Short-run Risk, Business Cycle, and the Value Premium

We jointly explain the variations of the equity and value premium in a model with both short-run (SRR) and long-run (LRR) consumption risk. In our preliminary empirical analysis, we find that SRR varies with the business cycle and it has a substantial predictive power for market excess returns and the value premium|both in-sample and out-of-sample. The LRR component also differs significantly from zero, and value stocks have a larger exposure to both LRR and SRR than growth stocks. To explain these patterns in asset returns, we propose an extended and analytically tractable LRR model.

The value premium refers to the phenomenon that stocks with lower price-to-fundamentals ratios will generate excess returns over those with high ratios. This differential in returns constitutes a puzzle because the return differential cannot be accounted for by CAPM, as documented in Fama and French (1992). A large body of research has tried to reconcile data with theory. A prominent example is the long-run risk (henceforth LRR) model proposed in Bansal and Yaron (2004), which can explain the magnitude of both the equity and value premium. 1 Through a constant leverage parameter, value stocks load more on the LRR component than growth stocks. Therefore, investors require compensation for bearing more LRR, thus generating the value premium. 2 However, the assumption of a constant leverage parameter is a serious drawback of the LRR approach because it fails to explain the variation of equity and value premiums over business cycles. Moreover, the model's design is complicated by the commonly found evidence that the equity premium is pro-cyclical while the value premium is counter-cyclical. 3 Our paper makes two contributions. The first contribution is to construct non-parametric measures of short-run risk (SRR) to formally study the covariation with the transient consumption risk as indicators of the business cycle. These measures are motivated to capture the priced risk of transient consumption growth, which is overlooked in most LRR literature. The definition of SRRs stems from the specification of consumption and cash flow dynamics in the LRR framework, although they do not depend on the equilibrium solutions. Hence, the empirical SRR measures correspond precisely to the residual consumption risk apart from LRR. We define the SRR in dividends as the short-run covariance with the consumption growth. Similarly, we define the SRR in consumption growth as its short-run variance.
The SRRs fluctuate substantially with business cycles and can even switch sign: the SRR in value stocks appears counter-cyclical, while the SRR in growth stocks seems pro-cyclical. By running predictive regressions of future returns on the estimated SRRs, we find that the SRRs in consumption, growth stocks, and value stocks explain 17.6% of the variations in the future one-year market excess returns, and 11.5% of the variations in the future three-year return differentials in value and growth stocks (value-minus-growth returns henceforth). In particular, the SRR in consumption negatively predicts future market returns but positively predicts future valueminus-growth returns, which echoes the evidence that the value premium is counter-cyclical. The regression coefficients on the SRRs are statistically significant. These results are consistent with the interpretation that the predictive power of SRRs stems from the comovements of both SRRs and market returns with the business cycle.
The predictive power of SRR on future returns is not only statistically but also economically significant. Notwithstanding the criticism in Welch and Goyal (2008) that most predictive regressions cannot beat historical average out-of-sample (OOS), the predictive power of SRRs remains strong OOS. Based on predictive regressions, we construct a market-timing strategy that adjusts the positions on the market portfolio and the risk-free asset once every year. This strategy doubles the Sharpe ratio of the market excess returns.
A byproduct of LRR model motivated SRR measures is that we can jointly study SRR and LRR. In a generalized method of moments (GMM) estimation, the null hypothesis of no LRR is rejected at 99.9% significance level. We find the LRR is persistent but large in magnitude relative to consumption growth. The value stocks not only have more exposure to LRR than growth stocks, consistent with Bansal, Dittmar, and Lundblad (2005a) and Parker and Julliard (2005), but they also have larger SRRs than growth stocks.
The second contribution is to extend the LRR model to account for the relationship between SRRs and cyclical variation in the equity and value premium. We model an economy explicitly with a market portfolio and portfolios of growth and value stocks. Guided by Santos andVeronesi (2006, 2010) and Menzly, Santos, and Veronesi (2004), the market portfolio and the cross-section of stocks should be studied jointly to provide a consistent explanation for the stylized facts of asset returns. To account for the time-variation of the SRRs across business cycles, we model stochastic covariances explicitly as state variables. 4 The resulting dynamic covariance structure in the cross- 4 We model stochastic covariances using a Wishart process. The theoretical foundations of Wishart processes are laid out in Bru (1991) and introduced to finance by Gourieroux and Sufana (2003). Buraschi, Cieslak, and Trojani (2008) subsequently use a Wishart covariance process to study the term structure of interest rates. For derivative pricing, we refer to Gourieroux and Sufana (2004) and Gruber et al. (2015); and for portfolio choice, see Buraschi et al. (2010). More recently, Cieslak and Povala (2016a) exploit the properties of the Wishart process to reflect a section of assets is necessary to resolve the negative correlation between the market risk premium and SRR in consumption, which would otherwise be positive in a univariate volatility setting.
With the conventional log-linearization approximation whose errors are economically small, our extended LRR model can be solved in quasi-closed form up to Riccati equations and is analytically tractable. This allows us to calibrate the model to match the dynamics in the growth of consumption and dividends, the time-varying SRRs, and asset pricing patterns, such as the equity premium, the value premium, and the price-dividend ratios. Under the calibrations, the model matches market data reasonably well. In particular, the model replicates the predictive power of SRRs on the future market returns and value-minus-growth returns.
We also perform a series of robustness checks on our results. For the empirical studies, the results adopting alternative measures of cash flows by accounting for repurchases remain qualitatively the same. For the model, Pohl, Schmedders, and Wilms (2018) demonstrated that the potential errors induced by the log-linear approximation could be considerable. Consequently, we solve the model via the projection method and find that the errors do not materially affect the model's results. Furthermore, we present empirical results where the SRRs are constructed from the monthly industrial production index instead of monthly consumption. Nonetheless, the industrial production index cannot explain future excess returns. Our regression exercise reveals a fundamental difference in the nature of consumption and industrial production data, rather than measurement errors.
Our paper shares many features with Bansal and Yaron (2004), although with some differences. First, our model is specified in continuous time, but Bansal and Yaron (2004) introduces a discretetime LRR model. While our parameters are defined similarly to conventional LRR literature, they should be interpreted in the continuous model. Second, Bansal and Yaron (2004) specify the growth rate dynamics of the monthly aggregated consumption, but we specify the growth rate dynamics of the annually aggregated consumption. Thus, in our paper, the growth rates of annually aggregated consumption can be represented by integrals, which is more natural in our continuous-time model.
In contrast, in Bansal and Yaron (2004), the growth rates of annually aggregated consumption are approximated by the weighted average of monthly consumption growth rates. Our approach enables time-varying correlation between short-rate expectations and term premia, which is a difficult feature to achieve with traditional exponential affine models. and dividends are aggregated annually to keep them in line with annually updated macro variables, similar to Bansal and Yaron (2004) and Campbell and Cochrane (1999). The annual real log return is the sum of monthly real log returns. We calculate monthly dividends before seasonal adjustments from the difference of returns with and without dividends in the same way as Beeler and Campbell (2012). We adjust for dividends' seasonality by using the adjusted monthly dividend as the moving average of dividends in the previous 12 months. Dividend growth is calculated as the seasonally adjusted dividend in the current month divided by that in the previous month. To calculate the end-of-year price-dividend ratio, we divide the asset price by the sum of the last 12 months of unadjusted dividends. The nominal 3-month Treasury bill rate data are taken from CRSP Fama risk-free rates. Given that the future inflation is uncertain, we approximate the risk-free rate by the ex-ante real 3-month Treasury rate. Similar to Beeler and Campbell (2012), the ex-ante real 3-month Treasury rate is the fitted value by the regression of the ex-post real rate (deflated 3-month Treasury rate using realized inflation) to the nominal 3-month interest rate and the growth of CPI in the previous year.

II. Short-run and long-run risk
In this section, we formally define SRR and LRR and study their empirical properties.

A. Identifying short-run risk
To clarify the definition of SRR in our model, we start with a simplified long-run risk model for consumption and dividend growth dynamics. By denoting aggregated consumption by C t and by D i t the dividend of asset i, we specify where µ c and µ i , are constants, and σ c,t , and σ i,t are possibly time-varying volatilities. The Brownian motions B c t and B i t may be correlated. X t is the LRR component in consumption and dividend growth, which is not correlated with B c t and B i t . 7 The leverage parameter is φ i , which controls the exposure to long-run consumption risk in dividends.
The long-run consumption risk in Equations (1) and (2) has been the focus on the LRR literature. However, the transient consumption risk σ c,t dB c t , which is also a priced risk in equilibrium models, has received relatively little attention. By ignoring the effect of transient consumption risk, we miss an important channel between macroeconomic states and asset prices. We need a measure of exposure to the transient consumption risk, similar to the leverage parameter for the LRR.
To capture the transient consumption risk, we define the short-run risk component in consumption, SRR c , as the time-t realized integrated variance of consumption growth over one time period, i.e., Similarly, the short-run risk component in asset or portfolio i, SRR i , at time t is defined as the realized integrated covariance: For our analysis, we focus on the SRRs of growth stocks SRR g , middle BM stocks SRR mid , value stocks SRR v , and the whole stock market SRR m .
In our empirical analysis, we assume that one time period corresponds to one year. The above definitions of SRR reflect the (co)variation of the transient components in the consumption and dividend dynamics, as the following equations suggest: There are two advantages to this definition. First, we can estimate the SRRs from realized variances and covariances directly from data. Second, because the LRR component X t only enters the drift terms of the consumption and dividend dynamics, it does not interfere with the estimation of the SRRs. 7 We deliberately omit the specification for the dynamics of Xt at this stage because the definition and measurement of SRR do not depend on the dynamics of Xt.
We remark that we impose no structures on the covariation with transient consumption shocks σ c,s σ i,s dB c s dB i s . Apart from stochastic variances, the covariation of the Brownian motions in consumption and cash flows are also time-varying. Whereas in traditional LRR models, the instantaneous correlation between Brownian shocks is either constant or zero. This deficiency calls for a more advanced model of the stochastic covariance, which is proposed in Section III.
Given the availability of monthly consumption and dividend data, we have ample data to estimate the SRR empirically. However, we face one obstacle-monthly and quarterly aggregated consumption data are seasonally adjusted, whereas dividends are adjusted by taking the yearly moving average. Hence, contemporaneous shocks in dividends and consumption are not reflected in these adjusted time series, which leads to bias in the SRRs estimates. To circumvent this problem, we use as monthly consumption growth the growth rate of the 12-month moving-average of the monthly aggregated consumption, which has some advantages. First, it fits well our model specification in which C t represents the aggregated consumption in the past year. In particular, under this construction, the sum of the monthly consumption growth rates is equal to the growth rate of the annually aggregated consumption. Second, most of the literature uses seasonally adjusted dividends by taking the 12-month moving average. Seasonalizing consumption in a similar way enables us to calculate the covariance between consumption and dividend growth rates more accurately. We refer the reader to Appendix C for more details about the construction of monthly consumption growth. 8 Following Equations (3) and (4), we directly estimate SRRs from data: where h = 1/12, i = v, g denote value or growth stocks, and ∆c t,t+h = log(C t+h /C t ).
8 BEA only publishes seasonalized monthly consumption using X13-ARIMA-SEATS. In Appendix C, we also show this algorithm does not affect the 12-month moving average materially. This observation is consistent with Koijen et al. (2017). When the macroeconomic activity is low, value stocks strongly align with the business cycle and pay little dividends, but growth stocks perform relatively well. If consumption growth is a measure of economic activity, then the cash flows of value (growth) stocks should move in the same (opposite) direction of the consumption growth rates around recessions. In Panel B, the SRR in consumption shoots up around economic recessions. This observation is in line with the common conception that the consumption volatility is countercyclical.

B. Regression results
To study the link between the SRRs and the business cycle formally, we use SRRs to predict future returns at horizons most pertinent to the business cycle. If SRRs convey bad (good) news about the economy, they should predict negative (positive) future returns at business cycle horizons.
Because the value premium is counter-cyclical, we expect the SRRs to have opposite implications for the future value-minus-growth returns.
The predictive regressions are of the following form: where SRR c , SRR g , SRR mid and SRR v t−1,t are SRRs in consumption, growth, middle, and value stocks. Z t is a vector of additional optional predictors, which are included to determine the robustness of SRRs in predictive regressions, and t,t+h is the residual. The LHS of Equation (9) is either future market excess returns or future value-minus-growth returns.
To test whether the predictive power of SRRs in the regression (9) are already contained in the macroeconomic variables, we include additional macroeconomic variables as predictors. These variables include the approximate log consumption-wealth ratio (cay, Lettau and Ludvigson, 2001a), income-consumption ratio (I/C, Santos and Veronesi, 2006) and Cochrane-Piazessi factor (CP, Cochrane and Piazzesi, 2005). We also study the log price-dividend ratio log P D . Some of the macroeconomic variables are documented to relate to the LRR. If the predictive powers remain significant even with such macroeconomic variables, it suggests that SRRs capture new information not contained in the LRR.
The correlations of the independent variables are shown in Table I. The SRRs correlated with macroeconomic variables, which implies that SRRs contain information in macroeconomic states.
The variables CP and log P D are strongly correlated, and both are negatively correlated with the income-consumption ratio. These variables capture similar aspects of the business cycle: a positive economic outlook is associated with a large CP factor, a large price-dividend ratio, and a small income-consumption ratio, while for an adverse economic outlook, it is the opposite. The income-consumption ratio is negatively correlated with cay because both are related to the representative agent's consumption-wealth ratio. The cay is pro-cyclical, which is in line with the argument in Lettau and Ludvigson (2001a) that the representative agent consumes a larger share of her total wealth in anticipation of good portfolio returns. SRR c is positively correlated with the income-consumption ratio and is negatively correlated with cay and the price-dividend ratio, which indicates that SRR c is negatively correlated with the consumption-wealth ratio. The SRR in value stocks is negatively correlated with that in growth stocks, consistent with the observations that these two variables move in opposite directions during recessions. However, the business cycle risks captured by SRRs are not adequately represented in those macroeconomic variables, as suggested by the next section's regression results.
[ Table I about here.] B.1. In-sample predictability Figure 2 shows the adjusted R 2 of the predictive regressions at different horizons using Equation (9). For future market excess returns, forecasting regressions using past consumption, growth and value SRRs has the best predictive power at the four-quarter horizon, where the R 2 is 17.6%.
Beyond the business cycle horizons, predictability wears off. This supports our hypothesis that SRRs capture business cycle risks, which do not persist over the long term. For the value-minusgrowth returns, predictability increases over time, where R 2 continues to rise to 11.5% at 12-quarter horizon using consumption, growth, and value SRRs.
[ Figure 2  To test the significance of SRRs in the presence of other macroeconomic variables as predictors, we focus on the four-quarter horizon for the market excess returns and 12-quarter horizon for the value-minus-growth returns. The horizons are chosen to maximize predictability. Table II and   Table III report the regression results using SRRs and macroeconomic variables as predictors. 9 A large consumption and value stock SRR, or a small growth stock SRR, usually accompanies an adverse economic outlook in the future one-year, and the market excess return drops. The SRR of the middle bucket does not seem to have significant predictive power. The adjusted R 2 s using SRRs are even more significant than those of in-sample constructed cay in both samples. If we use both the SRRs and cay in prediction, then R 2 would increase more, and cay would remain significant. This result suggests that SRRs are not redundant, even when given predictors such as cay. Other predictors-including log( P D ), CP factor, and income-consumption ratio-manifest little forecasting power of the future market excess returns in the one-year horizon. Consistent with the negative sign of the income-consumption ratio, which is first documented in Santos and Veronesi (2006), SRR m is positively associated with the future market excess returns.
To supplement this study, we also ran predictive regressions using univariate SRRs. Univariate regressions of the future market excess returns on SRRs gave coefficients similar to those in the multivariate regressions. This observation indicates that SRRs carry orthogonal information for the equity risk premia. The invariance of regression coefficients in univariate and multivariate regressions provides additional confidence in our regression results' robustness. The finding that SRRs in consumption negatively correlate with future market excess returns seems counter-intuitive because conventional wisdom would suggest that the market risk premium is positively correlated with market return volatility. However, consumption volatility is not perfectly correlated with all of the covariance matrix components of the cross-section of assets. Indeed, the correlation between the SRR in growth stocks and the SRR in consumption is almost zero. In contrast, the correlation between the SRR in value stocks and the SRR in consumption is nontrivial. The regression results suggest that the information in the cross-section of SRRs plays a vital role beyond the univariate market volatility.
For the future value-minus-growth excess returns, the SRR in consumption is significant. The SRRs in value and growth stocks do not explain the variations in the value-minus-growth excess returns. The negative coefficient sign suggests the value premium is counter-cyclical, which is, for example, consistent with Zhang (2005). Indeed, the quarterly aggregated market excess returns and value-minus-growth returns have a negative correlation of −16.15%. The value-minus-growth returns tend to rise post-crisis and are typically associated with a spike in the SRRs in consumption.
This observation is in line with Avramov et al. (2013), such that the value premium is mainly derived from survived distressed firms that are valued lower and bounce back harder post-crisis.
Because recoveries last longer than recessions, the adjusted R 2 in forecasting the future valueminus-growth excess returns increases over time.

B.2. OOS predictability and market timing strategy
Welch and Goyal (2008) argue that most predictive regressions cannot beat the historical average in forecasting the OOS market excess returns. In contrast, we find that our predictive regressions using SRRs have out-of-sample explanatory power. This out-of-sample predictability gives rise to an out-of-sample market-timing strategy, which leads to an economically significant improvement in portfolio performance for mean-variance investors.
We consider the out-of-sample R 2 , Sharpe ratio and the cumulated excess returns corresponding to five kinds of predictive regressions-four univariate regressions using SRRs in consumption, growth, middle and value stocks, respectively; and a multivariate regression using all SRRs given above except middle stocks-to forecast the future one-year market excess returns. At the start of each year, we estimate the regression coefficients using data from the previous 35 years. 10 We then compare our forecasts and the historical average to calculate out-of-sample R 2 : wherer e t , measured at beginning of year t, is the historical average of the annual market excess returns in the past 35 years.
For each predictive regression, we construct a market-timing strategy. At the beginning of each year, the regression gives an out-of-sample estimate of the market excess return of the following year, and we set the estimate as the weight to adjust the position in the market equity premium. 11 To make returns in such zero net position strategies comparable to the market excess returns, we ex post scale weights of these strategies such that their returns have ex post the same volatility as the market excess returns.  1956-1980 and 1980-2005. For convenience, we added the results from their Table 3 for dividend-price ratio (D/P), earnings-price ratio (E/P), 10 We estimate the regression coefficients from a quarterly sample. To avoid using future information, we exclude observations after the beginning of last year.
11 Given CRRA utility and constant variance in market excess returns, and supposing that the investor chooses from a risk-free asset and a market portfolio to invest, her position in the market portfolio is proportional to its estimated return.
12 In particular, we choose the asymptotic ratio of out-of-sample and in-sample points π to be 0.4. The significance levels are taken from Table 5. smooth earnings-price ratio (Smooth E/P), and dividend-price ratio growth (D/P Growth), since those turned out to have the best OOS predictive power.
[ The out-of-sample R 2 s of predictive regressions using individual and all SRRs except those in middle BM stocks are positive in the 1959-2017 sample, which suggests that SRRs can robustly forecast the future one-year market excess returns. The OOS F -statistics are all significant at 95%level except SRR mid t−1,t (Mid). Compared with the Sharpe ratio of the market excess returns, the Sharpe ratio of any market-timing strategy based on consumption SRR is improved by about 60%, and the Sharpe ratio of the strategy using all SRRs is even doubled. The predictive power of SRRs is more pronounced in the 1959-2005 sample, where the R 2 for using SRR in consumption alone is more than 20%. Moreover, the F -statistics are significant at 99% for using SRRs in consumption, value, or a combination of consumption, value, and growth. As a comparison, the best performing predictor in Campbell and Thompson (2008) for the first subperiod, the dividend-price ratio, has an out-of-sample R 2 s of 9.46% (in their unconstrained case) and 6.88% in the second subperiod (in the case of fixed coefficients).
[ where the bid-ask spread could eat up a significant part of the profits, this market-timing strategy is almost free from such costs. Moreover, the coefficients on rolling predictive regressions always have the same sign as their in-sample counterparts. In summary, the predictive power remains strong, even in out-of-sample regressions, which leads to a consistent improvement in the asset allocation strategies for a mean-variance investor.

C. Estimating the LRR
In what follows, we study the properties of the LRR and the SRR jointly via GMM. 13 The results only depend on consumption and dividend growth rates and are thus independent of any assumption on the transient consumption component. Hence, the results in this section apply also to the continuous-time version of Bansal and Yaron (2004)  Although X t is highly persistent and changes little over shorter periods, X t could potentially vary a lot from year to year. From Equation (1), we obtain the unconditional variance of the annually aggregated consumption growth as, where the cross-term between X t and B c s is zero because they are uncorrelated. Thus, the variance of the annually aggregated consumption growth is the sum of the variance of integrated LRR and the expectation of SRR. Similarly, the contribution of LRR to the covariance between dividend and consumption growth is controlled by φ i , as illustrated in Equation (13) Cov Motivated by Equations (12) and (13), we can study the variance of integrated LRR by GMM.
At the end of each year, we can estimate the annually aggregated growth of consumption and 13 The LRR literature has developed methods to identify the unobservable state variables, the LRR component, and the stochastic volatility. In Bansal et al. (2016), the two-state variables LRR and stochastic volatility are backed out from observed risk-free rate and market price-dividend ratio. However, we construct SRRs empirically directly from cash flows without pricing data and additional assumptions on instantaneous cash flows. Recently, Schorfheide et al.
(2018) decomposes consumption dynamics by using a Bayesian approach. While the Bayesian approach identifies the persistent LRR and stochastic volatility directly from cash flows for Bansal and Yaron (2004) model, there is no non-trivial extension to time-varying covariance structure. Taking those into consideration, we choose the GMM to decompose consumption, and dividend cash flows like in Bansal et al. (2016), but additionally includes SRRs from realized covariance estimators.
s ds, and SRRs as the realized variances or covariances. For notational brevity, we denote the variance of the annually integrated LRR and the mean of SRRs in consumption by Hence, we can formulate the GMM according to the following moment conditions: where i = v, g, mid, m represents the value, growth, mid, or market portfolio.
[ The GMM estimation results are reported in Table V. We test all of the parameters with the null hypothesis that it equals zero against the alternative that it is larger than zero. The σ 2 X is significantly different from zero at the 90.0%-level, which suggests the existence of a nontrivial LRR component. Consistent with the previous literature, such as Bansal et al. (2005a), the value stocks load more LRR than the growth stocks. The leverage parameter of value stocks φ v is significantly larger than zero at 99%-level, but the leverage parameter of growth stocks φ g is not significantly larger than 0. Apart from LRR, our paper also identifies that value stocks have higher SRR than the growth stocks, with the mean of SRR in value stocks significantly larger than zero at the 99%-level.
[ To further study the properties of the LRR, we summarize the statistics of the SRRs and consumption growth in Panel A of Table VI. 14 The means of SRRs are smaller than the covariances between the annually aggregated consumption and cash flows growth rates, which confirms the existence of LRR in consumption and dividends of book-to-market portfolios. The first-order 14 The middle portfolio is excluded since it is not economically interesting.
autocorrelation of the monthly consumption growth rates over the whole sample is 92.4%. The high persistence could be explained by the persistent LRR component. The autocorrelation of the annually aggregated consumption growth rates is 47.9%, which is smaller than the autocorrelation for the monthly growth rates because the LRR X t varies more over a longer period.

III. Theoretical model
In this section, we introduce the model, and we derive solutions to generate the patterns in asset prices and SRRs.

A. Model setup
We formulate our economy in continuous time, and we equip our representative agent with the recursive utility function, as defined in Duffie and Epstein (1992). We depart from the previous literature on LRR models in how we incorporate fluctuating economic uncertainty.
To model the covariance structure of the transient shocks in consumption and dividend growth, we impose a matrix-valued Wishart process given by where B σ t ∈ R n×n is a matrix of independent Brownian motions. The constant matrices M ∈ R n×n and Q ∈ R n×n control the mean reversion and volatility of the Wishart process. 15 To maintain parsimony, we fix the long-term mean for Σ t to kQQ and we set the scalar k = n + 1. 16 As in Bansal and Yaron (2004), we let both dividend and consumption growth be characterized by a persistent LRR component X t , which follows a mean-reverting process with stochastic volatility, where α controls the speed of mean-reversion, δ x ∈ R n is a constant vector, and B X t ∈ R is a Brownian motion. Note that the volatility of the transient shock δ x Σ t δ x is univariate, despite being a function of the stochastic matrix Σ t ∈ R n×n .
Our economy models n portfolios jointly. Each portfolio pays out dividends D i t , i = 1, . . . , n with the following dynamics, where B t ∈ R n is a vector of Brownian motions shared by all firms, B i t is univariate Brownian motion for firm i, σ i is the volatility of firm-specific shock, µ i measures the mean of firm i's dividend growth process, δ i ∈ R n is a constant vector, and φ i measures its loading on the LRR component X t .
To generate a time-varying correlation between consumption growth and dividend growth, we link the stochastic covariance matrix to the consumption process C t . We assume that C t has the following dynamics: where B c t ∈ R is a Brownian motion independent of B t , µ c is the mean consumption growth rate, and the constant vector δ c ∈ R n together with Σ t controls the loading on the transient component of consumption growth. Our representative agent may not generate income solely from dividends, but may also generate income from other sources, such as labor. Hence, we add an additional source of risk in the consumption growth dynamics, σ c,t dB c t , which is not spanned by asset markets. We where Tr(·) denotes the trace of a square matrix. We assume that the Brownian motions B t , B X t , B i t and B c t are mutually independent.
Our specifications in Equations (20) and (21) are consistent with the LRR framework in Equations (1) and (2). However, we extend the LRR model with a Wishart process, which models the multivariate stochastic volatility structure for consumption and dividend growth rates. In particular, we have The vectors δ c and δ i determine how much the variances and covariances load on the different elements of the matrix Σ t . Furthermore, we can construct the theoretical counterparts of Equations (3) and (4) to accommodate the time-varying SRR in closed-form: The SRR in asset dividends only captures the common shocks between consumption and dividends through δ c and δ i . Nonetheless, the SRR in consumption includes an additional component from σ c,t as defined in Equation 22. In models with univariate dividend and consumption growth variances, a larger consumption growth variance accompanies larger expected returns. In our model, dividend growth has a stochastic covariance structure. The loadings χ c allow the consumption volatility to load flexibly on the components in the cash flow covariance matrix Σ t , which is crucial in replicating the negative relationship between SRR in consumption and future asset returns.

B. Model solutions
We follow Duffie and Epstein (1992) and assume that the representative agent has recursive preferences. The results in this section are subject to a log-linear approximation conventional in LRR literature.
The value function J satisfies where where γ denotes the risk aversion coefficient and ψ the intertemporal elasticity of substitution. We assume that the representative agent prefers early resolution of risk, such that γ > 1 and ψ > 1. 17 To solve the model, we make use of the log-linear approximation as in Campbell and Shiller (1988).
Thus, we obtain a quasi-closed-form solution up to generalized continuous-time algebraic Riccati equations (CARE). 18 PROPOSITION 1: The value function is given by and the consumption-wealth ratio is given by 17 We discuss the case ψ = 1 in Appendix B.
18 The potential errors introduced by the log-linear approximation have come under scrutiny in a recent paper by Pohl et al. (2018). We perform some robustness checks in Section V and we find that, for our setup, the errors induced by log-linear approximations are negligible.
where A ka = 1−ψ 1−γ A k , for k = 0, 1, 2. Furthermore, The term g 1 and the constant positive semidefinite symmetric matrix A 2 need to be solved by generalized CARE.
Some comments are in order here. First, the generalized CARE admits a positive semidefinite solution with reasonable computational efficiency. Hence, although some numerical calculations are required, the model is still highly tractable. Second, with γ > 1 and ψ > 1, we have A 1a < 0. Therefore, following the standard LRR model's interpretations, the representative agent reacts to positive news in long-term consumption growth X t by consuming less out of her wealth portfolio, thereby smoothing consumption. Consequently, the substitution effect dominates the income effect.
Third, A 2 is positive semidefinite. Therefore, the consumption-wealth ratio increases when an overall increase in variance occurs, similar to Bansal and Yaron (2004). However, because in our model, each element of the stochastic covariance matrix could affect the consumption-wealth ratio through A 2 , elements of the covariance matrix have mixed effects on the consumption-wealth ratio. Finally, the persistent component X t on the consumption-wealth ratio A 1 increases with the persistence of X t , which is inversely related to the mean reversion coefficient α.
PROPOSITION 2: The state price deflator follows the dynamics with Furthermore, the risk-free interest rate is given as where the expressions for the coefficients r 0 , r x , and r Σ are given in equations (B.26) to (B.28).
From Equation (33), we can identify four components for the market price of risk in our model.
The first two components, Λ and Λ c , are the market prices of risk on transient consumption shocks, where Λ c arises from the additional source of risk that is not spanned by the asset market. These two components are proportional to the risk aversion coefficient γ, and they do not depend on the intertemporal elasticity of substitution ψ. The third component, Λ X , is the market price of risk for exposure to innovations in LRR. The fourth component, Λ σ , represents the market price of risk for innovations in the Wishart covariance process.
Our specification of the market price of risk extends the previous LRR models in that we account not only for the variance risk as in Zhou and Zhu (2015) but we also account for the covariance risk.
The off-diagonal elements of the covariance matrix are needed to match the time-varying returns in the assets' cross-section.
Our LRR model generates a risk-free interest rate in Equation (34) as an affine function of the LRR component X t and elements of Σ t . This specification of the risk-free rate is similar to the term structure models in Buraschi et al. (2008) and Cieslak and Povala (2016b). However, our focus is on the dynamics of the cross-section of equity returns instead of the risk-free rate term structure. 19 PROPOSITION 3: The dividend-price ratio for asset i has the following form A 0i and A 1i are given by and A 2i is a symmetric positive semidefinite matrix of coefficients. The expressions for g 1i and A 2i need to be solved by generalized CARE. The equity risk premium for asset i is 19 Note that if ψγ = 1, then the utility function reduces to CRRA form. Under CRRA, uncertainty in the future utility arising from uncertainty in the consumption growth process is no longer priced, so Λ X and Λσ are zero. Furthermore, we would obtain rΣ = 0, which shuts down the major channel of variation in risk-free interest rate because Xt moves only slowly.
A 1i and A 2i play similar roles to A 1 and A 2 . If φ i > 1/ψ, then we have A 1i < 0 and an increase in the LRR component X t drives up the valuation of asset i. In other words, the substitution effect dominates the income effect. Because A 2i is positive semidefinite, an increase in overall volatility in Σ t drives down the valuation of asset i.
The equity risk premium in Equation (38) comprises three parts. The first part is determined by the covariance of dividend growth and consumption growth, scaled by the risk aversion coefficient.
A higher covariance implies a higher risk premium. The second part is the contribution of the variation in the LRR component. Higher LRR volatility or intertemporal elasticity of substitution leads to a larger equity premium. As Bansal and Yaron (2004) show, sufficiently high persistence in the LRR component dynamics helps generate a large equity premium. Under our assumption that the representative agent prefers early resolution of risk, γ > 1 and ψ > 1. Hence, a high persistence ( a low value for α), leads to the large product A 1 A 1i . Furthermore, because Σ t is positive definite, δ x Σ t δ x is positive. Therefore, the risk premium part arising from long-run risk is always positive.
The third part arises from the exposure to the innovations in transient consumption shocks, which captures the compensation for the SRR. In models without stochastic covariation, the correlation between shocks is constant, and the only variation in this part stems from the stochastic volatility, which lacks the flexibility to model the compensation from the SRR. In contrast, the Wishart process enables SRR to manifest its importance in the risk premium.
To avoid over-parametrization, we impose additional restrictions on the model. We assume that there are three portfolios in the economy: the market portfolio, the portfolio of growth stocks, and the portfolio of value stocks. We further assume that the stochastic covariance matrix is a 2 × 2 Wishart process, which has three free components because any covariance matrix is symmetric.
Under these restrictions, the risk premium for asset i in Equation (38) is the linear combination of the three components in the Wishart process. Recall from Equations (23) and (24) for some β 0 , β c , β cg , β cv chosen to match the four dimensions in the 2 × 2 Wishart process and the constant. Hence, the model implies that the equity risk premia of an asset can be explained by the instantaneous variance of the consumption growth and the instantaneous covariance between the growth of consumption and the dividends in growth stocks and value stocks.
Equation (39) shows that the representative agent takes into account the time-varying covariance in the cash flows of assets, which leads to the time-varying equity risk premia. Given that SRRs are realized variances and covariances, the model could replicate the relation between SRRs and asset returns.
PROPOSITION 4: The model-implied regression coefficients β 0 , β c , β cg , βcv can be derived in closedform for predictive regressions of future returns on SRRs where Q denotes quarters, SRR c t−1,t and SRR g t−1,t (SRR v t−1,t ) are defined in Equations (3),(4), t,t+ Q 4 is the residual. We refer to Section B.4 for further details.

IV. Quantitative model results
In this section, we aim to calibrate the parameters to match moments in the sample from 1959-2017, 20 and we will study the quantitative results implied by our model. More details about the calibrations and the derivations of model-implied moments are given in Appendix B.6. 21 To reflect the dynamics of cash flows, we match the unconditional mean, the first-order autocorrelation, and the volatility of growth rates of consumption and dividends of value stocks, growth stocks, and the market portfolio. To ensure that our model captures asset returns patterns, we 20 Monthly aggregated consumption data is not available before 1959. 21 We remark that we only use the value and growth portfolios for calibration. As a robustness check and for completeness, we also report the results when we also include the middle portfolio in the calibration exercise. These results are summarized in Appendix F. also match the unconditional mean, the volatility, and the first-order autocorrelation of the riskfree rate, the aggregated market equity premia, and equity premia of value and growth stocks, as well as their price-dividend ratios. In particular, to verify the additional pricing channel of SRR, we match the theoretical moments of SRRs with sample moments of SRRs. Moreover, we match regression coefficients in predictive regressions of future returns on SRRs. The model is calibrated by matching these quantities jointly.
For a Wishart process of dimension n, Q has n(n + 1)/2 free parameters while M has n 2 parameters. To reduce the number of parameters, we set n = 2. To avoid over-identification and further reduce the number of parameters, w.l.o.g. we specify 22 where δ m , δ g , δ v correspond to the market, growth and value portfolio, respectively. We restrict M to be lower triangular, which further reduces the number of parameters.

A. Consumption and cash-flow growth
First, we study the dynamics of the growth rates of consumption and dividends under the joint calibration. In Panel A of Table VIII,  those of dividend growth, although the model-implied AC1(∆c) seems larger than the empirical value.
[ The leverage parameter φ v of value stocks is estimated to be 8.17, which is much higher than that of growth stocks φ g = 4.68. Growth stocks have less exposure to the LRR. The differential exposure to the LRR affects the correlations with the growth rates of the annually aggregated consumption.
In our sample, the correlation between the growth rates of the annually aggregated consumption and dividends of value stocks Corr(∆c, ∆d v ) is 0.588, while that between consumption and growth stocks Corr(∆c, ∆d g ) is 0.323. The higher loadings replicate the higher correlation with value stocks on the LRR in the dividend growth rate dynamics. Our model generates SRRs similar to their empirical levels. We find that most variations consumption growth comes from LRR, whereas SRRs account for most variations in dividend growth. Although the volatility of the annually aggregated consumption's growth rates is 0.865%, the mean of SRR is only about √ 0.0283% ≈ 0.168%.

B. Asset Returns
In this part, we study the model-implied asset returns, particularly the channels through which the agent's preferences and consumption risks determine asset prices. In Table IX, we report the model-implied and sample moments of asset returns. The model replicates the risk-free rate dynamics closely, matching its mean, volatility, and first-order autocorrelation. The model generates realistic equity risk premium levels averaging 5.461%, compared with 4.803% in data. [ [ A few comments are in order here. First, in the baseline calibration, we get γ = 2.4899. Hence, the risk aversion lies in a reasonable range between one and ten documented in Mehra and Prescott (1985). Moreover, the risk aversion γ is smaller than in Bansal and Yaron (2004). The EIS ψ is ψ = 1.0325 > 1 γ , so that the representative agent has a preference for the early resolving of risk. The EIS is also smaller than in Bansal and Yaron (2004). Consequently, the representative agent requires less compensation for the LRR. For the model to generate realistic levels of the risk premia in the cross-section, the compensation for the SRR must be sufficiently large. Second, although the leverage parameter of value stocks φ v is larger than growth stocks φ g , the difference in LRR alone is not sufficient to account for the value premium. Therefore, SRRs in value and growth stocks contribute a significant proportion to the observed value premium.

C. Predictability
In Section II.B, we demonstrated that SRR could predict the future market excess returns and value-minus-growth returns. The predictability could be partially explained by the business cycle: SRR in consumption is counter-cyclical, and SRR in growth (value) stocks is pro-cyclical (counter-cyclical). Our model replicates the link between the SRR and asset returns. In the data, the predictability for the market excess returns peaks at the four-quarter horizon-also, the predictability for the value-minus-growth returns increases in horizons of up to 12 quarters.
Our model focuses on those horizons where SRR has the most predictability. In addition, we incorporate the correlation structure between the cyclical SRRs documented in Section II.A. The loading δ c δ c + χ c of transient consumption volatility on the components corresponding to growth stocks are negative, while those on the components corresponding to value stocks are positive.
These loadings mimic the small correlation between SRR in consumption and growth stocks and the large correlation between SRR in consumption and value stocks. While a univariate volatility structure cannot generate a negative correlation between market risk premium and volatility, our model with a dynamic covariance structure resolves the negative correlation. 24 The details for the derivations of the model-implied regression coefficients are given in Appendix B.4.
[  Table XI reports the estimated coefficients of the predictive regressions. Our model implies a negative coefficient using the SRR in consumption to predict the future market excess returns, which lies within one standard error from the sample estimate at the four-quarter horizon. The SRR in growth stocks has a positive model-implied coefficient in predicting the market excess returns, albeit smaller than the sample counterpart. The model-implied regression coefficient of the SRR in growth stocks is less than one standard error away from sample estimates, where the coefficients are positive both in-sample and in-model. The model-implied coefficients of the SRR in value stocks are slightly outside the one-standard-deviation interval but have the same negative sign as in the 24 Although the negative values on χc raise the potential concern that σc,t could turn negative, the probability for σc,t to reach 0 in our calibration is around 0.00002. In a simulation of 2,400,000 months, σc,t only turned negative in 118 months. data. Overall, our model does a good job replicating the forecasting patterns of SRRs for the future market excess returns.
We have shown that to predict the future value-minus-growth returns, the SRR in consumption is the only significant predictor among SRRs. Hence, we focus only on the SRR in consumption for the forecasts. We find that the model-implied coefficient is less than one standard error away from the sample estimate at the 12-quarter horizon.

V. Robustness
In this section, we check our results' robustness to an alternative measurement of cash flows and log-linear approximation. We also empirically investigate the results replacing consumption by the industrial production index.

A. Dividends adjusted for repurchases
As a first robustness check, we adjust dividends to account for equity repurchases by the method proposed in Bansal et al. (2005a). Details about the adjustment method can be found in Appendix I.
[ Figure 4 Figure 4, SRRs in adjusted dividends also fluctuate with the business cycle. We run the regression (9) with adjusted cash flows to confirm this, see Table XII. Similar to the case using cash dividends, the SRRs in consumption and value stocks negatively predict the future equity premia at the fourquarter horizon, and the predictability declines as the horizon expands. The adjusted R 2 at the four-quarter horizon is 10.5%, which remains reasonably large. The SRR in consumption positively predicts the future value premia, and the predictive power scales up with the horizon. Meanwhile, SRRs in growth and value stocks are less significant in predicting future value premia. In summary, in line with the case of cash dividends, SRRs fluctuate with the business cycle and carry similar signals in predicting future returns.

B. Errors in log-linearization
In this part, we quantify the impact of approximation via the log-linearization. Pohl, Schmedders, and Wilms (2018) finds that log-linearization ignores the higher-order effects in long-run risk models. However, those higher-order effects can lead to "a strong impact on key financial statistics." To solve the equilibrium in the LRR model, we adapt the projection method proposed in Pohl, Schmedders, and Wilms (2018). More technical details can be found in Appendix E. Pohl, Schmedders, and Wilms (2018) suggests that the projection method is sufficiently accurate to solve the general equilibrium numerically and requires less computational cost than other methods such as Tauchen and Hussey (1991).
[  Table XIII is based on simulation, the results obtained through log-linearization could differ from theoretical moments. Compared with the more accurate solution by the projection method, the log-linearized solution provides moments with economically negligible errors, while providing better tractability by admitting quasi-closed-form solutions.
Pohl, Schmedders, and Wilms (2018) study several models in the LRR framework. They find that the log-linearization approximation induces large errors, especially when the risk aversion γ and the EIS φ are large, or the LRR component X t is highly persistent. Although the LRR component remains persistent in our study, our model implies a calibration with a small risk aversion γ around three and an EIS close to one, which are both smaller than in models studied in Pohl, Schmedders, and Wilms (2018). Therefore, solving our model with log-linearization approximation induces smaller errors than those analyzed in Pohl, Schmedders, and Wilms (2018), and the approximated solution suffices for our analysis.

C. Results using the industrial production index
In this part, we look into the short-run industrial product index risk (henceforth SRIR), which are defined similarly to SRR but with the production index replacing the role of consumption. We estimate SRIRs empirically, and we then run the predictive regressions of future returns on SRIRs. Other than these periods, variation in industrial production is disconnected from economic outlook.
The variation in industrial production index growth is less informative about the business cycle than the variation in consumption growth.
In Table XIV, predictive regressions SRIRs explain little variation in future market excess returns and value-minus-growth returns. The discrepancy in predictive power between consumption and industrial production is unlikely to be merely due to measurement errors in monthly consumption data. It is also unlikely that the pure noises in monthly consumption predict future returns.
A more plausible explanation for the discrepancy involves investigating the economic differences in industrial production index and consumption composition. We leave this aspect to further studies.

VI. Conclusions
This paper studies the relationship between SRRs in consumption, Book-to-Market portfolios, the business cycle, and asset returns. The SRRs vary with the business cycle. The SRR in growth stocks predicts the future equity premia negatively, while the SRRs in growth stocks and consumption predict the future equity premia positively. For the future one-year (three-year) horizon excess market (value-minus-growth) returns, the adjusted R 2 of forecasting regression is 17.6% (11.5%).
This predictability remains robust in out-of-sample regressions.
To capture the cyclical variations in SRR and asset returns, we propose an LRR model where a Wishart process models the stochastic covariance process. The model reproduces the growth dynamics in consumption and dividends, the cross-sectional asset pricing moments (particularly the value premium), and the predictive regressions' coefficients. Both SRR and LRR components contribute to the equity and value risk premia.

Appendix A. Wishart process
In what follows, we summarize some essential properties of the Wishart process for solving our model. More details can be found, for example, in Gourieroux et al. (2009).

A.1. Moments and autocovariances
In this section, we will give the first two moments of the Wishart process without detailed derivations. For l ∈ R + , denote A l := exp(lM ) and Ξ l := Let h 1 , h 2 be n × n constant symmetric matrices, l > 0, the second moments of the Wishart process are given as To solve the model, we also need calculate the second moments of the integrated Wishart process  (1978)).

A.2. Quadratic variation of matrix SDE
Here, we study the quadratic variation of traces of matrix stochastic processes. Denote the quadratic variation by , .
Lemma A.1: Assume W t is a n × n matrix Brownian Motion, and A t ,Ā t are predictable n × n matrix processes. Then, Tr( A . dW . ), Tr( Ā . dW . ) t = Tr(A tĀ t ). Proof.
Given that we work with Wishart process, the following corollary comes in handy.

B.1. Proof of proposition 1
Given the affine structure of the underlying problem, we guess the following exponential affine form for the value function: Because Σ t is symmetric, w.l.o.g. we can assume A 2 to be a symmetric matrix. From the optimization problem in (27), we obtain the Hamilton-Jacobi-Bellman (HJB) equation as: where A c is the infinitesimal generator associated with state variables (W t , X t , Σ t ). The first order condition of the HJB equation for consumption C t is For notational convenience, we define Then, from the first-order condition (B.3), we obtain the consumption-wealth ratio Ct Wt as: where A ka = 1−ψ 1−γ A k , for k = 0, 1, 2. The consumption-wealth ratio is an exponential affine function of the state variables. Note that if ψ = 1, consumption-wealth ratio Ct Wt is constant and equal to β. By substituting (B.5) back into (B.1), we get Then, for the case ψ = 1, the HJB equation can be rewritten as: and, for ψ = 1, To obtain the coefficients of the representation in (B.1) for the case when φ = 1, we adopt the standard log-linear approximation of Campbell and Shiller (1988). Defining c t := log C t and w t := log W t , we approximate the consumption-wealth ratio as where g 1 = exp(E(c t − w t )). Because C t /W t depends on A 0 , A 1 , A 2 , which in turn depend on g 1 , it is not possible to give an analytical expression of g 1 . Hence, g 1 must be calculated numerically.
We refer to Appendix B.5 for details. Then, the log-linearized HJB equation is For the case of ψ = 1, no approximation is needed since the log-linearization is exact. The resulting HJB equation is Now we solve for A 0 , A 1 , and A 2 . Irrespective of the value of ψ, A 1 satisfies − g 1 ψA 1 − αψA 1 + (1 − γ) = 0, (B.13) If ψ = 1, then g 1 = β. For ψ > 1, A 0 satisfies θ(g 1 − g 1 log g 1 + g 1 ψ log β) − βθ − g 1 ψA 0 + ψ Tr(A 2 ΩΩ ) + (1 − γ)µ c + (1 − γ)(−γ) 2σ 2 c = 0. (B.14) For ψ = 1, we have To obtain A 2 , we first note that the terms involving Σ t in the HJB equation (B.11) should sum up to zero: If we denote the matrix left-multiplying Σ t inside the trace operator by L, then L must satisfy L + L = 0 because Σ t is symmetric. L does not have to be a zero matrix. Thus, where A, Q and E are square matrices of the same dimension. Furthermore, Q and R are symmetric matrices. Hence, in our case,

B.2. Proof of proposition 2
To derive the state price deflator, we take partial derivatives of f (C, J) and use identities (B.5) and (B.6) to obtain: The expression for SDF under recursive utility is, according to Duffie and Epstein (1992), By plugging in expressions of f J (C t , J t ) and f C (C t , J t ), we obtain the dynamics of π t , where Λ σ , Λ, Λ X and Λ c are the prices of risk, which we can identify as We can read off risk-free interest rate directly from SDF. The risk-free interest rate can be decomposed in the following: where by matching constants and coefficients on X t and Σ t , we obtain

B.3. Proof of proposition 3
Assume that the dividend-price ratio has the following exponential affine form, where A 2i is a symmetric matrix. The instantaneous return of asset i is We perform a log-linear approximation as in Campbell and Shiller (1988).
Pt is the risk premium of the asset. The formula for the risk premium is the following: where we used Proposition A.1 to calculate the quadratic variation of Wishart diffusions. By comparing coefficients in Equations (B.30) and (B.32), we find that A 0i must satisfy Similarly, for A 1i : (B.35) and for A 2i : Hence, A 2i is a solution to: To obtain A 2i numerically, we can again cast it into the form of a generalized continuous time algebraic Riccati equation (B.17). In this case, Given proper technical conditions, a positive semidefinite solution A 2i exists.

B.4. Proof of proposition 4
In what follows, we derive model-implied regression coefficients, in which we regress future asset return on SRRs. We want to study the regressions of the form Thus, we derive the model-implied coefficients for the following regression: where r e is excess stock return over risk-free rate in the period from t to t + Q 4 , h i ∈ R 3×3 for i = 1, 2, ..m. Denote by β := (β 0 , β 1 , · · · , β m ) the vector of model-implied regression coefficients.
For convenience, we denote the right-hand-side independent variables by RHSVAR := 1, Tr Regression coefficients are therefore given by B.5. Numerically solving g 1 and g 1i Here we develop the algorithm to calculate g 1 and g 1i numerically. Recall Equation (30), Then, Note that this equation holds for ∀t > 0. Obviously e −αt X 0 converges to zero in probability as t → ∞, so Because δ x Σ t δ x has the long term mean δ x Σ(∞)δ x , we approximate g 1 by Laplace transform of W (K, 0, Ξ(∞)) is given in Gourieroux et al. (2009): Thus, we numerically solve for g 1 from Similarly for asset i, its stationary mean of dividend-price ratio g 1i is solved from

B.6. Theoretical moments
This section gives the analytical expressions for moments used in GMM estimation. There are 36 moments in total. Under our assumptions, M is negative definite and lower triangular, Q = qI n and δ x = ηδ c , the expressions of the following moments can be further simplified. The parameters are: where i = m, 1, 2, 3 represents market portfolio and three Fama-French portfolios respectively.
There are 26 parameters to be estimated. To make the estimations easier, we restrict µ i to be the sample mean of the corresponding mean of cash flow growth, which leaves us with 21 parameters to match 31 moments. To estimate the over-identified system, we introduce the weight matrix W , which we specify as a diagonal matrix that adjusts for the magnitudes of moments. Bansal et al. (2016) argues that the decision interval for the long-run risk model should be a month. To reflect the more frequent decision making (i.e., more than once per year), we model the agent to make decisions dynamically and continuously. Because observations are only available yearly in aggregate, we calculate theoretical moments at yearly aggregations. ∆c t,t+1 := t+1 t dCt Ct , and ∆d i t,t+1 , r i e,t,t+1 , r f,t,t+1 are similarly defined. Therefore, We plug in Equations (A.2) and (A.3) to calculate moments of Wishart process, expressions for the model implied moments are shown in Table XV. [

Appendix C. Monthly consumption growth
We define consumption as the sum of nondurable goods and services, where the data are from the U.S. Bureau of Economic Analysis (BEA). Per capita annual consumption data range from 1927 to 2017 in real terms. 26 The monthly consumption data are available from January 1959 to December 2017, as the national aggregate and in nominal dollar amounts. 27 We construct the monthly per capita consumption in real terms to make the consumption data at monthly and yearly frequency consistent. We divide aggregate nominal monthly consumption by population and the personal consumption deflator to get personal consumption in real terms. Given that the population is only measured quarterly, we linearly interpolate the quarterly population to get a monthly estimate of population level. We also linearly interpolate quarterly personal consumption deflator to get a monthly personal consumption deflator, where the quarterly personal consumption deflator is the ratio of nondurable consumption plus services in nominal terms divided by those in chained dollars.
To compare consumption and dividend growth rates and estimate their covariance, one should be cautious about the different constructions of seasonal adjusted consumption and dividends. While monthly personal consumption data from BEA are seasonally adjusted by removing the seasonal component, 28 dividends are seasonal adjusted simply by calculating the yearly moving average.
Consequently, any macroeconomic shock has an immediate impact on consumption growth but affects dividend growth data only after several quarters. Therefore, to make consumption and dividends comparable, we calculate seasonally adjusted consumption as its moving average in the last 12 months. We let one unit of time interval corresponding to one year, and we set h = 1/12.
We denote the personal real consumption before seasonal adjustment between time t and time t + h by C t,t+h , the contemporary seasonal component by S t,t+h . Hence, the seasonal adjusted consumption, corresponding to the data from BEA, is given by C SA t,t+h = C t,t+h − S t,t+h . We then calculate the moving average of consumption C M A t,t+h between time t and t + h: The approximation in the last step holds as long as the seasonal components derived from X13-ARIMA-SEATS within a year sum up to a small value close to 0. In principle, we would prefer to calculate the moving average of consumption directly from an unadjusted time series of monthly consumption data instead of using C SA , but no such data is available as of now. We confirm insensitivity to seasonalization by X13-ARIMA-SEATS in Panel A of Figure 6, where we plot the annual consumption growth directly calculated from the ratio of consecutive annually aggregated consumption, and that calculated from summing up the monthly changes in the 12-month moving average 28 See, https://www.bea.gov/faq/index.cfm?faq_id=123, where X13-ARIMA-SEATS is implemented.
of the monthly aggregated consumption. For the years where both data are available, the two series are almost identical with a high correlation at about 99.6%. 29 Panel B of Figure 6 plots the fluctuations of (rescaled) monthly consumption growth around their annual mean. Yearly aggregated consumption C t,t+1 is the sum of the monthly consumption within the year 12 i=1 C M A t+(i−1)h,t+ih . Then, we construct the monthly consumption growth in a way similar to how we construct the growth of dividend moving average. Log monthly consumption growth ∆c t,t+h is This monthly consumption growth is different from, for example, Bansal et al. (2016), in which the monthly consumption growth is log In their setup, the annually aggregated consumption growth is not the sum of the monthly consumption growth rates within the year.
In contrast, our construction of monthly consumption growth reflects the monthly changes in the annual consumption, which sum up to the annual consumption growth: [ Figure 6 about here.]

Appendix D. Simulation details in Table VI
The consumption growth is generated from the following specification: where the unit of time is one month. We simulate 1,000 years of monthly consumption growth rates and use the last 900 years for estimation to minimize the effect of the choice of initial value of long-run risk x 0 . We refer to the results in Panel B of Table VI. To study the effect of long-run risk, we study different calibration setups: baseline parameters (ρ = 0.975, σ x = 0.0237, σ c = 0.032, µ c = 0.16), and three alternatives with ρ = 0, σ x = 0 and µ c = 0 respectively.
The baseline parameters are chosen in a way to match the persistence of long-run risk in Bansal and Yaron (2004) and consumption growth dynamics. In the baseline case, we are able to generate similar short-and long-run consumption growth rates variance to real data, and the long-run risk component accounts for most of the variance annually aggregated consumption growth. Moreover, the within-year monthly autocorrelation is much smaller than the monthly autocorrelation estimated using the whole sample, which confirms that the persistent long-run consumption risk does not affect the within-year autocorrelation of the monthly aggregated consumption growth rates as much as in the autocorrelation of the annually aggregated consumption growth rates. For comparison, we shut down the LRR channel in other calibrations, and we find that without LRR, the dynamics of consumption growth behave distinctly from real consumption data. In the second column, we let the ρ = 0 so that x t is just another source of transitory risk. In this case, the variance of annually aggregated consumption is smaller due to zero persistence in x t , and the variance of monthly growth rates is almost identical to the variance of annual growth rates. All measures of autocorrelation are close to zero. In the third column we let σ x = 0 so that x t = 0 throughout.
Like the second column, the variance of monthly growth rates accounts for almost all of the annually aggregated consumption growth rates. All autocorrelation measures are close to zero. The fourth column studies the sensitivity to the mean of consumption growth, and we find the mean of consumption growth rates has no impact on the variance or autocorrelation measures.

Appendix E. Details of projection method
The projection method in this paper roughly follows the steps described in Pohl, Schmedders, and Wilms ( The degree of those polynomials is guided by the choice in Pohl, Schmedders, and Wilms (2018).
We use the collocation method to solve for G t . I.e., determine the polynomial coefficients in Appendix F. Quantitative model results with three BM portfolios In our analysis, we find that the middle book-to-market portfolio does not carry any useful financial information. Therefore, we have excluded it in our calibration exercise. Nevertheless, for completeness, we also report the results of the calibration using all three book-to-market portfolios.
To keep the model in line with our previous analysis and to reduce the model complexity, we stick to a two-dimensional Wishart process for modeling the SRRs. Note that the SRR of the middle portfolio is controlled by δ mid = (δ mid 1 , δ mid 2 ) .
The calibration results are reported in Tables XVI to XIX. The persistence of the monthly consumption growth remains large with α = 0.082. Value stocks have larger exposure to LRR than growth stocks, with φ v = 4.4065, φ g = 0.3391. The risk aversion parameter γ = 2.2466 and the EIS ψ = 1.0139 are similar to those in the base calibration. The decomposition of risk premiums similarly shows that the SRRs play an essential role in explaining the market risk premium and the value premium. As expected, the middle portfolio exhibits exposures to the LRR and SRR which lie between the exposures of the growth and value portfolios. Although the parameter values differ from those in the base calibration with two portfolios, the calibration using all three book-to-market portfolios conveys similar economic information and intuition. [ , the logarithm of price-dividend ratio (log P D ), the ratio of income over consumption (I/C) in Santos and Veronesi (2006), the cay in Lettau and Ludvigson (2001a) and the CP factor in Cochrane and Piazzesi (2005).
, the logarithm of price-dividend ratio (log P D ), the ratio of income over consumption (I/C) and cay. Newey-West standard errors with lag 8 are shown in parentheses, based on which the significance level is determined. SRRs are represented in squared percentage and returns in percentage terms.    1956-19801980-2005Predictor 1956-19801980-2005 -statistics (McCracken, 2007) for excess returns of market-timing strategies derive from predictive regressions in 1959-2017 sample. The predictors are resp., SRR c t−1,t (C), SRR g t−1,t (G), SRR mid t−1,t (Mid), SRR v t−1,t (V), and all the above SRRs except SRR mid t−1,t (All). Panel B reports the same results for the 1959-2005 sample for comparisons to (Campbell and Thompson, 2008). Panel C reports the out-of-sample R 2 s in 1956-1980and 1980-2005samples documented in Campbell and Thompson (2008, Table  3.  (15), (16) and (17). For each parameter, the p-value is calculated from the one-sided test of the parameter equal to 0 against larger than 0. The covariance matrix is estimated by Newey-West estimator. Growth rates are in percent and SRRs are in square percentage. Panel A summarizes the statistics of the SRRs, consumption growth (C), and the cash-flow growth of the market (M), growth (G), and value portfolios (V). The first row reports the averages of SRRs. In the second row, we list in parentheses the standard errors of the SRRs. The third row reports the unconditional variance of the annually aggregated consumption growth and the covariance between the annually aggregated consumption growth and cash-flow growth. The fourth row reports the average yearly observations of the first-order autocorrelations within the year of monthly aggregated growth rates, and the fifth row shows their standard errors in parentheses. The sixth row reports the first-order autocorrelations of the annual consumption growth and cash-flow growth. We report the first-order autocorrelation of the monthly consumption growth and cash-flow growth in the seventh row, both calculated using the full sample. Panel B lists the same statistics for simulated consumption growth under different dynamics specifications. We simulate monthly consumption growth for 1000 years and use the last 900 years to calculate statistics. The first column displays the baseline calibration (ρ = 0.975, σ x = 0.0237, σ c = 0.032, µ c = 0.16), the remaining columns correspond to cases where ρ = 0, σ x = 0 and µ c = 0. SRRs, variances, and covariances are expressed in squared percentage terms.   This table reports the model-implied (Model) and sample (Data) moments of the variables of interest, as well as their corresponding standard deviation (SE). In Panel A, we summarize the mean (E(·)), standard deviation (σ(·)), and first-order autocorrelation (AC1(·)) of the growth rates of the annually aggregated consumption and cash flows in value and growth stocks. In Panel B, we summarize the correlations of the growth rates in the annually aggregated consumption and dividends, the means of SRRs, and the means of short-run covariances between dividend growth rates. We consider the cash flows in the market portfolio (m), growth stocks (g,) and value stocks (v). Standard deviations are constructed by the delta method with Newey-West standard errors at eight lags. The growth rates are in percent. SRRs, variances, and covariances are in square percentage terms. This table reports the model-implied (Model) and sample (Data) moments of the asset price dynamics, including the means (E(r)), the standard deviations (σ(r)) and the first-order autocorrelations (AC1(r)) of the annually aggregated returns and the mean of price-dividend ratios (E(P/D)). The in-sample standard deviations (SE) are also reported. Standard deviations are constructed by the delta method with Newey-West standard errors at eight lags. The assets under consideration are risk-less asset (f ), market portfolio (m), growth stocks (g) and value stocks (v). Numbers are in percent. This table reports the decomposition of the risk premium in the market portfolio, and growth and value stocks. The risk premium can be attributed to three sources: the risk aversion times the instantaneous covariance between the growth rates of consumption and dividends (γ Cov t (∆c, ∆d)), the LRR and the SRR. Returns are in percent. This table reports the coefficients of the predictive regressions in model (Model) and data (Data), and standard errors of estimates in data (SE). In the first three columns, we predict 4-quarter horizon future market-excess returns using SRR in consumption (C), dividends of growth stocks (G), and value stocks (V). In the last column, the SRR in consumption (C) is used to predict 12-quarter horizon value-minus-growth returns. This table reports the results of the predictive regressions. The independent variables include SRR c t−1,t (C), SRR g t−1,t (G), SRR v t−1,t (V). The first (last) three columns report results from regression forecasting the future 4, 8, and 12 quarters market excess returns (value-minus-growth returns). Newey-West (NW) standard errors with lag 8 are shown in parentheses, based on which the significance level is determined. The variances and covariances are represented in square percentage terms, and returns are represented in percent.  This table reports the results of predictive regressions. Independent variables include SRIR c t−1,t (Ind), SRIR g t−1,t (G), SRIR v t−1,t (V). The first (last) three columns report results from regression forecasting future 4, 8, and 12 quarters market excess returns (value-minus-growth returns). Newey-West (NW) standard errors with eight lag are shown in parentheses, based on which significance level is determined. SRIRs are represented in square percentage terms, and returns are represented in percent.  This table shows the expressions of model-implied moments. For notational simplicity, we denote the coefficient on Σ t for excess returns of portfolio i by   This table reports the model-implied (Model) and sample (Data) moments of the variables of interest, as well as their corresponding standard deviation (SE) in sample. In Panel A, we summarize the mean (E(·)), standard deviation (σ(·)) and first-order autocorrelation (AC1(·)) of the growth rates of the annually aggregated consumption and cash flows in value and growth stocks. In Panel B, we summarize the correlations of the growth rates in the annually aggregated consumption and dividends, the means of SRRs, and the means of short-run covariances between dividend growth rates. We consider the cash flows in the market portfolio (m), growth stocks (g), middle stocks (M id) and value stocks (v). Standard deviations are constructed by the delta method with NW errors at eight lags. The growth rates are in percentage points. SRRs, variances, and covariances in square percentage terms. This table reports the model-implied (Model) and sample (Data) moments of the asset price dynamics, including the means (E(r)), the standard deviations (σ(r)) and the first-order autocorrelations (AC1(r)) of the annually aggregated returns and the mean of price-dividend ratios (E(P/D)). The in-sample standard deviations (SE) are also reported. Standard deviations are constructed by the delta method with NW errors at eight lags. The assets under consideration are risk-less asset (f ), market portfolio (m), growth stocks (g) and value stocks (v). Numbers are reported in percentage points. This table reports the decomposition of the risk premium in the market portfolio and growth, middle, and value stocks. The risk premium can be attributed to three sources: the risk aversion times the instantaneous covariance between the growth rates of consumption and dividends (γ Cov t (∆c, ∆d)), the LRR, and the SRR. Numbers are reported in percentage points. In Panel A, we plot the monthly observations of SRRs in value, middle, and growth stocks. In Panel B, we plot the monthly observations of SRRs in consumption. The shaded areas are NBER recorded recession periods. To construct value, growth and middle portfolios, we use the upper and lower 30 percentiles, and the middle 40 percentiles of the value-weighted portfolios formed on book-to-market of individual stocks. Our sample ranges from 1959 to 2017. In k-th month in year t, we calculate the SRRs according to Equations (7) and (8). SRRs are represented in squared percentage terms.

Market Excess Return Value Premium
This figure plots the adjusted R 2 s (in percent) of the predictive regression of future returns on SRRs specified by Equation (9). The dependent variables are the future market excess returns or the value-minus-growth returns at future Q-quarter horizons. The independent variables are the SRRs in consumption, value, and growth stocks. The black line shows the adjusted R 2 for predicting the future market excess returns, and the red line the adjusted R 2 for predicting the future value-minus-growth returns. We plot cumulative excess returns using different investment strategies starting at the end of 1995. In each such strategy, we multiply the position in the market excess returns by the predictions from the out-of-sample regressions. As predictors, we use SRR c t−1,t (C), SRR g t−1,t (G), SRR mid t−1,t (Mid), SRR v t−1,t (V), and all the above SRRs except SRR mid t−1,t (All). As a benchmark, we plot the cumulative market excess returns. We rescale the weights on the market excess returns ex post to ensure that all of the portfolios have the same volatility in returns. The out-of-sample predictive regressions estimate coefficients on a rolling basis from the past 35 years of data, and the weight on the market excess returns is updated at the beginning of each year.  In Panel A, we plot the monthly observations of SRIRs in the value, middle, and growth portfolios. In Panel B, we plot the monthly observations of SRIRs in the industrial production index growth. The shaded areas correspond to the NBER recorded recession periods. For the value, growth, and middle portfolios, we use the upper and lower 30 percentiles, and the middle 40 percentiles of the value-weighted portfolios formed on book-to-market. Our sample ranges from 1959-2017. We estimate the SRIRs similar to Equations (7) and (8), with consumption replaced by the industrial production index. SRIRs are represented in square percentage terms.  1960 1966 1972 1978 1984 1990 1996 2002