## Abstract

We show that several asset pricing models that rely on long-run risks imply that the state of the economy can be captured by factors derived from the price-dividend ratios of stock portfolios. We find two factors with small growth and large value tilts are important for this purpose, thereby relating the Fama-French model and the Bansal-Yaron and Merton intertemporal asset pricing models. As predicted by the model, these price-dividend ratio factors track consumption volatility and predict future consumption and stock dividends, and the covariance of returns with their innovations explains the cross-section of average returns of several stock portfolios. (JEL G19)

## Introduction

We show that long-run risk factors in the model introduced by Bansal and Yaron (2004) can be estimated through principal component analysis of the covariance matrix of log price-dividend ratios of a collection of stocks. Previous studies obtain the long-run risk factors by projecting the future consumption growth and its volatility on the market price-dividend ratio and the real risk-free rate (Bansal, Yaron, and Kiku 2007; Constantinides and Ghosh 2011; Ferson, Nallareddy, and Xie 2013). However, both consumption growth and the real risk-free rate are measured with considerable error. As we show using Monte Carlo simulations, this error could be large enough to result in the rejection of the long-run risk model too often even when it holds. Our approach shows a remarkable improvement as long as a reasonable number of portfolios are used to estimate the long-run risk factors. Our approach enables an answer to one of the important critiques of the long-run risk model raised by Marakani (2009), Beeler and Campbell (2012) and Constantinides and Ghosh (2011), among others, that the log market price-dividend ratio does not predict dividend or consumption growth.

Fama and French (1993) construct a set of twenty-five stock portfolios by sorting stocks based on their size and book-to-market ratios. We argue that results in the literature suggest that the log P/D ratios of these portfolios are likely to show wide variation in their exposure to the long-run risk factors, making them a suitable choice from an econometric point of view. We therefore extract two price-dividend ratio factors using time-series observations on the price-dividend ratios of these twenty-five stock portfolios. The first factor is of a level type with a small-growth firms tilt and the other has a form similar to large-value minus small-growth firms tilt as seen from Table 2. The two factors also predict future aggregate consumption and dividends, and track consumption volatility over time. The finding that the two P/D factors from the Fama-French portfolios are related to future consumption and dividends and their volatilities establishes a tighter link between Fama and French (1993), Merton (1973), and Bansal and Yaron (2004).

Table 1

Nonrejection frequencies for the hypothesis that the market price of risks of the estimated relevant factor innovations are zero (i.e. rejection frequencies for the long-run risk model) using both the traditional and our methodologies under the assumption that the long-run risk model holds and that reasonable measurement errors in consumption, the real risk-free rate and price-dividend ratios are taken into account

Nonrejection rate of hypothesis (i.e. rejection rate of model)

p = 0.10 p = 0.05 p = 0.01

Traditional methodology, hypothesis $λe∼,λw∼=0$

12.3% 24.0% 50.4%
Our methodology, hypothesis $λIF1∼,λIF∼2=0$
5 portfolios 8.7% 13.5% 24.5%
10 portfolios 0.4% 1.0% 3.6%
15 portfolios 0.5% 0.7% 2.2%
20 portfolios 0.3% 0.6% 1.8%
25 portfolios 0.4% 0.6% 1.5%
Nonrejection rate of hypothesis (i.e. rejection rate of model)

p = 0.10 p = 0.05 p = 0.01

Traditional methodology, hypothesis $λe∼,λw∼=0$

12.3% 24.0% 50.4%
Our methodology, hypothesis $λIF1∼,λIF∼2=0$
5 portfolios 8.7% 13.5% 24.5%
10 portfolios 0.4% 1.0% 3.6%
15 portfolios 0.5% 0.7% 2.2%
20 portfolios 0.3% 0.6% 1.8%
25 portfolios 0.4% 0.6% 1.5%
Table 2

The rotation matrix that relates the log annual price dividend ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio to their first and second principal components. F1 and F2 represent the first and second principal components of the annual log P/D ratios over the period 1943–2013

Rotation matrix for $F1≡FVol$

Growth

Value Avg

Small −0.329 −0.262 −0.208 −0.194 −0.176 −0.234
−0.340 −0.249 −0.205 −0.175 −0.145 −0.223
−0.305 −0.214 −0.185 −0.158 −0.144 −0.201
−0.231 −0.183 −0.162 −0.137 −0.110 −0.165
Large −0.142 −0.119 −0.109 −0.121 −0.125 −0.123
avg.  −0.269 −0.205 −0.174 −0.157 −0.140
Rotation matrix for $F1≡FVol$

Growth

Value Avg

Small −0.329 −0.262 −0.208 −0.194 −0.176 −0.234
−0.340 −0.249 −0.205 −0.175 −0.145 −0.223
−0.305 −0.214 −0.185 −0.158 −0.144 −0.201
−0.231 −0.183 −0.162 −0.137 −0.110 −0.165
Large −0.142 −0.119 −0.109 −0.121 −0.125 −0.123
avg.  −0.269 −0.205 −0.174 −0.157 −0.140
Rotation matrix for $F2≡FX$

Growth

Value Avg

Small −0.515 −0.152 −0.050 0.008 0.118 −0.118
−0.312 −0.016 0.097 0.180 0.207 0.031
−0.191 0.036 0.175 0.198 0.210 0.086
−0.064 0.110 0.170 0.286 0.234 0.147
Large −0.049 0.077 0.162 0.236 0.296 0.144
avg.  −0.220 0.011 0.111 0.182 0.213
Rotation matrix for $F2≡FX$

Growth

Value Avg

Small −0.515 −0.152 −0.050 0.008 0.118 −0.118
−0.312 −0.016 0.097 0.180 0.207 0.031
−0.191 0.036 0.175 0.198 0.210 0.086
−0.064 0.110 0.170 0.286 0.234 0.147
Large −0.049 0.077 0.162 0.236 0.296 0.144
avg.  −0.220 0.011 0.111 0.182 0.213

Fama and French (1993) interpret the SMB and HML factors as innovations to state variables in the Merton (1973) intertemporal capital asset pricing model. We show that the class of long-run risk models that we consider can be viewed as versions of the Merton (1973) model.1 The Fama and French interpretation implies that the price-dividend ratio factors whose innovations are similar to SMB and HML. We find this to be the case with the P/D factors extracted from the twenty-five Fama-French portfolios. We further find that the factor innovation betas explain the cross-section of average excess returns on the twenty-five assets and other assets as well. These observations are consistent with the predictions of the long-run risk models and help to relate the widely used Fama and French (1993) model and the intertemporal asset pricing models of Merton (1973) and Bansal and Yaron (2004). An important insight that follows from our analysis is that the cross-section of P/D ratios of stocks contains information about state variables relevant for valuing securities with risky payoffs.

To evaluate the robustness of our findings, we also extract two price-dividend ratio factors from a different collection of stock portfolios: ten stock portfolios formed by sorting stocks on their dividend yield. We find that they perform as well as the state variables in Petkova (2006) or the two factors constructed from the twenty-five portfolios discussed earlier in jointly predicting future consumption and dividends, and tracking consumption volatility. The betas on the factor innovations are also able to explain the cross-section of average returns of the various test assets, thereby providing robust support for the wide class of long-run risk models. These findings support the view that the P/D factors are the relevant state variables in the Merton (1973) model as well.2

Our conclusions are robust to the recent critique of factor models by Kleibergen (2010) and the lookahead bias critique of Ferson, Nallareddy, and Xie (2013). We analyze our results using the robust test statistics developed by Kleibergen (2009) in Appendix B and find that they perform well along that dimension. We also develop lookahead bias-free versions of the factors and find that they behave in the expected manner (see Tables 11, 12, and 13). Our asset pricing results are also robust to the critique of Lewellen, Nagel, and Shanken (2010) as our cross-sectional regression intercepts are generally very small and close to zero. The importance of the small intercept has been particularly emphasized by Jagannathan and Wang (2007). The fact that the expected model implied relation between the P/D factors and macroeconomic variables holds, even though the latter do not have any mechanical relationship with the size and book-to-market ratio factors, gives further credence to the point that our results are robust to the critique of Lewellen, Nagel, and Shanken (2010).

We find that the predictability of consumption growth does not extend to the period following the financial crisis and the Great Recession. This may be due to a structural break in the mean of consumption growth after the global financial crisis. We present evidence for this, but the hypothesis only can be properly verified in the future as time unfolds.

## 1. The Long-Run Risk Model

The long-run risk model introduced by Bansal and Yaron (2004) is expressed as

(1)
$Δct+1=μ+Xt+Vtηt+1,$

(2)
$Xt+1=ρXt+φxVtet+1,$

(3)

(4)
$Vt+1=V¯+ν(Vt−V¯)+σwt+1,$
where c is the log per capita consumption, dl is the log dividend of asset l,3 and the shocks $ηt+1, et+1$ and $wt+1$ are independent standard normals. The shock $ul,t+1$ is a vector of normally distributed shocks with covariance Ω, which is independent of $η,e$, and w.

The key elements of this model are the inclusion of

• a persistent factor X, which governs the growth of both expected consumption and expected dividends, and

• a persistent factor V, which characterizes the consumption growth volatility.

In Internet Appendix A, we extend and modify the above model to include

• multiple persistent X-type factors $Xi,t$ driving expected consumption and dividend growth and multiple persistent volatility-type or V-type factors $Vj,t$ driving consumption growth volatility, and

• a rich correlation structure between the shocks in the model, e, η, w, and u.

The first of these is theoretically important as such factors could exist but we empirically find only evidence of one X-type factor and one V-type factor in the sample of stock portfolios that we examine. Hence, we mostly stick to the simpler model with only one X-type and one V-type factor but will discuss the evidence for this within the richer structure of the extended model when we identify the number and type of factors.

The richer structure of the extended model means that it is also able to accommodate the specifications proposed by Bansal and Yaron (2004), Bansal, Yaron, and Kiku (2007), and Zhou and Zhu (2009) as special cases.4

Following Ferson, Nallareddy, and Xie (2013), we note that, under Epstein-Zin preferences (Epstein and Zin 1989; Weil 1990), the innovation of the log stochastic discount factor is a linear combination of the innovations to consumption growth and the X and V factors. As they note, this implies that the beta pricing relation is

(5)
$Et[rl,t+1−rf,t]+12Vart[rl,t+1−rf,t]≈βl,ΔcλΔc+βl,eλe+βl,wλw,$
where $βl,Δc, βl,e$, and $βl,w$ are the coefficients obtained on regressing $rl,t+1−rf,t$ on consumption growth,5 the innovations in the X factor e, and the innovations in the V factor w. When the model is extended to include multiple X- and V-type factors, the asset pricing relation becomes
(6)
$Et[rl,t+1−rf,t]+12Vart[rl,t+1−rf,t]≈βl,ΔcλΔc+∑i=1nβl,eiλei+∑j=1mβl,wjλwj,$
with the βs and λs being defined in an analogous manner.6

We follow Ferson, Nallareddy, and Xie (2013) in estimating the market prices of risk λ purely from the data without restricting them using estimates of the model parameters, unlike Bansal, Yaron, and Kiku (2007) and Constantinides and Ghosh (2011). This has the benefit of not tying the analysis to a particular form or parametrization of the long-run risk model.

To take this asset pricing relation to the data, we need to estimate the number of underlying factors and the innovations in their time series. The analysis of the methodology used for this purpose and its relation to the methodologies used in similar studies is the subject of the next two sections.

## 2. Factor Structure of Log P/D Ratios

Bansal and Yaron (2004) and Constantinides and Ghosh (2011) (among others) show that the long-run risk model implies that the log market price-dividend ratio and real risk-free rate are affine functions of the X and V factors within the long-run risk model. In general, as pointed by Eraker and Shaliastovich (2008), affine pricing models have an implication of this type.

In Internet Appendix A, we show, using an approach similar to that of Bansal and Yaron (2004), this continues to hold within the extended long-run risk model, where it is shown that

(7)
$log(Pl,tDl,t)=pl,t−dl,t=A0,l+∑i=1nA1,l,iXi,t+∑j=1mA2,l,jVj,t,$
where $Pl,t$ is the price of portfolio l, and the expressions for $A1,l,i,1≤i≤n$ and $A2,l,j,1≤j≤m$ are derived in Internet Appendix A.

The relation (7) can be inverted to express the state variables $Xi,1≤i≤n$, and $Vj,1≤j≤m$ in terms of m + n log P/D ratios (one of which could be the real risk-free rate) as

(8)
$Xi=∑k=1m+nA¯i,klog(P/D)k$

(9)
$Vj=∑k=1m+nB¯j,klog(P/D)k$
The key innovation of this paper is that this estimation can be done using a rich cross-section of assets rather than just the log market price-dividend ratio and the real risk-free rate. This has two advantages: Since log P/D ratios are not exact linear combinations of a small number of factors in the data, we use the following relation in our empirical work
(10)
$log(Pl,tDl,t)=pl,t−dl,t=A0,l+∑i=1nA1,l,iXi,t+∑j=1mA2,l,jVj,t+ϵl,t,$
where $ϵl,t∼N(0,Ve)$ are i.i.d. $ϵl,t$ can arise due to market imperfections, such as illiquidity or due to the existence of incompletely diversified idiosyncratic factors. In Section 5, we show that, given this error structure, principal component analysis (or singular value decomposition) can be used to estimate the linear subspace that the m + n factors span once m + n is specified. We use statistical methods suggested in the literature to estimate m + n from the data.

• it enables the number of factors to be estimated from the data rather than specified a priori; and

• it enables a more accurate estimation of the long-run risk factors rather than by just using the log market price-dividend ratio and real risk-free rate as is conventional in the literature. In Appendix A, we show that the resultant reduction in the type II error of the asset pricing tests can be very substantial when following this approach in contrast to the conventional one.

This differs from the methodology used by Bansal, Yaron, and Kiku (2007) and Ferson, Nallareddy, and Xie (2013), who project the realized long-term consumption growth and its volatility on the log market P/D ratio and the real risk-free rate. We show in Appendix 8, through Monte Carlo simulations, that our methodology produces much fewer spurious rejections of the model when reasonable measurement errors in consumption growth and the real risk-free rate are taken into account provided a reasonable number of portfolios are used in the estimation. In other words, the methodology used in this paper is robust to reasonable measurement errors and that, in their presence, has much lower type II error than the traditional methodology used by Bansal, Yaron, and Kiku (2007) and Ferson, Nallareddy, and Xie (2013). Further details about the magnitudes and effects of this measurement error can be found in Appendix A.

We emphasize again that our approach does not constrain either the market prices of risk or the coefficients of the functions estimating the factors with the use of the model parameters. In this regard, we follow the approach of Ferson, Nallareddy, and Xie (2013) as it enables us to examine the largest possible class of long-run risk models (or, in other words, the general idea of long-run risk), which ensures that the possibility of rejection simply due to incorrect implicit or explicit parametrization of the model is minimized. However, we differ from Ferson, Nallareddy, and Xie (2013) in the quantities used to estimate the model factors. Ferson, Nallareddy, and Xie (2013) use the projection of consumption growth and consumption growth volatility on the log market price-dividend ratio and the real risk-free rate to compute their estimates of the long-run risk factors. The downside is that the power of the empirical examination with respect to any fully parametrized model is lower than an approach that uses the parametrization to restrict the estimates of the market prices of risk and/or factors.

We empirically evaluate the asset pricing relation in Equation (5) mainly using cross-sectional regressions.7 The cross-sectional regression results are particularly useful in relating our results to those in the literature that use various linear beta pricing representations. The appropriate pricing relations for the cross sectional regressions are developed in the next section.

### 2.1 Selection of portfolios for estimating P/D factors

A natural question arises to which set of portfolios to use. Since we are estimating principal components, we would like to have sufficient cross-sectional variation in the exposure of the log P/D ratios to the long-run risk factors.

To better understand what this significant variation means, we recall the loglinear approximation for the log P/D ratio Campbell and Shiller (1988), which is

(11)
$log(PtDt)≈k+Et∑j=1∞κj−1(Δdt+j−rt+j),$
where k and κ are log-linearization constants. This expression shows that the two main quantities that determine the log P/D ratio are long-run dividend growth and long-run-expected returns. Hence, the problem of finding a set of portfolios whose log P/D ratios exhibit a wide variation in exposure to the long-run risk factors X and V reduces to finding a set of portfolios whose long-run dividend growth and long-run-expected returns exhibit a wide variation of exposure to these factors.

The findings of the consumption-cash-flow literature (including, but not limited to, the studies of Bansal, Dittmar, and Lundblad 2005; Da 2009) that size and book-to-market ratio-sorted portfolios have a wide variation in the exposure of their long-run dividend growth rates to the long-run component of consumption growth show that such portfolios will have a wide variation of exposure of their log P/D ratios to the X factor. This is because X is the long-run component of consumption growth as can be seen from (1). Hence, size and book-to-market ratio-sorted portfolios are ideal candidates for estimating this factor using our procedure.

We also argue that book-to-market ratio-sorted portfolios are also likely to show a wide variation in the exposure of their log P/D ratios to the V factor. To do so, we use the fact that, shown in Internet Appendix A, the discount rate is an affine function of V. Given this fact, a set of portfolios whose long-run-expected returns show wide variation in their exposure to the discount rate should be good choice for use in the estimation of V. This implies that a set of portfolios, such as those sorted on the book-to-market ratio, which exhibit a wide variation in duration, which captures this very exposure, is what is required. Further, Da (2009) finds that portfolios sorted along the value dimension show a lot of variation in their cash-flow durations, which is the dependence of their long-run cash-flow growths on the discount rate. This adds further support for the use of portfolios sorted along the value dimension, that is, the book-to-market ratio characteristic.

Further, size and book-to-market ratio-sorted portfolios have a wide variation in their expected returns, which, in turn, implies a wide variation in their exposure to the long-run risk factors, provided the model is correct.

Finally, we have to decide how many portfolios to use. For example, the literature commonly uses 6, 25, or 100 size and book-to-market ratio-sorted portfolios. In this regard, the Monte Carlo simulation results tabulated in Table 1 show that twenty-five portfolios show a remarkable improvement in results, compared with the traditional methodology and that ten should be sufficient to obtain most of that improvement. We tabulate the results obtained from Monte Carlo simulations using different numbers of portfolios in Table 1, which suggests that ten or twenty-five portfolios are sufficient for our purposes.

We therefore use the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio for our estimation.

In our robustness checks, we also use a set of ten portfolios formed on the basis of dividend yield and also find that they give good estimates of the long-run risk factors, which exhibit the expected behavior.

## 3. Data

We examine the cross-section of annual returns on several stock portfolios for the period 1943 to 2013, that is, 71 years. While we primarily use annual consumption data, we compute consumption growth volatility using quarterly consumption data to obtain more precise estimates. Consumption data are obtained from the National Income and Product Accounts (NIPA) tables available at the BEA Web site. Real annual per capita consumption is defined to be the nominal aggregate annual consumption of nondurables and services divided by the NIPA estimate of the midyear population and deflated by the implicit personal consumption deflator.8 Annual consumption growth is defined to be the first difference of the logarithm of this series. Quarterly seasonally adjusted consumption data are also obtained from NIPA, and their growth are defined in an analogous manner.

The proxy for the nominal risk-free rate is the Fama three-month T-bill rate obtained from CRSP. It is converted to two proxies of the real risk-free rate using the realized and past inflation as measured by the CPI. The CPI data for this are obtained from CRSP.

The stock market proxy (used to determine the relationship between the factors and future dividend growth and expected returns) is defined as the CRSP value-weighted index of all stocks listed on the NYSE, AMEX, and NASDAQ. The construction of portfolios based on size and book-to-market ratios is as in Fama and French (1993) and Fama and French (1996). Data on the twenty-five $(5×5)$ and six $(2×3)$ portfolios formed on the basis of both these characteristics, as well as ten portfolios formed on the basis of dividend yield and seventeen portfolios formed on the basis of industry, are obtained from Ken French’s Web site.

For testing the asset pricing relationships with portfolios other than the ones used to estimate the factors (we call this out of sample testing), we use three sets of ten portfolios each formed on the basis of long-term reversal, short-term reversal and the earnings to price (E/P) ratio. The long-term reversal portfolios are formed monthly on the basis of stock’s return over the past five years minus its return over the past year. In other words, they are formed at time t – 1 (time being indexed by month) by sorting stocks into ten portfolios according to their returns from t – 61 to t – 13. Similarly, short-term reversal portfolios are formed at time t – 1 by sorting stocks into ten portfolios based on their return from t – 2 to t – 1. The E/P-based portfolios are formed at the end of June of year t by sorting stocks into ten portfolios (using NYSE breakpoints) on the basis of their E/P ratios, where E is defined to be the earnings before extraordinary items during fiscal year t – 1 and P is defined to be the market capitalization at the end of December of year t – 1. Data on these thirty portfolios are also obtained from Ken French’s Web site.

Monthly dividends of these portfolios are calculated using the difference between the returns of the corresponding portfolios including and excluding dividends as reported in the data on Ken French’s Web site. The price-dividend ratios are then calculated by dividing the real price of the ex-dividend portfolio by the sum of the lagged twelve real monthly dividends. This procedure adjusts for the pronounced seasonality of the dividend series. The nominal prices and dividends are deflated by the CPI to get the real prices and dividends. As pointed by van Binsbergen and Koijen (2010), the effect of assumptions regarding the handling of dividends paid during the year on the price-dividend ratios is negligible with the correlation between the different measures being about 0.9999.

Real-time consumption data are obtained from the Web site of the Federal Reserve Bank of St. Louis and are described by Croushore and Stark (2001). The real consumption during a quarter is defined to be the sum of the real consumption of nondurables and services during that quarter. The real consumption during a year is defined to be the sum of the real consumptions during each quarter of that year. Real per capita consumption during a period is defined to be the real consumption during that period divided by the midperiod estimate of population. The real-time annual per capita consumption growth for year t is defined to be the difference between the logarithms of the real per capita consumptions during years t and t – 1, respectively, as calculated using data of vintage Q1 of year t + 1. To provide an example, the data set of Q1 1976 vintage is used to construct the real-time annual per capita consumption growth for 1975. It is constructed by adding the real nondurables and services consumptions of Q1–Q4 1974 and Q1–Q4 1975, dividing each of them by the midyear estimates of the population, and then taking the difference of the logarithms of the corresponding quantities.

## 4. Construction of the Principal Components and Their Innovations

Marakani (2009) finds that the parameters of long-run risk models could not have been the same before and after 1942. We reproduce the evidence in Internet Appendix C for the convenience of the interested reader. Hence, we only consider the post-1942 period in our analysis and assume that consumers assigned zero probability for the regime change.9

From (7), the problem of obtaining the factors of log P/D ratios is, for a fixed number of factors m + n, equivalent to the problem of finding time-series processes $Fi,tm+n$ to solve

(12)
$V(m+n)=minΛ,Fm+n1NT∑i=1N∑t=1T(PDi,t−Λim+nFi,tm+n)2,$
where $PDi,t$ are demeaned log P/D ratios of portfolio i at time t, N is the number of portfolios, T is the length of the time series, F are the factors, and Λ are the loadings of the individual log P/D ratios on them (the superscript m + n keeps track of the number of assumed factors). The equivalency of the two problems follows trivially from the assumption that the error terms are i.i.d and Gaussian. Hence, this problem is the same as the well-studied standard factor analysis problem.10 The assumptions regarding the error terms are not crucial for our results as they hold even if we perform the principal component analysis after first scaling the log P/D ratios to make them each have unit variance or after first scaling them each according to their residual variances. In other words, our results are robust to the use of different specifications for the error term.

Hence, the factors can be calculated by singular value decomposition of the matrix of de-meaned log P/D ratios. This is equivalent to the more usual method of using the eigenvectors of the covariance matrix or directly solving (12), but is preferred because it has greater numerical stability. The number of relevant factors $k=m+n$ is determined by using the information criterion

(13)
$argmink ICp2≡argmink(logV(k)+2k(N+TNT)logmin(N,T)),$
suggested by Bai and Ng (2008) and Connor and Korajczyk (2010). This method is known to be consistent when the number of quantities and the length of the time-series become large. As pointed out by Bai and Ng (2008), traditional methods usually overestimate the number of factors that are present in the data.

We carry out this procedure on the annual log P/D ratios of the twenty-five Fama French portfolios from 1943 (to account for the structural break). We find two significant factors in this series (as well as in the quarterly and monthly series of log P/D ratios; not reported).

We plot the information criterion as a function of the number of factors in Figure 1 and the variances explained by the principal components in Figure 2.

Figure 1

Information criterion as a function of the number of factors for the annual log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1943–2013.

Figure 1

Information criterion as a function of the number of factors for the annual log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1943–2013.

Figure 2

Variances explained by the first ten principal components of the annual log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1943–2013.

Figure 2

Variances explained by the first ten principal components of the annual log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1943–2013.

Using the same procedure, we find two factors in the first differences of the quarterly log P/D ratios of these portfolios. We also note that the plot of the variances explained by their principal components in Figure 3 unambiguously points to a two-factor structure.

Figure 3

Since a factor structure for the log P/D ratios also implies a similar factor structure for the first differences in the log P/D ratios, we check that the first differences of the log P/D ratios also shows a two-factor structure in the data. The above graph shows the variances explained by the first ten principal components of the first differences of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1943–2013. It shows that these first differences also exhibit the expected two-factor structure.

Figure 3

Since a factor structure for the log P/D ratios also implies a similar factor structure for the first differences in the log P/D ratios, we check that the first differences of the log P/D ratios also shows a two-factor structure in the data. The above graph shows the variances explained by the first ten principal components of the first differences of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1943–2013. It shows that these first differences also exhibit the expected two-factor structure.

We tabulate the rotations that relate the annual log P/D ratios of the twenty-five Fama-French portfolios to their first two principal components, denoted by $F1,2$ with the subscript the principal component, in Table 2. From the rotation matrices, we find that F1 loads negatively on all the portfolios and slightly more so on the small and growth stock portfolios. In contrast, F2 loads positively on large and value stocks and negatively on growth and small stocks. We thus expect F2 to be closely related to the cross-sectional differences among the portfolios.

We estimate the innovations of the two identified principal components as the ordinary least squares (OLS) residuals obtained on regressing them on n lags of themselves, n being the smallest value for which they are serially uncorrelated at the 10% level according to both the Ljung-Box and Durbin-Watson tests. n is always found to be one for the annual data and sometimes two for the quarterly data (data not reported). We denote these estimated innovations as IF1 and IF2.

An alternative way to calculate the long-run risk factors and their innovations is to use dynamic, rather than static, factor analysis. This carries the advantage of possibly being more robust given the relatively high autocorrelations of the log P/D ratios. We obtain similar results to those presented here when proceeding in this manner, except that the factors are not cleanly separable into X and V types. Since the results using our approach are similar and are easier to interpret and understand, we concentrate on them in this paper.11

## 5. Principal Components and the Long-Run Risk Factors

Since the Xi factors represent joint predictable components of consumption and dividend growth, a positive and significant coefficient should result on regressing future consumption and dividend growth against these factors. Similarly, since the Vj factors are components of the consumption growth volatility, regressing consumption growth volatility against them should also lead to a positive and significant coefficient. Since the principal component analysis only identifies affine transformations of the full set of long-run risk factors, we can, in general, expect to find that the principal components will be related to both the Xi and Vj factors and that both regressions above will lead to significant coefficients given that the long-run risk model holds. However, we find that only the volatility regression generates a significant coefficient for the first identified principal component F1 and that only the future consumption and dividend growth regression generates a significant coefficient for the second identified principal component F2. This implies that the first identified principal component is naturally identifiable as an affine function of the only V factor and that the second identified principal component is naturally identifiable as an affine function of the only X factor.

We now examine the volatility regression in some detail. To construct a consumption volatility series, we estimate the innovations of quarterly consumption growth $ϵv,t$ as the OLS residuals obtained on regressing it on m lags of itself, m being the smallest value for which they are uncorrelated at the 10% level according to both the Ljung-Box and Durbin-Watson tests. m is found to be three for this data series. Using these estimated innovations, the n period consumption volatility series is estimated as

(14)
$vtn=log∑i=1n|ϵv,⌊t+n/2−i⌋|n.$
This methodology suggested by Bansal, Khatchatrian, and Yaron (2005) and has been used in the context of long-run risk models by Beeler and Campbell (2012).

The results of regressing $vt24,vt12$ and $vt6$ on F1 and F2, the factors constructed using quarterly log P/D ratios, are summarized in Table 3. We also include the results of regressing detrended $vt24,vt12$ and $vt6$ on detrended F1 as a robustness check as both consumption growth volatility and the first principal component have a pronounced time trend in the data as seen in Figure 4. The results with detrending are an acid test as the fact that the trends coincide is, by itself, support for the model. They show that F1 is very closely related to consumption growth volatility with the R2 of the 24 quarter volatility regression being as high as 81%. Even the R2 for six quarter volatility regression, where the measurement error is likely to be high, is quite high at 45%. Further, the fact that the coefficients of F1 in the various regressions are very similar to each other (i.e., for volatilities estimated over several horizons) provides strong evidence that the relation is robust. In contrast, there is no evidence at all that F2 is related to consumption growth volatility. This result, when combined with the finding, detailed below, that F1 is unrelated to future consumption and dividend growth, leads to the conclusion that F1 is an affine function of a V-type factor. This conclusion follows because F1 satisfies the conditions we have identified for such a factor: it is an affine function of log P/D ratios; it tracks consumption growth volatility; and it does not predict future consumption or dividend growth.12

Figure 4

The first plot is that of the twelve quarter consumption growth volatility and F1, the first principal component of the log price-dividend ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio, for the period 1947–2013. The second plot is that of the forward real market dividend growth and F2, the second principal component of the log price-dividend ratios of the 25 Fama-French portfolios formed on the basis of size and book-to-market ratio, over the period 1947–2013.

Figure 4

The first plot is that of the twelve quarter consumption growth volatility and F1, the first principal component of the log price-dividend ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio, for the period 1947–2013. The second plot is that of the forward real market dividend growth and F2, the second principal component of the log price-dividend ratios of the 25 Fama-French portfolios formed on the basis of size and book-to-market ratio, over the period 1947–2013.

Table 3

Results of regressing volatility as defined in (14) against F1 and F2, the two significant principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio for the period 1947–2013. The t-statistics are based on standard errors, which are Newey-West corrected with the required number of lags estimated using the procedure of Newey and West (1994). The principal components of the quarterly log P/D ratios were used in these regressions. Use of the rotation matrix for the annual log P/D ratios to create quarterly versions of these principal components leads to very similar results. The detrending is carried out by regressing against a time trend.The first-order autocorrelation coefficients for F1 and F2 are 0.916 and 0.854 for the period 1943–2008 and 0.934 and 0.866 for the period 1943–2013

Regression of consumption growth volatility on F1 and F2

$F1≡FVol$ $F2≡FX$ R2
24 quarter volatility $0.102***$ (7.29) 0.058 (1.61) 80.8%
12 quarter volatility $0.116***$ (4.00) 0.019 (0.24) 60.6%
6 quarter volatility $0.118***$ (3.93) 0.032 (0.37) 44.7%
Regression of consumption growth volatility on F1 and F2

$F1≡FVol$ $F2≡FX$ R2
24 quarter volatility $0.102***$ (7.29) 0.058 (1.61) 80.8%
12 quarter volatility $0.116***$ (4.00) 0.019 (0.24) 60.6%
6 quarter volatility $0.118***$ (3.93) 0.032 (0.37) 44.7%
Regression of detrended consumption growth volatility on detrended $F1≡FVol$

$F1≡FVol$ R2
24 quarter volatility $0.081***$ (4.50) 38.4%
12 quarter volatility $0.091***$ (2.76) 18.7%
6 quarter volatility $0.092**$ (2.14) 10.3%
Regression of detrended consumption growth volatility on detrended $F1≡FVol$

$F1≡FVol$ R2
24 quarter volatility $0.081***$ (4.50) 38.4%
12 quarter volatility $0.091***$ (2.76) 18.7%
6 quarter volatility $0.092**$ (2.14) 10.3%

We now examine the dividend and consumption growth regressions in detail. The results of regressing annual real market dividend growth (i.e., growth of annual market dividends deflated by the CPI) on the lagged values of F1 and F2 are summarized in Table 4. We find, from them, that F2, but not F1, predicts market dividend growth. This predictive ability is robust to lagging twice to account for time aggregation with the coefficient for F2 being significant at the 5% level.13 While our results are robust to time aggregation of dividend growth along this dimension, we acknowledge that such aggregation also leads to biases in our estimates of the price-dividend ratios since we calculate them, as is conventional in the literature, using dividends aggregated on an annual basis to adjust for their pronounced seasonality. However, we also note that time aggregation of dividend growth, in contrast to time aggregation of consumption growth, is generally not considered a significant issue in the literature and that our model specification is annual rather than monthly.14

The results of regressing annual real-time consumption growth against the lagged values of F1 and F2 are also summarized in Table 4. We find, from them, for the period 1965–2008, that F2 also predicts real-time consumption growth as defined in the data section and that this predictive ability is robust to lagging twice to account for time aggregation. This is in accordance with the long-run risk hypothesis that dividend and consumption growth share the same persistent component(s) X.15

Table 4

Results of regressing real annual market dividend growth ($Δdm$, where dm is the log real annual market dividend with the inflation adjustment being done according to the CPI) and real-time consumption growth ($ΔcRT$) against lagged F1 and F2, the two significant principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio, and the lagged log market P/D ratio ($log(P/D)m$). The t-statistics are based on standard errors which are Newey-West corrected with the required number of lags estimated using the procedure of Newey and West (1994). The regressions using the log market P/D ratio are for the same time period as for the ones using F1 and F2. The first-order autocorrelation coefficients for F1 and F2 are 0.916 and 0.854 for the period 1943–2008 and 0.934 and 0.866 for the period 1943–2013

Regression of real market dividend growth on lagged F1 and F2, compared with that on the lagged log market P/D ratio

1944–2008
$Δdm,t+1$ $F1,t≡FVol,t$ $F2,t≡FX,t$ $log(P/D)m,t$ R2
−0.0004 (−1.29) $0.0317***$ (4.01)  16.0%
0.0028 (0.10) 0.0%
$Δdm,t+2$ 0.0005 (0.17) $0.0134*$ (1.68)  2.9%
−0.0058 (−0.23) 0.2%
$dm,t+3−dm,t$ −0.0008 (−0.06) $0.0571**$ (2.19)  11.6%
0.005 (0.05) 0.0%

1944–2013

$Δdm,t+1$ $F1,t≡FVol,t$ $F2,t≡FX,t$ $log(P/D)m,t$ R2
−0.0016 (−0.50) $0.0291***$ (4.04)  13.7%
0.011 (0.41) 0.4%
$Δdm,t+2$ 0.0001 (0.04) $0.0178**$ (1.98)  4.7%
−0.0040 (−0.14) 0.1%
$dm,t+3−dm,t$ −0.0010 (−0.09) $0.0526*$ (1.72)  10.4%
0.005 (0.05) 0.0%
Regression of real market dividend growth on lagged F1 and F2, compared with that on the lagged log market P/D ratio

1944–2008
$Δdm,t+1$ $F1,t≡FVol,t$ $F2,t≡FX,t$ $log(P/D)m,t$ R2
−0.0004 (−1.29) $0.0317***$ (4.01)  16.0%
0.0028 (0.10) 0.0%
$Δdm,t+2$ 0.0005 (0.17) $0.0134*$ (1.68)  2.9%
−0.0058 (−0.23) 0.2%
$dm,t+3−dm,t$ −0.0008 (−0.06) $0.0571**$ (2.19)  11.6%
0.005 (0.05) 0.0%

1944–2013

$Δdm,t+1$ $F1,t≡FVol,t$ $F2,t≡FX,t$ $log(P/D)m,t$ R2
−0.0016 (−0.50) $0.0291***$ (4.04)  13.7%
0.011 (0.41) 0.4%
$Δdm,t+2$ 0.0001 (0.04) $0.0178**$ (1.98)  4.7%
−0.0040 (−0.14) 0.1%
$dm,t+3−dm,t$ −0.0010 (−0.09) $0.0526*$ (1.72)  10.4%
0.005 (0.05) 0.0%
Regression of real-time annual consumption growth on lagged F1 and F2
1965–2008

$F1,t≡FVol,t$ $F2,t≡FX,t$ R2
$Δct+1RT$ $2×10−4(0.22)$ $0.0068*$ (1.84) 17.4%
$Δct+2RT$ $5×10−4(0.71)$ $0.0054**$ (2.57) 9.8%
$Δct+1RT+Δct+2RT$ $5×10−4(0.33)$ $0.0123**$ (2.46) 18.9%

1965–2013

$F1,t≡FVol,t$ $F2,t≡FX,t$ R2
$Δct+1RT$ $3.76×10−4(0.37)$ $−4.00×10−4(−0.09)$ 0.6%
$Δct+2RT$ $−3.17×10−4(−0.31)$ $−6.85×10−4(−0.23)$ 1.0%
$Δct+1RT+Δct+2RT$ $2.52×10−4(0.12)$ $−1.47×10−3(−0.18)$ 0.3%
Regression of real-time annual consumption growth on lagged F1 and F2
1965–2008

$F1,t≡FVol,t$ $F2,t≡FX,t$ R2
$Δct+1RT$ $2×10−4(0.22)$ $0.0068*$ (1.84) 17.4%
$Δct+2RT$ $5×10−4(0.71)$ $0.0054**$ (2.57) 9.8%
$Δct+1RT+Δct+2RT$ $5×10−4(0.33)$ $0.0123**$ (2.46) 18.9%

1965–2013

$F1,t≡FVol,t$ $F2,t≡FX,t$ R2
$Δct+1RT$ $3.76×10−4(0.37)$ $−4.00×10−4(−0.09)$ 0.6%
$Δct+2RT$ $−3.17×10−4(−0.31)$ $−6.85×10−4(−0.23)$ 1.0%
$Δct+1RT+Δct+2RT$ $2.52×10−4(0.12)$ $−1.47×10−3(−0.18)$ 0.3%

### 5.1 Predictability of real-time consumption growth and the global financial crisis

The regressions showing the predictability of real-time consumption growth in Table 4 only agree with the expected pattern until 2008 and do not do so when the data are extended to 2013. We argue that this is not because the log price-dividend ratio factor F2 does not behave as expected but is because the data for consumption growth before and after the global financial crisis period shows different behavior in having different means.

To support this assertion, we carry out a Bai-Perron structural break analysis Bai and Perron (1998) for a change in the mean of real-time consumption growth and tabulate the result in Table 5. From it, we see that the mean of real-time consumption growth fell sharply after 2006. We also carry out this analysis for the predictability regression of real-time consumption growth against lagged F2 and find that once this break in the mean is accounted for, the coefficient of lagged F2 remains almost unchanged before and after the 2006 break. This provides evidence that the relationship between future consumption growth and the second log price-dividend ratio factor does have the same structure before and after the global financial crisis.

Table 5

Structural break analysis of real-time consumption growth with regards to its mean and its predictability by FX, the second principal component of the log price-dividend ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1943–2013, carried out as prescribed by Bai and Perron (1998). The time period over which the tests were run was 1965–2013

Structural break analysis of real-time consumption growth

Regressors Bai-Perron test Coefficient before break Coefficient after break
Structural break test of the mean

1 break at 2006 0.0178 0.0038
90% C.I. (2004,2012)

Structural break test against lagged F2

Intercept 1 break at 2006 0.02205 −0.00534
Lagged FX 90% C.I. (2005, 2008) 0.00797 0.00737
Structural break analysis of real-time consumption growth

Regressors Bai-Perron test Coefficient before break Coefficient after break
Structural break test of the mean

1 break at 2006 0.0178 0.0038
90% C.I. (2004,2012)

Structural break test against lagged F2

Intercept 1 break at 2006 0.02205 −0.00534
Lagged FX 90% C.I. (2005, 2008) 0.00797 0.00737

The change in mean of real-time consumption growth documented above poses a challenge for the long-run risk model as it does not allow for such a change in the mean. This means that we cannot tie our results closely to the long-run risk model in this single aspect and that there is more to systematic risk than can be captured by long-run risk models. However, if our principal aim is to understand the type of quantities that are related to future consumption and dividend growth and use the long-run risk model as a guide, this analysis does indicate that the factor F2 has a positive and significant relation with both future dividend and real-time consumption growth though the change in the mean of the latter must be taken into account to fully determine this relationship.

It also should be noted that many other aspects of the data show differences after the global financial crisis (though these differences are much smaller than the ones before and after World War II documented in Internet Appendix C). These include Sharpe ratios of stock returns and real dividend growth rates, which have increased significantly after the global financial crisis as well as real and nominal risk-free rates, which have decreased sharply across the same period.

We also note that the results of the asset pricing tests documented in the next section are better when restricted to the period just before the global financial crisis. This suggests that financial crises pose a challenge for the long-run risk model.

### 5.2 Identification of the principal components as the long-run risk factors

From the results in Table 4, we conclude that F2 can be identified as an affine function of a X type factor as it satisfies the essential properties of such factors: it is an affine function of log P/D ratios, predicts dividend, and consumption growth, but not consumption growth volatility. The good fit to forward real market dividend growth is confirmed by the plot in Figure 4. This conclusion is further bolstered by our finding that the predictable dividend growth across the cross-section is also strongly related to F2.16 While this is not a necessary property of an X-type factor in our general version of the model, it is expected to hold in many long-run risk models in the literature such as the ones in Bansal and Yaron (2004), Kiku (2006), and Bansal, Yaron, and Kiku (2007).

Since the above regressions involve the whole sample and are subject to forward-looking bias, we investigate whether the predictability implications that lead to the identification of the first two principal components as affine functions of the long-run risk factors X and V hold good for lookahead bias-free versions of the factors and find that they do so. These lookahead bias-free factors are constructed from expanding window estimation so that only past data are used in their construction. This is tabulated in the section on robustness tests in Tables 11 and 12. We further find, as tabulated in Table 13, that the lookahead bias-free estimates of the factor innovations successfully explain the expected returns of three sets of ten portfolios formed on the basis of short-term reversal, long-term reversal and the earnings-to-price ratio.17 Hence, we see that our results are robust both to forward-looking bias and are not an artifact of testing the model with the same portfolios as those that are used to estimate the factors.

Another concern that can be raised is the use of the twenty-five Fama-French portfolios to construct the factors. We address this concern in the section on robustness checks by carrying out the same procedure using the log price-dividend ratios of ten portfolios formed on the basis of dividend yield. We find that the properties of the first two log price-dividend ratio factors are very similar to those constructed using the twenty-five Fama-French portfolios and that the innovations of these factors are also able to price both the twenty-five Fama-French portfolios and the three sets of ten portfolios formed on the basis of long- and short-term reversal and the earnings-to-price ratio. These results are documented in Table 16.

Given the strong and clear relationships between the factors and the consumption and dividend growth and volatility detailed above, we relabel F1 as FVol and F2 as FX for the rest of this paper.

## 6. Asset Pricing Tests

Since the dividends in this analysis have to be measured annually due to seasonality considerations, the asset pricing restrictions are only strictly correct at the annual time scale. Hence, we restrict ourselves to annual data in the following analysis.18

### 6.1 Cross-sectional regressions

When we investigate the pricing relationship (5) using the twenty-five Fama-French portfolios, we find that it performs well. The cross-sectional regression results are tabulated in Table 6 and plotted in Figure 5. The OLS R2 is moderate at 64.1% but provides good support for the model given that the estimate of the cross-sectional regression intercept is both economically small and statistically insignificant at -1.0%/year with an absolute t-statistic value of only -0.21.19 This, together with the insignificant intercepts and high R2 values in the other cross-sectional regressions detailed below provides support for our specification as the intercept must be zero for a correctly specified model, a fact emphasized by Jagannathan and Wang (2007). The coefficients of the innovations of the volatility and consumption/dividend growth factors also have the expected sign, and the latter is statistically significant. The relative difficulty in estimating the volatility innovations as pointed out in Appendix A could be reason for the insignificant volatility innovation coefficient. Another possibility is that the relatively unusual data after the global financial crisis has added noise to this estimate as we find that the coefficient is statistically significant in the cross-sectional regression until 2008.

Figure 5

Results of the cross-sectional regression of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio on their betas with respect to consumption growth, IFVol and IFX. FVol and FX are the first two principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio which, as we show, can be interpreted as the volatility and consumption/dividend growth factors. IFVol and IFX are their innovations.

Figure 5

Results of the cross-sectional regression of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio on their betas with respect to consumption growth, IFVol and IFX. FVol and FX are the first two principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio which, as we show, can be interpreted as the volatility and consumption/dividend growth factors. IFVol and IFX are their innovations.

Table 6

Results of the two pass cross-sectional regression, $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]$$=βIFVolλIFVol+βIFXλIFX+βΔcλΔc$, of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio and three sets of ten portfolios formed on the basis of long-term reversal, short-term reversal and the earnings-to-price ratio, on their betas with respect to consumption growth, IFVol and IFX. FVol and FX are the first two principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio which, as we show, can be interpreted as the volatility and consumption/dividend growth factors while IFVol and IFX are their innovations. For the OLS coefficients, the t values with and without the Shanken correction Shanken (1992); Shanken and Zhou (2007) are reported below the coefficient (the value without the correction is reported first) while for the WLS coefficients, only the t values with the correction are reported. The R2 adjusted for the number of variables is reported below the unadjusted R2

Results of the cross-sectional regression $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=α+βIFVolλIFVol+βIFXλIFX+βΔcλΔc$

for the 25 Fama-French portfolios formed on the basis of size and book-to-market ratio

1944–2008

1944–2013

Intercept $λIFVol$ $λIFX$ $λΔc$ R2 Intercept $λIFVol$ $λIFX$ $λΔc$ R2
OLS −0.023 −0.545 0.464 −0.0033 68.8% −0.010 −0.399 0.491 −0.00039 64.1%
(−0.70) (−2.47) (4.01) (−0.93) (64.4%) (−0.33) (−1.61) (4.29) (−0.098) (59.0%)
(−0.44) (−1.71) (2.66) (−0.62)  (−0.21) (−1.13) (2.91) (−0.07)
WLS −0.023 −0.577 0.417 0.0000  −0.013 −0.495 0.417 0.00155
(−0.48) (−1.94) (2.66) (0.01)  (−0.31) (−1.49) (2.75) (0.29)
Results of the cross-sectional regression $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=α+βIFVolλIFVol+βIFXλIFX+βΔcλΔc$

for the 25 Fama-French portfolios formed on the basis of size and book-to-market ratio

1944–2008

1944–2013

Intercept $λIFVol$ $λIFX$ $λΔc$ R2 Intercept $λIFVol$ $λIFX$ $λΔc$ R2
OLS −0.023 −0.545 0.464 −0.0033 68.8% −0.010 −0.399 0.491 −0.00039 64.1%
(−0.70) (−2.47) (4.01) (−0.93) (64.4%) (−0.33) (−1.61) (4.29) (−0.098) (59.0%)
(−0.44) (−1.71) (2.66) (−0.62)  (−0.21) (−1.13) (2.91) (−0.07)
WLS −0.023 −0.577 0.417 0.0000  −0.013 −0.495 0.417 0.00155
(−0.48) (−1.94) (2.66) (0.01)  (−0.31) (−1.49) (2.75) (0.29)
for 30 portfolios based on long-term reversal, short-term reversal and the earnings-to-price ratio

1952–2008

1952–2013

Intercept $λIFVol$ $λIFX$ $λΔc$ R2 Intercept $λIFVol$ $λIFX$ $λΔc$ R2
OLS −0.008 −0.412 0.437 −0.0030 76.1% 0.025 −0.041 0.402 0.0025 64.7%
(−0.39) (−2.15) (4.18) (−0.88) (73.4%) (1.19) (−0.15) (3.31) (0.68) (60.6%)
(−0.25) (−1.61) (2.98) (−0.61)  (0.82) (−0.11) (2.41) (0.49)
WLS −0.013 −0.450 0.443 −0.0028  0.011 −0.172 0.381 0.0036
(−0.39) (−1.83) (3.04) (−0.57)  (0.41) (−0.57) (2.52) (0.79)
for 30 portfolios based on long-term reversal, short-term reversal and the earnings-to-price ratio

1952–2008

1952–2013

Intercept $λIFVol$ $λIFX$ $λΔc$ R2 Intercept $λIFVol$ $λIFX$ $λΔc$ R2
OLS −0.008 −0.412 0.437 −0.0030 76.1% 0.025 −0.041 0.402 0.0025 64.7%
(−0.39) (−2.15) (4.18) (−0.88) (73.4%) (1.19) (−0.15) (3.31) (0.68) (60.6%)
(−0.25) (−1.61) (2.98) (−0.61)  (0.82) (−0.11) (2.41) (0.49)
WLS −0.013 −0.450 0.443 −0.0028  0.011 −0.172 0.381 0.0036
(−0.39) (−1.83) (3.04) (−0.57)  (0.41) (−0.57) (2.52) (0.79)

Since we obtained the P/D ratio factors using the twenty-five Fama and French portfolios, it is necessary to check that these factors can price other portfolios as well. For that purpose, we examine three sets of ten portfolios formed on the basis of short-term reversal, long-term reversal, and the E/P ratios. The cross-sectional regression results are tabulated in Table 6 and plotted in Figure 6. We again find that the R2 is reasonably high at 64.7% and that the intercept is not statistically different from zero at conventional significance levels with the absolute value of the t-statistic being only 0.82. We also again find that the coefficients of the innovations of the volatility and consumption/dividend growth factors also have the expected sign and that the latter is statistically significant.

Figure 6

Results of the cross-sectional regression of thirty portfolios (three sets of ten portfolios formed on the basis of long-term reversal, short-term reversal and the E/P ratio) on their betas with respect to consumption growth, IFVol and IFX for the period 1952–2013. FVol and FX are the first two principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio for the period 1943–2013 which, as we show, can be interpreted as the volatility and consumption/dividend growth factors. IFVol and IFX are their innovations.

Figure 6

Results of the cross-sectional regression of thirty portfolios (three sets of ten portfolios formed on the basis of long-term reversal, short-term reversal and the E/P ratio) on their betas with respect to consumption growth, IFVol and IFX for the period 1952–2013. FVol and FX are the first two principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio for the period 1943–2013 which, as we show, can be interpreted as the volatility and consumption/dividend growth factors. IFVol and IFX are their innovations.

A common concern when using the cross-sectional methodology is that the betas do not show sufficient cross-sectional variation. However, we find that this is not the case. The F test for the hypothesis that the factor innovation betas are the same for all the portfolios is strongly rejected for both factor innovations (p < 0.001).

### 6.2 Robustness tests

Since the excess returns of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio have a strong factor structure, it is important to use robust test statistics to eliminate the problem of useless factors being identified as useful (a problem forcefully brought out by Llewellen, Nagel, and Shanken 2010; Kleibergen 2009; Kleibergen 2010). Hence, we test the above cross-sectional regressions using the robust test statistics suggested by Kleibergen (2009) in Appendix B to ensure that the factors here are not useless and that the hypothesis of factor pricing is not rejected. As shown in detail in this Appendix, we find that both conditions are satisfied for our model specifications.

We also note that the number of time-series observations in our analysis is small due to our low-frequency data and that we find the betas of the assets to be significantly different from each other. As noted by Kan and Zhang (1999), these characteristics make it much less likely that a useless factor is spuriously found to be “useful” in a cross-sectional regression. Finally, we note that the cross-sectional regression intercept should be zero if the model is correctly specified, as emphasized by Jagannathan and Wang (2007), and that the intercepts that we obtain for all of the cross-sectional regressions examined in the previous subsection are indeed close to zero. In particular, these intercepts are all statistically indistinguishable from zero, and the absolute t-statistic value is always less than one.

As Ferson, Nallareddy, and Xie (2013) emphasize, it is important to check whether the results are robust to possible lookahead bias. We therefore investigate this and find that our results are not due to possible lookahead bias. In the presentation of the details of this investigation in the next section, we document that the bias-free estimates of the log P/D factors and their innovations, in fact, perform slightly better at predicting consumption and dividend growth.

### 6.3 The long-run risk, ICAPM and Fama-French three-factor model

In this section, we discuss the relationship between the long-run risk model, the ICAPM, and the Fama-French three-factor model in the context of the results we have found. In the long-run risk model, the shock to the log stochastic discount factor is given by

(15)
$mt+1−Et[mt+1]=−γ(Δct+1−Et[Δct+1]) −αx(Xt+1−Et[Xt+1]) −αv(Vt+1−Et[Vt+1]),$
where αx and αv are the market prices of risk for innovations to the long-run risk factors. The expressions for these quantities are provided in Internet Appendix A.

The return on wealth and consumption growth can be linked using the loglinear approximation of Campbell and Shiller (1988) as

(16)
$rw,t+1=ν0+ν1(wt+1−ct+1)−(wt−ct)+Δct+1,$
where κ0 and κ1 are loglinearization constants and where w stands for log wealth. In the long-run risk model, the log wealth to consumption ratio $wt−ct$ is given by the following relation
(17)
$wt−ct=A0+A1Xt+A2Vt,$
where the expressions for the constants A0, A1 and A2 are given in Internet Appendix A. Combining (16) and (17), we can write
(18)
$rw,t+1−Et[rw,t+1]=ν1A1(Xt+1−Et[Xt+1])+ν1A2(Vt+1−Et[Vt+1])+Δct+1−Et[Δct+1]$
or
(19)
$Δct+1−Et[Δct+1]=rw,t+1−Et[rw,t+1]−ν1A1(Xt+1−Et[Xt+1])−ν1A2(Vt+1−Et[Vt+1]).$
This further enables us to write the innovation in the log stochastic discount factor as
(20)
$mt+1−Et[mt+1]=−γ(rw,t+1−Et[rw,t+1]) −αx∼(Xt+1−Et[Xt+1]) −αv∼(Vt+1−Et[Vt+1]).$
This shows that the long-run risk model can be expressed in a manner similar to the ICAPM with the risk premium being determined by the innovations in the wealth return and the long-run risk state variables or factors.20

To further understand the relationship between the long-run risk and Fama-French three-factor models, we investigate the relation between the innovations of the log price-dividend ratio factors and the Fama-French factors (which are excess return factors). The latter have been proposed as proxies for future consumption growth by Parker and Julliard (2005) and consumption growth volatility by Boguth and Kuehn (2013). We summarize the results of regressing IFVol and IFX on the annual Fama-French factors in Table 7. We find that excess market returns are related to both consumption growth and consumption growth volatility, that SMB is related to consumption growth volatility, and that HML is related to future consumption and dividend growth.

Table 7

Results of regressing IFVol and IFX on the annual Fama-French factors. IFVol and IFX are the innovations of the first two principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1943–2013 which, as we show, can be interpreted as the volatility and consumption/dividend growth factors. The t-statistics are reported in parantheses

Regressions of IFVol and IFX on the Fama-French factors

Intercept $Rm−Rf$ SMB HML R2
IFVol $0.44***$ (4.40) $−4.17***$ (−7.32) $−2.11***$ (−3.30) –0.28 (−0.47) 64.3%
IFX $−0.18***$ (−3.60) $1.28***$ (7.53) –0.11 (−0.29) $1.35***$ (5.63) 45.0%
Regressions of IFVol and IFX on the Fama-French factors

Intercept $Rm−Rf$ SMB HML R2
IFVol $0.44***$ (4.40) $−4.17***$ (−7.32) $−2.11***$ (−3.30) –0.28 (−0.47) 64.3%
IFX $−0.18***$ (−3.60) $1.28***$ (7.53) –0.11 (−0.29) $1.35***$ (5.63) 45.0%
Table 8

Results of the constrained cross-sectional regression of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio and three sets of ten portfolios formed on the basis of earnings-to-price ratio, long-term reversal and short-term reversal, on their betas with respect to IFVol, IFX, and the Fama-French factors Mkt, SMB and HML. FVol and FX are the first two principal components of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio, which, as we show, can be interpreted as the volatility and consumption/dividend growth factors. IFVol and IFX are their innovations. The t values with and without the Shanken correction Shanken (1992); Shanken and Zhou (2007) are reported below the coefficient (the value without the correction is reported first)

Results of the constrained cross-sectional regression for the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio on the Fama-French factors, IFVol and IFX$E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVolλIFVol+βIFXλIFX+βMktλMkt+βSMBλSMB+βHMLλHML$.

1944–2008

1944–2013

$λIFVol$ $λIFX$ λMkt λSMB λHML $λIFVol$ $λIFX$ λMkt λSMB λHML
OLS −0.624 0.344 7.73 2.46 6.77 −0.786 0.368 8.62 2.78 5.97
(−3.76) (3.72) (3.33) (1.47) (4.07) (−3.97) (4.04) (3.89) (1.76) (3.80)
(−3.12) (2.84) (3.26) (1.44) (3.97) (−3.08) (2.99) (3.81) (1.72) (3.69)
Results of the constrained cross-sectional regression for the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio on the Fama-French factors, IFVol and IFX$E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVolλIFVol+βIFXλIFX+βMktλMkt+βSMBλSMB+βHMLλHML$.

1944–2008

1944–2013

$λIFVol$ $λIFX$ λMkt λSMB λHML $λIFVol$ $λIFX$ λMkt λSMB λHML
OLS −0.624 0.344 7.73 2.46 6.77 −0.786 0.368 8.62 2.78 5.97
(−3.76) (3.72) (3.33) (1.47) (4.07) (−3.97) (4.04) (3.89) (1.76) (3.80)
(−3.12) (2.84) (3.26) (1.44) (3.97) (−3.08) (2.99) (3.81) (1.72) (3.69)
Results of the constrained cross-sectional regression for thirty portfolios formed on the basis of the earnings-to-price ratio, long-term reversal and short-term reversal on the Fama-French factors, IFVol and IFX $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVolλIFVol+βIFXλIFX+βMktλMkt+βSMBλSMB+βHMLλHML$

1952–2008

1952–2013

$λIFVol$ $λIFX$ λMkt λSMB λHML $λIFVol$ $λIFX$ λMkt λSMB λHML
OLS −0.556 0.318 7.19 0.20 7.05 −0.709 0.265 8.18 0.26 6.77
(−3.62) (3.97) (2.88) (0.08) (3.13) (−3.84) (2.93) (3.42) (0.11) (3.25)
(−3.18) (3.25) (2.83) (0.06) (2.70) (−3.24) (2.34) (3.37) (0.09) (2.82)
Results of the constrained cross-sectional regression for thirty portfolios formed on the basis of the earnings-to-price ratio, long-term reversal and short-term reversal on the Fama-French factors, IFVol and IFX $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVolλIFVol+βIFXλIFX+βMktλMkt+βSMBλSMB+βHMLλHML$

1952–2008

1952–2013

$λIFVol$ $λIFX$ λMkt λSMB λHML $λIFVol$ $λIFX$ λMkt λSMB λHML
OLS −0.556 0.318 7.19 0.20 7.05 −0.709 0.265 8.18 0.26 6.77
(−3.62) (3.97) (2.88) (0.08) (3.13) (−3.84) (2.93) (3.42) (0.11) (3.25)
(−3.18) (3.25) (2.83) (0.06) (2.70) (−3.24) (2.34) (3.37) (0.09) (2.82)

This also means that the innovation to the long-run risk model’s stochastic discount factor (20) is very much similar to that of the Fama-French three-factor model provided one can interpret the return of wealth as the excess market return.21 Hence, there is a close relation between the Fama-French three-factor model and the long-run risk model with the SMB and HML factors naturally being identified with innovations to the long-run risk state variables. The relation can also be interpreted in an ICAPM-like manner as noted above.

We can further understand the relation between the price-dividend ratio factor innovations and the excess return factors using the following expression, which follows from the loglinear approximation of Campbell and Shiller (1988)

(21)
$rl,t+1−rf,t≈Et[rl,t+1−rf,t]+ν1,l(pl,t+1−dl,t+1−Et[pl,t+1−dl,t+1])︸price−dividend ratio innovation +(dl,t+1−dl,t−Et[dl,t+1−dl,t])︸dividend growth innovation$
From it, we see that the main difference between the two types of factors arises in the dividend growth innovation term which could have their own factor structure (with the expected return term contributing to a lesser extent). If these factors are second order in importance, the innovations of the log price-dividend ratio factors and the excess return factors will do equally well. Hence, to the extent the long-run risk model and the Fama and French three-factor model do equally well, they may be viewed as alternative representations of the same underlying phenomenon. The fact that one does not drive out the other suggests that both models may be missing different aspects of reality.

### 6.4 Price-dividend ratio factors and the Fama-French factors

We undertake further analysis to clarify the link between the long-run risk model and the Fama-French three-factor model by looking at the P/D “factors” that result from using portfolios analogous to those used in defining the SMB and HML quantities. We need to do this as using the Fama-French factors directly in predictive regressions of these quantities, unsurprisingly, does not work well as seen from the results in Table 9. This is because the Fama-French factors are innovations of underlying factors or state variables and not the state variables themselves. Since we know that the market price-dividend ratio does not predict future dividend growth from both Table 4 and the study of Cochrane (2008), we concentrate on the price-dividend ratio factor corresponding to SMB and HML. Since SMB is defined to be SMB = $13$ (Small Value (sv) + Small Neutral (sn) + Small Growth (sg)) - $13$ (Big Value (bv)+ Big Neutral (bn)+ Big Growth(bg)) (these portfolios being defined with respect to the six Fama-French portfolios formed on the basis of size and book-to-market ratio), we define the P/D analogue corresponding to SMB as

(22)
$FSMB=13(logP/Dsv+logP/Dsn+logP/Dsg)−13(logP/Dbv+logP/Dbn+logP/Dbg)$
Similarly, since HML is defined as $12$ (sv + bv) - $12$ (sg + bg), we define the define the P/D analogue corresponding to HML as
(23)
$FHML=12(logP/Dsv+logP/Dbv)−12(logP/Dsg+logP/Dbg)$
From Table 10, we see that FSMB and the log market price-dividend ratio are both negatively related to consumption growth volatility. This is consistent with the relationship that we find between the innovations of the volatility P/D factor and the Fama-French factors in Table 7. Further, the strong negative relationship between the log market price-dividend ratio and consumption growth volatility that we find is consistent with the long-run risk literature (Bansal, Yaron, and Kiku 2007, 2010; Beeler and Campbell 2012).

Table 9

Results of predictive regressions for market dividend (dm) and real-time consumption (cRT) growth using the three Fama-French factors and the identified X-type P/D factor, FX, which is the second principal component of the log P/D ratios of the twenty-five Fama-French portfolios. The t-statistics are based on standard errors, which are Newey-West corrected with the required number of lags estimated using the procedure of Newey and West (1994)

Regression of real market dividend growth and real-time consumption growth on lagged FX and Fama-French factors

$Mkt−Rft×1000$ $SMBt×1000$ $HMLt×1000$ $FX,t$ R2
$Δdm,t+1$ $0.92*$ (1.67) −0.28 (−0.45) 0.44 (0.81) $0.0235***$ (2.83) 18.6%
$dm,t+2−dm,t$ 0.32 (0.28) −0.07 (−0.06) 1.31 (1.46) $0.0468*$ (1.62) 17.3%
$ct+2RT−ctRT$ (1965–2008) 0.20 (1.67) 0.20 (0.91) 0.12 (0.75) $0.0103**$ (2.06) 26.1%
$ct+2RT−ctRT$ (1965–2013) $0.37*$ (1.68) 0.22 (1.00) −0.83 (−0.75) −0.0036 (−0.59) 10.3%
Regression of real market dividend growth and real-time consumption growth on lagged FX and Fama-French factors

$Mkt−Rft×1000$ $SMBt×1000$ $HMLt×1000$ $FX,t$ R2
$Δdm,t+1$ $0.92*$ (1.67) −0.28 (−0.45) 0.44 (0.81) $0.0235***$ (2.83) 18.6%
$dm,t+2−dm,t$ 0.32 (0.28) −0.07 (−0.06) 1.31 (1.46) $0.0468*$ (1.62) 17.3%
$ct+2RT−ctRT$ (1965–2008) 0.20 (1.67) 0.20 (0.91) 0.12 (0.75) $0.0103**$ (2.06) 26.1%
$ct+2RT−ctRT$ (1965–2013) $0.37*$ (1.68) 0.22 (1.00) −0.83 (−0.75) −0.0036 (−0.59) 10.3%
Table 10

Results of predictive regressions for market dividend (dm) and real-time consumption (cRT) growth using lagged FSMB and FHML, which are the price-dividend ratio analogs of the factors underlying SMB and HML. FSMB is defined as $13(logP/Dsv+logP/Dsn+logP/Dsg)−13(logP/Dbv+logP/Dbn+logP/Dbg)$ where sg, sn, sv, bg, bn, and bv stand for small growth, small neutral, small value, big growth, big neutral and big value respectively. FHML is similarly defined as $12(logP/Dsv+logP/Dbv)−12(logP/Dsg+logP/Dbg)$ The t-statistics are based on standard errors which are Newey-West corrected with the required number of lags estimated using the procedure of Newey and West (1994)

Regression of consumption growth volatility on $log(P/D)mkt$, FSMB, and FHML

$log(P/D)mkt$ FSMB FHML R2
24 quarter volatility $−0.318***$ (−3.72) $−0.594***$ (−3.46) 0.045 (0.29) 82.4%
12 quarter volatility $−0.325*$ (−1.74) $−0.871**$ (−2.35) −0.152 (−0.44) 62.4%
6 quarter volatility –0.326 (−1.37) $−1.043**$ (−2.58) −0.344 (−1.03) 46.9%
Regression of consumption growth volatility on $log(P/D)mkt$, FSMB, and FHML

$log(P/D)mkt$ FSMB FHML R2
24 quarter volatility $−0.318***$ (−3.72) $−0.594***$ (−3.46) 0.045 (0.29) 82.4%
12 quarter volatility $−0.325*$ (−1.74) $−0.871**$ (−2.35) −0.152 (−0.44) 62.4%
6 quarter volatility –0.326 (−1.37) $−1.043**$ (−2.58) −0.344 (−1.03) 46.9%
Regression of real market dividend growth and real-time consumption growth on lagged FSMB and FHML

$FSMB,t$ $FHML,t$ R2 (Adj. R2
$Δdm,t+1$ $0.0978***$ (3.04) $0.1106***$ (4.77) 20.1% (17.7%)
$dm,t+2−dm,t$ $0.169***$ (2.91) $0.199**$ (4.98) 26.3% (24.1%)
$ct+2RT−ctRT$ (1965-2008) 0.0101 (0.98) 0.0185 (1.09) 6.5% (1.7%)
$ct+2RT−ctRT$ (1965-2013) −0.0039 (−0.26) −0.0007 (−0.04) 0.2% (−4.3%)
Regression of real market dividend growth and real-time consumption growth on lagged FSMB and FHML

$FSMB,t$ $FHML,t$ R2 (Adj. R2
$Δdm,t+1$ $0.0978***$ (3.04) $0.1106***$ (4.77) 20.1% (17.7%)
$dm,t+2−dm,t$ $0.169***$ (2.91) $0.199**$ (4.98) 26.3% (24.1%)
$ct+2RT−ctRT$ (1965-2008) 0.0101 (0.98) 0.0185 (1.09) 6.5% (1.7%)
$ct+2RT−ctRT$ (1965-2013) −0.0039 (−0.26) −0.0007 (−0.04) 0.2% (−4.3%)

We also see from Table 10 that both FSMB and FHML predict real dividend growth very well but do not predict real-time consumption growth even up to the financial crisis (we do not use the market price-dividend ratio for this regression as that is not related to future dividend or consumption growth as can be seen from Table 4 and from the study of Cochrane 2008). This is to be expected, since the focus of Fama and French (1993) was to explain the cross-section of returns, and the model does well along the dimension by capturing whatever is missing in the class of long-run risk models we examine. For example, the aggregate dividend process may be more complex, and interesting, the HML and SMB P/D factors forecast future dividends surprisingly well, which may be more important for pricing stocks.

To address the look ahead bias critique of Ferson, Nallareddy, and Xie (2013), we also undertake a complete reanalysis of the model where we remove the possibility of lookahead bias. We do so by recalculating the rotation matrix for the principal components, and therefore the factors and their innovations, every year with these components and their innovations being calculated based on the rotation matrix estimated with data up to the previous year (i.e., using expanding windows). We ensure that a minimum of twenty-three years of data is used for the calculation of each rotation matrix. We denote these no lookahead bias version of the factors as $F1nlb$ and $F2nlb$.

We find that these estimated no lookahead bias factors, labeled $F1nlb$ and $F2nlb$, generally track consumption growth volatility and predict market dividend and real-time consumption growth in a manner similar to that documented for the in sample factors. This is documented in Tables 11 and 12. The predictability of real-time consumption growth, as for the standard factors, does not hold when the data are extended to 2013. A Bai-Perron structural break analysis shows, just as it does for the standard factors, that the coefficient of $F2nlb$ does not change across the structural break at 2006.22 The signs in both the volatility and dividend and consumption growth regressions are also as expected.

Table 11

Results of regressing annualized 12 and 24 quarter consumption growth volatility against $F1nlb$ and $F2nlb$, the lookahead bias free estimates of the first and second log price-dividend ratio factors of the 25 Fama-French portfolios formed on the basis of size and book-to-market ratio. The regressions are for the period 1965–2013. The t-statistics are based on standard errors, which are Newey-West corrected with the required number of lags estimated using the procedure of Newey and West (1994). The detrending is done by regressing against a time trend

Regression of annualized 12 and 24 quarter consumption growth volatility on $F1nlb$ and $F2nlb$

$F1nlb≡FVolnlb$ $F2nlb≡FXnlb$ R2
$vt24$ $0.102***$ (6.80) $0.060*$ (1.71) 74.9%
Detrended $vt24$ $0.0538**$ (2.39) 0.114 (1.14) 26.3%
$vt12$ $0.090***$ (2.81) –0.025 (−0.35) 43.1%
Regression of annualized 12 and 24 quarter consumption growth volatility on $F1nlb$ and $F2nlb$

$F1nlb≡FVolnlb$ $F2nlb≡FXnlb$ R2
$vt24$ $0.102***$ (6.80) $0.060*$ (1.71) 74.9%
Detrended $vt24$ $0.0538**$ (2.39) 0.114 (1.14) 26.3%
$vt12$ $0.090***$ (2.81) –0.025 (−0.35) 43.1%
Table 12

Results of regressing real annual market dividend growth and real-time consumption growth ($ΔcRT$) against lagged $F1nlb$ and $F2nlb$, the lookahead bias-free estimates of the first and second log price-dividend ratio factors of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio. The t-statistics are based on standard errors, which are Newey-West corrected with the required number of lags estimated using the procedure of Newey and West (1994)

Regression of real market dividend growth on $F1nlb$ and $F2nlb$ and the log market price-dividend ratio

$F1nlb≡FVolnlb$ $F2nlb≡FXnlb$ $log(P/D)m$ R2
1 yr. Mkt div. growth 0.0049 (0.79) $0.0458***$ (3.20)  20.5%
(1966-2008)   0.024 (0.71) 2.8%
1 yr. Mkt div. growth 0.0076 (1.38) $0.0317*$ (1.78)  15.5%
(1966-2013)   0.035 (1.00) 4.1%
Regression of real market dividend growth on $F1nlb$ and $F2nlb$ and the log market price-dividend ratio

$F1nlb≡FVolnlb$ $F2nlb≡FXnlb$ $log(P/D)m$ R2
1 yr. Mkt div. growth 0.0049 (0.79) $0.0458***$ (3.20)  20.5%
(1966-2008)   0.024 (0.71) 2.8%
1 yr. Mkt div. growth 0.0076 (1.38) $0.0317*$ (1.78)  15.5%
(1966-2013)   0.035 (1.00) 4.1%
Regression of real-time annual consumption growth on lagged values of $F1nlb$ and $F2nlb$

$F1nlb≡FVolnlb$ $F2nlb≡FXnlb$ R2
$Δct+1RT$ $6.0×10−4(0.43)$ 0.0049 (1.36) 10.4%
(1965–2008)  $0.0036**$ (2.25) 9.7%
$Δct+1RT$ −0.0015 (−1.00) −0.0026 (−0.90) 4.8%
(1965–2013)
$Δct+1RT+Δct+2RT$ 0.00138 (0.66) $0.0092*$ (1.77) 10.7%
(1966–2008)  $0.0060**$ (2.14) 9.4%
$Δct+1RT+Δct+2RT$ −0.0025 (−0.74) −0.0059 (−0.79) 5.7%
(1966–2013)  −0.0019 (−0.30) 1.3%
Regression of real-time annual consumption growth on lagged values of $F1nlb$ and $F2nlb$

$F1nlb≡FVolnlb$ $F2nlb≡FXnlb$ R2
$Δct+1RT$ $6.0×10−4(0.43)$ 0.0049 (1.36) 10.4%
(1965–2008)  $0.0036**$ (2.25) 9.7%
$Δct+1RT$ −0.0015 (−1.00) −0.0026 (−0.90) 4.8%
(1965–2013)
$Δct+1RT+Δct+2RT$ 0.00138 (0.66) $0.0092*$ (1.77) 10.7%
(1966–2008)  $0.0060**$ (2.14) 9.4%
$Δct+1RT+Δct+2RT$ −0.0025 (−0.74) −0.0059 (−0.79) 5.7%
(1966–2013)  −0.0019 (−0.30) 1.3%

For the sake of brevity, we only report the results for the cross-sectional regression of the thirty portfolios based on long-term reversal, short-term reversal and the earnings-to-price ratio on lookahead bias-free versions of the factor innovations as it also removes the concern that the factors are estimated with the same portfolios as those being tested. These results are summarized in table 13 and are plotted in Figure 7. The results are supportive for the model as the R2 is reasonably high and that the intercept of 2.3%/year is statistically insignificant.

Figure 7

Results of the cross-sectional regression of thirty portfolios (three sets of ten portfolios formed on the basis of long-term reversal, short-term reversal and the E/P ratio) on their betas with respect to $IFVolnlb, IFXnlb$, and $Δc$ innovations. $FVolnlb$ and $FXnlb$ are the volatility and consumption/dividend growth factors constructed in a lookahead bias-free manner and $IFVolnlb$ and $IFXnlb$ are their innovations. The cross-sectional regression is for the period 1966–2013.

Figure 7

Results of the cross-sectional regression of thirty portfolios (three sets of ten portfolios formed on the basis of long-term reversal, short-term reversal and the E/P ratio) on their betas with respect to $IFVolnlb, IFXnlb$, and $Δc$ innovations. $FVolnlb$ and $FXnlb$ are the volatility and consumption/dividend growth factors constructed in a lookahead bias-free manner and $IFVolnlb$ and $IFXnlb$ are their innovations. The cross-sectional regression is for the period 1966–2013.

Table 13

Results of the cross-sectional regression, including pricing errors, of thirty portfolios (three sets of ten portfolios formed on the basis of long-term reversal, short-term reversal and the E/P ratio) on their betas with respect to $IFVolnlb$, $IFXnlb$ and consumption growth. $FVolnlb$ and $FXnlb$ are the volatility and consumption/dividend growth factors estimated in a manner free of lookahead bias and $IFVolnlb$ and $IFXnlb$ are their innovations also estimated in a manner free of lookahead bias. For the OLS coefficients, the t values with and without the Shanken correction Shanken (1992); Shanken and Zhou (2007) are reported below the coefficient (the value without the correction is reported first), while for the WLS coefficients, only the t values with the correction are reported. The R2 adjusted for the number of variables is reported below the unadjusted R2

Results of the cross-sectional regression $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVolλIFVol+βIFXλIFX+βΔcλΔc$ using a lookahead bias-free approach for thirty portfolios based on long- and short-term reversal and the E/P ratio

1966–2008

1966–2013

Intercept $λIFnlbVol$ $λIFXnlb$ $λΔc$ R2 Intercept $λIFnlbVol$ $λIFXnlb$ $λΔc$ R2
OLS −0.006 −0.351 0.401 −0.0036 69.2% 0.023 0.128 0.398 0.00086 58.3%
(−0.19) (−1.36) (2.76) (−1.09) (65.6%) (0.85) (0.41) (2.64) (0.23) (53.5%)
(−0.13) (−1.01) (1.97) (−0.80)  (0.59) (0.31) (1.94) (0.18)
WLS −0.014 −0.387 0.386 −0.0017  0.006 −0.059 0.385 0.0018
(−0.35) (−1.28) (2.16) (−0.44)  (0.17) (−0.18) (2.05) (0.41)
Results of the cross-sectional regression $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVolλIFVol+βIFXλIFX+βΔcλΔc$ using a lookahead bias-free approach for thirty portfolios based on long- and short-term reversal and the E/P ratio

1966–2008

1966–2013

Intercept $λIFnlbVol$ $λIFXnlb$ $λΔc$ R2 Intercept $λIFnlbVol$ $λIFXnlb$ $λΔc$ R2
OLS −0.006 −0.351 0.401 −0.0036 69.2% 0.023 0.128 0.398 0.00086 58.3%
(−0.19) (−1.36) (2.76) (−1.09) (65.6%) (0.85) (0.41) (2.64) (0.23) (53.5%)
(−0.13) (−1.01) (1.97) (−0.80)  (0.59) (0.31) (1.94) (0.18)
WLS −0.014 −0.387 0.386 −0.0017  0.006 −0.059 0.385 0.0018
(−0.35) (−1.28) (2.16) (−0.44)  (0.17) (−0.18) (2.05) (0.41)
Pricing errors $×100$
1966–2008

bottom         top
10
LTR −0.58 0.81 0.27 −1.38 0.21 0.66 1.24 0.89 −1.11 0.15
STR −1.03 2.94 1.69 0.56 0.25 −1.87 −1.47 −0.34 −0.85 −1.81
E/P 0.09 −1.06 −0.47 −0.75 0.30 0.26 1.19 0.68 0.40 0.13

1966–2013

bottom         top
10
LTR −0.34 1.24 0.23 0.49 −1.02 0.63 1.32 0.80 −2.06 0.07
STR −1.68 2.47 1.38 0.05 0.07 −1.58 −0.96 0.22 −0.67 −2.00
E/P −0.55 −0.89 −0.75 −0.72 −0.28 −0.40 1.75 0.37 1.21 1.58
Pricing errors $×100$
1966–2008

bottom         top
10
LTR −0.58 0.81 0.27 −1.38 0.21 0.66 1.24 0.89 −1.11 0.15
STR −1.03 2.94 1.69 0.56 0.25 −1.87 −1.47 −0.34 −0.85 −1.81
E/P 0.09 −1.06 −0.47 −0.75 0.30 0.26 1.19 0.68 0.40 0.13

1966–2013

bottom         top
10
LTR −0.34 1.24 0.23 0.49 −1.02 0.63 1.32 0.80 −2.06 0.07
STR −1.68 2.47 1.38 0.05 0.07 −1.58 −0.96 0.22 −0.67 −2.00
E/P −0.55 −0.89 −0.75 −0.72 −0.28 −0.40 1.75 0.37 1.21 1.58

### 6.6 Factors derived from other portfolios

To address the concern that the results derive purely from a fortunate choice of portfolios to estimate the factors from, we carry out the same procedure using a different set of portfolios, namely ones formed on the basis of dividend yield. We again find two significant factors with the first being related to consumption growth volatility and the second to future consumption and dividend growth. The regressions documenting these facts are tabulated in Tables 14 and 15.

Table 14

Results of regressing volatility as defined in (14) against $F1dy$ and $F2dy$, the two significant principal components of the log P/D ratios of ten portfolios formed on the basis of dividend yield for the period 1947–2013. The t-statistics are based on standard errors which are Newey-West corrected with the required number of lags estimated using the procedure in Newey and West (1994). The principal components of the quarterly log P/D ratios were used in these regressions. Use of the rotation matrix for the annual log P/D ratios to create quarterly versions of these principal components leads to very similar results. The detrending is carried out by regressing against a time trend

Regression of consumption growth volatility on $F1dy$ and $F2dy$

$F1dy≡FVoldy$ $F2dy≡FXdy$ R2
24 quarter volatility $0.217***$ (5.29) 0.133 (1.34) 77.2%
12 quarter volatility $0.238***$ (3.13) 0.008 (004) 55.4%
6 quarter volatility $0.239***$ (3.46) −0.119 (−0.47) 39.9%
Regression of consumption growth volatility on $F1dy$ and $F2dy$

$F1dy≡FVoldy$ $F2dy≡FXdy$ R2
24 quarter volatility $0.217***$ (5.29) 0.133 (1.34) 77.2%
12 quarter volatility $0.238***$ (3.13) 0.008 (004) 55.4%
6 quarter volatility $0.239***$ (3.46) −0.119 (−0.47) 39.9%
Regression of detrended consumption growth volatility on detrended $F1dy≡FVoldy$

$F1dy≡FVoldy$ R2
24 quarter volatility $0.147***$ (3.76) 35.5%
12 quarter volatility $0.152**$ (2.24) 14.7%
6 quarter volatility $0.143*$ (1.74) 6.9%
Regression of detrended consumption growth volatility on detrended $F1dy≡FVoldy$

$F1dy≡FVoldy$ R2
24 quarter volatility $0.147***$ (3.76) 35.5%
12 quarter volatility $0.152**$ (2.24) 14.7%
6 quarter volatility $0.143*$ (1.74) 6.9%
Table 15

Results of regressing real annual market dividend growth ($Δdm$ where dm is the log real annual market dividend) and real-time consumption growth ($ΔcRT$) against lagged $F1dy$ and $F2dy$, the two significant principal components of the log P/D ratios of ten portfolios formed on the basis of dividend yield, and the lagged log market P/D ratio ($log(P/D)m$). The t-statistics are based on standard errors which are Newey-West corrected with the required number of lags estimated using the procedure of Newey and West (1994)

Regression of real market dividend growth on lagged $F1dy$ and $F2dy$, compared with that on the lagged log market P/D ratio

$F1,tdy≡FVol,tdy$ $F2,tdy≡FX,tdy$ $log(P/D)m,t$ R2
1944–2008

$Δdm,t+1$ −0.0011 (−0.21) $0.0685***$ (5.44)  16.4%
0.0028 (0.10) 0.0%
$Δdm,t+2$ 0.0022 (0.38) $0.0251*$ (1.73)  2.6%
−0.0058 (−0.23) 0.2%
$dm,t+3−dm,t$ 0.0002 (0.01) $0.112**$ (2.84)  10.3%
0.005 (0.05) 0.0%

1944–2013

$Δdm,t+1$ −0.0020 (−0.38) $0.0808***$ (4.75)  18.2%
0.011 (0.41) 0.4%
$Δdm,t+2$ 0.0024 (0.46) $0.0252**$ (2.05)  4.7%
−0.0040 (−0.14) 0.1%
$dm,t+3−dm,t$ 0.0037 (0.15) $0.110**$ (2.31)  8.8%
0.005 (0.05) 0.0%
Regression of real market dividend growth on lagged $F1dy$ and $F2dy$, compared with that on the lagged log market P/D ratio

$F1,tdy≡FVol,tdy$ $F2,tdy≡FX,tdy$ $log(P/D)m,t$ R2
1944–2008

$Δdm,t+1$ −0.0011 (−0.21) $0.0685***$ (5.44)  16.4%
0.0028 (0.10) 0.0%
$Δdm,t+2$ 0.0022 (0.38) $0.0251*$ (1.73)  2.6%
−0.0058 (−0.23) 0.2%
$dm,t+3−dm,t$ 0.0002 (0.01) $0.112**$ (2.84)  10.3%
0.005 (0.05) 0.0%

1944–2013

$Δdm,t+1$ −0.0020 (−0.38) $0.0808***$ (4.75)  18.2%
0.011 (0.41) 0.4%
$Δdm,t+2$ 0.0024 (0.46) $0.0252**$ (2.05)  4.7%
−0.0040 (−0.14) 0.1%
$dm,t+3−dm,t$ 0.0037 (0.15) $0.110**$ (2.31)  8.8%
0.005 (0.05) 0.0%
Regression of real-time annual consumption growth on lagged $F1dy$ and $F2dy$

$F1,tdy≡FVol,tdy$ $F2,tdy≡FX,tdy$ R2
1965–2008

$Δct+1RT$ $−9.1×10−4(−0.43)$ $0.0091*$ (1.69) 11.0%
$Δct+2RT$ $2.8×10−4(0.13)$ $0.0085**$ (2.07) 6.3%
$Δct+1RT+Δct+2RT$ 0.0011 (0.25) $0.0171**$ (2.06) 11.8%

1965–2013

$Δct+1RT$ $7.56×10−4(0.31)$ 0.0026 (0.38) 1.6%
$Δct+2RT$ 0.0011 (0.39) 0.0017 (0.28) 1.1%
$Δct+1RT+Δct+2RT$ $−5.68×10−6(0.00)$ 0.0033 (0.29) 0.4%
Regression of real-time annual consumption growth on lagged $F1dy$ and $F2dy$

$F1,tdy≡FVol,tdy$ $F2,tdy≡FX,tdy$ R2
1965–2008

$Δct+1RT$ $−9.1×10−4(−0.43)$ $0.0091*$ (1.69) 11.0%
$Δct+2RT$ $2.8×10−4(0.13)$ $0.0085**$ (2.07) 6.3%
$Δct+1RT+Δct+2RT$ 0.0011 (0.25) $0.0171**$ (2.06) 11.8%

1965–2013

$Δct+1RT$ $7.56×10−4(0.31)$ 0.0026 (0.38) 1.6%
$Δct+2RT$ 0.0011 (0.39) 0.0017 (0.28) 1.1%
$Δct+1RT+Δct+2RT$ $−5.68×10−6(0.00)$ 0.0033 (0.29) 0.4%

The innovations of the principal components of the log price-dividend ratios are also able to price both the twenty-five Fama-French portfolios and the three sets of ten portfolios formed on the basis of long- and short- term reversal and the earnings-to-price ratio reasonably well. The R2 of the cross-sectional regressions documented in Table 16 are reasonably high and, more importantly, the intercepts are economically small and statistically insignificant with both intercepts being below 1%/year.

Table 16

Results of the two pass cross-sectional regressions, $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVoldyλIFVoldy+βIFXdyλIFXdy+βΔcλΔc$, of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio, and three sets of ten portfolios formed on the basis of long-term reversal, short-term reversal and the earnings-to-price ratio, on their betas with respect to consumption growth, $IFVoldy$ and $IFXdy$. $FVoldy$ and $FXdy$ are the first two principal components of the log P/D ratios of ten portfolios formed on the basis of dividend yield, which, as we show, can be interpreted as volatility and consumption/dividend growth factors, while $IFVoldy$ and $IFXdy$ are their innovations. For the OLS coefficients, the t values with and without the Shanken correction Shanken (1992); Shanken and Zhou (2007) are reported below the coefficient (the value without the correction is reported first), while for the WLS coefficients, only the t values with the correction are reported. The R2 adjusted for the number of variables is reported below the unadjusted R2

Results of the cross-sectional regression $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVoldyλIFVoldy+βIFXdyλIFXdy+βΔcλΔc$

for the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio

1944–2008

1944–2013

Intercept $λIFVoldy$ $λIFdyX$ $λΔc$ R2 Intercept $λIFVoldy$ $λIFdyX$ $λΔc$ R2
OLS −0.0070 −0.0865 0.394 −0.0129 62.1% 0.0067 −0.151 0.318 0.00176 50.6%
(−0.19) (−0.55) (3.76) (−2.22) (56.7%) (0.20) (−0.98) (3.75) (0.42) (43.5%)
(−0.08) (−0.28) (1.79) (−1.06)  (0.12) (−0.66) (2.47) (0.28)
WLS −0.0211 −0.183 0.346 −0.0102  −0.0100 −0.258 0.264 0.0027
(−0.28) (−0.66) (1.85) (−1.02)  (−0.20) (−1.22) (2.33) (0.47)
Results of the cross-sectional regression $E[ri,t+Δt−rf,t]+12Var[ri,t+Δt−rf,t]=βIFVoldyλIFVoldy+βIFXdyλIFXdy+βΔcλΔc$

for the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio

1944–2008

1944–2013

Intercept $λIFVoldy$ $λIFdyX$ $λΔc$ R2 Intercept $λIFVoldy$ $λIFdyX$ $λΔc$ R2
OLS −0.0070 −0.0865 0.394 −0.0129 62.1% 0.0067 −0.151 0.318 0.00176 50.6%
(−0.19) (−0.55) (3.76) (−2.22) (56.7%) (0.20) (−0.98) (3.75) (0.42) (43.5%)
(−0.08) (−0.28) (1.79) (−1.06)  (0.12) (−0.66) (2.47) (0.28)
WLS −0.0211 −0.183 0.346 −0.0102  −0.0100 −0.258 0.264 0.0027
(−0.28) (−0.66) (1.85) (−1.02)  (−0.20) (−1.22) (2.33) (0.47)
for thirty portfolios based on long-term reversal, short-term reversal and the earnings-to-price ratio

1952–2008

1952–2013

Intercept $λIFVoldy$ $λIFXdy$ $λΔc$ R2 Intercept $λIFVoldy$ $λIFXdy$ $λΔc$ R2
OLS −0.0108 −0.167 0.314 −0.0041 73.7% 0.0049 −0.148 0.305 −0.0023 61.2%
(−0.53) (−1.37) (4.05) (−1.38) (70.7%) (0.24) (−1.23) (3.91) (−0.71) (56.8%)
(−0.33) (−0.96) (2.70) (−0.94)  (0.16) (−0.90) (2.71) (−0.50)
WLS −0.0096 −0.151 0.297 0.0002  0.0067 −0.150 0.258 0.0021
(−0.33) (0.95) (2.87) (0.05)  (0.27) (−1.04) (2.69) (0.62)
for thirty portfolios based on long-term reversal, short-term reversal and the earnings-to-price ratio

1952–2008

1952–2013

Intercept $λIFVoldy$ $λIFXdy$ $λΔc$ R2 Intercept $λIFVoldy$ $λIFXdy$ $λΔc$ R2
OLS −0.0108 −0.167 0.314 −0.0041 73.7% 0.0049 −0.148 0.305 −0.0023 61.2%
(−0.53) (−1.37) (4.05) (−1.38) (70.7%) (0.24) (−1.23) (3.91) (−0.71) (56.8%)
(−0.33) (−0.96) (2.70) (−0.94)  (0.16) (−0.90) (2.71) (−0.50)
WLS −0.0096 −0.151 0.297 0.0002  0.0067 −0.150 0.258 0.0021
(−0.33) (0.95) (2.87) (0.05)  (0.27) (−1.04) (2.69) (0.62)

## 7. Conclusion

We show that a broad class of long-run risk models, that subsume those of Bansal and Yaron (2004), Bansal, Yaron, and Kiku (2007), and Zhou and Zhu (2009), imply that the log P/D ratios of financial assets have a factor structure when the intertemporal budget constraint of the marginal investor can be well approximated by the loglinear method of Campbell and Shiller (1988). We demonstrate that these factors are related to per capita aggregate consumption growth, market dividend growth, and consumption growth volatility. We find that the log P/D ratios have two significant factors, one of which predicts consumption growth volatility, and the other predicts future dividend and consumption growth rates.

We explain the Beeler and Campbell (2012) finding that the log P/D ratio of the stock market index portfolio does not help predict future dividends or aggregate consumption growth. However, this is in and by itself, not evidence against long-run risk models. That is because the log P/D ratio of the stock market index portfolio weights mostly on the volatility price-dividend ratio factor, and not on the factor that is related to expected future consumption and dividends.

The innovations of these two P/D factors perform well in explaining the cross-section of excess returns of the twenty-five Fama-French portfolios but also three sets of ten portfolios based on long-term reversal, short-term reversal and the earnings-to-price ratio. The coefficients obtained from the cross-sectional regressions are statistically and economically significant and have the right sign. Further, the cross-sectional regression intercepts are generally economically small and not statistically significantly different from zero. The latter is an important test that must be satisfied by a well-specified model as emphasized by Jagannathan and Wang (2007).

Our findings link the Fama and French (1993) three-factor model, that has become the workhorse of the academic empirical finance literature, and the more recent long-run risk models that relate the equity premium, the risk-free rate, and their dynamics to underlying macroeconomic conditions. This follows from our analysis showing that the Fama-French factors are closely related to the innovations of the long-run risk factors. In an economy with perfect markets and investors with rational expectations, the intertemporal asset pricing model of Merton (1973), as well as a wide class of long-run risk models, will in general, hold at the same time, and the state variables summarizing the dynamics of the investment opportunity sets and timing of cash flows available to investors can be represented as the P/D ratios of a few cleverly constructed stock portfolios. Our findings suggest that the Fama and French (1993) three-factor model and a wide class of long-run risk models are alternative representations of the same underlying phenomena.

#### Appendix A Testing Long-Run Risk Models: Monte Carlo Evidence

##### A.1 The Model

For the purpose of analyzing the performance of the asset pricing tests, we use the long-run risk model of Bansal and Yaron (2004). In this model, the per capita consumption and dividend growth rates $Δc$ and $Δd$ (for M assets indexed by l) and their common persistent component x are assumed to follow the processes assumed by Bansal and Yaron (2004)

(A.1)
$Δct+1=μ+Xt+Vtηt+1,$

(A.2)
$Xt+1=ρXt+φxVtet+1,$

(A.3)
$Δdl,t+1=μl,d+ϕlXt+φl,dV¯ul,t+1,1≤l≤M,$

(A.4)
$Vt+1=V¯+ν(Vt−V¯)+σwt+1,$
where the shocks $et+1, ηt+1$, and $wt+1$ are taken to be independent standard normals for parsimony. $ul,t+1$ is a vector of normally distributed shocks with covariance Ω, which is independent of $e,η$, and w. In the simulations, Ω is set so as to fit the factor structure of returns. (Note that we follow the convention that lowercase characters stand for the logarithm of quantities denoted by the corresponding uppercase characters.)

Consumers in the model have Epstein-Zin-Weil preferences (Epstein and Zin 1989; Weil 1990)

(A.5)
$Ut=((1−δ)Ct1−γθ+δEt[Ut+11−γ]1θ)θ1−γ,$
with $γ>1/ψ$, where γ is the relative risk aversion and ψ is the elasticity of intertemporal substitution. This implies that they prefer early resolution of uncertainty and that persistent consumption and volatility shocks have a high market price of risk. This high price of risk results in a high equity premium and low risk-free rate. With these preferences, asset returns satisfy
(A.6)
$Et[δθ(Ct+1Ct)−θ/ψRa,t+1−(1−θ)Ri,t+1]=1,$
where C is per capita consumption, Ra is the gross return on an asset that pays a dividend of per capita consumption, Ri is the asset return, $0<δ<1$ is the time discount factor, and θ is defined to be
(A.7)
$θ=1−γ1−1ψ$
The log P/D ratios of assets in this economy have a factor structure (within the loglinear approximation) with the factors being Xt and Vt. In other words, if $zi,t$ is the log P/D ratio of asset i, we have
(A8)
$zi,t=A0,i+A1,iXt+A2,iVt,$
This is shown for this particular model by Bansal and Yaron (2004) and similar results for related models are shown by Bansal, Yaron, and Kiku (2007), Drechsler and Yaron (2011), Zhou and Zhu (2009), Ferson, Nallareddy, and Xie (2013), Eraker and Shaliastovich (2008) and in Internet Appendix A. Since the dividend processes of the assets are specified in this model, the relation above gives the time-series of their prices for a given realization of the random variables. Hence, the prices and other quantities of interest in this economy are readily simulated.

##### A.2 Monte Carlo Simulation of the Model

The global parameters summarized in Table 17 while the asset-specific parameters are chosen to replicate the factor structure of log price-dividend ratios and returns.23 We first note that these parameters generate economic moments (calculated from 500 simulations of the long-run risk economy), which are roughly in line with the values observed in the 1943–2008 U.S. consumption and return data as shown in table 19. The time period takes into account the strong structural break in the long-run risk model documented by Marakani (2009) and Internet Appendix C and the uncertainty regarding how well the model works after the global financial crisis.24 When realistic noise is added to the log P/D ratios as described below, they are also compatible with the predictability of real-time consumption growth in the data as seen from the numbers in Table 20. One moment that does not match well is the standard deviation of the real risk-free rate, which is much smaller in the simulations than in the data. This, however, as argued by Beeler and Campbell (2012), points to a strength rather than a weakness of the long-run risk model as most models struggle to make this quantity low enough. Further, as we argue in the next section, this quantity is very noisily measured, meaning that the reported standard deviation would be significantly larger than the actual one.25

Table 17

Global parameters for the simulation (the time unit is one year). μ represents the unconditional mean of consumption growth, σ its conditional volatility, ρ the first-order autocorrelation of the long-run risk state variable x, $φx$ the conditional volatility of x in relation to that of consumption growth, ν the first order autocorrelation of volatility, σw the volatility of volatility, γ the relative risk aversion, ψ the elasticity of intertemporal substitution, and δ the time preference

Parameter Value
μ 0.02
σ 0.012
ρ 0.85
$φx$ 0.45
ν 0.99
σw $10−5$
γ 25
ψ 1.5
δ 0.994
Parameter Value
μ 0.02
σ 0.012
ρ 0.85
$φx$ 0.45
ν 0.99
σw $10−5$
γ 25
ψ 1.5
δ 0.994
Table 18

Asset-specific parameters for the simulation. The assets are indexed by l. $μl,d$ represents the unconditional mean of the dividend growth for asset l, $ϕl$ the dependence of predictable dividend growth on the long-run risk state variable x and $φl,d$ the idiosyncratic volatility of dividend growth

Parameters for the asset dividend growths

l $μl,d$ $ϕl$ $φl,d$
–0.0286 1.7834 19.1677
0.0889 3.7689 21.7081
0.0160 3.2545 19.4655
0.0456 3.4405 23.5766
0.0471 2.6758 24.0000
0.0907 4.6342 16.6065
0.0778 5.8088 16.3543
0.0457 2.4918 8.5237
0.0928 9.5089 24.0000
10 –0.0145 5.5979 24.0000
11 –0.0012 4.8912 24.0000
12 0.0821 8.5459 22.0032
13 0.0556 10.9271 8.9635
14 0.0272 6.0810 21.8607
15 0.0926 5.1230 24.0000
16 0.0454 5.1540 6.0000
17 0.0327 3.0965 21.1709
18 0.0317 3.3548 16.4485
19 0.0147 3.5232 23.0091
20 0.0619 3.3028 6.6980
21 0.0167 2.5690 12.5081
22 0.0421 10.8271 6.0000
23 0.0901 3.7845 11.6097
24 0.0436 2.5953 24.0000
25 0.0788 3.7323 11.0877
Parameters for the asset dividend growths

l $μl,d$ $ϕl$ $φl,d$
–0.0286 1.7834 19.1677
0.0889 3.7689 21.7081
0.0160 3.2545 19.4655
0.0456 3.4405 23.5766
0.0471 2.6758 24.0000
0.0907 4.6342 16.6065
0.0778 5.8088 16.3543
0.0457 2.4918 8.5237
0.0928 9.5089 24.0000
10 –0.0145 5.5979 24.0000
11 –0.0012 4.8912 24.0000
12 0.0821 8.5459 22.0032
13 0.0556 10.9271 8.9635
14 0.0272 6.0810 21.8607
15 0.0926 5.1230 24.0000
16 0.0454 5.1540 6.0000
17 0.0327 3.0965 21.1709
18 0.0317 3.3548 16.4485
19 0.0147 3.5232 23.0091
20 0.0619 3.3028 6.6980
21 0.0167 2.5690 12.5081
22 0.0421 10.8271 6.0000
23 0.0901 3.7845 11.6097
24 0.0436 2.5953 24.0000
25 0.0788 3.7323 11.0877
Table 19

Moments implied by the model and from the data. The model implied moments are obtained from 500 simulations for a time period equal to that of the data (65 years). The model was actually simulated for 165 years with the first 100 years of data being discarded to ensure that arbitrary initial values did not have any impact on the results. The data period for the comparison is taken to be 1944–2008

Moment Data Simulation mean 5th percentile 95th percentile
$E[Δct]$ 0.0199 0.0200 0.0153 0.0246
$Std[Δct]$ 0.0136 0.0151 0.0105 0.0194
$AC(1)[Δct]$ 0.243 0.320 0.148 0.488
$E[rf,t]$ 0.0059 0.0035 −0.0012 0.0079
$Std[rf,t]$ 0.0343 0.0067 0.0045 0.0089
$Min[rl,t−rf,t]$ 0.010 0.018 −0.012 0.049
$Max[rl,t−rf,t]$ 0.133 0.209 0.131 0.292
$Min E[Δdl,t]$ −0.023 −0.030 −0.062 0.002
$Max E[Δdl,t]$ 0.105 0.104 0.070 0.149
$Min Std[Δdl,t]$ 0.087 0.085 0.075 0.095
$Max Std[Δdl,t]$ 0.385 0.306 0.279 0.333
Moment Data Simulation mean 5th percentile 95th percentile
$E[Δct]$ 0.0199 0.0200 0.0153 0.0246
$Std[Δct]$ 0.0136 0.0151 0.0105 0.0194
$AC(1)[Δct]$ 0.243 0.320 0.148 0.488
$E[rf,t]$ 0.0059 0.0035 −0.0012 0.0079
$Std[rf,t]$ 0.0343 0.0067 0.0045 0.0089
$Min[rl,t−rf,t]$ 0.010 0.018 −0.012 0.049
$Max[rl,t−rf,t]$ 0.133 0.209 0.131 0.292
$Min E[Δdl,t]$ −0.023 −0.030 −0.062 0.002
$Max E[Δdl,t]$ 0.105 0.104 0.070 0.149
$Min Std[Δdl,t]$ 0.087 0.085 0.075 0.095
$Max Std[Δdl,t]$ 0.385 0.306 0.279 0.333

The scaled eigenvalues of the covariance matrix of the post-1942 continuously compounded excess returns of the 25 Fama-French portfolios sorted on the basis of size and book-to-market ratio are tabulated in Table 21, together with the mean and 5th and 95th percentiles of the corresponding values obtained in 500 simulations of the above model for the same time period (65 years).26 Because the first few eigenvalues, which are of principal interest, are very similar to those in the data, the model replicates the observed factor structure of excess returns quite well.

Table 20

Predictability of consumption growth in the model and in the data. For the data, we use real-time consumption growth as a measure of consumption growth. The results for the model are derived from 1000 simulations over 165 years with the data for the first 100 years being dropped so as to limit the impact of initial values on the numbers

R2 obtained on regressing consumption growth against the lagged values of the first two principal components of the log P/D ratios

Data Simulation mean 5th percentile 95th percentile
17.4% 32.6% 10.6% 55.2%
R2 obtained on regressing consumption growth against the lagged values of the first two principal components of the log P/D ratios

Data Simulation mean 5th percentile 95th percentile
17.4% 32.6% 10.6% 55.2%
Table 21

Eigenvalues of the covariance matrix of the continuously compounded excess returns of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1944–2008 as well as those obtained by simulating the model for a time span equal to that of the data (65 years). The model was actually simulated for 165 years with the first 100 years of data being discarded to ensure that arbitrary initial values did not have any impact on the results

Eigenvalues of the covariance matrix of excess returns

Data Simulation mean 5th percentile 95th percentile
1.00000 1.00000 1.00000 1.00000
0.06052 0.06171 0.04705 0.07889
0.04741 0.03926 0.02994 0.04970
0.01280 0.01135 0.00871 0.01403
0.00823 0.00807 0.00637 0.01010
0.00626 0.00667 0.00531 0.00824
0.00535 0.00573 0.00455 0.00711
0.00389 0.00497 0.00399 0.00613
0.00339 0.00433 0.00351 0.00541
0.00316 0.00375 0.00302 0.00460
0.00288 0.00331 0.00267 0.00403
0.00231 0.00294 0.00240 0.00359
0.00207 0.00263 0.00214 0.00324
0.00200 0.00236 0.00191 0.00289
0.00149 0.00213 0.00170 0.00265
0.00142 0.00191 0.00152 0.00234
0.00132 0.00171 0.00136 0.00212
0.00108 0.00151 0.00118 0.00186
0.00099 0.00132 0.00106 0.00164
0.00097 0.00112 0.00087 0.00139
0.00074 0.00076 0.00059 0.00095
0.00067 0.00058 0.00045 0.00075
0.00056 0.00040 0.00030 0.00050
0.00045 0.00030 0.00023 0.00038
0.00043 0.00023 0.00017 0.00029
Eigenvalues of the covariance matrix of excess returns

Data Simulation mean 5th percentile 95th percentile
1.00000 1.00000 1.00000 1.00000
0.06052 0.06171 0.04705 0.07889
0.04741 0.03926 0.02994 0.04970
0.01280 0.01135 0.00871 0.01403
0.00823 0.00807 0.00637 0.01010
0.00626 0.00667 0.00531 0.00824
0.00535 0.00573 0.00455 0.00711
0.00389 0.00497 0.00399 0.00613
0.00339 0.00433 0.00351 0.00541
0.00316 0.00375 0.00302 0.00460
0.00288 0.00331 0.00267 0.00403
0.00231 0.00294 0.00240 0.00359
0.00207 0.00263 0.00214 0.00324
0.00200 0.00236 0.00191 0.00289
0.00149 0.00213 0.00170 0.00265
0.00142 0.00191 0.00152 0.00234
0.00132 0.00171 0.00136 0.00212
0.00108 0.00151 0.00118 0.00186
0.00099 0.00132 0.00106 0.00164
0.00097 0.00112 0.00087 0.00139
0.00074 0.00076 0.00059 0.00095
0.00067 0.00058 0.00045 0.00075
0.00056 0.00040 0.00030 0.00050
0.00045 0.00030 0.00023 0.00038
0.00043 0.00023 0.00017 0.00029

The model also replicates the observed factor structure of log P/D ratios fairly well. This is best seen from the normalized eigenvalues for the covariance matrix of the log P/D ratios of the assets, both from the data and from the simulations, which are tabulated in Table 22. The model’s two-factor structure is highly evident here as all the eigenvalues after the second one are zero. To better reflect the data and investigate the possible consequences of the inclusion of small, irrelevant factors into the long-run risk model, we added white noise with a variance of 20% of the simulated values to the log P/D ratios. The introduction of this noise can also be thought of as representing measurement error in the prices or dividends brought about due to liquidity issues or other market imperfections. The normalized eigenvalues after adding this noise are summarized in Table 23. From it, we see that the model is able to replicate the key elements of this factor structure after adding the noise.27

Table 22

Eigenvalues of the covariance matrix of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1944–2008 as well as those obtained by simulating the model for an equivalent time period (65 years). The model was actually simulated for 165 years with the first 100 years of data being discarded to ensure that arbitrary initial values did not have any impact on the results

Eigenvalues of the covariance matrix of log P/D ratios

Data Simulation mean 5th percentile 95th percentile
1.00000 1.00000 1.00000 1.00000
0.06041 0.03598 0.01272 0.07067
0.01669 0.00000 0.00000 0.00000
0.01169 0.00000 0.00000 0.00000
0.00627 0.00000 0.00000 0.00000
0.00522 0.00000 0.00000 0.00000
0.00494 0.00000 0.00000 0.00000
0.00318 0.00000 0.00000 0.00000
0.00245 0.00000 0.00000 0.00000
0.00238 0.00000 0.00000 0.00000
0.00215 0.00000 0.00000 0.00000
0.00168 0.00000 0.00000 0.00000
0.00137 0.00000 0.00000 0.00000
0.00101 0.00000 0.00000 0.00000
0.00094 0.00000 0.00000 0.00000
0.00085 0.00000 0.00000 0.00000
0.00072 0.00000 0.00000 0.00000
0.00063 0.00000 0.00000 0.00000
0.00052 0.00000 0.00000 0.00000
0.00049 0.00000 0.00000 0.00000
0.00046 0.00000 0.00000 0.00000
0.00040 0.00000 0.00000 0.00000
0.00028 0.00000 0.00000 0.00000
0.00022 0.00000 0.00000 0.00000
0.00018 0.00000 0.00000 0.00000
Eigenvalues of the covariance matrix of log P/D ratios

Data Simulation mean 5th percentile 95th percentile
1.00000 1.00000 1.00000 1.00000
0.06041 0.03598 0.01272 0.07067
0.01669 0.00000 0.00000 0.00000
0.01169 0.00000 0.00000 0.00000
0.00627 0.00000 0.00000 0.00000
0.00522 0.00000 0.00000 0.00000
0.00494 0.00000 0.00000 0.00000
0.00318 0.00000 0.00000 0.00000
0.00245 0.00000 0.00000 0.00000
0.00238 0.00000 0.00000 0.00000
0.00215 0.00000 0.00000 0.00000
0.00168 0.00000 0.00000 0.00000
0.00137 0.00000 0.00000 0.00000
0.00101 0.00000 0.00000 0.00000
0.00094 0.00000 0.00000 0.00000
0.00085 0.00000 0.00000 0.00000
0.00072 0.00000 0.00000 0.00000
0.00063 0.00000 0.00000 0.00000
0.00052 0.00000 0.00000 0.00000
0.00049 0.00000 0.00000 0.00000
0.00046 0.00000 0.00000 0.00000
0.00040 0.00000 0.00000 0.00000
0.00028 0.00000 0.00000 0.00000
0.00022 0.00000 0.00000 0.00000
0.00018 0.00000 0.00000 0.00000
Table 23

Eigenvalues of the covariance matrix of the log P/D ratios of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio as well as those obtained by simulating the model for an equivalent period (65 years) and adding some noise to the result. The model was actually simulated for 165 years with the first 100 years of data being discarded to ensure that arbitrary initial values did not have any impact on the results

Eigenvalues of the covariance matrix of noisy log P/D ratios

Data Simulation mean 5th percentile 95th percentile
1.00000 1.00000 1.00000 1.00000
0.06041 0.04536 0.02144 0.08128
0.01669 0.01451 0.01323 0.01577
0.01169 0.01337 0.01234 0.01442
0.00627 0.01251 0.01160 0.01352
0.00522 0.01179 0.01095 0.01269
0.00494 0.01114 0.01046 0.01183
0.00318 0.01057 0.00989 0.01132
0.00245 0.01003 0.00936 0.01071
0.00238 0.00952 0.00877 0.01022
0.00215 0.00904 0.00842 0.00972
0.00168 0.00859 0.00800 0.00918
0.00137 0.00813 0.00757 0.00872
0.00101 0.00771 0.00712 0.00837
0.00094 0.00730 0.00677 0.00786
0.00085 0.00692 0.00639 0.00751
0.00072 0.00652 0.00603 0.00706
0.00063 0.00615 0.00567 0.00664
0.00052 0.00579 0.00533 0.00627
0.00049 0.00541 0.00496 0.00588
0.00046 0.00506 0.00464 0.00554
0.00040 0.00470 0.00429 0.00514
0.00028 0.00433 0.00388 0.00477
0.00022 0.00394 0.00350 0.00437
0.00018 0.00345 0.00295 0.00390
Eigenvalues of the covariance matrix of noisy log P/D ratios

Data Simulation mean 5th percentile 95th percentile
1.00000 1.00000 1.00000 1.00000
0.06041 0.04536 0.02144 0.08128
0.01669 0.01451 0.01323 0.01577
0.01169 0.01337 0.01234 0.01442
0.00627 0.01251 0.01160 0.01352
0.00522 0.01179 0.01095 0.01269
0.00494 0.01114 0.01046 0.01183
0.00318 0.01057 0.00989 0.01132
0.00245 0.01003 0.00936 0.01071
0.00238 0.00952 0.00877 0.01022
0.00215 0.00904 0.00842 0.00972
0.00168 0.00859 0.00800 0.00918
0.00137 0.00813 0.00757 0.00872
0.00101 0.00771 0.00712 0.00837
0.00094 0.00730 0.00677 0.00786
0.00085 0.00692 0.00639 0.00751
0.00072 0.00652 0.00603 0.00706
0.00063 0.00615 0.00567 0.00664
0.00052 0.00579 0.00533 0.00627
0.00049 0.00541 0.00496 0.00588
0.00046 0.00506 0.00464 0.00554
0.00040 0.00470 0.00429 0.00514
0.00028 0.00433 0.00388 0.00477
0.00022 0.00394 0.00350 0.00437
0.00018 0.00345 0.00295 0.00390

Albuquerque, Eichenbaum, and Rebelo (2012) have recently pointed out that some versions of the long-run risk model imply very high consumption-return correlations over long time scales. We plot this correlation for our simulated model as a function of the time frame in Figure 8. We see from it that our simulated model is compatible with a low value for this correlation even over long time frames for the given length of the data (65 years).28

Figure 8

Consumption growth-return correlations as a function of the time frame for the model simulated in this Appendix. The simulations are for the same period as the data (65 years). The model was actually simulated for 165 years with the first 100 years of data being discarded to ensure that arbitrary initial values did not have any impact on the results.

Figure 8

Consumption growth-return correlations as a function of the time frame for the model simulated in this Appendix. The simulations are for the same period as the data (65 years). The model was actually simulated for 165 years with the first 100 years of data being discarded to ensure that arbitrary initial values did not have any impact on the results.

We thus see that the long-run risk model being simulated here is compatible not only with many of the important observed moments of macroeconomic quantities but also with the observed factor structure of excess returns and P/D ratios. Given this, it is interesting to examine the performance of different asset pricing tests for long-run risk models within the context of these simulations. This will enable the study of the effect of finite sample size and measurement noise on the efficacy of these tests and will point to the choice of test to be used in this paper. Since we are particularly interested in examining the impact of measurement noise on these tests, we first turn to the task of establishing a reasonable estimate for the size of this noise for two important quantities in long-run risk models, the consumption growth, and the real risk-free rate.

##### A.3 Measurement Noise

We do so by analyzing the degree of correlation between different measures for the same fundamental macroeconomic quantities. For consumption growth, we use the estimates of consumption growth derived from the continuously revised NIPA tables, as well as those from the real-time database maintained by the Federal Reserve Bank of St. Louis (described in detail by Croushore and Stark (2001)). Regressing these estimates against each other leads to the results in Table 24. The R2 of 67% or about two-thirds indicates that the variance of measurement noise in consumption growth is about half of the variance of actual consumption growth. We thus simulate measured consumption growth as actual consumption growth plus i.i.d. noise with half its realized variance in that simulation.

Table 24

Regression of the conventional revised measure of consumption growth ($Δct$) on the corresponding real-time measure ($ΔctRT$) over the period 1965–2008

Regression of $Δct$ against $ΔctRT$

Coefficient Estimate (t-statistic)
Intercept $0.0060***$ (3.16)
Real-time consumption growth $0.838***$ (9.11)
R2 67.0%
Regression of $Δct$ against $ΔctRT$

Coefficient Estimate (t-statistic)
Intercept $0.0060***$ (3.16)
Real-time consumption growth $0.838***$ (9.11)
R2 67.0%

The real risk-free rate has been a problematic quantity to measure accurately since true real risk-free assets have not been available in the market until recent times (further, even these recent innovative instruments have limited liquidity). This quantity has therefore usually been approximated by subtracting some measure of expected inflation from the nominal risk-free rate. Two implicit assumptions are made in doing so. The first is that the measure of expected inflation being used is reasonable and the second is that inflation uncertainty risk is unpriced. This first issue is a serious one for data over long time frames since inflation survey data are typically unavailable over the entire period. The general convention in this regard has been to use lagged or realized inflation. Hence, it is important to understand the possible size of error that can be caused by using lagged or realized inflation. We therefore regress the measures of the real risk-free rate made using lagged, realized and expected inflation on each other to estimate the amount of measurement noise. We tabulate the results in Table 25. We see that the R2 of each of the regressions is quite low with the average being under 33%. This indicates that the measurement noise in the reported real risk-free rate has about twice the variance of the underlying quantity. This is in line with the size of inflation forecast errors as reported by Croushore (2010). Hence, for the simulations, we model the measured real risk-free rate as the actual risk-free rate plus i.i.d. noise with twice its realized variance.

Table 25

Regression of three measures of the real risk-free rate on each other. The three measures are computed using the lagged, realized, and expected inflation. The regressions are restricted to the post-1946 period as expected inflation data are only available for it

Regression of $rf,tlagged$ against $rf,trealized$

Coefficient Estimate (t-statistic)
Intercept 0.0046 (1.64)
$rf,trealized$ $0.454***$ (4.28)
R2 23.6%
Regression of $rf,tlagged$ against $rf,trealized$

Coefficient Estimate (t-statistic)
Intercept 0.0046 (1.64)
$rf,trealized$ $0.454***$ (4.28)
R2 23.6%
Regression of $rf,tlagged$ against $rf,texpected$

Coefficient Estimate (t-statistic)
Intercept −0.0023 (−0.77)
$rf,texpected$ $0.890***$ (6.14)
R2 38.6%
Regression of $rf,tlagged$ against $rf,texpected$

Coefficient Estimate (t-statistic)
Intercept −0.0023 (−0.77)
$rf,texpected$ $0.890***$ (6.14)
R2 38.6%
Regression of $rf,trealized$ against $rf,texpected$

Coefficient Estimate (t-statistic)
Intercept −0.0007 (−0.20)
$rf,texpected$ $0.859***$ (5.08)
R2 30.4%
Regression of $rf,trealized$ against $rf,texpected$

Coefficient Estimate (t-statistic)
Intercept −0.0007 (−0.20)
$rf,texpected$ $0.859***$ (5.08)
R2 30.4%

We also note that the dispersion in twelve-month inflation forecasts by the participants in the Livingston survey is quite large. The average standard deviation for the inflation forecast is 0.96% over the period 1946–2003 which is very comparable with the standard deviation in the real risk-free rate of 2.48% over the same period. Any temporal variation in the inflation risk premium will only increase the importance of the measurement error as the standard way of calculating the real risk-free rate from the nominal rate assumes that there is no such risk premium.

We do not investigate the second issue (the assumption of zero price of risk for inflation risk uncertainty) but note that its inclusion will only strengthen the results in our favor as the stock portfolios are real assets and their P/D ratios are real yields and thus are not directly exposed to inflation risk.

##### A.4 Type II Error of Asset Pricing Tests with Respect to the Long-Run Risk Model

We now analyze the performance of tests of four different asset pricing restrictions of the long-run risk model to determine which is the most reasonable one to use in the analysis in this paper.

The asset pricing restrictions that we consider are related to the one analyzed by Ferson, Nallareddy, and Xie (2013).29 It is30

(A.9)
where the returns $rl,t$ are continuously compounded, while $e∼$ and $w∼$ are the estimated values of the innovations of the X and V processes. They, together, with X and V, are estimated in the same manner as by Bansal, Yaron, and Kiku (2007) and Ferson, Nallareddy, and Xie (2013), that is, by the use of the following regressions
(A.10)
$Δct+1=α0+α1zm,t+α2rf,t+V∼tηt+1,$

(A.11)
$X∼t=α0−μ+α1zm,t+α2rf,t,$

(A.12)
$X∼t+1=ρX∼t+V∼tIX˜t+1,$

(A.13)
$Vtηt+12=β0+β1zm,t+β2rf,t+ωt+1,$

(A.14)
$V∼t=β0+β1zm,t+β2rf,t,$

(A.15)
$V∼t+1=V¯+ν(V∼t−V¯)+σVIV˜t+1,$
where $zm,t$ is the log market P/D ratio (taken to be the log P/D ratio of the first asset in the simulations) and $rf,t$ is the real risk-free rate.

The alternate asset pricing restrictions that we consider are analogous, but, use the two largest estimated log P/D ratio factors instead of the log market P/D ratio and the real risk-free rate as they should also span X and V. The principal idea behind this approach is that given the null, they should be more accurately estimated in the presence of measurement error since they are estimated using multiple assets. The asset pricing restriction analogous to (A9) is then given by

(A16)
$E[rl,t+Δt−rf,t]+12Var[rl,t+Δt−rf,t]≈∑i=12βl,IFiλIFi,$
where Fi and IFi are the ith principal components of the log P/D ratios of the assets and their estimated innovations respectively (the latter are estimated by fitting the former to an AR(1) process).

The above asset pricing relations do not include the $βl,V2λV$ term, which, as we have shown in theorem 1 of Internet Appendix A, should be included to make it exact. Excluding it, however, has very minor impact on the results both because the short time series in the data and the high persistence of the long-run risk processes makes these βs difficult to estimate precisely and also because this term captures the variance in expected excess returns which is much smaller than the expected excess returns. The latter is true because the order of magnitude of the expected excess returns is 0.1, while that of the variance of the expected excess returns is the square of this number, that is, 0.01. Another reason we exclude this term is to ensure that we are comparing our procedure with the methodology in the extant literature.

We examine whether the hypothesis that the factors being considered are useless is rejected by the cross-sectional regression methodology. This is done using the Wald test for the risk premia of the factors with their covariance matrix being estimated in the standard manner (see for, e.g., Shanken 1992; Shanken and Zhou 2007). The nonrejection frequencies for each of these tests (i.e., rejection frequencies for the long-run risk model) in 1000 simulations are reported in Table 1. The results show that the test of the asset pricing restriction involving the log P/D ratio factors (which also include noise calibrated to fit the observed factor structure of log P/D ratios) display much greater power than those involving the estimated long-run risk processes as long as a reasonable number of portfolios are used to estimate them.

We also find, from the simulation results, that the effects of measurement error are compounded by the persistence of the long-run risk processes, making estimation of the innovations in these variables liable to be error prone. In this regard, we find, from the Monte Carlo simulations, that our methodology performs much better than the conventional approach. In particular, we find that the median correlation between the true innovations of X and its projection onto the subspace spanned by the estimates of the innovations of the two (noisy) log price-dividend ratio factors is 0.866, while the median correlation between the true X innovations and the innovations estimated in the conventional manner is only 0.149. The corresponding figures for the volatility innovations are 0.559 and 0.162. Thus, the innovations are much more precisely identified by our methodology, which explains the superiority of the type II error observed in the Monte Carlo simulation results.

From the above analysis, it is seen that our proposed methodology offers superior performance compared to the traditional methodology once measurement errors are taken into account. Hence, we use this proposed methodology in our paper.

##### A.5 Conclusion

In this Appendix, we simulate a twenty-five asset long-run risk economy with parameters chosen so as to match key economic and financial moments with those in U.S. economic and financial data. We analyze the type II error of different asset pricing tests within this economy and find, when realistic measurement noise is introduced into it, that tests using estimates of the long-run risk components derived from projections of consumption growth onto the log market price-dividend ratio and real risk-free rate display high type II error while those estimating the same components using the principal components of the log price-dividend ratios of the assets do not do so. This implies that the latter type of tests have a more desirable profile. Hence, we use such tests in this paper.

#### Appendix B Robust Test Statistics

Since the excess returns of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio have a strong factor structure, it is important to use robust test statistics to eliminate the problem of useless factors being identified as useful (a problem forcefully brought out by Kleibergen 2009, 2010). Hence, we use the robust test statistics suggested by Kleibergen (2009) to ensure that the factors here are not useless.

We find that these robust test statistics reject the joint hypothesis that $λIFVol=λIFX=0$ (nonrejection of the hypothesis would indicate that the pricing factors are useless) and do not reject either the hypothesis of factor pricing or that of $λIFVol=λ^IFVol,λIFX=λ^IFX$ for many values of $(λ^IFVol,λ^IFX)$, including those estimated using the cross-sectional regressions (rejection of this would indicate that the model is rejected by the data). Figure 9 contains the plot of the p-values of the FAR test statistic for many different values of $(λ^IFVol,λ^IFX)$. This statistic tests the joint hypothesis of factor pricing and of $λIFVol=λ^IFVol,λFX=λ^IFX$. It shows that the joint hypothesis is rejected at $λ^IFVol=λ^IFX=0$ and also that it is not rejected for many other values of $λ^IFVol$ and $λ^IFX$, including those in Table 6. Further, the region identified by p > 0.1 excludes $λIFX=0$, but not $λIFVol=0$.

Figure 9

p-value plot of the test of the joint hypothesis of factor pricing together with $(λIFVol,λIFX)=(λ^IFVol,λ^IFX)$ using the FAR statistic proposed by Kleibergen (2009) for the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1944–2013. $λIFVol$ and $λIFX$ are the factor risk premia for IFVol and IFX. FV ol and FX are the volatility and consumption/dividend growth factors and IFVol and IFX are their innovations.

Figure 9

p-value plot of the test of the joint hypothesis of factor pricing together with $(λIFVol,λIFX)=(λ^IFVol,λ^IFX)$ using the FAR statistic proposed by Kleibergen (2009) for the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1944–2013. $λIFVol$ and $λIFX$ are the factor risk premia for IFVol and IFX. FV ol and FX are the volatility and consumption/dividend growth factors and IFVol and IFX are their innovations.

The JGLS statistic, which tests the hypothesis of factor pricing for a given value of $λIFVol$ and $λIFX$, is plotted in Figure 10. Since it tests a weaker hypothesis, it is not surprising that it rejects fewer values of $λIFVol$ and $λIFX$ than the FAR statistic. When combined with the GLS-LM statistic, also plotted in Figure 10, which tests the hypothesis that $λIFVol=λ^IFVol,λFX=λ^IFX$ given that factor pricing is correct, it gives very similar results to those given by the FAR statistic.

Figure 10

p-value plot of the test of the hypothesis of factor pricing given $(λIFVol,λIFX)=(λ^IFVol,λ^IFX)$ using the JGLS and GLS-LM statistics proposed by Kleibergen (2009) for the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1944–2013. $λIFVol$ and $λIFX$ are the factor risk premia for IFVol and IFX. FVol and FX are the volatility and consumption/dividend growth factors and IFVol and IFX are their innovations.

Figure 10

p-value plot of the test of the hypothesis of factor pricing given $(λIFVol,λIFX)=(λ^IFVol,λ^IFX)$ using the JGLS and GLS-LM statistics proposed by Kleibergen (2009) for the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio over the period 1944–2013. $λIFVol$ and $λIFX$ are the factor risk premia for IFVol and IFX. FVol and FX are the volatility and consumption/dividend growth factors and IFVol and IFX are their innovations.

Hence, we can conclude that the robust test statistics show that we cannot reject the hypothesis of factor pricing with the innovations of the two identifid log P/D factors However, these results do cast some doubt on the significance of $λIFVol$.

For their valuable comments and suggestions, we would like to thank Ravi Bansal, George Constantinides, Du Du, Dana Kiku, Ernst Schaumburg, Jonathan Parker, Robert Korajczyk, Annette Vissing-Jørgensen, Arvind Krishnamurthy, Tatjana Xenia-Puhan, Bernard Dumas, seminar participants at INSEAD, the City University of Hong Kong, the Indian School of Business, the Western Finance Association 2011 Annual Meeting, and the 2012 City University International Conference on Corporate Finance, and the editor, Wayne Ferson. Earlier versions of the paper appeared under the titles “Long run risks, the factor structure of price-dividend ratios and the cross-section of stock returns” and “Long-run risks and P/D factors.” Supplementary data can be found on The Review of Asset Pricing Studies web site.

1Petkova (2006) uses the short-term interest rate, yield spread, credit spread, aggregate dividend yield as state variables that describe the investment opportunities set, and finds that HML and SMB factor returns are correlated with innovations to these state variables, consistent with the Fama and French (1993) three-factor model being a version of the ICAPM. Our approach is more direct. When the long-run risk model holds, and the HML and SMB factors are innovations to state variables summarizing the state of the economy, then the price-dividend ratios on the underlying stock portfolios should predict future consumption and dividends and track consumption volatility.
2Merton (1973) used time separable utility function and did not consider investors' preferences over the timing of resolution of uncertainty. Campbell (1993) showed that the Merton (1973) ICAPM relation will continue to hold in a model economy populated by a representative investor with Epstein-Zin utility function with preference over timing of uncertainty resolution. In that model, with conditional homoscedasticty, the state variables predict expected future returns on the aggregate wealth portfolio. Campbell, Giglio, Polk, and Turley (2013) generalize the results to an economy with stochastic volatility. In contrast, we establish the relationship between the state variables in the Merton (1973) ICAPM relation and the state variables in the Bansal and Yaron (2004) type long run risk models.
3 In the original model, only the market dividends were modeled. We make the straightforward extension to multiple assets and discuss the reasonableness of the assumption later.
4 Note that the volatility process has to be modified to an Ornstein-Uhlenbeck one to accommodate the first two specifications. This modification does not affect any of the fundamental theoretical results or empirical analysis.
5 While strictly speaking, this should be the consumption growth innovation, the regression coefficient obtained with consumption growth is the same within this model
6 When the volatility process is set to be a Bessel process to avoid negative values, as in Internet Appendix A, an additional term involving the square of the beta with respect to the lagged volatility factor enters into the asset pricing relation. We find that this term has little empirical significance. Results including this term are available upon request.
7 We do not present the GMM results for brevity. They are consistent with our findings.
8 Since we make use of data expressed in terms of chained dollars, we use a Tornqvist-type index Whelan (2000) to construct the implicit consumption deflator.
9 The use of post-1945 or post-1950 data does not significantly change our results.
10 For an excellent review, see Connor and Korajczyk (2010).
11 The dynamic factor analysis results are available upon request.
12 When analyzing the results obtained after detrending, the reader should keep in mind that removing this time trend removes important information as neither quantity is expected to have a time trend on a theoretical basis. This is probably the reason for the lower R2 obtained after the trends are removed. The most important aspect of these results, which is that the coefficient of detrended F1 is significant and has the same sign as without detrending, still provides support to our conclusion.
13 It is interesting that F2, which weights the value portfolios more heavily, predicts future market dividend growth better than the log market P/D ratio (whose inability to predict dividend growth is well known Cochrane (2005)). We hypothesize that this is because value stocks have a low duration which makes their P/D ratios depend more on dividend growth than on future expected excess returns.
14 In this study, the convention used in the calculation of the price-dividend ratio is equivalent to the assumption that dividends received during the year are consumed immediately and that the agent is completely indifferent to the timings of these dividends during the year. Our results are robust to the use of either of the other conventional assumptions regarding the investment of dividends received during the year (the first being that such dividends are invested in nominal cash until the end of the year and the second being that they are invested in the asset itself until the end of the year).
15 We note that while the use of this measure of consumption is not standard, it is more relevant for the current analysis as it better matches the information structure of the consumers in the economy. (It is also possible that real-time data capture the sentiment of consumers as they reflect their current view of the state of the economy.)
16 We do not provide the details as it is not a necessary implication of the model we specify in this paper. They are available upon request.
17 They also explain the expected excess returns of the twenty-five Fama-French portfolios formed on the basis of size and book-to-market ratio.
18 In unreported results, we find that the model also exhibits good performance at the quarterly time scale.
19 The pricing relationship (5) is also empirically supported at the quarterly frequency. Although we do not report detailed results at this frequency for brevity, we note that the cross-sectional regression intercept is not significantly different from zero.
20 This is, strictly speaking, slightly different from the ICAPM as X is only related to the future cash flow of the market but does not determine the investment opportunity set. The key point made, however, is that the relation is very much in the spirit of the ICAPM.
21 This is, strictly speaking, not true of the long-run risk model as noted by Koijen, Lustig, Van Nieuwerburgh, and Verdelhan (2010), but there is a large strand of literature that makes this identification.
22 A similar structural break to the standard factors is found at 2006. In addition, the test also identifies a less important structural break at 1987. The coefficient of $F2nlb$ again hardly changes at this break with the significant change being in the mean.
23 They are available upon request from the authors.
24 Extending the sample to 2013 gives very similar Monte Carlo results except that the model implied predictability of consumption growth is not satisfied in the data.
25 Measurement error (in either inflation or dividends) can also account for the somewhat low standard deviation of real dividend growth of the portfolios in the simulations.
26 The model was actually simulated for 165 years with the data for the first 100 years being discarded so as to minimize the effect of the assumed initial values of the dynamic quantities.
27 Note that it is not necessary to replicate the features of the small factors as these represent a very small fraction of the variance and are not economically interesting.
28 Further details are available on request.
29Ferson, Nallareddy, and Xie (2013) use GMM with the Euler moment restrictions in the SDF framework. We use the beta representation which is approximate but quite accurate when dealing with continuously compounded returns.
30 Note that we do not need a $βΔc$ term as there is no contemporaneous correlation between the dividend growth and consumption growth innovations in this model.

## References

Albuquerque
R
Eichenbaum
M
Rebelo
S
Understanding the equity-premium and correlation puzzles
2012

Private communication
Bai
J
Ng
S
Large dimensional factor analysis
Foundations and Trends in Econometrics
,
2008
, vol.
3
(pg.
89
-
168
)
Bai
J
Perron
P
Estimating and testing linear models with multiple structural changes
Econometrica
,
1998
, vol.
66
(pg.
47
-
78
)
Bansal
R
Dittmar
R F
C
Consumption, dividends, and the cross-section of equity returns
Journal of Finance
,
2005
, vol.
60
(pg.
1639
-
72
)
Bansal
R
Khatchatrian
V
Yaron
A
Interpretable asset markets?
European Economic Review
,
2005
, vol.
49
(pg.
531
-
60
)
Bansal
R
Yaron
A
Risks for the long run: A potential resolution of asset pricing puzzles
Journal of Finance
,
2004
, vol.
59
(pg.
1481
-
509
)
Bansal
R
Yaron
A
Kiku
D
Risks for the long run: Estimation and inference
2007

Working Paper.
Bansal
R
Yaron
A
Kiku
D
Long run risks, the macroeconomy, and asset prices
American Economic Review
,
2010
, vol.
100
(pg.
542
-
46
)
Beeler
J
Campbell
J Y
The long-run risks model and aggregate asset prices: An empirical assessment
Critical Finance Review
,
2012
, vol.
1
(pg.
141
-
82
)
Boguth
O
Kuehn
L A
Consumption volatility risk
Journal of Finance
,
2013
, vol.
68
(pg.
2589
-
615
)
Campbell
J Y
Shiller
R J
The dividend-price ratio and expectations of future dividends and discount factors
Review of Financial Studies
,
1988
, vol.
1
(pg.
195
-
227
)
Campbell
J Y
Intertemporal asset pricing without consumption data
American Economic Review
,
1993
, vol.
83
(pg.
487
-
512
)
Campbell
J Y
Giglio
S
Polk
C
Turley
R
An intertemporal CAPM with stochastic volatility
2013

Working Paper.
Cochrane
J H
Asset Pricing
,
2005
Princeton

Princeton University Press
Cochrane
J H
The dog that did not bark: A defense of return predictability
Review of Financial Studies
,
2008
, vol.
21
(pg.
1533
-
75
)
Connor
G
Korajczyk
R A
Factor models of asset returns
Encyclopedia of Quantitative Finance
,
2010
Constantinides
G M
Ghosh
A
Asset pricing tests with long-run risks in consumption growth
Review of Asset Pricing Studies
,
2011
, vol.
1
(pg.
96
-
136
)
Croushore
D
Philadelphia Fed forecasting surveys: Their value for research
,
2010
, vol.
Q3
(pg.
1
-
11
)
Croushore
D
Stark
T
A real-time data set for macroeconomists
Journal of Econometrics
,
2001
, vol.
105
(pg.
111
-
30
)
Da
Z
Cash flow, consumption risk, and the cross-section of stock returns
Journal of Finance
,
2009
, vol.
64
(pg.
923
-
56
)
Drechsler
I
Yaron
A
What’s vol got to do with it
Review of Financial Studies
,
2011
, vol.
24
(pg.
1
-
45
)
Eraker
B
Shaliastovich
I
An equilibrium guide to designing affine pricing models
Mathematical Finance
,
2008
, vol.
18
(pg.
519
-
43
)
Fama
E F
French
K R
Common risk factors in the returns on bonds and stocks
Journal of Financial Economics
,
1993
, vol.
33
(pg.
3
-
56
)
Fama
E F
French
K R
Multifactor explanations of asset pricing anomalies
The Journal of Finance
,
1996
, vol.
51
(pg.
55
-
84
)
Ferson
W
Nallareddy
S
Xie
B
The out-of-sample performance of long-run risk models
Journal of Financial Economics
,
2013
, vol.
107
(pg.
537
-
556
)
Jagannathan
R
Wang
Y
Lazy investors, discretionary consumption, and the cross-section of stock returns
Journal of Finance
,
2007
, vol.
62
(pg.
1623
-
61
)
Kan
R
Zhang
C
Two-pass tests of asset pricing models with useless factors
Journal of Finance
,
1999
, vol.
54
(pg.
203
-
35
)
Kiku
D
Is the value premium a puzzle?
2006

USC FBE Seminar Series Paper
Kleibergen
F
Tests of risk premia in linear factor models
Journal of Econometrics
,
2009
, vol.
149
(pg.
149
-
73
)
Kleibergen
F
Statistical anomalies in “support” of factor pricing
2010

Working Paper
Koijen
R S
Lustig
H
Van Nieuwerburgh
S
Verdelhan
A
Long run risk, the wealth-consumption ratio, and the temporal pricing of risk
American Economic Review
,
2010
, vol.
100
(pg.
552
-
56
)
Lewellen
J
Nagel
S
Shanken
J
A skeptical appraisal of asset pricing tests
Journal of Financial Economics
,
2010
, vol.
96
(pg.
175
-
94
)
Marakani
S
Long run consumption risks: Are they there?
2009

Working Paper
Merton
R C
An intertemporal capital asset pricing model
Econometrica
,
1973
, vol.
41
(pg.
867
-
87
)
Newey
W K
West
K D
Automatic lag selection in covariance matrix estimation
Review of Economic Studies
,
1994
, vol.
61
(pg.
631
-
53
)
Parker
J
Julliard
C
Consumption Risk and Cross-Sectional Returns
Journal of Political Economy
,
2005
, vol.
113
(pg.
185
-
222
)
Petkova
R
Do the Fama–French factors proxy for innovations in predictive variables?
Journal of Finance
,
2006
, vol.
61
(pg.
581
-
612
)
Shanken
J
On the estimation of beta-pricing models
Review of Financial Studies
,
1992
, vol.
5
(pg.
1
-
33
)
Shanken
J
Zhou
G
Estimating and testing beta pricing models: Alternative methods and their performance in simulations
Journal of Financial Economics
,
2007
, vol.
84
(pg.
40
-
86
)
van Binsbergen
J
Koijen
R S
Predictive regressions: A present-value approach
Journal of Finance
,
2010
, vol.
65
(pg.
1439
-
71
)
Whelan
K
A guide to the use of chain aggregated NIPA data
2000

Note
Zhou
G
Zhu
Y
A long-run risks model with long- and short-Run volatilities: Explaining predictability and volatility risk premium
2009
SSRN eLibrary