducted with pooled panel data regressions. Yet, it is well known that pooling the data may lead to more powerful methods, which is particularly relevant when studying stock return predictability since any predictable component will always be small relative to the overall variance in the returns process.

The aim of this paper is twofold. First, by considering a large global data set, I provide the most comprehensive picture of stock return predictability to date. The data contain over 20,000 monthly observations from 40 countries, including markets in 24 developed economies.² The longest data series is for the U.K. stock-market and dates back to 1836 while data for eight other markets date back to before 1935. Second, I develop and apply new results for pooled forecasting regressions, utilizing the panel structure of the data.

Since an international data set of stock returns and forecasting variables provides a panel, the theory part of this paper analyzes econometric inference in predictive regressions in a panel data setting, when the regressors are nearly persistent and endogenous.³ As is well known (Stambaugh (1999)), OLS inference in the corresponding time-series predictive regressions is generally biased and various bias and size correction procedures have been proposed.

In the panel case, it turns out that the pooled estimator is unbiased as long as no fixed effects are included. The intuition behind this is that when pooling the data, independent cross-sectional information dilutes the endogeneity effects that cause the Stambaugh bias in the time-series case. That is, the Stambaugh bias only arises when the predictors are both persistent and endogenous; by pooling the data, the endogeneity is, in a sense, removed, and hence also the bias. Furthermore, the standard pooled estimator has an asymptotically normal distribution and normal inference can therefore be performed.

The intuition just described for the standard pooled estimator no longer holds when fixed effects are allowed for, and the asymptotic properties of the pooled estimator with fixed effects are very different from those of the pooled estimator with a common intercept. The time-series demeaning of the data, which is implicit in a fixed effects estimation, causes the fixed effects estimator to suffer from a second order bias that invalidates inference from standard test statistics. When demeaning each time-series in the panel, information after time t is used to form the time t regressor, and information before time t is used to form the time t returns. This induces a correlation between the lagged value of the demeaned regressor and the error term in the forecasting equation, which gives rise to the second order bias in the fixed effects estimator. Thus, in contrast to the case with a common intercept, the regressors no longer act as if they were exogenous. To correct for this bias, I develop an estimator based on the idea of recursive demeaning (e.g. Moon and Phillips (2000), and Sul et al. (2005)). By using information only after time $t$ in the demeaning of the returns and the non-demeaned regressor as an instrument, the distortive effects arising from standard demeaning are eliminated.

The overall conclusion from the theoretical results and the supporting Monte Carlo simulations is that, in the typical panel data case with fixed effects, persistent and endogenous regressors will cause standard inference to be biased. While this result is well established in the time-series case (e.g. Stambaugh (1999)), the results in this paper show that equal caution is required when working with panel data.

In the empirical analysis, I conduct time-series regressions for individual countries as well as pooled regressions. In both types of analyses, I estimate regressions for four of the most commonly used forecasting variables: the dividend- and earnings-price ratios, the short interest rate, and the term spread. In the pooled regressions, countries are either all grouped together in a global panel or split up into groups of developed and emerging markets.

The results indicate that the short interest rate and the term spread are both fairly robust predictors of (excess) stock returns in developed markets. The null of no predictability is clearly rejected in the pooled regressions for developed markets as well as in a number of individual time-series regressions. These results are generally in line with those found by Campbell and Yogo (2006) with U.S. data and with the limited international results of Ang and Bekaert (2007). In contrast to the interest rate variables, no strong or consistent evidence of predictability is found when considering the earnings- and dividend-price ratios as predictors. In particular, neither predictor yields any consistent predictive power for the developed markets and, as seen in plots of the regression coefficient over time, this is especially true for the dividend-price ratio.

The rest of the paper is organized as follows. Sections II and III describe the empirical model and derive the main asymptotic properties of the pooled estimators. The finite sample properties of the procedures developed in this paper are analyzed through Monte Carlo experiments in Section IV. The data are described in Section V and the empirical results, including out-of-sample exercises, are provided in Section VI. Section VII concludes and technical assumptions and proofs are found in the Appendix.

II  Pooled Estimation in Predictive Regressions

II.A  Model and Assumptions

Consider a panel model with dependent variables $y_{i,t}$, $i=1,...,n$, $t=1,...,T$, and the corresponding vector of regressors, $x_{i,t}$, where $x_{i,t}$ is an $m\times1$ vector. In this paper, $y_{i,t}$ is the stock return in country $i$, and $x_{i,t}$ are the corresponding predictor variables. The behavior of $y_{i,t}$ and $x_{i,t}$ are modelled as follows:

$$y_{i,t} =\alpha_{i}+\beta_{i}^{\prime}x_{i,t-1}+\gamma_{i}^{\prime} f_{t}+u_{i,t},$$ (1)

$$x_{i,t} =x_{i,t}^{0}+\Gamma_{i}^{\prime}z_{t},$$ (2)

$$x_{i,t}^{0} =A_{i}x_{i,t-1}^{0}+v_{i,t},$$ (3)

$$z_{t} =A_{g}z_{t-1}+g_{t}.$$ (4)

That is, stock returns $y_{i,t}$ are a function of the past values of the predictor variables plus two factors representing country specific $\left( u_{i,t}\right)$ and global $\left( f_{t}\right)$ innovations. In the typical time-series predictive regression using, for instance, aggregate U.S. data, these two error terms are generally not distinguishable, and in terms of econometric inference, it makes no difference whether the shocks are U.S. specific or global in some sense. However, when pooling data from several countries, it becomes important to control for whether innovations to returns are due to country specific shocks or shocks that are common to all countries in the sample. Intuitively, if one ignores the presence of common factors in the error terms, the total amount of (independent) variation in the pooled data is overstated, and the econometric inference will be biased.

The vector of predictor variables, $x_{i,t}$, is also assumed to be the sum of country specific $\left( x_{i,t}^{0}\right)$ and global $\left( z_{t}\right)$ terms. Both $x_{i,t}^{0}$ and $z_{t}$ follow $AR\left( 1\right)$ processes. More precisely, the auto-regressive roots of both of these processes are parameterized as being local-to-unity, such that $A_{i}=I+C_{i}/T$ and $A_{g}=I+C_{g}/T$, where both $A_{i}$ and $A_{g}$ are $m\times m$ matrices. This captures the near unit-root, or highly persistent, behavior of many predictor variables, but is less restrictive than a pure unit-root assumption. The near unit-root construction, where the autoregressive root drifts closer to unity as the sample size increases, is used as a tool to enable an asymptotic analysis where the persistence in the data remains large relative to the sample size, even when the sample size increases to infinity. That is, if the auto-regressive roots are treated as fixed and strictly less than unity, then as the sample size grows, the regressors will behave as strictly stationary processes asymptotically, and the standard first order asymptotic results will not provide a good guide to the actual small sample properties of the model. If the roots are exactly equal to unity, the usual unit-root asymptotics apply to the model, but this is clearly a restrictive assumption for most potential predictor variables. Instead, by using the near unit-root construction, the effects from the high persistence in the regressor will appear also in the asymptotic results, but without imposing the strict assumption of a unit root.

Finally, the regressors $x_{i,t}$ can be endogenous in the sense that $u_{i,t}$ and $v_{i,t}$ are contemporaneously correlated; $f_{t}$ and $g_{t}$ may be contemporaneously correlated as well, and can, in fact, be identical. The model specification is completed in Appendix A with some additional formal assumptions. Unless otherwise noted, all variables appearing in the asymptotic distributions derived below are defined in Appendix A.

II.B  Motivations for Pooling

II.B.1  Practical and Econometric Considerations

The theoretical part of this paper analyzes the pooled estimation of the slope coefficient in equation (1). That is, by pooling data from several countries, an estimate of a joint slope coefficient $\beta$ is obtained. If the individual slope coefficients are all identical, such that $\beta _{i}=\beta$ for all $i=1,...,n$, the pooled estimator will converge to this common parameter. In addition, the pooled estimator can either impose a common intercept $\alpha$, or allow for individual intercepts, or fixed effects, $\alpha_{i}$. When the restrictions $\beta_{i}=\beta$, and potentially $\alpha_{i}=\alpha$, hold for all $i$, pooling the data should lead to more precise estimates than time-series estimation of each individual $\beta_{i}$.

When the slope coefficients $\beta_{i}$ are not all identical, pooled estimation may still be useful. In this case, the pooled estimator will converge to a well-defined average slope coefficient. The pooled estimate, and related tests, thus makes a statement about the average predictive relationship in the panel, which provides a useful tool for interpreting and understanding the empirical results, especially if the individual time-series regressions deliver mixed results. Furthermore, and as importantly, the pooled estimate may in some respects provide at least as good an estimate of $\beta_{i}$ for a given $i$, by providing a possibly less noisy estimate than the time-series one. That is, if the $\beta_{i}s$ are not identical, the pooled estimator will generally not provide an unbiased estimate for a given $\beta_{i}$, but in a bias-variance trade-off it may still dominate the time-series estimate of $\beta_{i}$. This bias-variance trade-off is illustrated by out-of-sample forecasts at the end of this paper where it is shown that the forecasts based on the pooled estimator often dominate those based on the time-series estimates.

II.B.2  Economic Rationale

Is it likely, from the perspective of economic theory, that the $\alpha_{i}s$ and $\beta_{i}s$ are identical across $i$? That is, can one justify pooling the data from an economic perspective, and if so, should fixed effects be included?

Consider first the question of fixed effects. Under the null of no predictability, such that $\beta_{i}=0$ for all $i$, the restriction of $\alpha_{i}=\alpha$ for all $i$ imposes the same expected excess return in all countries. Although it is very difficult to obtain precise estimates of average returns, detailed empirical studies such as Jorion and Goetzman (1999) strongly suggest that the equity premium varies across countries. In addition, if an international world CAPM applies, identical $\alpha_{i}s$ in the absence of predictability imply that the world CAPM beta for each country is identical. The restriction of identical CAPM betas is strongly rejected in previous studies such as Ferson and Harvey (1994), who report international CAPM betas in the range from $0.4$ to 1.3, and Harvey (1995), who shows that the world CAPM betas for some emerging markets are negative. Although the world CAPM does not offer a complete model of stock returns, it does capture a sizeable amount of the variation in international stock returns (Ferson and Harvey, 1994). Model predictions that strongly contradict it, such as identical CAPM betas for all countries, should thus be seen as a warning sign of misspecification. Therefore, given the importance of having a model that is correctly specified under the null hypothesis, fixed effects should generally be included.⁴

In order to understand the economic constraints that are imposed by identical $\beta_{i}s$, one needs to analyze a model that implies predictability in stock returns. Menzly et al. (2004) explicitly analyze cross-sectional differences in time-series return predictability. They use an external habit model similar to Campbell and Cochrane (1999) and show that the dividend-price ratio predicts excess stock returns. The slope coefficient in this predictive regression varies across assets as a function of the properties of the assets' cash-flow share of overall income; in an international asset pricing framework, with integrated markets, each country portfolio can be viewed as an individual asset, as in the international CAPM. The model in Menzly et al. (2004) thus implies that, in general, the slope coefficients $\beta_{i}$ in the predictive regression in equation (1) may not be identical across $i$. However, the model says little about how disperse the slope coefficients actually are in practice. That is, even though it is unlikely that the $\beta_{i}s$ are identical across countries or assets, as is true for most parameters that may be estimated in economics or finance, what is of primary importance for the empirical scope of this paper is whether they are similar enough that it may be beneficial from an econometric point of view to treat them as equal.⁵ From this empirical perspective, the implications of the Menzly et al. (2004) model are essentially silent, and it is unlikely that other models of return predictability would deliver any stronger practical implications.

The results in the current paper suggest that pooling the data, and thus imposing a common slope coefficient, is in fact quite often empirically justified in the sense that the null hypothesis of a common slope coefficient can often not be rejected in formal statistical tests and the forecasts based on the pooled estimates often tend to outperform those based on the individual time-series estimates in out-of-sample exercises. Thus, even though economic theory does not generally predict that the $\beta_{i}s$ are all identical, it cannot be a priori rejected that they are similar enough for there to be benefits from pooling the data, which ties back to the discussion on the practical motivations in the section above.

In general, it is quite reasonable to conjecture that countries that share many common characteristics are more likely to have similar predictability patterns than those that do not. One of the most natural splits along these lines in international data is to distinguish between developed and emerging markets. Previous literature, such as Harvey (1995), also shows that emerging markets tend to have different return characteristics than developed markets, and different patterns of predictability. To the extent that stock markets in different countries are more likely to have similar predictability if they are priced globally in integrated financial markets, rather than locally in segregated markets, the group of developed markets is also likely to better satisfy this requirement. The empirical analysis separately analyzes developed and emerging market panels, and includes a test of slope homogeneity that shows that these two groups of countries appear more homogenous than all countries combined.

II.C  Pooled Estimation

To understand the basic properties of pooled estimators of a common slope coefficient $\beta$ in equation (1), it is instructive to start with analyzing the case when there are no common factors in the data. That is, let $\gamma_{i}\equiv0$ and $\Gamma_{i}\equiv0$, for all $i$. This assumption will be maintained throughout the remainder of Section II and the effects of common factors are analyzed in Section III. Unless otherwise noted, it is assumed that the slope coefficients $\beta_{i}$ are identical and equal to $\beta$ for all $i$.

II.C.1  The Standard Pooled Estimator without Fixed Effects

To estimate the parameter $\beta$, consider first the traditional pooled estimator when there are no individual effects, i.e. when $\alpha_{i} \equiv\alpha$ for all $i$. Although the previous discussion strongly suggested the use of individual intercepts in the international analysis performed in the current paper, there may be other cases when a common intercept can be justified. In addition, a comparison of the pooled estimator with and without fixed effects highlights some important differences and helps form an understanding of the effects of pooling the data. The pooled estimator with a common intercept is given by

$$\hat{\beta}_{Pool}=\left( \sum_{i=1}^{n}\sum_{t=1}^{T}\tilde{x}_{i,t-1} \tilde{x}_{i,t-1}^{\prime}\right) ^{-1}\left( \sum_{i=1}^{n}\sum_{t=1} ^{T}\tilde{y}_{i,t}\tilde{x}_{i,t-1}\right) ,$$ (5)

where

$$\tilde{y}_{i,t}=y_{i,t}-\frac{1}{nT}\sum_{i=1}^{n}\sum_{t=1}^{T}y_{i,t}, \tilde{x}_{i,t}=x_{i,t}-\frac{1}{nT}\sum_{i=1}^{n}\sum_{t=1}^{T}x_{i,t}.$$ (6)

Following the work of Phillips and Moon (1999), asymptotic results for the panel estimators are derived using sequential limits, which implies first keeping the cross-sectional dimension, $n$, fixed and letting the time-series dimension, $T$, go to infinity, and then letting $n$ go to infinity. Such sequential convergence is denoted $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$.⁶ As mentioned before, the definitions of the variables that appear in the theorems and derivations below are all found in Appendix A, unless otherwise noted.

Theorem 1:  With $\gamma_{i}\equiv0$, $\Gamma_{i}\equiv0$, $\alpha _{i}\equiv\alpha$, and $\beta_{i}\equiv\beta$ for all $i$, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}},$

$$\sqrt{n}T\left( \hat{\beta}_{Pool}-\beta\right) \Rightarrow N\left( 0,\Omega_{xx}^{-1}\Phi_{ux}\Omega_{xx}^{-1}\right) .$$ (7)

The pooled estimator of $\beta$ is thus asymptotically normally distributed; summing up over the cross-section eliminates the usual near unit-root asymptotic distributions found in the time-series case. The rate of convergence is also faster in the pooled case $\left( \sqrt{n}T\right)$ compared to the time-series case $\left( T\right)$, which again is a result of the additional cross-sectional information. The limiting distribution depends on $\Omega_{xx}$ and $\Phi_{ux}$ and, in order to perform inference, estimates of these parameters are required. Let $\hat{u}_{i,t}=\tilde{y} _{i,t}-\hat{\beta}_{Pool}\tilde{x}_{i,t-1}$ , $\hat{\Phi}_{ux}=\frac{1}{n} \sum_{i=1}^{n}\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{s=1}^{T}\left( \hat{u} _{i,t}\tilde{x}_{i,t-1}\right) \left( \hat{u}_{i,s}\tilde{x}_{i,s-1}\right) ^{\prime}$ , and $\hat{\Omega}_{xx}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{T^{2} }\sum_{t=1}^{T}\tilde{x}_{i,t-1}\tilde{x}_{i,t-1}^{\prime}$ . The estimator $\hat{\Phi}_{ux}$ is thus the panel equivalent of HAC (heteroskedasticty and auto-correlation consistent) estimators for long-run variances.

Standard tests can now be performed. For instance, the null hypothesis $\beta_{\left( k\right) }=\beta_{\left( k\right) }^{0}$, for some $k=1,...,m$, where $\beta=\left( \beta_{\left( 1\right) },...,\beta _{\left( m\right) }\right) ^{\prime}$ , can be tested using a $t-$test. Let $\hat{\Sigma}=\hat{\Omega}_{xx}^{-1}\hat{\Phi}_{ux}\hat{\Omega}_{xx}^{-1}$ . Using the results derived above, it follows easily that under the null-hypothesis,

$$t_{k}=\frac{\hat{\beta}_{\left( k\right) ,Pool}-\beta_{\left( k\right) }^{0}}{\sqrt{\left. a^{\prime}\hat{\Sigma}a\right/ \left( nT^{2}\right) } }\Rightarrow N\left( 0,1\right) ,$$ (8)

as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$, where $a$ is an $m\times1$ vector with the $k$'th component equal to one and zero elsewhere, and $\hat{\beta}_{\left( k\right) ,Pool}$ is the $k$'th component of $\hat{\beta}_{Pool}$. More general linear hypotheses can be evaluated using a Wald test.

II.C.2  Fixed Effects

Let $\underline{y}_{i,t}$ and $\underline{x}_{i,t}$ denote the time-series demeaned data. That is, $\underline{x}_{i,t}=x_{i,t}-\frac{1}{T}\sum_{t=1} ^{T}x_{i,t-1}$ and $\underline{y}_{i,t}=y_{i,t}-\frac{1}{T}\sum_{t=1} ^{T}y_{i,t}$. The fixed effects pooled estimator, which allows for individual intercepts, is then given by

$$\hat{\beta}_{FE}=\left( \sum_{i=1}^{n}\sum_{t=1}^{T}\underline{x} _{i,t-1}\underline{x}_{i,t-1}^{\prime}\right) ^{-1}\left( \sum_{i=1}^{n} \sum_{t=1}^{T}\underline{y}_{i,t}\underline{x}_{i,t-1}\right) ,$$ (9)

and

$$\hat{\beta}_{FE}-\beta=\left( \frac{1}{n}\sum_{i=1}^{n}\frac{1}{T^{2}} \sum_{t=1}^{T}\underline{x}_{i,t-1}\underline{x}_{i,t-1}^{\prime}\right) ^{-1}\left( \frac{1}{n}\sum_{i=1}^{n}\frac{1}{T^{2}}\sum_{t=1}^{T} \underline{u}_{i,t}\underline{x}_{i,t-1}\right) .$$ (10)

Clearly, the estimator is still consistent. Its asymptotic distribution, however, will be affected by the demeaning. For fixed $n$, as $T\rightarrow \infty$,

$$T\left( \hat{\beta}_{FE}-\beta\right) \Rightarrow\left( \frac{1}{n} \sum_{i=1}^{n}\int_{0}^{1}\underline{J}_{i}\underline{J}_{i}^{\prime}\right) ^{-1}\left( \frac{1}{n}\sum_{i=1}^{n}\int_{0}^{1}dB_{1,i}\underline{J} _{i}\right) ,$$ (11)

where $\underline{J}_{i}$ and $dB_{1,i}$ are the limiting processes of $\underline{x}_{i,t}$ and $u_{i,t}$, respectively, as defined in Appendix A; the limiting process for $v_{i,t}$ is denoted $dB_{2,i}$. Let $\omega_{21}=\lim_{n\rightarrow\infty}n^{-1}\sum\omega_{21i}$ denote the average covariance vector between $u_{i,t}$ and $v_{i,t}$, and observe that

$$E\left[ \int_{0}^{1}dB_{1,i}\underline{J}_{i}\right] =E\left[ \int _{0}^{1}dB_{1,i}\left( r\right) J_{i}\left( r\right) -\int_{0}^{1} dB_{1,i}\left( s\right) \int_{0}^{1}J_{i}\left( r\right) dr\right]$$

$$=-\int_{0}^{1}\int_{0}^{1}\int_{0}^{r}E\left[ e^{\left( r-s\right) C_{i} }\right] E\left[ dB_{1,i}\left( s\right) dB_{2,i}\left( q\right) \right] dsdr$$

$$=-\left( \int_{0}^{1}\int_{0}^{r}E\left[ e^{\left( r-s\right) C_{i} }\right] dsdr\right) \omega_{21},$$ (12)

which is different from zero whenever $\omega_{21}\neq0$. Thus, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$,

$$T\left( \hat{\beta}_{FE}-\beta\right) \rightarrow_{p}-\underline{\Omega }_{xx}^{-1}\left( \int_{0}^{1}\int_{0}^{r}E\left[ e^{\left( r-s\right) C_{i}}\right] dsdr\right) \omega_{21},$$ (13)

and the estimator suffers from a second order bias from the demeaning process.⁷

The differences in sample properties between the standard pooled estimator with a common intercept and the fixed effects estimator are rather striking. Mechanically, the standard pooled estimator works well because for each $i$, the terms in the numerator of the estimator have mean zero and are independently distributed across $i$. As they are summed up over $n$, the central limit theorem applies and an asymptotically normally distributed estimator is obtained. More intuitively, when pooling the data, independent cross-sectional information dilutes the endogeneity effects that cause the Stambaugh (1999) bias in the time-series case. The same result does not hold for the fixed effects estimator because the numerator terms no longer have a zero mean as a consequence of the time-series demeaning of the data, which leads to a correlation between the innovation processes $u_{i,t}$ and the demeaned regressors $\underline{x}_{i,t-1}$ whenever the regressor is endogenous. Thus, unlike in the case with a common intercept, the pooling does not remove the endogeneity effects and the estimator suffers from a second order bias.

More generally, from the perspective of panel data econometrics, the natural way of understanding the detrimental impact of fixed effects is to view them as an instance of the incidental parameter problem, which was originally raised by Neyman and Scott (1948) and discussed in a panel data context by Nickell (1981). That is, as the panel grows larger asymptotically, the number of (incidental) fixed effects that need to be estimated also goes to infinity, as the cross-sectional dimension grows. Thus, although more and more data becomes available asymptotically, the number of parameters to estimate also increases. In the traditional (dynamic) panel setup studied by Nickell (1981), where $T$ is fixed as $n\rightarrow\infty$, inclusion of fixed effects causes the standard estimator of the slope coefficient to become inconsistent. Here, where both $n$ and $T$ tend to infinity, the fixed effects estimator remains consistent but with a second order bias.

II.D  Recursive Demeaning

The second order bias in the fixed effects estimator arises because the demeaning process induces a correlation between the innovation processes $u_{i,t}$ and the demeaned regressors $\underline{x}_{i,t-1}$. Intuitively, $u_{i,t}$ and $\underline{x}_{i,t-1}$ are correlated because, in the demeaning of $x_{i,t-1}$, information available after time $t-1$ is used. Or, equivalently, because in the demeaning of the dependent variable, $y_{i,t}$, information before time $t$ is used. One solution is therefore to use recursive demeaning of $x_{i,t}$ and $y_{i,t}$ (e.g. Moon and Phillips, (2000), and Sul et al. (2005)). In particular, I will consider a `forward demeaned' equation. That is, define

$$\underline{y}_{i,t}^{dd}=y_{i,t}-\frac{1}{T-t+1}\sum_{s=t}^{T}y_{i,s}, \underline{x}_{i,t}^{dd}=x_{i,t}-\frac{1}{T-t+1}\sum_{s=t} ^{T}x_{i,s}.$$ (14)

Observe that

$$\underline{y}_{i,t}^{dd}=y_{i,t}-\frac{1}{T-t+1}\sum_{s=t}^{T}y_{i,s} =\beta^{\prime}\left( x_{i,t-1}-\frac{1}{T-t+1}\sum_{s=t}^{T}x_{i,s-1} \right) +u_{i,t}-\frac{1}{T-t+1}\sum_{s=t}^{T}u_{i,s}=\beta^{\prime }\underline{x}_{i,t-1}^{dd}+\underline{u}_{i,t}^{dd},$$

and consider the following pooled estimator, using the recursively demeaned data,

$$\hat{\beta}_{RD}=\left( \sum_{i=1}^{n}\sum_{t=1}^{T}\underline{x} _{i,t-1}^{dd}x_{i,t-1}^{\prime}\right) ^{-1}\left( \sum_{i=1}^{n}\sum _{t=1}^{T}\underline{y}_{i,t}^{dd}x_{i,t-1}\right) .$$ (15)

In $\hat{\beta}_{RD}$, the non-demeaned regressors $x_{i,t-1}$ are used as instruments, and the dependent variable, $\underline{y}_{i,t}^{dd}$, is formed using data dated only after time $t$. Since $\underline{u}_{i,t}^{dd}$ and $x_{i,t-1}$ are now independent of each other, unlike $\underline{u}_{i,t}$ and $\underline{x}_{i,t-1}$, the estimator $\hat{\beta}_{RD}$ will not suffer from the same second order bias as the standard fixed effects estimator. This is stated formally in the following theorem.

Theorem 2:  With $\gamma_{i}\equiv0$, $\Gamma_{i}\equiv0$, and $\beta _{i}\equiv\beta$ for all $i$, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}},$

$$\sqrt{n}T\left( \hat{\beta}_{RD}-\beta\right) \Rightarrow N\left( 0,\left( \underline{\Omega}_{xx}^{RD}\right) ^{-1}\underline{\Phi}_{ux}^{RD}\left( \underline{\Omega}_{xx}^{RD}\right) ^{-1}\right) ,$$ (16)

where $\underline{\Phi}_{ux}^{RD}$ and $\underline{\Omega}_{xx}^{RD}$ are defined in the proof of the theorem.

To perform inference, let $\hat{u}_{i,t}^{dd}=\underline{y}_{i,t}^{dd} -\hat{\beta}_{RD}\underline{x}_{i,t-1}^{dd}$ , $\underline{\hat{\Phi}} _{ux}^{RD}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{T^{2}}\sum_{t=1}^{T}\sum _{s=1}^{T}\left( \hat{u}_{i,t}^{dd}x_{i,t-1}\right) \left( \hat{u} _{i,s}^{dd}x_{i,s-1}\right) ^{\prime}$ , and $\underline{\hat{\Omega}} _{xx}^{RD}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{T^{2}}\sum_{t=1}^{T}\underline {x}_{i,t-1}^{dd}x_{i,t-1}^{\prime}$ . The $t-$test and Wald test based on $\underline{\hat{\Phi}}_{ux}^{RD}$and $\underline{\hat{\Omega}}_{xx}^{RD}$ will satisfy the usual properties. Observe that the forward demeaning of the data introduces a moving average component in the returns process, which is reflected in the limiting distribution derived in the proof of Theorem 2. The variance-covariance matrix estimator that was just proposed automatically accounts for this by calculating the long-run variance using the forward demeaned residuals and the panel equivalent of a HAC estimator.

The recursive demeaning procedure gives up some efficiency by relying on a somewhat inefficient method for demeaning the data. However, there are no clear-cut alternatives in the general case when the autoregressive roots $C_{i}$ (or equivalently, $A_{i}$) are unknown. If the $C_{i}s$ were known, the bias term in equation (13) could be directly estimated and a bias-corrected fixed effects estimator could be constructed. More ambitiously, for known $C_{i}s$, a panel version of fully modified estimation could be considered, as suggested by Phillips and Moon (1999) in the pure unit-root case. However, although such procedures are likely more efficient than the recursive demeaning proposed here, they are not feasible in practice since the $C_{i}s$ are unknown.⁸

II.E  Relaxing the Pooling Assumption

II.E.1  Properties of the Pooled Estimators when the $\beta_{i}s$ Are Not Identical

So far, the focus has been on the problems raised by fixed effects. However, it is also possible that the slope coefficients $\beta_{i}$ may vary across $i$. In this section, I therefore discuss the properties of the pooled estimator when the $\beta_{i}s$ are not identical.⁹ To start with, suppose $\beta_{i} =\beta+\theta_{i}$, where $\left\{ \theta_{i}\right\} _{i=1}^{n}$ is $iid$ with mean zero.

Theorem 3:   Let $\gamma_{i}\equiv0$, $\Gamma_{i}\equiv0$, and $\beta_{i}=\beta+\theta_{i}$ for all $i$.
      (a) If $\theta_{i}$ is orthogonal to $x_{i,t}$ for all $i$ and $t$, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}},$

$$\sqrt{n}\left( \hat{\beta}_{FE}-\beta\right) \Rightarrow N\left( 0,\underline{\Omega}_{xx}^{-1}\underline{\Phi}_{xx}^{\theta}\underline{\Omega }_{xx}^{-1}\right) ,$$ (17)

where $\underline{\Phi}_{xx}^{\theta}$ is defined in the proof of the theorem.
      (b) If $\theta_{i}$ is not orthogonal to $x_{i,t}$, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}},$

$$\hat{\beta}_{FE}\rightarrow_{p}\beta+\underline{\Omega}_{xx}^{-1}E\left[ \left( \int_{0}^{1}\underline{J}_{i}\underline{J}_{i}^{\prime}\right) \theta_{i}\right] .$$ (18)

Analogous results also hold in the case without fixed effects.

In the case where the distribution of the slope coefficients is independent of the regressors, it follows that the pooled estimator converges to the average parameter $\beta\equiv E\left[ \beta_{i}\right]$. The rate of convergence is much slower than in the homogenous case, however, and the fixed effects estimator no longer suffers from a small sample bias. These results stem from the fact that when the $\beta_{i}s$ are non-identical, the residuals in the regression are now given by $\theta_{i}^{\prime}x_{i,t-1}+u_{i,t}$. Since $x_{i,t-1}$ is a near integrated process, it will dominate the asymptotic properties of the residuals, and will therefore slow down the rate of convergence and also render the second order bias term in the fixed effects estimator irrelevant. However, when the deviations $\theta_{i}$ are small, the second order bias term is still a concern. Results from Monte Carlo simulations, which are not presented here, show that for most potentially relevant values of $\beta$ and $\theta_{i}$ in a stock return predictability context, the bias in the fixed effects estimator is still highly relevant. Likewise, when the deviations $\theta_{i}$ are small, the slow-down in the rate of convergence will not be as drastic as in Theorem 3.¹⁰

If the $\theta_{i}s$ are correlated with the regressors, the pooled estimator does not converge to the average slope coefficient $\beta$. However, as discussed at length in Phillips and Moon (1999, 2000), the average of the individual parameters $\beta_{i}$ is not necessarily the natural way of defining an average relationship between $y_{i,t}$ and $x_{i,t-1}$. Phillips and Moon note that in a framework with persistent variables, one can define the individual regression coefficients $\beta_{i}$ as $\beta_{i} =\Omega_{xx,i}^{-1}\Omega_{yx,i}$, where $\Omega_{xx,i}$ is the long-run variance for $x_{i,t}$ and $\Omega_{yx,i}$ is the long-run covariance between $y_{i,t}$ and $x_{i,t-1}$. They then define the long-run average relationship between $y_{i,t}$ and $x_{i,t-1}$ as $\beta_{LRA}\equiv E\left[ \Omega_{xx,i}\right] ^{-1}E\left[ \Omega_{yx,i}\right]$ , rather than $E\left[ \Omega_{xx,i}^{-1}\Omega_{yx,i}\right] =\beta$, and show that the pooled estimator, with or without fixed effects, will converge to $\beta _{LRA}$ under very general conditions; in the special case when $\beta _{i}=\beta+\theta_{i}$ and $\theta_{i}$ is independent of $x_{i}$, it follows that $\beta_{LRA}=\beta$. Thus, $\hat{\beta}_{FE}$ and $\hat{\beta}_{Pool}$ converge to a well defined average relationship under very general circumstances, although not necessarily to $\beta=\lim_{n\rightarrow\infty }\frac{1}{n}\sum_{i=1}^{n}\beta_{i}$ .

II.E.2  A Test of Slope Homogeneity

The analysis above shows that the pooled estimators are robust to deviations from the assumption of homogenous slope coefficients, and will converge to a well-defined average coefficient when the $\beta_{i}s$ are non-identical. In many cases, it is still of interest, however, to evaluate whether the slope coefficients are in fact all equal.

I adopt a version of a test originally proposed by Swamy (1970) and further developed by Pesaran (2007). The basic idea is to analyze a weighted sum of squared differences between the unrestricted time-series estimates of the individual $\beta_{i}s$ and the fixed effects pooled estimate, which imposes a common slope coefficient.

Define the following weighted fixed effects estimator,

$$\hat{\beta}_{WFE}=\left( \sum_{i=1}^{n}\sum_{t=1}^{T}\frac{\underline {x}_{i,t-1}\underline{x}_{i,t-1}^{\prime}}{\hat{\omega}_{11i}}\right) ^{-1}\left( \sum_{i=1}^{n}\sum_{t=1}^{T}\frac{\underline{y}_{i,t} \underline{x}_{i,t-1}}{\hat{\omega}_{11i}}\right) ,$$ (19)

where $\hat{\omega}_{11i}$ is an estimate of the variance of $u_{i,t}$ $\left( \omega_{11i}\right)$; the standardization by $\omega_{11i}$ leads to a natural reduction in nuisance parameters in the asymptotic distribution of the below test statistic. Further, let

$$S_{\beta}=\sum_{i=1}^{n}\left( \hat{\beta}_{i}-\hat{\beta}_{WFE}\right) ^{\prime}\left( \sum_{t=1}^{T}\frac{\underline{x}_{i,t-1}\underline {x}_{i,t-1}^{\prime}}{\hat{\omega}_{11i}}\right) \left( \hat{\beta}_{i} -\hat{\beta}_{WFE}\right) ,$$ (20)

where $\hat{\beta}_{i}$ is the OLS estimate of the slope coefficient for country $i$.

Theorem 4:   With $\gamma_{i}\equiv0$, $\Gamma_{i}\equiv0$, and under $H_{0}:\beta_{i}=\beta$ for all $i$, as $\left( T,n\rightarrow \infty\right) _{\operatorname{seq}}$,

$$\Delta_{\beta}=\sqrt{n}\left( \frac{\frac{1}{n}S_{\beta}-\mu_{Z}}{\sigma_{Z} }\right) \Rightarrow N\left( 0,1\right) .$$ (21)

where $\mu_{Z}$ and $\sigma_{Z}$ are defined in the proof of the theorem.

Given $\mu_{Z}$ and $\sigma_{Z}$, $\Delta_{\beta}$ provides an asymptotically normally distributed test of slope homogeneity. Unfortunately, $\mu_{Z}$ and $\sigma_{Z}$ are functions of the unknown nuisance parameters $\left\{ C_{i}\right\} _{i=1}^{n}$; they are also functions of the average correlation $\left( \delta\right)$ between the innovations $u_{i,t}$ and $v_{i,t}$, but this value can easily be estimated.

Through simulations, it easy to show that $\mu_{Z}$ changes fairly slowly with the values of the $C_{i}s$, whereas $\sigma_{Z}$ can vary substantially from small changes in the $C_{i}s$. In order to obtain a feasible test with approximately correct size, I therefore propose to use $\mu_{Z}$, evaluated for a common value of $C_{i}=\tilde{C}$ for all $i$, where $\tilde{C}$ is given by the average of the median unbiased estimates of each $C_{i}$. As originally shown by Stock (1991), median unbiased, although inconsistent, estimates of each $C_{i}$ can be obtained by inverting a unit-root test statistic. Further, $\sigma_{Z}$ is replaced by an empirical estimate that is consistent under the null hypothesis of $\beta_{i}=\beta$ for all $i$. Write $S_{\beta}\equiv\sum_{i=1}^{n}Z_{i,n,T}$ where $Z_{i,n,T}$ represents the expression in (20). From the proof of Theorem 4, an estimate of $\sigma_{Z}$ is obtained by calculating the sample standard deviation of $Z_{i,n,T}$. Under the alternative, when the $\beta_{i}s$ are not all identical, this estimate will be upward biased for $\sigma_{Z}$, and some power will therefore be lost. However, given a lack of knowledge of the $C_{i}s$, and the strong dependence of $\sigma_{Z}$ on the values of the $C_{i}s$, this seems like a preferable approach.

In terms of practical implementation, the median unbiased estimates for $C_{i}$ are obtained by inverting the DF-GLS unit-root test statistic, as described in detail in Campbell and Yogo (2006). In the case when $x_{i,t}$ is a vector, the same procedure can be applied to each of the component processes of $x_{i,t}$ and, with the extra restriction that $C_{i}$ is a diagonal matrix, one can proceed exactly as in the scalar case. The variances $\left( \omega_{11i}\right)$ of $u_{i,t}$ and the average correlation $\left( \delta\right)$ between $u_{i,t}$ and $v_{i,t}$ are estimated from the residuals of the time-series regressions of equations (1) and (2). The values for $\mu_{Z}$ are obtained by direct simulation of the asymptotic expression given in the proof of Theorem 4; these values are available from the author upon request. The null hypothesis is rejected for large positive values of $\Delta_{\beta}$; e.g. a five percent test would reject for values larger than 1.65.

III  Cross-Sectional Dependence

III.A  The Effects of Common Factors

I now return to the general setup with common factors in the data. The following theorem summarizes the asymptotic properties of both the standard pooled estimator and the fixed effects estimator when there are common factors. Again, the $\beta_{i}s$ are assumed to be identical unless otherwise noted.

Theorem 5:   (a) With $\alpha_{i}\equiv\alpha$ and $\beta_{i}\equiv\beta$ for all $i$, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}},$

$$T\left( \hat{\beta}_{Pool}-\beta\right) \Rightarrow\left( \Omega _{xx}+\Omega_{zz}\right) ^{-1}\left[ \int_{0}^{1}\left( \gamma^{\prime }dB_{f}\right) \left( \Gamma^{\prime}J_{g}\right) \right] .$$ (22)

(b) With $\beta_{i}\equiv\beta$ for all $i$, as $\left( T,n\rightarrow \infty\right) _{\operatorname{seq}},$

$$T\left( \hat{\beta}_{FE}-\beta\right) \Rightarrow\left( \underline{\Omega }_{xx}+\underline{\Omega}_{zz}\right) ^{-1}\left[ \int_{0}^{1}\left( \gamma^{\prime}dB_{f}\right) \left( \Gamma^{\prime}\underline{J}_{g}\right) -\left( \int_{0}^{1}\int_{0}^{r}E\left[ e^{\left( r-s\right) C_{i} }\right] dsdr\right) \omega_{21}\right] .$$ (23)

Thus, in the presence of the general factor structure outlined in Section II, the standard pooled estimator exhibits a non-standard limiting distribution, although it is still consistent; standard tests can therefore not be used. Similarly, the limiting behavior of the fixed effects estimator is determined by the bias term arising from the time-series demeaning of the data, as well as an additional term that stems from the common factors in the data. Note that the term $\int_{0}^{1}\left( \gamma^{\prime}dB_{f}\right) \left( \Gamma^{\prime}J_{g}\right)$ is random and can take on both negative and positive values. Thus, correcting for it will have an ambiguous effect on the outcome of the estimation and test results.

III.B  Robust Estimators

Based on the methods of Pesaran (2006), I propose an estimator that is more robust to cross-sectional dependence in the data. Pesaran's (2006) idea is to project the data onto the space orthogonal to the common factors, thereby removing the cross-sectional dependence from the data used in the estimation. However, since the factors are not observed in practice, an indirect approach is required. Pesaran suggests using the cross-sectional means of the dependent and independent variable as proxies for the common factors. A similar approach is adopted below, but only the cross-sectional means of the regressors are used to control for the common factors. This is done because of the different orders of integration between the error terms and the regressors. For $\beta\neq0$, the stochastic behavior of $y_{i,t}$ is dominated by that of $x_{i,t-1}$, and the matrix $T^{-1/2}\left( \bar{y}_{\cdot,t},\bar{x} _{\cdot,t-1}\right) ^{\prime}$ would be asymptotically singular.

Thus, consider the following estimator of $\beta$,

$$\hat{\beta}_{Pool}^{+}=\left( \sum_{i=1}^{n}\mathbf{X}_{i,-1}^{\prime }\mathbf{M}_{\mathbf{\bar{H}}}\mathbf{X}_{i,-1}\right) ^{-1}\left( \sum_{i=1}^{n}\mathbf{X}_{i,-1}^{\prime}\mathbf{M}_{\mathbf{\bar{H}} }\mathbf{Y}_{i}\right)$$ (24)

where $\mathbf{Y}_{i}$ denotes the $T\times1$ matrix of the observations for the dependent variable and $\mathbf{X}_{i}$ the $T\times m$ matrix of regressor observations. $\mathbf{M}_{\mathbf{\bar{H}}}=\mathbf{I} -\mathbf{\bar{H}}\left( \mathbf{\bar{H}}^{\prime}\mathbf{\bar{H}}\right) ^{-1}\mathbf{\bar{H}}^{\prime}$ is a $T\times T$ matrix and $\mathbf{\bar{H}}$ is the $T\times m$ matrix of observations of $\bar{H}_{t}=\frac{1}{n} \sum_{i=1}^{n}x_{i,t-1}=\bar{x}_{\cdot,t-1}.$

The estimator $\hat{\beta}_{Pool}^{+}$ is obtained by applying the pooled estimator to the residuals from a projection of the original data onto the cross-sectional averages of the regressors. The intuition behind this is that the cross-sectional average of $x_{i,t}$ is close to the common stochastic trend $z_{t}$, since the cross-sectional averages of the cross-sectionally independent data may be expected to be close to zero; the projection onto the compliment of the cross-sectional means will therefore remove the effects of the common factors in the regressors. As is shown in the proof of the following theorem, it is sufficient to remove the factors from the regressors (and not from the innovations to the regressand) in order to achieve a mixed normal distribution.

Theorem 6:   With $\alpha_{i}\equiv\alpha$ and $\beta_{i}\equiv\beta$ for all $i$, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$,

$$\sqrt{n}T\left( \hat{\beta}_{Pool}^{+}-\beta\right) \Rightarrow MN\left( 0,\Omega_{xx}^{-1}\left( \Phi_{ux}+\Phi_{fx}\right) \Omega_{xx}^{-1}\right) .$$ (25)

where $MN\left( \cdot\right)$ denotes a mixed normal distribution.

The estimator $\hat{\beta}_{Pool}^{+}$ thus achieves a $\sqrt{n}T-$convergence rate and an asymptotic mixed normal distribution. The mixed normality in this case arises from the common factors, which leads to a mixed normal distribution rather than the normal distribution seen above in the no common factors case. That is, the limiting distribution is effectively a normal distribution with a random variance-covariance matrix that is a function of the common shocks; conditional on the realization of the common factors, the distribution is thus normal. For practical purposes, the mixed normal distribution allows for standard inference in that the $t-$tests and Wald tests will have asymptotically standard distributions. Allowing for fixed effects in the arguments, it is easy to show the following result.

Corollary 1:   Let $\widehat{\mathbf{X}}_{i,-1}=\mathbf{M} _{\mathbf{\bar{H}}}\mathbf{X}_{i,-1}$ and $\widehat{\mathbf{Y}}_{i} =\mathbf{M}_{\mathbf{\bar{H}}}\mathbf{Y}_{i}$ and define

$$\hat{\beta}_{FE}^{+}=\left( \sum_{i=1}^{n}\underline{\widehat{\mathbf{X}} }_{i,-1}^{\prime}\underline{\widehat{\mathbf{X}}}_{i,-1}\right) ^{-1}\left( \sum_{i=1}^{n}\underline{\widehat{\mathbf{X}}}_{i,-1}^{\prime}\underline {\widehat{\mathbf{Y}}}_{i}\right) ,$$ (26)

and

$$\hat{\beta}_{RD}^{+}=\left( \sum_{i=1}^{n}\widehat{\mathbf{X}}_{i,-1} ^{dd\prime}\widehat{\mathbf{X}}_{i,-1}\right) ^{-1}\left( \sum_{i=1} ^{n}\widehat{\mathbf{X}}_{i,-1}^{\prime}\widehat{\mathbf{Y}}_{i}^{dd}\right)$$ (27)

where $\underline{\widehat{\mathbf{X}}}_{i,-1}$and $\underline{\widehat{\mathbf{Y}}}_{i}$ represent the time-series demeaned versions of $\widehat{\mathbf{X}}_{i,-1}$ and $\widehat{\mathbf{Y}}_{i}$, and $\widehat{\mathbf{X}}_{i,-1}^{dd}$ and $\widehat{\mathbf{Y}}_{i}^{dd}$ are the recursively demeaned variables. Then, with $\beta_{i}\equiv\beta$ for all $i$, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$,

$$T\left( \hat{\beta}_{FE}^{+}-\beta\right) \rightarrow_{p}-\underline{\Omega }_{xx}^{-1}\left( \int_{0}^{1}\int_{0}^{r}e^{\left( r-s\right) C_{i} }dsdr\right) \omega_{21},$$ (28)

and

$$\sqrt{n}T\left( \hat{\beta}_{RD}^{+}-\beta\right) \Rightarrow MN\left( 0,\left( \underline{\Omega}_{xx}^{RD}\right) ^{-1}\left( \underline{\Phi }_{ux}^{RD}+\underline{\Phi}_{fx}^{RD}\right) \left( \underline{\Omega} _{xx}^{RD}\right) ^{-1}\right) ,$$ (29)

where $\underline{\Phi}_{fx}^{RD}$ is defined analogously with $\Phi_{fx}$ and $\underline{\Phi}_{ux}^{RD}$.

$\hat{\beta}_{Pool}^{+}$ and $\hat{\beta}_{RD}^{+}$ thus provide pooled estimators for predictive regressions that are asymptotically mixed normally distributed in the presence of common factors and with the allowance for fixed effects in the latter. Standard $t-$tests and Wald tests can therefore be used; by simply using the defactored data, the variance-covariance matrix can be estimated in a manner analogous to that described above for the no common factors case. The practical implementation of these estimators is thus very simple: Premultiply the data by $\mathbf{M}_{\mathbf{\bar{H}}}$, and use the resulting variables in the original estimation procedures.

As shown in the simulations below, the $\Delta_{\beta}-$test of slope homogeneity appears robust to the presence of factors -unlike the pooled $t-$tests- and I do not attempt to modify it to control for common factors.

IV  Finite Sample Evidence

IV.A  No Cross-Sectional Dependence

To evaluate the small sample properties of the panel data estimators proposed in this paper, a Monte Carlo study is performed. In particular, I focus on the size and power properties of the pooled ttests. Equations (1) and (3) are simulated for the case with a single regressor. The innovations $\left( u_{i,t},v_{i,t}\right)$ are drawn from normal distributions with mean zero, unit variance, and correlations δ = 0, -0.4, -0.7, and -0.95; there is no cross-sectional dependence. The local-to-unity parameters $C_{i}$ are drawn from a uniform distribution with support [-20, -2]. In analyzing the power properties, the slope coefficient β varies between -0.05 and 0.05, and is identical for all i. The sample size is given by T = 100, n = 20. The intercepts $\alpha_{i}$ are normally distributed with mean and standard deviation equal to 0.005. All results are based on 10,000 repetitions. The $t-$test based on the fixed effects estimator using standard demeaning, $\hat{\beta}_{FE}$, and that based on the recursively demeaned pooled estimator, $\hat{\beta}_{RD}$, are considered. Throughout the simulation study, the normal distribution is used to determine significance; i.e. the null is rejected for absolute test values greater than 1.96.

Panel A in Table 1 shows the average rejection rates for the nominal five percent two sided t-tests under the null hypothesis of β = 0. Panels A1 and A2 in Figure 1 show the corresponding power curves of the tests for the cases of δ = 0 and δ = -0.95. Table 1 and the power curves in Figure 1 clearly show the effects of the second-order bias in the fixed effects estimator; the test based on the standard fixed effects estimator severely over rejects under the null hypothesis for δ = -0.95. The test based on the recursively demeaned estimator has rejection rates close to the nominal size under the null, while maintaining decent power properties.¹¹

IV.B  Common Factors

In this section, I repeat the Monte Carlo experiment above for the case when there is a common factor in the innovations. In particular, equations (1)-(4) are now simulated with a single regressor and a single common factor $f_{t}=g_{t}$, drawn from a standard normal distribution. The factor loadings, $\gamma_{i}$ and $\Gamma_{i}$, are also normally distributed with means of minus one and plus one, respectively, and standard deviations equal to 2^-1/2 in both cases. The innovations in the returns and regressor processes are formed as $2^{-1/2}\left( \gamma_{i}^{\prime }f_{t}+u_{i,t}\right)$ and $2^{-1/2}\left( \Gamma_{i}^{\prime}f_{t} +v_{i,t}\right)$, respectively, where $\left( u_{i,t},v_{i,t}\right)$ are drawn from standard normal distributions; the scaling by 2^-1/2 is performed in order to achieve an approximate unit variance in the innovations, which enables easier comparison with the cross-sectionally independent case. As before, the correlation between $u_{i,t}$ and $v_{i,t}$ is set δ = 0, -0.4, -0.7 and -0.95.

The results are shown in Panels B and C of Table 1 and in Panels B1, B2, C1, and C2 in Figure 1. Panel B in Table 1 and Panels B1 and B2 in Figure 1 show the outcomes of the Monte Carlo experiments when the model generated with common factors is estimated using the estimators $\hat{\beta}_{FE}$ and $\hat{\beta}_{RD}$, which do not control for cross-sectional dependence. It is clear that when the common factors are ignored in the estimation process, the actual size of the corresponding $t-$tests is very far from the nominal size of $5$ percent, with rejection rates above $30$ percent under the null.

Panel C in Table 1 and Panels C1 and C2 in Figure 1 show the same results for the estimators $\hat{\beta}_{FE}^{+}$ and $\hat{\beta }_{RD}^{+}$, which do control for the common factors. The $t-$test based on the recursively demeaned data, $\hat{\beta}_{RD}^{+}$, now possesses good size and power properties. As before, the standard fixed effects estimator exhibits a finite sample bias and extremely poor size properties.

IV.C  Test of Slope Homogeneity

The final set of simulations analyzes the finite sample properties of the $\Delta_{\beta}-$test of slope homogeneity. The setup is identical to that used above, with the exception of the slope coefficients. Three different scenarios are considered. In the first, the null hypothesis is imposed and $\beta_{i}=\beta=0$. In the two other cases, the slope coefficients exhibit heterogeneity and are generated as $\beta_{i}=\beta+\beta\theta_{i}$, where $\theta_{i}$ is a standard normal random variable, independently distributed across $i$, and $\beta$ is equal to $0.05$ and $0.1$, respectively. The cases with and without common factors are considered, although, as described previously, no adjustment to the test is made when there are common factors. The test is evaluated as a one-sided test with a nominal size of five percent; i.e. the null is rejected when $\Delta_{\beta}$ is greater than 1.65.

Panel A of Table 2 shows the results without any common factors and Panel B shows the results with common factors. Under the null, the test is somewhat under sized, and marginally more so when there are common factors. Unlike the $t-$tests analyzed above, the test of slope homogeneity is thus not particularly sensitive to common factors in the data. Under the first alternative, with $\beta=0.05$, the power of the test is around 45 percent without common factors, and around 35 percent with common factors. Under the second alternative, with $\beta=0.1$, the power rises to above 90 percent in both cases. The relatively low power under the first alternative reflects the fact that it is difficult to distinguish between such small absolute differences between the $\beta_{i}s$, even though the relative differences are reasonably large; one should keep in mind that the test compares across the cross-section of the data, which in the simulations only amounts to $n=20$ observations. Nevertheless, the test can serve as a useful diagnostic of panel homogeneity.

V  Data Description

All of the data come from the Global Financial Data database and are on a monthly frequency. Total returns, including direct returns from dividends, on market wide indices in 40 countries were obtained, as well as the corresponding dividend- and earnings-price ratios and measures of the short and long interest rates.

With the exception of Spain, the dividend-price ratio data is available over the same sample period as the total stock returns. But, the other predictor variables are typically not available during the whole sample of total stock returns. Due to the two world wars, France, Germany, Japan, and the U.K. have some years during which no observations are available. Further, Spain's total returns data start in 1940, but no dividends data is available during 1968-1983. Thus, in the time-series analysis, separate regressions are fitted for each sample period for these five countries, and in the pooled estimation separate intercepts are estimated. In Table 3, which presents the pooled results, the row listing the number of 'countries' in each panel can therefore include more than one count of some countries.

As is conventional in the literature, the dividend-price ratio is defined as the sum of dividends during the past year divided by the current price and the earnings-price ratio is defined as the current price divided by the latest 12 months of available earnings. Short interest rate measures come from Global Financial Data and use rates on 3-month T-bills when available or, otherwise, private discount rates or interbank rates. The long rate is measured by the yield on long-term government bonds. When available, a 10-year bond is used; otherwise, I use that with the closest maturity to 10 years. The term spread is defined as the log difference between the long and short rates. Excess stock returns are defined as the return on stocks, in the local currency, over the local short rate. This provides the international analogue of the typical forecasting regressions estimated for U.S. data. All regressions are run at the one-month frequency using log-transformed variables with the log excess returns over the domestic short rate as the dependent variable.

Countries are pooled into a global panel, as well as developed and emerging stock market panels, according to the MSCI classifications.¹²

VI  Empirical Results

In the empirical analysis, I conduct pooled regressions as well as time-series regressions for individual countries. The results from the pooled regressions and summaries of the time-series results are presented in Table 3. The time-series results for individual countries are given in Table 4. Each table contains multiple panels, which correspond to the different forecasting variables. For the pooled regressions, results from both the standard fixed effects estimator, $\hat{\beta}_{FE}$, and the corresponding $t-$statistic, $t_{FE}$, as well as the estimator using recursively demeaned data, $\hat{\beta}_{RD}$ and $t_{RD}$, are documented. Separate results are shown for the case when common factors are controlled for and when they are not. As discussed at length above, the $t_{FE}-$test is not robust to the endogeneity and persistence of the regressors, but provides an interesting illustration of the potential pitfalls of not addressing these issues. The short interest rate and the term spread are generally less endogenous and inference based on the fixed effects estimator for these two variables will be fairly accurate (see also the discussion on this topic in Campbell and Yogo (2006)); however, the fixed effects $t-$test will tend to greatly over reject the null for the dividend- and earnings-price ratio. In Table 3, significant results at the one-sided five percent level based on robust test-statistics from which proper inference can be drawn, i.e. the $t-$statistic corresponding to $\hat{\beta}_{RD}^{+}$, are indicated with a $^{\ast}$.¹³

The results from the individual time-series regressions, shown in Table 4, are presented in a similar manner to the pooled regressions. Since normal inference based on the OLS $t-$statistic will generally be biased, inference based on a robust 90 percent confidence interval for $\beta_{i}$, using the methods of Campbell and Yogo (2006), are also provided. If viewed as a test, this confidence interval can be seen as a five percent one-sided test and a rejection of the null hypothesis of no predictability is indicated with a $^{\ast}$ next to the coefficient estimate; for brevity, the actual confidence intervals are not shown. In Table 3, the number of individual time-series regressions that yield significant coefficients according to the Campbell and Yogo test is indicated in the column labeled CY $_{\text{sig}}$.

VI.A  Testing for Slope Homogeneity

Before considering the empirical results for the different predictor variables, it is useful to briefly analyze the homogeneity of the slope coefficients in the pooled predictive regressions. Table 3 shows the outcome of the $\Delta_{\beta}-$test of slope homogeneity; it is a one-sided test, and a value greater than 1.65 indicates that the null hypothesis of $\beta_{i}=\beta$ for all $i$ is rejected at the five percent level. As is seen, slope homogeneity can always be rejected for the global panel. For the developed panel, homogeneity is only rejected in the dividend-price ratio regression using the full sample, which spans a much larger range of time than the other panels since the dividend-price ratio is available further back than any of the other predictor variables; when data before 1950 is dropped, slope homogeneity can no longer be rejected. For the emerging market panels, the null of slope homogeneity is only rejected in the regression with the short interest rate. These results thus support the notion that countries within the groups of developed and emerging markets tend to be more homogenous in terms of predictability than countries across these groups.

As shown previously, the pooled analysis is valid also when the slope coefficients are not homogenous. However, the estimates from the global panels, and in a couple of instances from the developed and emerging panels, are best interpreted as average relationships, and the corresponding tests as tests of whether there is predictability on average in the data.

VI.B  The Earnings-Price Ratio

The results for the earnings-price ratio are presented in Panel A of Tables 3 and 4. There is minimal evidence of a positive predictive relationship. Specifically, pooling the data at either the global or developed market levels does not yield a significant coefficient, regardless of whether one controls for common factors; however, there is evidence of a predictive relationship when pooling at the emerging market level and controlling for common factors. To ensure that the developed market results are not driven by the longer earnings-price ratio time-series available for the U.K. and the U.S., I also estimate these pooled regressions when restricting the sample to observations after 1950. The individual country time-series results confirm the lack of evidence of a predictive relationship in the pooled regressions. In particular, in the post-1950 sample, only four of the 38 time-series regressions (Argentina, Jordan, South Africa, and the U.K.) yield any significant coefficients.

There is thus rather weak evidence that the earnings-price ratio predicts stock returns; the majority of evidence that does exist is for emerging economies. It is noteworthy that the null of no predictability would have been rejected in all of the pooled regressions if one relied on non-robust methods that fail to control for the endogeneity and persistence of the regressors, as well as common factors in the data. Controlling for common factors appears to be of potentially great importance. It is interesting to note that doing so does not necessarily weaken the results.

VI.C  The Dividend-Price Ratio

Panel B in Table 3 shows the results from pooled regressions with the dividend-price ratio as the regressor. The results are generally somewhat stronger than for the earnings-price ratio. Specifically, when controlling for common factors, the coefficient is significant when pooling both at the post-1950 global level and the developed market level, as well as in the full sample emerging panel. The overall picture depicted by the individual time-series regressions shown in Panel B of Table 4, however, is still fairly weak, although evidence of predictability is observed for post-1950 Australia, Chile, post-WWII Japan, Jordan, Mexico, Taiwan, the U.K., and post-1950 U.S. The time-series evidence thus presents no clear pattern of predictability and the evidence that exists is distributed fairly equally between developed and emerging markets. As in the case of the earnings-price ratio, the null of no predictability would have been often rejected when using non-robust tests.

VI.D  The Short Interest Rate

In light of the empirical evidence of a predictive relationship seen in U.S. data, one would expect there to be a negative relationship between the current short rate and future stock returns. The data used in all interest rate regressions are restricted to start in 1952 or after, following the convention used in studies with U.S. data.¹⁴ The pooled results for the short interest rate are presented in Panel C of Table 3.

The null of no predictability is strongly rejected in the pooled sample of developed markets. In contrast to this strongly significant negative relationship, the pooled relationship in the emerging markets is not significant. Given the rather capricious character of interest rates in many emerging economies (e.g. Argentina), I focus strictly on the developed market results. As seen in Panel C of Table 4, this finding of predictability is supported by the results of the individual time-series regressions for the developed markets. In particular, a significant predictive relationship is found in eight out of 23 developed markets, including: Canada, Germany, the Netherlands, New Zealand, Portugal, Spain, Switzerland, and the U.S.. In addition, a closer look at the individual country level results further strengthens this pattern; in particular, it reveals that the estimates for 15 of the developed markets are more than one standard deviation away from zero while the slope coefficient estimate is negative for 20 of the 23 countries.

VI.E  The Term Spread

Based on the U.S. experience, one would expect there to be a positive predictive relationship, if any, between the term spread and stock returns. As in the case of the short interest rate, I find a positive significant predictive relationship only in developed markets. As shown in Panel D of Table 3, there is strong evidence of a predictive relationship when pooling the developed economies. As this relationship is not evident for the emerging markets, I once again focus on the results for the developed markets. As seen in Panel D of Table 4, this finding of predictability is supported strongly by the results of the individual time-series regressions for the developed markets. For 10 of 23 individual time-series regressions, there is a positive and significant predictive relationship: Canada, France, Germany, Italy, the Netherlands, New Zealand, Norway, Spain, Switzerland, and the U.S.. Furthermore, 14 countries have a coefficient that is more than one standard deviation from zero.

VI.F  Stability over Time

How robust are these patterns of predictability to different sample periods? To analyze this, I consider pooled regressions with expanding windows of observations for the developed markets. I focus on the developed panel since the time-series in that panel typically have longer samples available. A new country is added to the expanding window regression when there are five years of observations available; no 'old' observations are ever dropped from the estimation window and the estimates at each point in time are thus based on all observations available up to that date.¹⁵ Confidence intervals, with a nominal 90 percent coverage rate, are calculated in a manner analogous to the test statistics shown in Table 3, based on the normal distribution. The left column of Figure 2 shows the results from using the standard fixed effects estimator, without controlling for common factors. The right column shows the results from the estimator using recursively demeaned data and controlling for common factors. The confidence intervals in the left column are thus typically biased and will generally not have an actual coverage rate of 90 percent; however, these results further illustrate the importance of controlling for endogeneity and common factors.

The results presented in Figure 2 mostly reflect those discussed above based on the complete sample. However, the results for the dividend-price ratio $\left( d-p\right)$, which generally appeared somewhat stronger than those for the earnings-price ratio $\left( e-p\right)$, now appear very weak when viewed over time. Overall, the support of any stable predictive ability in either of these two variables is very weak. The term spread $\left( y-r_{s}\right)$ coefficient fluctuates around zero until the late 1970s, after which the lower bound of the confidence interval typically hovers above zero, although the coefficient is only consistently significant after 1993. It is only for the short rate $\left( r_{s}\right)$ that the coefficient is significantly different from zero during the whole sample period that is analyzed. The overall instability of the regression coefficients over time is in line with the results of the formal test procedures of Pettenuzzo and Timmermann (2005) and Paye and Timmermann (2006).

Given the relative strength of the findings for the short interest rate and the term spread found so far, I also present expanding window regression estimates for individual country time-series regressions for these two variables. The time-series OLS estimates for the short interest rate, with corresponding robust 90 percent confidence intervals calculated using the Campbell and Yogo method, are shown in Figure 3, for the twelve developed markets with the longest sample periods.¹⁶ For all of the countries considered, except Japan, a very similar pattern is evident. After around 1980, the estimated coefficients and confidence intervals stabilize; in most cases, this occurs on or below zero, indicating a significant or near significant negative relationship. The analogous results for the term spread are shown in Figure 4. These show a similar pattern to that found for the short interest rate, but overall the results are perhaps somewhat weaker.

VI.G  Out-of-Sample Evidence

Finally, the out-of-sample predictability of stock returns is considered, using the forecasting variables discussed above. To allow for a sufficient sample size in the out-of-sample analysis, which requires an initial 'training-sample' to obtain the estimates on which the first round of forecasts is based, I exclude all countries with less than 40 years of data; this allows for a 20-year training period and a minimum of a 20-year forecasting period. In each period following the first twenty years, the coefficients are re-estimated, with the latest observations included. Next period's returns are forecasted based on the estimated regression equation. These `conditional' forecasts, based on the regression model, are compared to the `unconditional' forecasts, which in each period are identical to the sample mean of the then available past returns. To compare the conditional and unconditional forecasts, an out-of-sample $R^{2}$ is calculated,

$$R_{i,OS}^{2}=1-\frac{\sum_{t=s}^{T}\left( y_{i,t}-\hat{y}_{i,t}\right) ^{2} }{\sum_{t=s}^{T}\left( y_{i,t}-\bar{y}_{i,t}\right) ^{2}},$$ (30)

where $\hat{y}_{i,t}$ and $\bar{y}_{i,t}$ are the conditional and unconditional forecasts, respectively, and $s$ is the length of the training sample. The $R_{i,OS}^{2}$ statistic will be positive when the conditional forecast outperforms the unconditional one. I consider out-of-sample forecasts that are based on both the time-series and pooled fixed effects estimates of the slope coefficients in the forecasting regressions. It is possible that the pooled estimate yields better out-of-sample performance if the time-series estimate is imprecise due to fewer available observations, even though the fitting of individual coefficients for each country allows for more freedom.

The forward demeaning used in the recursively demeaned estimator makes it less suitable for out-of-sample exercises. The standard fixed effects estimator is therefore used, even though it will tend to produce biased estimates for the valuation ratios, which may limit the benefits of using pooled estimates for these variables when forming forecasts.

Although the strongest in-sample results were found for the interest rate variables, it is still worth briefly considering the out-of-sample performance of the valuation ratios. As seen in Panels A and B of Table 5, there are three countries where the earnings-price ratio is a significant predictor according to the Campbell and Yogo test, and two countries where the dividend-price ratio is significant. For each of the regressions, which delivered significant in-sample results with the earnings-price ratio as a predictor, the out-of-sample $R^{2}$ is positive when the forecasts are based on either the time-series estimates or the pooled estimates. The results for the dividend-price ratio are overall weaker, with the out-of-sample $R^{2}$ most often negative.

As seen in Panel C of Table 5, there are five countries for which the domestic short rate has a significant negative predictive relationship. For these five countries, the out-of-sample $R^{2}$ are positive in three cases when using the conditional forecasts that are based on the pooled estimates; using the time-series based forecasts, the out-of-sample $R^{2}$ for Spain alone is positive. In total, there are 10 countries for which the estimated coefficients are more than one standard deviation away from zero (with the right sign), and for six of these the out-of-sample $R^{2}$ are positive when using the pooled forecasts, but only for one when using the time-series based forecasts.

Panel D of Table 5 presents the results for the term spread. In this case, a significant in-sample predictive relationship is seen in seven countries, according to the Campbell and Yogo test, and ten countries have (positive) coefficients that are more than one standard deviation away from zero. When basing the forecasts on the pooled estimates, the conditional forecast beats the unconditional one in all seven of the countries with significant in-sample results, and in nine out of ten of the countries with a $t-$statistic greater than one. For the time-series based forecasts, the unconditional forecast is beaten only in four out of the seven significant countries and in five out of ten of the countries with a $t-$statistic greater than one.

Overall, the results in Table 5 provide some evidence that a significant in-sample relationship also tends to be associated with out-of-sample predictive power, particularly for the term spread.¹⁷ In addition, forecasts based on the pooled estimates typically out-perform forecasts based on the time-series estimates. Thus, it is possible that pooled methods could be useful in reducing risk when using conditional forecasts in portfolio choice.

VII  Conclusion

I analyze stock return predictability in a large global data set with 40 different markets and develop new econometric methods for predictive regressions with panel data. The theoretical results provide an important extension to the existing literature on time-series methods for predictive regressions and show that a careful analysis of the impact of nearly persistent and endogenous regressors is required also in the panel data case.

The empirical analysis delivers two main findings: (i) Traditional valuation measures such as the dividend- and earnings-price ratios have very limited predictive ability in international data. It is evident, however, that using methods that do not account for the persistence and endogeneity of these variables would lead one to vastly misjudge their predictive powers. (ii) Interest rate variables are more robust predictors of stock returns, although their predictive power is mostly evident in developed markets.

The international results for the interest rate variables are similar to those of the U.S. while the overall findings for the earnings- and dividend-price ratios are substantially weaker. In summary, the results presented in this paper provide strong evidence that there is a predictable component in stock returns, which is captured at least partially by interest rate variables.

Appendix

A  Formal Assumptions

Assumption 1   (Autoregressive roots)  The auto-regressive roots $A_{i}$, or equivalently the local-to-unity parameters $C_{i}$, are $iid$ random variables independently distributed of other random elements in the model.

Assumption 2    (Factor loadings)
1.  The factor loadings $\gamma_{i}$ $\left( l\times1\right)$ and $\Gamma_{i}$ $\left( k\times m\right)$ are random coefficients that are $iid$ across $i$ and independently distributed of the specific errors, $u_{j,t}$ and $v_{j,t}$, and the common factors $f_{t}$ and $g_{t}$, for all $i,j,$ and $t$, with fixed means $\gamma$ and $\Gamma$, and finite variances
2.  Rank $\left( \Gamma\right) =k$.

The rank condition on $\Gamma$ is required for identification in the estimation procedures that control for the common factors. It essentially states that all information regarding the common factors in the regressors can potentially be recovered from the data on the regressors.

Assumption 3   (Innovations)  Let $w_{i,t}=\left( u_{i,t},v_{i,t},f_{t} ,g_{t}\right) ^{\prime}$ and $\varpi_{i,t}=\left( u_{i,t},\epsilon _{i,t},f_{t},\eta_{t}\right) ^{\prime}$ with the corresponding filtration $\mathcal{F}_{t}=\left\{ \left. \varpi_{i,s}\right\vert s\leq t,i=1,...,n\right\}$ . Then, for all $i=1,...,n,$ and $t=1,...,T$ $:$
1.   $E\left[ \left. \varpi_{i,t}\right\vert \mathcal{F}_{t-1}\right] =0$ , $E\left[ \varpi_{i,t}\varpi_{i,t}^{\prime}\right] =\Sigma_{i}$, and $E\left[ \left\vert \left\vert \varpi_{i,t}\right\vert \right\vert ^{4}\right] <\infty$ .
2.  The innovations to the regressor, $v_{i,t}$ and $g_{t}$, are general linear processes that satisfy $v_{i,t}=D_{i}^{\epsilon}\left( L\right) \epsilon_{i,t}=\sum_{j=0}^{\infty}D_{i,j}^{\epsilon}\epsilon_{i,t-j}$ with $\bar{D}_{j}^{\epsilon}\equiv\sup_{i}\left\vert \left\vert D_{i,j}^{\epsilon }\right\vert \right\vert <\infty$ and $\sum_{j=0}^{\infty}j\left\vert \left\vert \bar{D}_{j}^{\epsilon}\right\vert \right\vert <\infty$ , and $g_{t}=D^{\eta}\left( L\right) \eta_{t}=\sum_{j=0}^{\infty}D_{j}^{\eta} \eta_{t-j}$ with $\sum_{j=0}^{\infty}j\left\vert \left\vert D_{j}^{\eta }\right\vert \right\vert <\infty$ .
3.   The long-run variance-covariance matrix of $w_{i,t}$ is given by $\Omega_{i}=\left[ \left( \Omega_{uv,i},0\right) ,\left( 0,\Omega _{fg}\right) \right] ^{\prime},$ where $\Omega_{uv,i}=\left[ \left( \omega_{11i},\omega_{12i}\right) ,\left( \omega_{21i},\Omega_{22i}\right) \right] ^{\prime}$ .
4.   $\Omega=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}\Omega_{i}$ .
5.   $E\left[ \left( u_{i,t},v_{i,t}\right) \left( u_{j,s},v_{j,s}\right) ^{\prime}\right] =0$ for all $t,s$ and $i\neq j$.

The innovations to the dependent variable, $u_{i,t}$ and $f_{t}$, satisfy martingale difference sequences (mds) and the innovations to the regressor, $v_{i,t}$ and $g_{t}$, are assumed to follow general linear processes. The idiosyncratic innovations $\left( u_{i,t},v_{i,t}\right)$ are independent of the common factors $\left( f_{t},g_{t}\right)$, and cross-sectionally independent.

By standard arguments, $\frac{1}{\sqrt{T}}\sum_{t=1}^{\left[ Tr\right] }w_{i,t}\Rightarrow B_{i}\left( r\right) =BM\left( \Omega_{i}\right) \left( r\right)$ , where $B_{i}\left( \cdot\right) =\left( B_{1i}\left( \cdot\right) ,B_{2i}\left( \cdot\right) ,B_{f}\left( \cdot\right) ,B_{g}\left( \cdot\right) \right) ^{\prime}$ denote a $1+m+l+k-$dimensional Brownian motion. Further, by the results in Phillips (1987, 1988), it follows that as $T\rightarrow\infty$, $\frac{x_{i,t}}{\sqrt{T}}=\frac{x_{i,t}^{0} }{\sqrt{T}}+\Gamma_{i}^{\prime}\frac{z_{it}}{\sqrt{T}}\Rightarrow J_{i}\left( r\right) +\Gamma_{i}^{\prime}J_{g}\left( r\right)$ , where $J_{i}\left( r\right) =\int_{0}^{r}e^{\left( r-s\right) C_{i}}dB_{2,i}\left( s\right)$ and $J_{g}\left( r\right) =\int_{0}^{r}e^{\left( r-s\right) C_{g}} dB_{g}\left( s\right)$ . Analogous results hold for the time-series demeaned data, $\underline{x}_{i,t}=x_{i,t}-\frac{1}{T}\sum_{t=1}^{T}x_{i,t}$, with $J_{i}$ replaced by $\underline{J}_{i}=J_{i}-\int_{0}^{1}J_{i}$; when there is no risk of confusion, the dependence of $J_{i},J_{g}$ and $B_{i}$ on $r$ will be suppressed. Finally, the following two lemmas summarize the key asymptotic results that are used to prove the main results in the paper.

Lemma 1:   As $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}},$
(a)   $n^{-1/2}\sum_{i=1}^{n}T^{-1}\sum_{t=1}^{T}u_{i,t}x_{i,t-1}\Rightarrow MN\left( 0,\Phi_{ux}+\Phi_{uz}\right)$ where $\Phi_{ux}\equiv E\left[ \left( \int_{0}^{1}dB_{1,i}J_{i}\right) \left( \int_{0}^{1}dB_{1,i} J_{i}\right) ^{\prime}\right]$ , $\Phi_{uz}\equiv\Gamma^{\prime}\left( E\left[ \left. \left( \int_{0}^{1}dB_{1,i}J_{g}\right) \left( \int _{0}^{1}dB_{1,i}J_{g}\right) ^{\prime}\right\vert \mathcal{C}\right] \right) \Gamma$ , and $\mathcal{C}$ is the $\sigma-$field generated by $\left\{ f_{t},g_{t}\right\} _{t=1}^{\infty}$.
(b)   $n^{-1}\sum_{i=1}^{n}T^{-1}\sum_{t=1}^{T}\gamma_{i}^{\prime}f_{t} x_{i,t-1}\Rightarrow\int_{0}^{1}\left( \gamma^{\prime}dB_{f}\right) \left( \Gamma^{\prime}J_{g}\right) .$
(c)   $n^{-1/2}\sum_{i=1}^{n}T^{-1}\sum_{t=1}^{T}\gamma_{i}^{\prime} f_{t}x_{i,t-1}^{0}\Rightarrow MN\left( 0,\Phi_{fx}\right)$ where $\Phi _{fx}=\gamma^{\prime}E\left[ \left. \left( \int_{0}^{1}dB_{f}J_{i}\right) \left( \int_{0}^{1}dB_{f}J_{i}\right) ^{\prime}\right\vert \mathcal{C} \right] \gamma$ .
(d)   $n^{-1}\sum_{i=1}^{n}T^{-2}\sum_{t=1}^{T}x_{i,t}x_{i,t}^{\prime }\Rightarrow\Omega_{xx}+\Omega_{zz}$ where $\Omega_{xx}\equiv E\left[ \int_{0}^{1}J_{i}J_{i}^{\prime}\right]$ and $\Omega_{zz}\equiv\Gamma ^{\prime}\left( \int_{0}^{1}J_{g}J_{g}^{\prime}\right) \Gamma.$
(e)   $n^{-1}\sum_{i=1}^{n}T^{-1}\sum_{t=1}^{T}u_{i,t}\underline{x} _{i,t-1}\rightarrow_{p}-\left( \int_{0}^{1}\int_{0}^{r}E\left[ e^{\left( r-s\right) C_{i}}\right] dsdr\right) \omega_{21}.$
(f)   $n^{-1}\sum_{i=1}^{n}T^{-1}\sum_{t=1}^{T}\gamma_{i}^{\prime}\underline {f}_{t}\underline{x}_{i,t-1}\Rightarrow\int_{0}^{1}\left( \gamma^{\prime }dB_{f}\right) \left( \Gamma^{\prime}\underline{J}_{g}\right) .$
(g)   $n^{-1}\sum_{i=1}^{n}T^{-2}\sum_{t=1}^{T}\underline{x}_{i,t}\underline {x}_{i,t}^{\prime}\Rightarrow\underline{\Omega}_{xx}+\underline{\Omega}_{zz}$ where $\underline{\Omega}_{xx}\equiv E\left[ \int_{0}^{1}\underline{J} _{i}\underline{J}_{i}^{\prime}\right]$ and $\underline{\Omega}_{zz} \equiv\Gamma^{\prime}\left( \int_{0}^{1}\underline{J}_{g}\underline{J} _{g}^{\prime}\right) \Gamma.$

Lemma 2:   The following orders of magnitudes hold:
1.   $\frac{\mathbf{\bar{X}}_{\cdot,-1}^{0\prime}\mathbf{\bar{X}}_{\cdot,-1} ^{0}}{T^{2}}$ is of order $O_{p}\left( \frac{1}{n}\right) .$
2.   $\frac{\mathbf{Z}_{-1}^{\prime}\mathbf{\bar{X}}_{\cdot,-1}^{0}}{T^{2} },\frac{\mathbf{u}_{i}^{\prime}\mathbf{\bar{X}}_{\cdot,-1}^{0}}{T},$ $\frac{\mathbf{f}^{\prime}\mathbf{\bar{X}}_{\cdot,-1}^{0}}{T},$ and $\frac {1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{0\prime}\mathbf{\bar{X} }_{\cdot,-1}^{0}}{T^{2}}$ are of order $O_{p}\left( \frac{1}{\sqrt{n} }\right) .$

B  Technical Proofs

Proof of Lemma 1.  Only part (a) is shown since the rest follow in a similar manner. By standard results and the continuous mapping theorem (CMT), for a fixed $i$ as $T\rightarrow\infty,$

$$\frac{1}{T}\sum_{t=1}^{T}u_{i,t}x_{i,t-1}=\frac{1}{T}\sum_{t=1}^{T}\left( u_{i,t}x_{i,t-1}^{0}+u_{i,t}\Gamma_{i}^{\prime}z_{t-1}\right) \Rightarrow \int_{0}^{1}dB_{1,i}J_{i}+\Gamma_{i}^{\prime}\left( \int_{0}^{1}dB_{1,i} J_{g}\right) .$$

Since $u_{i,t}$, and hence $B_{1,i}$, are cross-sectionally independent, by the central limit theorem (CLT) as $n\rightarrow\infty$,

$$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{1}dB_{1,i}J_{i}\Rightarrow N\left( 0,E\left[ \left( \int_{0}^{1}dB_{1,i}J_{i}\right) \left( \int_{0} ^{1}dB_{1,i}J_{i}\right) ^{\prime}\right] \right) .$$

Conditional on $J_{g}$ (or $\mathcal{C}$), $\int_{0}^{1}dB_{1,i}J_{g}$ is $iid$ with mean zero and, using similar arguments as in Andrews (2005),

$$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\Gamma_{i}^{\prime}\left( \int_{0} ^{1}dB_{1,i}J_{g}\right) \Rightarrow MN\left( 0,E\left[ \left. \Gamma _{i}^{\prime}\left( \int_{0}^{1}dB_{1,i}J_{g}\right) \left( \int_{0} ^{1}dB_{1,i}J_{g}\right) ^{\prime}\Gamma_{i}\right\vert \mathcal{C}\right] \right) .$$

Proof of Lemma 2.  As $T\rightarrow\infty$,

$$\frac{\mathbf{\bar{X}}_{\cdot,-1}^{0\prime}\mathbf{\bar{X}}_{\cdot,-1}^{0} }{T^{2}}=\frac{1}{T^{2}}\sum_{t=1}^{T}\bar{x}_{\cdot,t-1}^{0}\bar{x} _{\cdot,t-1}^{0\prime}=\frac{1}{n}\frac{1}{n}\sum_{i=1}^{n}\sum_{k=1}^{n} \frac{1}{T^{2}}\sum_{t=1}^{T}x_{i,t-1}^{0}x_{j,t-1}^{0\prime}\Rightarrow \frac{1}{n}\left( \frac{1}{n}\sum_{i=1}^{n}\sum_{k=1}^{n}\int_{0}^{1} J_{i}J_{j}^{\prime}\right) =O_{p}\left( \frac{1}{n}\right) O_{p}\left( 1\right) ,$$

since $\int_{0}^{1}J_{i}J_{k}^{\prime}$ is $iid$ with mean zero for all $i\neq j$. The rest follow in an analogous manner.

Proof of Theorem 1.  Note that

$$\frac{\tilde{x}_{i,t=\left[ Tr\right] }}{\sqrt{T}}=\frac{x_{i,t}}{\sqrt{T} }-\frac{1}{n}\sum_{i=1}^{n}\frac{1}{T^{3/2}}\sum_{t=1}^{T}x_{i,t} =\frac{x_{i,t}}{\sqrt{T}}+O_{p}\left( \frac{1}{\sqrt{n}}\right) \Rightarrow J_{i}\left( r\right) ,$$

so that the demeaning has no asymptotic effects (as opposed to in the fixed effects case). The result therefore follows directly by (a) and (c) in Lemma 1, with $\Phi_{uz}=0$ and $\Omega_{zz}=0$, and the CMT.

Proof of Theorem 2.  Note first that, by the continuous mapping theorem, as $T\rightarrow\infty$ for a fixed $i$,

$$\frac{\underline{x}_{i,t}^{dd}}{\sqrt{T}}=\frac{x_{i,t}}{\sqrt{T}}-\left( \frac{T-t+1}{T}\right) ^{-1}\frac{1}{T}\sum_{s=t}^{T}\frac{x_{i,s}}{\sqrt{T} }\Rightarrow J_{i}\left( r\right) -\left( 1-r\right) ^{-1}\int_{r} ^{1}J_{i}\left( u\right) du=\underline{J}_{i}^{dd}\left( r\right) .$$

Now, for fixed $n$, as $T\rightarrow\infty$,

$$\sqrt{n}T\left( \hat{\beta}_{RD}-\beta\right) =\left( \frac{1}{n} \sum_{i=1}^{n}\frac{1}{T^{2}}\sum_{t=1}^{T}\underline{x}_{i,t-1}^{dd} x_{i,t-1}^{\prime}\right) ^{-1}\left( \frac{1}{\sqrt{n}}\sum_{i=1}^{n} \frac{1}{T}\sum_{t=1}^{T}\underline{u}_{i,t}^{dd}x_{i,t-1}\right)$$

$$\Rightarrow\left( \frac{1}{n}\sum_{i=1}^{n}\int_{0}^{1}\underline{J} _{i}^{dd}J_{i}^{\prime}\right) ^{-1}\left( \frac{1}{\sqrt{n}}\sum_{i=1} ^{n}\int_{0}^{1}\left[ dB_{1,i}\left( r\right) -\left( 1-r\right) ^{-1}\Delta_{r}B_{1,i}\left( 1\right) dr\right] J_{i}\right) ,$$

where $\Delta_{r}B_{1,i}\left( 1\right) \equiv B_{1,i}\left( 1\right) -B_{1,i}\left( r\right)$ , since

$$\frac{1}{T}\sum_{t=1}^{T}\underline{u}_{i,t}^{dd}x_{i,t-1} =\frac{1}{T} \sum_{t=1}^{T}\left( u_{i,t}-\frac{1}{T-t+1}\sum_{s=t}^{T}u_{i,s}\right) x_{i,t-1}$$

$$=\frac{1}{T}\sum_{t=1}^{T}u_{i,t}x_{i,t-1}-\frac{1}{T}\sum_{t=1}^{T}\left( \frac{T-t+1}{T}\right) ^{-1}\left( \frac{1}{\sqrt{T}}\left( \sum_{s=1} ^{T}u_{i,s}-\sum_{s=1}^{t-1}u_{i,s}\right) \right) \frac{x_{i,t-1}}{\sqrt {T}}$$

$$\Rightarrow\int_{0}^{1}dB_{1,i}\left( r\right) J_{i}\left( r\right) -\int_{0}^{1}\left( 1-r\right) ^{-1}\left( B_{1,i}\left( 1\right) -B_{1,i}\left( r\right) \right) J_{i}\left( r\right) dr,$$ (31)

$$\equiv\int_{0}^{1}\left[ dB_{1,i}\left( r\right) -\left( 1-r\right) ^{-1}\Delta_{r}B_{1,i}\left( 1\right) dr\right] J_{i}\left( r\right)$$

by standard arguments. By the independent increments property of the Brownian motion, it follows that the expectation of (31) is equal to zero. Denote $\underline{\Omega}_{xx}^{RD}=E\left[ \int_{0}^{1}\underline{J}_{i}^{dd} J_{i}^{\prime}\right]$ , and

$$\underline{\Phi}_{ux}^{RD}=E\left[ \left( \int_{0}^{1}J_{i}\left( r\right) \left[ dB_{1,i}\left( r\right) -\left( 1-r\right) ^{-1}\Delta_{r} B_{1,i}\left( 1\right) dr\right] \right) \left( \int_{0}^{1}J_{i}\left( r\right) \left[ dB_{1,i}\left( r\right) -\left( 1-r\right) ^{-1} \Delta_{r}B_{1,i}\left( 1\right) dr\right] \right) ^{\prime}\right] ,$$

and the result follows from similar arguments as before.

Proof of Theorem 3.  Since $\beta_{i}=\beta +\theta_{i}$, it follows as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$,

$$\sqrt{n}\left( \hat{\beta}_{FE}-\beta\right) =\left( \frac{1}{n} \sum_{i=1}^{n}\frac{1}{T^{2}}\sum_{t=1}^{T}\underline{x}_{i,t-1}\underline {x}_{i,t-1}^{\prime}\right) ^{-1}\left( \frac{1}{\sqrt{n}}\sum_{i=1} ^{n}\frac{1}{T^{2}}\sum_{t=1}^{T}\underline{x}_{i,t-1}\left( \underline {x}_{i,t-1}^{\prime}\theta_{i}+u_{i,t}\right) \right)$$

$$\Rightarrow N\left( 0,\underline{\Omega}_{xx}^{-1}E\left[ \left( \int _{0}^{1}\underline{J}_{i}\underline{J}_{i}^{\prime}\right) \theta_{i}\left( \left( \int_{0}^{1}\underline{J}_{i}\underline{J}_{i}^{\prime}\right) \theta_{i}\right) ^{\prime}\right] \underline{\Omega}_{xx}^{-1}\right) \equiv N\left( 0,\underline{\Omega}_{xx}^{-1}\underline{\Phi}_{xx}^{\theta }\underline{\Omega}_{xx}^{-1}\right) ,$$

since the $u_{i,t}$ term is asymptotically irrelevant. In part (b), $E\left[ \left( \int_{0}^{1}\underline{J}_{i}\underline{J}_{i}^{\prime }\right) \theta_{i}\right] \neq0$ , and the result follows.

Proof of Theorem 4.  Under the null of $\beta _{i}=\beta$ for all $i$, using the matrix notation of Section III,

$$S_{\beta}=\sum_{i=1}^{n}\left\{ \left( \frac{\mathbf{U}_{i}^{\prime }\underline{\mathbf{X}}_{i-1}}{T\sqrt{\hat{\omega}_{11i}}}\right) \left( \frac{\underline{\mathbf{X}}_{i-1}^{\prime}\underline{\mathbf{X}}_{i-1}} {T^{2}}\right) ^{-1}\left( \frac{\underline{\mathbf{X}}_{i-1}^{\prime }\mathbf{U}_{i}}{T\sqrt{\hat{\omega}_{11i}}}\right) -\left( \frac {\mathbf{U}_{i}^{\prime}\underline{\mathbf{X}}_{i-1}}{T\sqrt{\hat{\omega }_{11i}}}\right) \left( \sum_{i=1}^{n}\frac{\underline{\mathbf{X}} _{i,-1}^{\prime}\underline{\mathbf{X}}_{i,-1}}{T^{2}}\right) ^{-1}\left( \sum_{i=1}^{n}\frac{\underline{\mathbf{X}}_{i,-1}^{\prime}\mathbf{U}_{i} }{T\sqrt{\hat{\omega}_{11i}}}\right) \right\} .$$

Write $\Omega_{22i}=L_{i}^{\prime}L_{i},$ so that $J_{i}=L_{i}^{\prime} J_{i}^{W}=\int_{0}^{r}e^{\left( r-s\right) C_{i}}dW_{2,i}\left( s\right)$ and $B_{1,i}=\sqrt{\omega_{11i}}W_{1,i}$; $W_{1,i}$ and $W_{2,i}$ are thus standardized Brownian motions with correlation $\delta_{i}$. That is, $\delta_{i}$ is the correlation between $u_{i,t}$ and $v_{i,t}$, and let $\delta=\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\delta_{i}$ be the average correlation. For a fixed $n$, as $T\rightarrow\infty$,

$$\frac{S_{\beta}}{n} \Rightarrow\frac{1}{n}\sum_{i=1}^{n}\left\{ \left( \int_{0}^{1}dW_{1,i}\underline{J}_{i}^{W\prime}\right) \left( \int_{0} ^{1}\underline{J}_{i}^{W}\underline{J}_{i}^{W\prime}\right) ^{-1}\left( \int_{0}^{1}\underline{J}_{i}^{W}dW_{1,i}\right) \right.$$

$$\left. -\left( \int_{0}^{1}dW_{1,i}\underline{J}_{i}^{W\prime} L_{i}\right) \left( \frac{1}{n}\sum_{i=1}^{n}\int_{0}^{1}L_{i}^{\prime }\underline{J}_{i}^{W}\underline{J}_{i}^{W\prime}L_{i}\right) ^{-1}\left( \frac{1}{n}\sum_{i=1}^{n}\int_{0}^{1}L_{i}^{\prime}\underline{J}_{i} ^{W}dW_{1,i}\right) \right\} \equiv\frac{1}{n}\sum_{i=1}^{n}Z_{i,n} \equiv\bar{Z}_{i,n}$$

Define $\mu_{Z}\equiv E\left[ \bar{Z}_{i,n}\right]$, $\left. \sigma _{Z}^{2}\right/ n\equiv Var\left( \bar{Z}_{i,n}\right)$ and it follows by the CLT, as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$, $\Delta_{\beta}=\sqrt{n}\left( \left. \left( \frac{1}{n}\hat{S}_{\beta} -\mu_{Z}\right) \right/ \sigma_{Z}\right) \Rightarrow N\left( 0,1\right) .$ Write $W_{1,i}=W_{1\cdot2,i}+\delta_{i}^{\prime}W_{2,i}$, where $W_{1\cdot2,i}$ is orthogonal to $W_{2,i}$. It follows easily, as $n\rightarrow\infty$,

$$E\left[ \bar{Z}_{i,n}\right] \rightarrow_{p}E\left[ \left( \int_{0} ^{1}dW_{1\cdot2,i}\underline{J}_{i}^{W\prime}\right) \left( \int_{0} ^{1}\underline{J}_{i}^{W}\underline{J}_{i}^{W\prime}\right) ^{-1}\left( \int_{0}^{1}\underline{J}_{i}^{W}dW_{1\cdot2,i}\right) \right]$$

$$+\delta^{\prime}E\left[ \left( \int_{0}^{1}dW_{2,i}\underline{J} _{i}^{W\prime}\right) \left( \int_{0}^{1}\underline{J}_{i}^{W}\underline {J}_{i}^{W\prime}\right) ^{-1}\left( \int_{0}^{1}\underline{J}_{i} ^{W}dW_{2,i}^{\prime}\right) \right] \delta$$

$$-\delta^{\prime}E\left[ \int_{0}^{1}dW_{2,i}\underline{J}_{i}^{W\prime }\right] E\left[ \int_{0}^{1}\underline{J}_{i}^{W}\underline{J}_{i} ^{W\prime}\right] ^{-1}E\left[ \int_{0}^{1}\underline{J}_{i}^{W} dW_{2,i}^{\prime}\right] \delta.$$

Proof of Theorem 5.  The results follow immediately from Lemma 1 and the CMT.

Proof of Theorem 6.  Note that,

$$\sqrt{n}T\left( \hat{\beta}_{Pool}^{+}-\beta\right) =\left( \frac{1}{n} \sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M}_{\mathbf{\bar{H}} }\mathbf{X}_{i,-1}}{T^{2}}\right) ^{-1}\left[ \left( \frac{1}{\sqrt{n}} \sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M}_{\mathbf{\bar{H}} }\mathbf{f}\gamma_{i}}{T}\right) +\left( \frac{1}{\sqrt{n}}\sum_{i=1} ^{n}\frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M}_{\mathbf{\bar{H}}} \mathbf{u}_{i}}{T}\right) \right] .$$

By the rank condition on $\Gamma$, $\mathbf{M}_{\mathbf{Q}}\mathbf{X} _{i,-1}=\mathbf{M}_{\mathbf{G}}\mathbf{X}_{i,-1}\mathbf{=X}_{i,-1}^{0}$ , since $\mathbf{Z}_{-1}\mathbf{\subset G}$. From Lemma 2,

$$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime} \mathbf{M}_{\mathbf{\bar{H}}}\mathbf{f}\gamma_{i}}{T}=\frac{1}{\sqrt{n}} \sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M}_{\mathbf{Q} }\mathbf{f}\gamma_{i}}{T}+O_{p}\left( \frac{1}{\sqrt{n}}\right) =\frac {1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{0\prime}\mathbf{f} \gamma_{i}}{T}+O_{p}\left( \frac{1}{\sqrt{n}}\right) ,$$

and as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$, $\frac {1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M} _{\mathbf{\bar{H}}}\mathbf{f}\gamma_{i}}{T}\Rightarrow MN\left( 0,\Phi _{fx}\right) .$ Similarly,

$$\frac{1}{\sqrt{n}T}\sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime} \mathbf{M}_{\mathbf{\bar{H}}}\mathbf{u}_{i}}{T}=\frac{1}{\sqrt{n}T}\sum _{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M}_{\mathbf{Q}} \mathbf{u}_{i}}{T}+O_{p}\left( \frac{1}{\sqrt{n}}\right) =\frac{1}{\sqrt {n}T}\sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{0\prime}\mathbf{u}_{i}}{T} +O_{p}\left( \frac{1}{\sqrt{n}}\right) ,$$

and as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$, $\frac {1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M} _{\mathbf{Q}}\mathbf{u}_{i}}{T}\Rightarrow N\left( 0,\Phi_{ux}\right) .$ Finally,

$$\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M} _{\mathbf{\bar{H}}}\mathbf{X}_{i,-1}}{T^{2}}=\frac{1}{n}\sum_{i=1}^{n} \frac{\mathbf{X}_{i,-1}^{\prime}\mathbf{M}_{\mathbf{Q}}\mathbf{X}_{i,-1} }{T^{2}}+O_{p}\left( \frac{1}{\sqrt{n}}\right) =\frac{1}{n}\sum_{i=1} ^{n}\frac{\mathbf{X}_{i,-1}^{0\prime}\mathbf{X}_{i,-1}^{0}}{T^{2}} +O_{p}\left( \frac{1}{\sqrt{n}}\right) \rightarrow_{p}E\left[ \int_{0} ^{1}J_{i}J_{i}^{\prime}\right] ,$$

as $\left( T,n\rightarrow\infty\right) _{\operatorname{seq}}$. Summing up, $\sqrt{n}T\left( \hat{\beta}_{Pool}^{+}-\beta\right) \Rightarrow MN\left( 0,\Omega_{xx}^{-1}\left( \Phi_{ux}+\Phi_{fx}\right) \Omega_{xx}^{-1}\right) .$ $\qedsymbol$

Proof of Corollary 1.  The result follows in an identical manner to above.

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Table 1:  Size Results from the Monte Carlo Study - Panel A: No Common Factor

The table shows the average rejection rates under the null of β = 0, for the t-tests corresponding to the respective estimators; the nominal size of the tests are 5 percent. The differing values of δ are given in the top row of the table and the results are based on 10,000 repetitions. The sample size used is T = 100 and n = 20. Panel A shows the results when no common factors are included in the data generating process, Panel B shows the effects of including common factors in the data but ignoring them in the estimation, and Panel C shows the results when common factors are included and accounted for in the estimation.

Estimator	δ = 0.0	δ = -0.4	δ = -0.7	δ = -0.95
$\hat{\beta}_{FE}$	0.078	0.177	0.379	0.563
$\hat{\beta}_{RD}$	0.069	0.071	0.071	0.074

Table 1: Size Results from the Monte Carlo Study - Panel B: Common Factor with no Correction

Estimator	δ = 0.0	δ = -0.4	δ = -0.7	δ = -0.95
$\hat{\beta}_{FE}$	0.430	0.503	0.559	0.604
$\hat{\beta}_{RD}$	0.353	0.348	0.356	0.347

Table 1: Size Results from the Monte Carlo Study - Panel C: Common Factor Using Correction

Estimator	δ = 0.0	δ = -0.4	δ = -0.7	δ = -0.95
$\hat{\beta}_{FE}^{+}$	0.026	0.065	0.157	0.284
$\hat{\beta}_{RD}^{+}$	0.029	0.031	0.039	0.052

Table 2:  Size and Power for the Test of Slope Homogeneity - Panel A: No Common Factor

The table shows the average rejection rates for the Δ_β-test of slope homogeneity under the null of β_i = β = 0 for all i, as well as under two different alternative hypotheses; the nominal size of the test is 5 percent. The alternative hypotheses are given by β_i = β + βθ_i , where θ_i is a standard normal random variable, independent across i and β is equal to 0.05 and 0.1 under Alternative 1 and Alternative 2, respectively. The differing values of δ are given in the top row of the table and the results are based on 10,000 repetitions. The sample size used is T = 100 and N = 20. Panel A shows the results when no common factors are included in the data generating process and Panel B shows the effects of including common factors in the data.

Hypothesis	δ = 0.0	δ = -0.4	δ = -0.7	δ = -0.95
Null	0.014	0.015	0.014	0.016
Alternative 1	0.453	0.450	0.445	0.439
Alternative 2	0.958	0.955	0.956	0.951

Table 2: Size and Power for the Test of Slope Homogeneity - Panel B: Common Factor

Hypothesis	δ = 0.0	δ = -0.4	δ = -0.7	δ = -0.95
Null	0.007	0.008	0.007	0.009
Alternative 1	0.367	0.347	0.346	0.324
Alternative 2	0.923	0.922	0.916	0.913

Table 3:  Pooled Results - Panel A: The Earnings-Price Ratio

The first column indicates which panel is being used, with (1950.1-) denoting that all observations before 1950 are dropped from that panel. The next two columns give the number of individual time series and total number of observations in the panel. The column labeled Δ_β shows the outcome of the test of slope homogeneity. The following four columns report the results from the estimation procedures that do not control for common factors. The pooled estimate based on recursive demeaning of the data, the standard fixed effects estimate, and the two corresponding t-statistics are shown. The next four columns give the corresponding results when controlling for common factors. The last column gives the number of significant coefficients in the individual time-series regressions performed on each of the time-series in the panel, based on the Campbell and Yogo methods. The $t_{RD}^{+}$-statistics and the Δ_β-statistics that are significant at the one-sided five percent level (i.e. greater than 1.65) are shown with a * next to them.

Panel	n	# obs	Δβ	No Control for Common Factors: $\hat{\beta}_{RD}$	No Control for Common Factors: $\hat{\beta}_{FE}$	No Control for Common Factors: tRD	No Control for Common Factors: tFE	Controlling for Common Factors: $\hat{\beta}_{RD}^{+}$	Controlling for Common Factors: $\hat{\beta}_{FE}^{+}$	Controlling for Common Factors: $t_{RD}^{+}$	Controlling for Common Factors: $t_{FE}^{+}$	CYsig
Global	38	13,610	2.234*	0.020	0.010	0.334	4.079	0.002	0.001	1.126	1.518	5
Global (1950.1-)	38	12,400	1.984*	0.035	0.010	0.391	3.905	0.002	0.001	1.172	1.817	4
Developed	23	10,151	1.399	-0.008	0.005	-0.751	2.866	-0.001	0.000	-0.802	-0.805	2
Developed (1950.1-)	23	8,941	1.030	-0.003	0.004	-0.353	2.637	-0.001	0.000	-0.688	-0.029	1
Emerging	16	3,518	-1.638	-0.018	0.024	-1.441	6.252	0.011	0.007	2.438*	2.415	4

Table 3: Pooled Results - Panel B: The Dividend-Price Ratio

Panel	n	# obs	Δβ	No Control for Common Factors: $\hat{\beta}_{RD}$	No Control for Common Factors: $\hat{\beta}_{FE}$	No Control for Common Factors: tRD	No Control for Common Factors: tFE	Controlling for Common Factors: $\hat{\beta}_{RD}^{+}$	Controlling for Common Factors: $\hat{\beta}_{FE}^{+}$	Controlling for Common Factors: $t_{RD}^{+}$	Controlling for Common Factors: $t_{FE}^{+}$	CYsig
Global	46	20,594	2.485*	0.006	0.006	1.216	3.352	0.001	0.000	0.628	0.953	6
Global (1950.1-)	41	14,947	1.905*	0.007	0.009	1.354	4.742	0.002	0.001	2.334*	3.318	7
Developed	30	17,001	1.733*	0.001	0.004	0.556	2.149	0.000	0.000	-0.344	-0.142	2
Developed (1950.1-)	25	11,354	1.263	0.001	0.007	0.679	4.583	0.002	0.001	2.402*	4.307	3
Emerging	16	3,593	1.156	-0.108	0.013	-0.371	2.814	0.005	0.004	1.658*	2.238	4

Table 3: Pooled Results - Panel C: The Short Rate

Panel	n	# obs	Δβ	No Control for Common Factors: $\hat{\beta}_{RD}$	No Control for Common Factors: $\hat{\beta}_{FE}$	No Control for Common Factors: tRD	No Control for Common Factors: tFE	Controlling for Common Factors: $\hat{\beta}_{RD}^{+}$	Controlling for Common Factors: $\hat{\beta}_{FE}^{+}$	Controlling for Common Factors: $t_{RD}^{+}$	Controlling for Common Factors: $t_{FE}^{+}$	CYsig
Global	39	15,260	2.666*	0.224	0.100	1.103	0.494	0.098	0.086	0.476	0.462	11
Developed	23	11,468	1.341	-0.525	-0.980	-1.858	-4.164	-0.958	-0.687	-4.494*	-3.958	8
Emerging	16	3,792	1.846*	0.250	0.137	1.115	0.604	0.107	0.087	0.484	0.434	3

Table 3: Pooled Results - Panel D: The Term Spread

Panel	n	# obs	Δβ	No Control for Common Factors: $\hat{\beta}_{RD}$	No Control for Common Factors: $\hat{\beta}_{FE}$	No Control for Common Factors: tRD	No Control for Common Factors: tFE	Controlling for Common Factors: $\hat{\beta}_{RD}^{+}$	Controlling for Common Factors: $\hat{\beta}_{FE}^{+}$	Controlling for Common Factors: $t_{RD}^{+}$	Controlling for Common Factors: $t_{FE}^{+}$	CYsig
Global	35	13,048	2.310*	0.223	0.902	0.736	1.247	0.051	0.113	0.185	0.423	12
Developed	23	10,784	1.368	2.133	1.957	3.906	3.869	1.685	1.794	2.809*	3.318	10
Emerging	12	2,264	0.612	0.028	0.318	0.147	0.353	-0.159	-0.089	-0.712	-0.469	2

Table 4:  Country Level Results - Panel A: The Earnings-Price Ratio

The first and second columns indicate the country and sample period, respectively, on which the estimates are based. The third and fourth columns show the OLS estimates and corresponding t-statistics, respectively. A * next to the coefficient estimate indicates that the coefficient is significantly different from zero, with the expected sign, according to the robust 90 percent confidence interval obtained from the Campbell and Yogo procedure; this is equivalent to a one-sided rejection of the null hypothesis at the five percent level.

Country	Sample	$\hat{\beta}_{i}$	$t_{i}$
Argentina	1988.1 - 2004.6	0.041*	2.277
Australia	1962.1 - 2004.6	0.010	1.468
Austria	1981.11 - 2004.6	0.002	0.423
Belgium	1969.9 - 2004.6	0.012	1.305
Brazil	1988.3 - 2004.5	0.022	1.632
Canada	1956.3 - 2004.6	0.000	0.064
Chile	1988.3 - 2004.6	0.024	2.324
Denmark	1970.1 - 2004.6	0.001	0.165
Finland	1988.3 - 2004.6	-0.005	-0.615
France	1971.11 - 2004.6	-0.001	-0.112
Germany	1969.9 - 2004.6	0.008	1.359
Greece	1977.3 - 2004.6	-0.003	-0.302
Hong Kong	1973.1 - 2004.6	0.060	3.610
Hungary	1993.3 - 2004.6	0.001	0.088
India	1988.3 - 2003.12	0.025	1.640
Ireland	1990.7 - 2004.6	0.034	1.731
Italy	1981.3 - 2004.6	0.005	0.959
Japan	1956.3 - 2004.6	0.006	1.864
Jordan	1988.3 - 2003.2	0.025*	1.787
Malaysia	1973.1 - 2004.6	0.022	1.553
Mauritius	1996.3 - 2002.12	0.003	0.225
Mexico	1988.1 - 2004.6	0.017	0.959
Netherlands	1969.9 - 2004.6	0.002	0.439
New Zealand	1988.3 - 2004.6	-0.007	-0.979
Norway	1970.1 - 2001.9	-0.004	-0.764
Philippines	1982.3 - 2004.5	0.011	0.837
Poland	1992.3 - 2004.5	0.025	1.398
Portugal	1988.3 - 2004.6	0.010	0.832
Singapore	1973.1 - 2004.6	0.041	2.821
South Africa	1960.3 - 2004.6	0.020*	2.277
Spain	1980.1 - 2004.6	0.018	1.405
Sweden	1969.9 - 2004.6	0.002	0.343
Switzerland	1969.9 - 2004.6	-0.006	-0.879
Taiwan	1988.3 - 2004.1	0.043	1.761
Thailand	1975.6 - 2004.6	0.013	1.300
Turkey	1986.3 - 2004.6	0.035	2.706
UK	1928.1 - 2004.6	0.010*	2.471
UK	1950.1 - 2004.6	0.011*	2.513
USA	1871.3 - 2004.6	0.011*	3.398
USA	1950.1 - 2004.6	0.007	1.874

Table 4: Country Level Results - Panel B: The Dividend-Price Ratio

Country	Sample	$\hat{\beta}_{i}$	$t_{i}$
Argentina	1988.3 - 2004.6	-0.003	-0.224
Australia	1882.12 - 2004.6	0.005	1.416
Australia	1950.1 - 2004.6	0.013*	1.916
Austria	1970.2 - 2004.6	0.001	0.196
Belgium	1952.1 - 2004.6	0.003	0.694
Brazil	1988.3 - 2004.5	0.000	-0.016
Canada	1934.3 - 2004.6	0.006	1.592
Canada	1950.1 - 2004.6	0.007	1.483
Chile	1983.3 - 2004.6	0.019*	2.234
Denmark	1970.2 - 2004.6	0.000	0.071
Finland	1962.3 - 2004.6	-0.004	-0.742
France	1898.1 - 1914.7	0.048	1.681
France	1919.2 - 1940.3	0.018	0.813
France	1941.5 - 2004.6	-0.001	-0.353
France	1950.1 - 2004.6	0.004	0.836
Germany	1872.9 - 1942.3	0.035	2.123
Germany	1953.2 - 2004.6	0.003	0.451
Greece	1977.3 - 2004.6	0.003	0.505
Hong Kong	1973.1 - 2004.6	0.070	4.108
Hungary	1993.12 - 2004.6	0.017	0.874
India	1988.3 - 2003.12	0.036	2.213
Ireland	1990.7 - 2004.6	0.047	2.775
Israel	1994.1 - 2004.6	0.018	1.139
Italy	1925.3 - 2004.6	-0.006	-1.623
Italy	1950.1 - 2004.6	0.006	0.988
Japan	1922.1 - 1942.1	-0.008	-0.665
Japan	1949.7 - 2004.6	0.006*	2.474
Jordan	1988.2 - 2003.2	0.013*	1.802
Luxembourg	1985.2 - 1994.12	-0.007	-0.512
Malaysia	1973.1 - 2004.6	0.042	2.776
Mauritius	1997.1 - 2002.12	0.004	0.515
Mexico	1988.3 - 2004.6	0.037*	2.273
Netherlands	1969.9 - 2004.6	0.004	0.720
New Zealand	1987.1 - 2004.6	0.039	2.290
Norway	1970.2 - 2001.9	0.008	0.937
Philippines	1982.3 - 2004.5	0.003	0.635
Poland	1993.12 - 2004.5	0.045	2.451
Portugal	1988.4 - 2004.6	-0.004	-0.367
Singapore	1973.1 - 2004.6	0.034	2.714
South Africa	1960.4 - 2004.6	0.018	2.063
Spain	1940.6 - 1968.12	0.002	0.230
Spain	1981.3 - 2004.6	0.003	0.638
Sweden	1919.2 - 2004.6	-0.002	-0.532
Sweden	1950.1 - 2004.6	0.006	1.140
Switzerland	1966.4 - 2004.6	0.001	0.091
Taiwan	1988.3 - 2004.1	0.024*	2.188
Thailand	1976.1 - 2004.6	0.004	0.620
Turkey	1986.4 - 2004.6	0.041	2.687
UK	1836.1 - 1916.12	0.006	1.585
UK	1924.2 - 2004.6	0.018*	2.970
UK	1950.1 - 2004.6	0.026*	3.550
USA	1871.3 - 2004.6	0.004	1.105
USA	1950.1 - 2004.6	0.009*	2.294

Table 4: Country Level Results - Panel C: The Short Interest Rate

Country	Sample	$\hat{\beta}_{i}$	$t_{i}$
Argentina	1988.1 - 2004.5	2.034	5.349
Australia	1952.1 - 2004.3	-0.711	-1.260
Austria	1970.1 - 2004.5	-1.739	-1.179
Belgium	1952.1 - 2004.6	-0.514	-0.682
Brazil	1988.1 - 2004.5	-0.042	-0.417
Canada	1952.1 - 2004.5	-1.489*	-2.540
Chile	1983.1 - 2004.5	0.770	1.418
Denmark	1970.1 - 2004.5	-0.528	-0.917
Finland	1962.1 - 2004.5	-1.369	-1.541
France	1952.1 - 2004.5	-1.022	-1.399
Germany	1953.1 - 2004.5	-3.201*	-2.572
Greece	1977.1 - 2004.5	1.689	1.440
Hong Kong	1970.1 - 2004.5	-3.076	-1.355
Hungary	1991.3 - 2004.5	-0.922	-0.698
India	1988.1 - 2003.12	-1.368	-0.794
Ireland	1988.3 - 2004.5	-0.338	-0.292
Israel	1993.1 - 2004.5	2.489	1.219
Italy	1952.1 - 2004.5	-0.886	-1.512
Japan	1952.1 - 2004.5	1.182	1.178
Jordan	1988.1 - 2003.2	-6.669*	-3.393
Malaysia	1972.12 - 2004.5	-4.135	-1.091
Mauritius	1989.9 - 2004.5	-7.398*	-3.635
Mexico	1988.1 - 2004.6	-0.011	-0.030
Netherlands	1952.1 - 2004.5	-1.932*	-2.213
New Zealand	1986.8 - 2004.3	-4.346*	-3.915
Norway	1970.1 - 2001.9	-1.589	-1.223
Philippines	1982.1 - 2004.5	-0.173	-0.175
Poland	1991.6 - 2004.5	0.039	0.033
Portugal	1988.3 - 2004.5	-2.221*	-1.986
Singapore	1970.1 - 2004.5	-0.308	-0.153
South Africa	1960.3 - 2004.5	-0.991	-1.354
Spain	1952.1 - 2004.5	-1.609*	-3.045
Sweden	1952.1 - 2004.5	0.475	0.651
Switzerland	1966.3 - 2004.6	-2.492*	-2.010
Taiwan	1988.1 - 2004.1	-4.339	-0.798
Thailand	1975.6 - 2004.6	-5.321*	-3.540
Turkey	1986.3 - 2004.6	0.518	0.595
UK	1952.1 - 2004.5	-0.475	-0.629
USA	1952.1 - 2004.5	-1.825*	-2.558

Table 4: Country Level Results - Panel D: The Term Spread

Country	Sample	$\hat{\beta}_{i}$	$t_{i}$
Argentina	1997.1-2004.6	2.682*	2.169
Australia	1952.1 - 2004.6	1.428	1.056
Austria	1970.1 - 2004.6	-0.976	-0.345
Belgium	1952.1 - 2004.6	0.822	0.366
Brazil	1994.1 - 2004.5	0.006	0.005
Canada	1952.1 - 2004.6	3.387*	2.467
Denmark	1970.1 - 2004.6	0.591	0.520
Finland	1962.1 - 2004.6	2.559	1.345
France	1952.1 - 2004.6	2.902*	1.646
Germany	1953.1 - 2004.6	3.402*	1.780
Greece	1993.3 - 2004.6	-4.889	-1.212
Hong Kong	1994.11 - 2004.6	0.271	0.029
Hungary	1997.4 - 2004.6	3.692	0.308
India	1988.1 - 2003.12	1.813	0.936
Ireland	1988.3 - 2004.6	0.111	0.069
Italy	1952.1 - 2004.6	2.656*	1.659
Japan	1952.1 - 2004.6	-3.154	-1.997
Malaysia	1972.12 - 2004.6	3.579	0.911
Mexico	1995.3 - 2004.6	-0.488	-0.227
Netherlands	1952.1 - 2004.6	2.922*	1.824
New Zealand	1986.8 - 2004.6	10.240*	4.099
Norway	1970.1 - 2001.9	3.022*	1.497
Philippines	1994.11 - 2004.5	-0.966	-0.331
Poland	1994.4 - 2004.5	3.553	0.529
Portugal	1988.3 - 2004.6	5.605	1.142
Singapore	1988.1 - 2004.6	11.227	1.415
South Africa	1960.3 - 2004.6	2.240	1.425
Spain	1952.1 - 2004.6	3.678*	3.555
Sweden	1952.1 - 2004.6	-0.106	-0.061
Switzerland	1966.3 - 2004.6	3.234*	1.769
Taiwan	1995.3 - 2004.1	-4.869	-0.388
Thailand	1977.2 - 2004.6	6.253*	2.275
Turkey	1997.11 - 2004.6	-1.474	-1.065
UK	1952.1 - 2004.6	0.941	0.711
USA	1952.1 - 2004.6	4.946*	2.826

Table 5:  Out-of-Sample Results - Panel A: The Earnings-Price Ratio

The first and second columns indicate the country and the sample period that are used, respectively. The next three columns show the in-sample standard t-statistic, the full sample R² expressed in percent, and whether the in-sample coefficient estimate is found signficant according to the Campbell and Yogo test. The following two columns show the out-of-sample R² expressed in percent, based on the time-series estimates and the pooled estimates, respectively.

Country	Sample	In-Sample: $t_{i}$	In-Sample: $100\times R_{i}^{2}$	In-Sample: CYsig	Time-Series: $100\times R_{i,OS}^{2}$	Pooled: $100\times R_{i,OS}^{2}$
Australia	1962.1 - 2004.6	1.468	0.423	NO	-0.910	-0.138
Canada	1956.3 - 2004.6	0.064	0.001	NO	-0.529	-1.719
Japan	1956.3 - 2004.6	1.864	0.598	NO	0.289	0.606
South Africa	1960.3 - 2004.6	2.277	0.969	YES	0.899	0.644
UK	1928.1 - 2004.6	2.471	0.662	YES	0.396	0.627
USA	1871.3 - 2004.6	3.398	0.718	YES	0.435	0.570

Table 5: Out-of-Sample Results - Panel B: The Dividend-Price Ratio

Country	Sample	In-Sample: $t_{i}$	In-Sample: $100\times R_{i}^{2}$	In-Sample: CYsig	Time-Series: $100\times R_{i,OS}^{2}$	Pooled: $100\times R_{i,OS}^{2}$
Australia	1882.12 - 2004.6	1.416	0.138	NO	-0.328	0.012
Belgium	1952.1 - 2004.6	0.694	0.077	NO	-0.493	-0.047
Canada	1934.3 - 2004.6	1.592	0.300	NO	-0.160	-0.294
Finland	1962.3 - 2004.6	-0.742	0.109	NO	-0.584	-0.097
France	1941.5 - 2004.6	-0.353	0.016	NO	-0.186	0.151
Germany	1953.2 - 2004.6	0.451	0.033	NO	-0.268	-0.032
Italy	1925.3 - 2004.6	-1.623	0.276	NO	-1.280	-0.661
Japan	1949.7 - 2004.6	2.474	0.921	YES	-0.355	0.247
South Africa	1960.4 - 2004.6	2.063	0.798	NO	0.808	0.052
Sweden	1919.2 - 2004.6	-0.532	0.028	NO	-0.530	-0.044
UK	1924.2 - 2004.6	2.970	0.908	YES	0.203	-0.241
USA	1871.3 - 2004.6	1.105	0.076	NO	-0.310	-0.571

Table 5: Out-of-Sample Results - Panel C: The Short Interest Rate

Country	Sample	In-Sample: $t_{i}$	In-Sample: $100\times R_{i}^{2}$	In-Sample: CYsig	Time-Series: $100\times R_{i,OS}^{2}$	Pooled: $100\times R_{i,OS}^{2}$
Australia	1952.1 - 2004.3	-1.260	0.254	NO	-0.990	-0.828
Belgium	1952.1 - 2004.6	-0.682	0.074	NO	-0.812	-0.569
Canada	1952.1 - 2004.5	-2.540	1.019	YES	-0.286	-0.383
Finland	1962.1 - 2004.5	-1.541	0.466	NO	-0.024	0.278
France	1952.1 - 2004.5	-1.399	0.311	NO	-0.375	0.048
Germany	1953.1 - 2004.5	-2.572	1.064	YES	-0.141	0.702
Italy	1952.1 - 2004.5	-1.512	0.363	NO	-1.881	-0.548
Japan	1952.1 - 2004.5	1.178	0.221	NO	-0.355	-0.948
Netherlands	1952.1 - 2004.5	-2.213	0.775	YES	-0.777	-0.033
South Africa	1960.3 - 2004.5	-1.354	0.345	NO	-0.890	0.382
Spain	1952.1 - 2004.5	-3.045	1.457	YES	0.093	2.970
Sweden	1952.1 - 2004.5	0.651	0.068	NO	-0.894	-1.898
UK	1952.1 - 2004.5	-0.629	0.063	NO	-1.705	-1.855
USA	1952.1 - 2004.5	-2.558	1.032	YES	-1.134	0.902

Table 5: Out-of-Sample Results - Panel D: The Term Spread

Country	Sample	In-Sample: $t_{i}$	In-Sample: $100\times R_{i}^{2}$	In-Sample: CYsig	Time-Series: $100\times R_{i,OS}^{2}$	Pooled: $100\times R_{i,OS}^{2}$
Australia	1952.1 - 2004.6	1.056	0.177	NO	-0.404	-0.158
Belgium	1952.1 - 2004.6	0.366	0.021	NO	-0.870	0.020
Canada	1952.1 - 2004.6	2.467	0.960	YES	0.229	0.459
Finland	1962.1 - 2004.6	1.345	0.355	NO	0.199	0.511
France	1952.1 - 2004.6	1.646	0.430	YES	0.244	0.266
Germany	1953.1 - 2004.6	1.780	0.512	YES	0.342	0.503
Italy	1952.1 - 2004.6	1.659	0.436	YES	-0.213	0.546
Japan	1952.1 - 2004.6	-1.997	0.631	NO	-0.599	-0.110
Netherlands	1952.1 - 2004.6	1.824	0.527	YES	-0.242	0.349
South Africa	1960.3 - 2004.6	1.425	0.382	NO	-0.753	0.476
Spain	1952.1 - 2004.6	3.555	1.973	YES	1.939	1.775
Sweden	1952.1 - 2004.6	-0.061	0.001	NO	-0.616	-0.495
UK	1952.1 - 2004.6	0.711	0.081	NO	-1.176	-0.264
USA	1952.1 - 2004.6	2.826	1.256	YES	-0.544	0.557

Figure 1:  Power Results from the Monte Carlo Study

The graphs show the average rejection rates for a two-sided 5 percent t-test of the null hypothesis of β = 0, for samples with T = 100, and n = 20. The x-axis shows the true value of the parameter β, and the y-axis indicates the average rejection rate. The left-hand column gives the results for the case of exogenous regressors (δ = 0), and the right-hand column gives the results for the case of highly endogenous regressors (δ = -0.95). In the top two panels, A1 and A2, there are no common factors in the data, and the results for the t-tests corresponding to the standard fixed effects estimator, $\hat{\beta}_{FE}$, and the estimator based on recursive demeaning,$\hat{\beta}_{RD}$, are given by the long dashed lines and the short dotted lines, respectively. In the middle panels, B1 and B2, there is a common factor in the data but the estimators $\hat{\beta}_{FE}$ and $\hat{\beta}_{RD}$ that do not control for common factors, are still used. In the bottom panels, C1 and C2, the common factor is controlled for by using the estimators $\hat{\beta}_{FE}^{+}$ and $\hat{\beta }_{RD}^{+}$, and the results for the corresponding t-tests are shown. The flat lines indicate the 5% rejection rate. All results are based on 10,000 repetitions.

Figure 2:  Expanding Window Regression Estimates for the Developed Panel

The left hand graphs depict the fixed effects estimates without controlling for common factors (i.e. $\hat{\beta }_{FE}$), and the corresponding confidence bounds with a nominal coverage rate of 90% that are obtained when pooling at the developed market level all observations available up until the year in the plot. The right hand graphs depict the corresponding results for the recursive demeaning estimator, when also controlling for common factors (i.e. $\hat{\beta}_{RD}^{+}$). A time-series is added to the panel when five years of observations become available. The flat solid lines indicate a value of zero.

Figure 3:  Expanding Window Regression Estimates for the Short Interest Rate

Each graph depicts the point estimate and the Campbell and Yogo 90 percent confidence interval that result from regressing excess returns, in the country indicated, on the lagged value of the short interest rate. The samples used in the estimation include data up till the year shown in the graph. The flat solid lines indicate a value of zero.

Figure 4:  Expanding Window Regression Estimates for the Term Spread

Each graph depicts the point estimate and the Campbell and Yogo 90 percent confidence interval that result from regressing excess returns, in the country indicated, on the lagged value of the term spread. The samples used in the estimation include data up till the year shown in the graph. The flat solid lines indicate a value of zero.

Footnotes

*  This paper is forthcoming in the Journal of Financial and Quantitative Analysis. The current working paper version contains some additional technical material that will not appear in the published article. Return to text

†  I am very grateful to Peter Phillips and Robert Shiller for providing much useful advice. Other helpful comments have been provided by Don Andrews, Jon Faust, Lennart Hjalmarsson, Randi Hjalmarsson, Yuichi Kitamura, George Korniotis, Ugur Lel, Edith Liu, Vadim Marmer, Alex Maynard, Taisuke Otsu, Kevin Song, Jon Wongswan, Jonathan Wright, Pär Österholm, as well as participants in the summer workshop and econometrics seminar at Yale University, the international finance seminar at the Federal Reserve Board, the finance seminar at Göteborg University, and the European Summer Meeting of the Econometric Society in Vienna. Tel.: +1-202-452-2426; fax: +1-202-263-4850; email: [email protected]. The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.  Return to text

1.  See, for instance, Harvey (1991, 1995), Ferson and Harvey (1993), Campbell (2003), Polk et al. (2004), Paye and Timmermann (2006), and Ang and Bekaert (2007). Return to text

2.  Included in the sample are the stock-markets in Hong Kong and Taiwan. Since Hong Kong is part of China and Taiwan is not a formally recognized sovereign state, the use of the term country for these markets is not entirely correct, but is used for convenience throughout the paper. Return to text

3.  A predictive regressor is generally referred to as endogenous if the innovations to the returns are contemporaneously correlated with the innovations to the regressor. When the regressor is strictly stationary, such endogeneity has no impact on the properties of the estimator, but when the regressor is persistent in some manner, the properties of the estimator will be affected; see, for instance, Mankiw and Shapiro (1986), Cavanagh et al. (1995), Stambaugh (1999), Amihud and Hurvich (2004), Lewellen (2004), Campbell and Yogo (2006), and Jansson and Moreira (2006). Return to text

4.  By imposing a common intercept α, it is also implicitly assumed that the mean of x_i,t is the same for all i. In the international data used in this paper, this is often a very restrictive assumption. For instance, the nominal short interest rate will on average be much higher in countries that have experienced high inflation over the sample period, but it is typically too strong an assumption to assume that the stock returns in these countries therefore have been below their long-run average for most of the sample period, as a predictive regression with a common intercept and negative slope coefficient would imply. Return to text

5.  One could make a similar argument for the treatment of fixed effects, although the observed differences in equity premia across countries immediately makes the argument rather weak. Importantly, however, omitting fixed effects in the analysis will lead to estimates and test statistics that are only consistent when the assumption of a common intercept holds. In contrast, the pooled estimator is robust to heterogeneity in the β_is as mentioned in the previous section, and tests on the slope coefficient will provide valid inference on the (average) predictability in the panel. Return to text

6.  Subject to potential rate restrictions, such as n/T → 0, these results can generally be shown to hold as n and T go to infinity jointly; technical proofs of such joint convergence is not pursued in the current study, however. Return to text

7.  In the special case of ω₁₂ = 0, it follows easily that $\hat{\beta}_{FE}$ is also asymptotically normally distributed with convergence rate $\sqrt{n}T$, and inference can proceed in a manner analogous to the pooled case with a common intercept. Return to text

8.  The test of slope homogeneity developed below does rely on knowledge of the C_is, and an approximate solution is proposed. However, such an approach, which is not proven to be asymptotically correct, seems more justified in a test of slope homogeneity, which is of a more diagnostic nature and of second order importance, as compared to the actual test of return predictability. Return to text

9.  Note that this section focuses on estimating a common parameter (β), when the individual β_is are not identical. This is in contrast to the fixed effects α_i, which are in fact estimated for each i. However, estimating individual β_is as well would simply reduce the problem to individual time-series regressions. Return to text

10.  Although the asymptotic distributions for $\hat{\beta}_{Pool}$ (and $\hat{\beta}_{FE}$) differ depending on whether the β_is are identical or not, it is easy to show that inference based on standard test statistics will be self-standardizing. Thus, no prior knowledge on the homogeneity of the β_is is required to perform inference on the pooled estimate. Return to text

11.  Although not shown here, additional simulations also illustrate the asymptotic normality and unbiased nature of the standard pooled estimator when α_i = α for all i, as predicted by Theorem 1. Return to text

12.  According to the MSCI classification scheme, there are actually three different groups of markets: developed, emerging, and frontier markets. Here, I group together the emerging and frontier markets and refer to them as simply emerging markets. The MSCI classifictions can be found at http://www.mscibarra.com/products/indices/intl.jsp; the classifications used in this paper are as of February 2008. The group of emerging markets includes Argentina, Brazil, Chile, Hungary, India, Israel, Jordan, Malaysia, Mauritius, Mexico, the Philippines, Poland, South Africa, Taiwan, Thailand, and Turkey; the other countries shown in Table 4 are classified as developed. Return to text

13.  Since the panels used in the estimation are typically unbalanced, i.e. not all time-series are of the same length, the estimation procedures are modified in a straightforward manner to allow for this; details are available upon request. Return to text

14.  In the U.S., the interest rate was pegged by the Federal Reserve before this date. Of course, in other countries, deregulation of the interest rate markets occurred at different times, most of which are later than 1952. As seen in the international finance literature (e.g. Kaminsky and Schmukler (2002)), however, it is often difficult to determine the exact date of deregulation. And, if one follows classification schemes, such as those in Kaminsky and Schmukler (2002), then most markets are not considered to be fully deregulated until the 1980s, resulting in a very small sample period to study. Thus, the extent to which observed interest rates reflect actual market rates is hard to determine and one should keep this caveat in mind when interpreting the results. Return to text

15.  The term 'expanding window regression' is used instead of 'recursive regression' as to avoid confusion with the pooled estimator using recursive demeaning. Return to text

16.  Since the confidence intervals are not based directly on the OLS estimate, they are not necessarily symmetric around the point estimate. In fact, the OLS point estimate need not be inside the confidence interval, as is the case for Spain in Figure 3. Return to text

17.  Campbell and Thompson (2004) argue that by imposing some weak restrictions on stock return forecasts, their performance can be greatly improved. Results not presented in this paper show that by imposing the sign restrictions proposed by Campbell and Thompson, the out-of-sample results provided here can be substantially strengthened. Return to text

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