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Asset Prices and Capital Share Risks: Theory and Evidence
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Leading asset pricing theories frequently assume a single representative agent when seeking to model expected returns. For instance, Breeden's (breeden1979intertemporal) consumption based approach adopts a representative agent, allowing aggregate consumption growth to systematically price returns. However, stock returns are considerably more volatile than aggregate consumption growth. This empirical observation is a cornerstone of the equity premium puzzle, see mehra1985equity and breeden2014consumption. When endeavoring to explain the failure of consumption-based asset pricing models, one can relax the homogeneous agent assumption (see constantinides1996asset, chabi2014aggregation and lettau2019capital). Observed differences in stock market participation, resources and/or preferences justify heterogeneous agent asset pricing models. Indeed, campbell1993trading emphasize a focus upon heterogeneous agent asset price model, since it is shareholder consumption that matter for stock returns.
Important recent work by lettau2019capital adopts heterogeneous agents and proposes a capital share risk factor that maps shocks to the assets of high income stock holders' assets and consumption to explain U.S. returns. Capital risks account for limited stock market participation and proxy the concentration of wealth. lettau2019capital present evidence that a capital risk factor explains expected returns, and empirically dominate aggregate consumption growth and the Fama and French (fama1993common) factors. In developing a general equilibrium model with limited stock market participation and inequality, toda2018equity highlight that rising wealth holdings of the richest one percent predict excess stock returns. Their asset pricing model allows for heterogeneity in risk aversion or beliefs. See also theoretical asset pricing models accounting for heterogeneous beliefs under recursive preferences by borovivcka2020survival.
While explaining excess equity returns has been the focus of much academic research, elevated stock return variability has also been considered by homogeneous agent models. Seeking to resolve asset pricing puzzles associated with standard models, bansal2004risks develop a consumption representative agent model with long-run risks based upon the recursive preference utility framework of epstein1989substitution. The asset pricing puzzles highlighted by bansal2004risks include high conditional volatility of market returns and a negative risk premium on consumption volatility.
In this paper we develop a heterogeneous agent stock market model and test its predictions. Going beyond the homogeneous framework developed by bansal2004risks, our theoretical work generalizes the heterogeneous agent assumption of lettau2019capital and allows us to reconsider their empirical evidence, accounting in particular for time-varying risk prices and equity return volatility.Our asset pricing model has multiple economic agents: in particular, high and low income stockholders with different consumption patterns. Using United States wealth distribution data from saez2016wealth, lettau2019capital identify that high and low income stockholders' consumption behaviour responds differently to capital share growth. Given that high income stockholders consume primarily out of their wealth, capital share growth can also explain the elevated consumption variability of the richest cohorts. Therefore, our theory captures heterogeneity in the volatility of stockholder consumption growth, and this drives the relationship between capital share growth and equity returns. The impact of consumption volatility of wealthy stockholders on the whole market is captured by capital share change.
One novelty of our model is that the same risk factor is analysed separately using conditional and unconditional expectations. Inspired by campbell2000explaining, the capital share risk factor in our model is linked to stock returns under both unconditional and conditional expectations. The contrasting impact of the capital share factor on equity returns under conditional and unconditional settings also serves as a potential explanation of the weakness of the consumption-based CAPM under conditional estimation campbell2000explaining. Capital share growth is priced only under unconditional expectations, as found by lettau2019capital, and capital share variability is proposed in our paper to capture long-run market volatility. Therefore, our framework posits that capital risks enter the equity return variance equation under conditional estimation, while they enter the mean equation under unconditional estimation.
Motivated by our theoretical framework in which capital shares impact not only the mean but also the variance of return dynamics, we test the pricing power of capital risks in a more general setting than that of lettau2019capital. To avoid firm effects, we first use bootstrapped cross-sectional regressions to estimate the level and volatility effects of capital share risks in asset prices. The capital share risk price is adopted as a benchmark for pricing power in this case. To then test the conditional equity premium dynamics, we investigate the conditional capital share risk prices using the rolling-window Fama-MacBeth procedure used by lewellen2006conditional and the Bayesian asset price estimation from bianchi2017macroeconomic. Additionally, our paper estimates the impact of capital share growth on return volatility using rolling-window multiplicative GARCH. Finally, we test the capital risk variability as the long-run risk factor in the mean equation for U.S. stock returns using a bootstrap procedure, as we do for testing the capital share factor. This alternative capital variability factor empirically dominates the standard capital share growth factor of lettau2019capital.
The paper is structured as follows: Section 2 reviews standard asset pricing models and the capital share risk factor proposed by lettau2019capital. Section 3 presents a theoretical asset pricing model with recursive preferences and heterogeneous agents, in which consumption volatility operates through capital share risk factors. Section 4 sets out empirical methodologies for estimating overall pricing power, and the level and volatility effects of the capital share factor. Section 5 presents the data and Section 6 presents evidence on the impact of capital share risks. Section 7 proposes and tests the capital share variability factor. Finally, section 8 concludes on the capital share variability and the empirical evidence.
Modern asset pricing models describe the relationship between risk exposure and expected returns. Expected returns equal the sum of the risk free rate and the excess returns of associated risk factors. In capital asset pricing models, the stochastic discount factor (SDF) links the present value and the future cash flow of an asset, and the price of an asset can be computed by the expectation of total future cash flows discounted by that discount factor.
An asset pricing model can be seen as a special case of the following relationship:
where $M_{t+1}$ denotes the discount factor, $r_{t+1}$ denotes asset excess returns and $E_t$ is the conditional expectation given information at time $t$. The form of the SDF relies heavily on the assumptions made by different CAPMs cochrane2009asset. Equation ((ref)) is operationalized by agents' expectation formation and their utility or preferences function(s).
The consumption‐based capital asset pricing model (CCAPM) developed by breeden1979intertemporal states that, with a representative agent assumption, the SDF is based on the marginal rate of substitution over aggregate household consumption. The CCAPM assumes the SDF is equal to the time-discount factor ($\delta$) multiplied by the ratio of the marginal utility of aggregate consumption tomorrow $U'(C_{t+1})$ and today $U'(C_{t})$ as shown in equation ((ref)), where $U$ denotes the utility function of the representative agent, and $C_t$ denotes consumption at time $t$.
With an homogeneous agent and power utility, expected returns can be priced by aggregate consumption growth. Agents are however more reasonably considered to be heterogeneous, due to imperfect risk-sharing, concentrated wealth and limited stock holder participation. And the CCAPM may not perform well empirically, see breeden2014consumption. According to lettau2019capital, for the wealthiest households, relative to the least wealthy, aggregate consumption volatility multiplied by income share is considerably high. Stock market wealth is highly concentrated, since the top 5% of the wealth distribution owns over 70% of stocks. The least wealthy typically own no equity, and their consumption comes almost entirely from labor income. Therefore, aggregate consumption growth also fails to capture redistribution risk.\footnote{Wealth re-distributions between stockholders and workers.} Capital share growth better reflects the consumption of stockholders, while accounting for stockholder heterogeneity and redistributive shocks (see greenwald2014origins and lettau2019capital). The income shares, and therefore consumption patterns, of wealthy capital owners is well represented by the capital share.
lettau2019capital then proposes a linear approximation of the asset price SDF using the capital share factor:
Equation ((ref)) describes the richest household case in a stylised economy where workers are assumed to be absent from risky asset markets. $KS_t$ denotes capital share, $a$ is related to the time-discount factor (i.e. $\alpha = 1 + ln(\delta)$), while model parameters $b_{1}$ and $b_{2}$ are related to the risk aversion of consumers. Evidence presented by lettau2019capital indicates that capital share growth explains U.S. asset prices and restricting aggregate consumption growth to have no effect (i.e. $b_1=0$) is a reasonable assumption. Approximation error is denoted by $\mu$ and explains other factors.
The model proposed by lettau2019capital does not allow for time variation of the capital share parameters. Empirically risk factor loadings however may vary over time, and conditional asset pricing models can be justified theoretically (see jensen1968performance, jagannathan1996conditional and lewellen2006conditional). For instance, the static CCAPM fails to capture the effect of time-varying investment opportunities lettau2001resurrecting. The non-zero unconditional price anomalies do not necessarily indicate non-zero conditional alphas, given time-varying factor loadings that are correlated with the equity premium or market volatility lewellen2006conditional. Our paper generalises the role of capital share by testing both unconditional and conditional approaches.
Capital share can be motivated as an asset pricing risk factor in a stylized economy with heterogeneous agents, see lettau2019capital. In this economy, capital investors own the entire corporate sector, while workers do not participant in the stock market. Capital share is assumed to play a role through its influence on stockholders' consumption. Therefore, in this case, the stochastic discount factor is represented by the utility function of investors of the top wealth distribution.
Our capital share asset pricing model relaxes the Lettau et al.'s (lettau2019capital) assumption that capital share only impacts the top wealth distribution of stockholders to derive a more general case. Here capital share growth is assumed to influence both the high and the low income stockholder groups, since the stock market and the wealth weighted participation rates are not identical (see lettau2019capital). Our model assumes three income groups, which are high income stockholders, low income stockholders and labour workers. Labour workers are assumed to be absent from the stock market as in lettau2019capital. We assume for simplicity a constant relative share of high versus low income stockholders.\footnote{Individual can move between income groups, but the population sizes to move into and out from each income group are the same.} The population weight of the high income stockholders is denoted by $w^H$ and that of the low income stockholders is denoted by $w^L$. We then introduce these high and low income stockholders into a recursive asset pricing model.
Over an infinite horizon, the CCAPM in equation ((ref)) is nested as a special case in asset return models derived from a recursive preference utility framework proposed by epstein1989substitution. This recursive framework permits risk attitudes to be disentangled from the degree of intertemporal substitutability, and addresses the importance of consumption uncertainty in asset pricing epstein1989substitution. Epstein and Zin (epstein1989substitution) assume agents are homogeneous in the market, and our capital share model relaxes this assumption. In our model, the homogeneous assumptions for the recursive framework are satisfied within each stockholder group.
Going beyond the homogeneous agent model of bansal2004risks, the stockholder consumption growth $G_{t+1}$ is underpinned by the particular consumption patterns of different investors.\footnote{In our paper, $G^n_t$ denotes the consumption growth of agent $n$ calculated from $G^n_t=C^n_{t+1}/C^n_t$, where $C^n_t$ is the consumption at time $t$ of agent $n$. } According to lettau2019capital, the consumption growth rate of the high income stockholders is more volatile than that of those who derive income from wages. Intuitively, the top of the wealth distribution has a larger discretionary consumption on luxury goods linked to volatile asset prices, while workers spend a larger proportion on the same essential goods each month. The capital share growth rate is strongly and positively correlated with the consumption growth rate of the high income group, while strongly negative correlated with that of the low income group lettau2019capital. Accordingly, we define $\bar{G_t}$ as the weighted average of consumption growth of labour workers and the growth of the high and low income stockholders:
where $r^p$ is the stock market participation rate, and $G_t^{W}$ is the consumption of labour workers without stocks. The aggregate consumption growth $\bar{G}$ is independent from how the income groups are defined. Our heterogeneous model focuses upon the high ($G^H_t$) and low ($G^L_t$) income stockholder consumption growth. The former departs from the aggregate consumption growth ($\bar{G_t}$) based upon the capital share and excess volatility, as follows:\footnote{The stock market participation rate is about 50% lettau2019capital. We assumes that the low income stockholders have the aggregate consumption growth volatility. The consumption growth volatility of the low income stockholders is, therefore, higher than that of labour workers and lower than that of high income stockholders. Excess volatility of high income stockholder consumption growth is absorbed by labour workers and does not affect $\bar G_t$.}
where the stochastic term $\xi_t\sim N_{i.i.d}(0,\Sigma)$ captures the excess volatility of the consumption growth rate of the high income group compared to the low income group.\footnote{Our model does not make an autocorrelation assumption for $\xi_t$ to avoid possible explosive growth. $\Sigma$ denotes a constant variance for $\xi_t$.} The consumption growth volatility shocks of the high income stockholders are absorbed by the labour workers, and the aggregate consumption growth $\bar{G_t}$ remains independent from such shocks according to equation ((ref)). The $\xi_t$ term therefore defines the variance of the consumption distribution of the economy. Based on the data correlations identified by lettau2019capital, we formulate $f^H_{KS,t}$ as a monotonic increasing function, and $f^L_{KS,t}$ as a monotonic decreasing function, of the capital share growth rate. The volatility of $G^H_t$ is bounded due to limited resources and productivity growth. If the volatility of $G^L_t$ equals zero, the volatility of aggregate consumption growth $\bar{G_t}$ must also be zero when $f^L_{KS,t}$ is non-zero.
Since labour workers do not participate in the stock market, our model only focuses on the partial equilibrium of stockholders. The average stockholder consumption growth $G^S_t$ can be approximated by the weighted average of high and low income stockholder groups:
Substituting disaggregate consumption in equations ((ref)) and ((ref)) into stockholder consumption growth in equation ((ref)):
Given evidence from saez2016wealth, lettau2019capital assume that the stockholder consumption equals to the product of the aggregate consumption and capital share. Inspired by lettau2019capital, we assume that high (low) income stockholders' consumption growth is positively (negatively) related to the capital share growth, such that $f^H_{KS,t}=1+F_{KS,t}$ and $f^L_{KS,t}=1-F_{KS,t}$.\footnote{We define capital share growth as capital share factor $F_{KS,t}$ to be consistent with the notation used by lettau2019capital.}\footnote{The empirical evidence presented by lettau2019capital points out that bottom 90% wealth distribution is strongly negatively correlated with capital share growth. Using saez2016wealth data, lettau2019capital Table 2 identifies that capital share has a negative and statistically significant resource impact upon low income U.S. stock owners (OLS coefficient = -1.27, t-statistic = - 6.82). In comparison capital share has a positive resources impact upon high income stock owners which is approximately equal and opposite (OLS coefficient = 1.20, t-statistic = 7.34).} Therefore, the consumption of each stockholder group contains a persistent component as in bansal2004risks.\footnote{We assume two stockholder groups and each group satisfies assumptions of bansal2004risks independently.}
Thus, stockholder consumption growth in equation ((ref)) can be written as:\footnote{In equation ((ref)), $w^H \in [0, 1]$ by definition. About 95% of $F_{KS}$ falls in the range between -4% and 4% (within two standard deviations) as shown by the empirical results in Table (ref). The bounded volatility of $C^H_t$ also implies a bounded excess volatility $\xi_t$. Therefore, the $\bar{G_t}[w^H(1+ F_{KS,t})]\xi_t$ term is bounded and the stockholder consumption does not witness an explosive growth in this model.}
Any percentage at the top can be used to illustrate how the concentration of wealth affects the intensive margin of the stock market lettau2019capital.\footnote{The wealth-weighted participation rate is lower than the aggregate participation rate, regardless which quantile of wealth distribution is selected as a benchmark.} We assume that stockholder wealth and population wealth in the economy are drawn from the same distribution, to solve the population sizes of each stockholder income group. Our model therefore approximates the high income stockholder population weight $w^H$ by the stock market participation rate ($w^H\approx r^p$), and low income stockholder population weight $w^L$ by labour worker population weight in the economy ($w^H\approx1-r^p$). Given the average stock market participation rate is close to 50% over time lettau2001resurrecting, we can simply assume $w^H = w^L$ for simplicity of the calculation.
Rewriting aggregate consumption growth in equation ((ref)), given a relatively small $F_{KS,t}$ and the fact that $w^H-w^L= 0$, the positive impact of capital share growth on the high income group and the negative impact of capital share growth cancel one another out. At time $t$, taking conditional expectations of ((ref)) and, therefore, setting the stochastic term to zero, we have the expected stockholder consumption growth as:
Notice that in equation ((ref)), when the dynamics of low and high income stockholder consumption growth $f^L_{KS,t}$ and $f^H_{KS,t}$ have different functional forms, their mutual effect on the level of expected stockholder consumption growth can be tested can be tested by having the capital share factor in the excess return equation. In contrast, equation ((ref)) contains the $\bar{G_t}[w^H(1+ F_{KS,t})]\xi_t$ term, which indicates the stockholder consumption growth volatility operates through capital share growth. The magnitude of aggregate stockholder consumption growth volatility also associates with the population size of the high income group $w^H$, and the excess volatility $\xi_t$. Equations ((ref)) and ((ref)) are consistent with the empirical findings of lettau2019capital in that the consumption growth of the top wealth distribution is more volatile than that of the rest of the population, but the expected consumption growth rate of the richest individuals is at around the same level as that of the whole economy.
To model stockholder consumption growth, we further assume the aggregate consumption growth rate $g_t$ contains the persistent expected growth rate component $x_t$ proved by bansal2004risks to define the centre of the consumption growth distribution for the economy. We define the aggregate consumption growth rate $g_{t+1}=log\bar{G}_{t+1}=\mu+x_{t}+\sigma\eta_{t+1}$, and $x_{t}$ is the predictable term following bansal2004risks.\footnote{$\bar{G}_{t+1}=1+g_{t+1}$} According to the dynamics of stockholder consumption growth described by equation ((ref)) and the aggregate consumption growth $\bar{G}$, the stockholder consumption growth rate $g_{t+1}$ can be written as the following function:\footnote{The $w^H(1+F_{KS,t+1})\xi_{t+1}$ term is relatively small given the range of $w^H$ and $F_{KS,t+1}$, and the definition of $\xi_{t+1}$. Therefore, we use a Taylor approximation here.}
Therefore, the time-varying volatility of stockholder consumption growth rate $\sigma_{t+1}$ is defined as $w^H(1+F_{KS,t+1})\xi_{t+1}+\sigma\eta_{t+1}$.
The intuition behind the consumption growth rate in equation ((ref)) is as follows. According to saez2016wealth, wealth is highly concentrated at the top of the wealth distribution. For example, the richest 5% of the population owns more than half of the aggregate wealth in the economy. Capital share change does not affect the conditional expectation of stock owner consumption growth, see equation ((ref)). However, when the capital share growth is positive, the wealth increase of individuals in the high income stockholders is higher than in the low income stockholders, as addressed by gabaix2016dynamics.\footnote{This statement does not conflict with assumed high and low income stockholder consumption growth in equations ((ref)) and ((ref)): the aggregate consumption $\bar{G_t}$ is also increased when capital share increases, as they are positively correlated.} The consumption of the top of the wealth distribution will have a larger impact on aggregate consumption growth when capital share increases. Therefore, stockholder consumption growth volatility is positively correlated with capital share growth.
Also relevant to our model is the consumption volatility risk (CVR) factor derived by boguth2013consumption. In the theoretical motivation of their volatility risk factor, the consumption growth is assumed to switch between high and low volatility states. In our model, instead of assuming a Markov switching process based upon changing beliefs, volatility is explicitly modeled using a capital share factor. In boguth2013consumption, consumption growth volatility $\sigma_t$ is assumed to be a time varying function of high ($\sigma^H$) and low ($\sigma^L$) volatility states:
In our model, according to the innovation of consumption growth in equation ((ref)), the volatility of consumption growth is assumed to be:
The excess volatility of the high state to the low state ($\sigma^H-\sigma^L$) and $\sigma^L$ in equation ((ref)) are explained in our model as excess volatility of the high to the low income stockholders ($\xi_t$) and $\sigma\eta_{t+1}$ in equation ((ref)) respectively. Therefore, in equation ((ref)), the counterpart of belief $b_t$ is $w^H(1+F_{KS,t+1})$, see equation ((ref)).
In line with the assumption by lettau2019capital, the labour workers do not influence equity prices and, consequently, they are independent from the stock market and their participation is not modeled. Equity returns are linear functions of the equity premium according to the Capital Asset Pricing Model cochrane2009asset. To solve the relationship between equity returns and capital share growth, our paper extends the model of bansal2004risks to derive the equity premium explicitly. The system stated in our paper is a hybrid system of the constant volatility case (Case I) and the time-varying volatility case (Case II) of bansal2004risks. The volatility of the log stockholder consumption growth contains both a constant element $\sigma$ and a time-varying part $w^H(1+F_{KS,t+1})\xi_{t+1}$. The dividend growth volatility is correlated with consumption growth volatility, as suggested by bansal2004risks. Therefore, $\sigma_{d,t+1}$ is assumed to be partially correlated with both $F_{KS,t+1}\xi_{t+1}$ and $\sigma$ in our model.\footnote{The specification of $\sigma_{d,t+1}$ also relaxes the setting by bansal2004risks Case II, in which $g_{t+1}$ and $g_{d,t+1}$ are cointergated, to be consistent with empirical literature campbell1999force. }
In our model, the stock market is driven by a persistent growth component ($x_{t+1}$), capital share and stochastic high income volatility shocks ($\xi_{t+1}$) based upon equation ((ref)) as follows:
where the $g_{d,t+1}$ is the log dividend growth rate, and $\rho$ is the persistence of the expected growth rate process. Parameters $\mu$ and $\mu_d$ are the constant component of $g_{t+1}$ and $g_{d,t+1}$, respectively. $\phi_e>1$ and $\phi_d>1$ allow for parameter calibration. The parameter $\phi$ can be interpreted as the leverage ratio on expected consumption growth, see bansal2004risks and abel1999risk. The stochastic error terms $e_{t+1}$, $u_{t+1}$, and $\eta_{t+1}$ are independent from each other bansal2004risks. $\sigma$ is a constant which captures the volatility of $x_{t+1}$ and $g_{t+1}$.\footnote{bansal2004risks Case II adds time varying volatility and fluctuating economic uncertainty into their model through a general error term. Our model does not assume a stochastic innovation of $\sigma$ in order to isolate the volatility effect generalized by the introduction of the capital share factor.} The innovation of $g_{d,t+1}$ which is found to be more volatile than $g_{t+1}$ campbell1999asset is tackled by $\phi_d$. Our model therefore formalises uncertainty in terms of the impact of high income consumption variability, rather than a generic uncertainty as set out by bansal2004risks.
Based upon the recursive preference utility function, the asset pricing restrictions for gross return $R_{i,t+1}$ satisfy
where $\theta=(1-\gamma)/(1-\frac{1}{\psi})$. In equation ((ref)), $G_{t+1}$ denotes the aggregate consumption growth rate, and $R_{a,t+1}$ is the gross return on an asset that generates dividends that cover the aggregate stockholder consumption. $0<\delta<1$ is the time discount factor, $\gamma\geq0$ is the risk-aversion parameter, and $\psi\geq0$ is the intertemporal elasticity of substitution (IES).
Given the asset pricing constraint in equation ((ref)), the intertemporal marginal rate of substitution (IMRS):
where $g_{t+1}$ and $r_{a,t+1}$ are the natural logarithm of $G_{t+1}$ and $R_{a,t+1}$, respectively.
We also adopt the standard approximation proposed by campbell1988stock to derive the functional form of the equity premium. The innovation of log gross consumption $r_{a,t+1}$ and log market return $r_{m,t+1}$ are assumed to follow:
where $z_t$ is the log price-consumption ratio ($log(\frac{P_t}{C_t})$) and $z_{m,t}$ is the log price-dividend ratio ($log(\frac{P_t}{D_t})$).\footnote{$D_t$ denotes the dividend.} Therefore, $z_t$ and $z_{m,t}$ are assumed to satisfy $z_t=A_0+A_1x_t+A_{2,t}\xi_t$ and $z_{m,t}=A_{m,0}+A_{m,1}x_t+A_{2,m,t}\xi_t$.\footnote{$A_0$ and $A_{m,0}$ are constants; $A_1$ and $A_{m,1}$ are parameters of the persistent consumption growth component $x_t$; $A_{2,t}$ and $A_{2,m,t}$ are parameters of the excess volatility $\xi_t$} The relevant state variables in solving for the equilibrium are $x_t$ and $\xi_t$. We modify the functional form of the log price-consumption and log price-dividend ratios assumed by bansal2004risks to include the time-varying part of stockholder consumption growth volatility.\footnote{ See bansal2004risks Case II.}
In addition, we need the dynamics of capital share growth to solve the equity premium. According to lettau2019capital, the capital share growth follows an AR(1) process:\footnote{The constant is not significant according to our AR(1) estimation.}
where $e^{KS}_{t+1}$ captures unexpected shocks in capital share growth.
Since the log consumption growth $g_{t}$, log dividends growth $g_{d,t}$ , and the capital share growth are exogenous processes in our system, the functional form of the innovation of consumption return, the pricing kernel, and equity returns in this economy can be derived explicitly using equations ((ref))- ((ref)).\footnote{Detailed proofs are provided in the Appendix.}
We first solve the parameters of the persistent consumption growth $x_t$ and excess volatility $\xi_t$ on price-consumption and price-dividend ratios, which track expected risk prices campbell2000explaining. In our model, the resulting $A_{1}$ and $A_{1,m}$ are identical to bansal2004risks. The sensitivity of the price-consumption (and price-dividend) ratio to the excess volatility $\xi_{t}$ is constant over time. $A_{2,t}$ (and $A_{2,m,t}$) are constants when we hold $w^H$, $\rho_{KS}$ and $F_{KS,t}$ constant:\footnote{$A_{2,t}$ and $A_{2,m,t}$ are derived in Appendix.}
According to the parameters of excess volatility in equations ((ref)) and ((ref)), given the stochastic nature of $\xi_{t}$, the capital share growth does not have an impact on the magnitude but affects the uncertainty of the price-consumption and price-dividend ratios. Therefore, due to constant excess volatility between two adjacent periods, capital share growth does not shift the expected rate of return under short-run (conditional) expectations. However, in the long-run, volatility shocks fail to feature in expectations and the increased uncertainty of returns generate redistribution risks between high and low income stockholders.
We now set out equity returns conditionally and unconditionally. The difference between these two settings is due to the difference between conditional and unconditional expectations of $\xi_{t+1}$. Conditioning on information at $t$, $E_t(\xi_{t+1})=\xi_t$ due to smoothed consumption, while the unconditional expectation of $\xi_{t+1}$ is $0$.
We now derive equity premiums under conditional and unconditional expectations, respectively. \footnote{Full details are in the Appendix.}
Conditional on information at time $t$, all shocks alter agents' expectations. The conditional innovation of the pricing kernel $m_{t+1}$ is:
The conditional innovation of market return $r_{m,t+1}$ is:
In equations ((ref)) and ((ref)), $\lambda_{m,e}$, $\lambda_{\eta}$ and $\lambda_{e}$ are constants, while $\lambda_{m,\xi,t+1}$ and $\lambda_{\xi,t+1}$ are functions of $e^{KS}_{t+1}$.\footnote{See the Appendix.} Therefore, the conditional pricing kernel innovation in equation ((ref)) is only correlated to unexpected capital share growth $e^{KS}_t$, but the conditional market return innovation is correlated with capital share growth through $\sigma_{d,t+1}$.
Following bansal2004risks, the continuous equity premium in the presence of time-varying economic uncertainty is
At time $t$, the conditional expectation $E_t(\xi_{t+1})=\xi_t$, so the effect of predictable capital share growth is omitted in equation ((ref)). As shown by equation ((ref)), the conditional equity premium is constant and has one source of systematic risk that relates to fluctuations in expected consumption growth $\sigma^2$. However, the capital share factor enters the innovation of market return in equation ((ref)). Hence, the excess volatility of the high income group, through capital share growth, is linked to the variability of equity returns. We estimate the conditional equity premium in equation ((ref)) using the short-window regression as suggested by lewellen2006conditional and the Bayesian approach proposed by bianchi2017macroeconomic.
Under unconditional expectations, we do not allow unexpected shocks of parameters. The unconditional innovation of the pricing kernel is as follows:
The unconditional innovation of market return is:
Detailed functional forms of the parameters in equations ((ref)) and ((ref)) are in the Appendix. Using equations ((ref)) and ((ref)), the unconditional equity premium is calculated as:
where $E[\lambda^u_{r,\xi,t+1}\lambda^u_{\xi,t+1}-0.5(\lambda^u_{r,\xi,t+1})^2]$ is positively correlated with $E(F^2_{KS,t+1})$.\footnote{See the Appendix.} Under unconditional expectations, the equity premium is a function of fluctuations in expected consumption growth $\sigma^2$ and capital share variability $E(F^2_{KS,t+1})$.
The intuition behind $E(F^2_{KS,t+1})$ as an unconditional risk factor is as follows. The variance of consumption volatility, captured by $e^{KS}_t$ in our model, is very small and gets magnified under unconditional expectations because of the long-lasting nature of the volatility shock bansal2004risks. Intuitively, the ratio of the conditional risk premium to the conditional volatility of the market portfolio fluctuates with consumption volatility bansal2004risks. The maximal Sharpe ratio approximated by volatility of the pricing kernel innovation also varies with consumption volatility. In our model, consumption volatility operates through capital share growth. Therefore, risk prices will rise as economic uncertainty represented by capital share variability rises. Conditional on both of the two stockholder groups surviving in the long-run, the magnitude of IES, $\psi$, is justified by the survival analysis of borovivcka2020survival which studies a two-agent model from the perspective of beliefs, where under different belief styles, IES is found to be greater than 1 to ensure the long-run coexistence of two heterogeneous agents. When $\psi>1$ holds, the negative coefficient of capital share growth in parameters $A_{2,t}$ in equation ((ref)) and $A_{2,m,t}$ in equation ((ref)) ensures that capital share growth is negatively correlated with the uncertainty in the price-consumption and the price-dividend ratio. In response to lower expected expected rates of return uncertainty, asset demand rises to generate positive risk price of the capital share variability in our model. The utility study of colacito2018volatility also highlights that increased macroeconomic volatility increases the stochastic discount factor under the recursive utility framework, thus raises expected returns and generates a positive volatility risk price. We estimate the unconditional equity premium in equation ((ref)) using a Fama-MacBeth approach suggested by lettau2019capital and capital share variability as a risk factor. Given our theoretical framework, the capital share variability risk price is expected to be positive.
The model specification of Boguth and Kuehn's (boguth2013consumption) consumption volatility risk (CVR) factor is a potential explanation of the nonlinear relationship between the equity premium and capital share growth in equation ((ref)) due to the following reasons. Changes in beliefs about consumption growth volatility are found important in explaining unconditional equity returns by boguth2013consumption, which indicates that the assumption of two volatility states is reasonable. Although the functional forms of equations ((ref)) and ((ref)) are similar, the change of volatility in equation ((ref)) is a smoothed process. Thus, our model can be alternatively explained by assuming infinite states of consumption growth volatility. At each time $t$, consumption growth volatility has only two latent states, but $\xi_t$ is an unknown stochastic variable and, hence, this is a setup that is consistent with the quadratic relationship between equity returns and capital share growth in our model.
To conclude, our theoretical model indicates that: under conditional expectations, the capital share factor captures the impact of consumption volatility from the high income group onto equity returns; while under unconditional expectations, capital share variability serves as a risk factor that captures long-run market volatility.
We employ both unconditional and conditional estimation approaches to examine the empirical importance of capital share risks and to test the predictions of our model. The unconditional estimation is a bootstrapped fama1973risk procedure, which corrects both cross-sectional correlations and the firm effect of equity returns. Conditional estimations include Lewellen and Nagel's (lewellen2006conditional) rolling-window regressions, the Bayesian time-varying beta with stochastic volatility (B-TVB-SV) estimation from bianchi2017macroeconomic, and a rolling-window multiplicative GARCH. The rolling-window and the B-TVB-SV approaches assume that a risk factor enters the mean equation of the stochastic discount factor as shown in equation ((ref)) to test the explanatory power of capital share growth on the level of equity returns. The rolling-window and the B-TVB-SV estimates are expected to generate statistically insignificant capital share factor loadings according to the conditional stochastic discount factor in equation ((ref)) and the conditional equity premium in equation ((ref)). The rolling-window multiplicative GARCH assumes that the capital share factor influences market volatility only, and is used to test the conditional innovation of market return in equation ((ref)).
The risk price measures the risk-reward relationship between factors and returns. Fama and MacBeth's (FMB) two-step procedure is widely used in estimating risk prices of factors and in testing asset pricing models when risk factors enter the mean equation of equity returns. In practice, the static or static based F-MB approach estimates both the mean effect and the variance effect of risk factors together. When a risk factor enters the mean equation of the true data dynamics, which is consistent with the assumption of the static F-MB approach, the risk price estimate of this factor will be significant. However, when a risk factor enters the variance equation of the true dynamics, the ordinary least squares (OLS) regression in the first step of the static F-MB approach will be biased due to heteroskedasticity problems. The risk price estimate of this factor will also be significant because of the change in the width of the factor loading distribution.
Our paper employs the F-MB bootstrap to test the pricing power of the capital share factor and estimate the unconditional equity premium of the capital share variability factor in equation ((ref)). The F-MB bootstrap is based upon the static F-MB procedure, and can be used to correct both cross-sectional correlations and firm effects lettau2019capital while it constraints the factor loadings to be constant over time as the static F-MB estimators.
When testing the importance of the capital share factor for U.S. asset returns, lettau2019capital adopt the non-overlapping block residual bootstrap for both steps of the F-MB procedure. Although it is argued that utilizing the overlapping bootstrap is a more robust method, andrews2004block compares overlapping and non-overlapping block bootstraps, and reaches the conclusion that although the former is often favored in applications, the latter generates similar numerical results. Our paper uses the non-overlapping bootstrap and the capital share dynamics assumed by lettau2019capital for the capital share factor.\footnote{In the first step time series regression, capital share growth is assumed to follow an AR(1) process to factor in the serial correlation.}
The optimal length of the bootstrap block should increase as the sample size increases to maintain the consistency of moments and distribution functions horowitz1997bootstrap. In the first step, our sample spans 526 months.\footnote{The bootstrap only estimate January 1974 to August 2018 which is consistent with sample span for B-TVB-SV approach.}.Therefore, the optimal block-length is $536^{(\frac{1}{5})}\approx 4$ following hall1995blocking. The second step involves 25 portfolio returns, and the optimal block-length is identical to lettau2019capital.
The F-MB bootstrap tackles both the cross-sectional correlation and serial correlation in estimation, but it can only serve as a rough check of pricing power of the capital share factor and cannot infer the functional form of the true dynamics of equity returns. When assuming that capital share growth is a risk factor in the mean equation, the time variation of factor loadings are not captured by simply estimating one time series regression in the first step of F-MB bootstrap approach, hence the risk prices are estimated unconditionally. Additionally, the non-linearity in the dynamics of equity returns is omitted, leading to biased F-MB bootstrap estimates of the capital share risk price.
According to the theoretical justification in equation ((ref)), the capital share factor explains the variance of equity returns under conditional expectations. Also, the true unconditional risk factor should be the capital share variability according to equation ((ref)). In our paper, the monthly F-MB bootstrap estimates are benchmarks of pricing power. The monthly risk price of the capital share factor is expected to be significant due to its multicollinearity with capital share variability $E(F^2_{KS,t})$ in a single capital share factor model.\footnote{$F_{KS,t}$ and $E(F_{KS,t})$ all contains the mean of $F_{KS,t}$ plus terms containing deviation from the mean. Therefore, $F_{KS}$ and $E(F_{KS,t})$ are correlated. } In a two factor model including both the capital share factor and capital share variability, the latter can be expected to dominate according to the unconditional equity premium in equation ((ref)).
As shown by equation ((ref)), the mean equation of the equity premium is independent from the capital share factor. We adopt rolling-window regressions to estimate factor loadings in a conditional manner as suggested by lewellen2006conditional. Our paper estimates the F-MB first step regression following lewellen2006conditional. The window length selected in the first step F-MB is 12 months. The second step is identical to the original cross-sectional regression of the F-MB approach. The results of the rolling-window regression serves as a benchmark for the true DGP of factor loadings under the assumption of a modest level of temporal variation.
A short window for estimation is adopted for the following reasons. Within each window, the regression using short horizon data can be viewed as an estimation that is robust to firm effects, especially since the autocorrelation of stock returns is weaker over a relatively short regression window fama1988permanent. Another function of the rolling-window regression is to serve as a volatility estimator. Volatility is constant within each window, but varies across windows.
The limitations of the rolling-window approach are widely known. The rolling-window F-MB is an appropriate approximation for time-varying factor loadings, only conditional on the assumption that there are no structural breaks present within each window. The time variations are still not fully captured due to the ad-hoc window length selection: robustness of the rolling-window approach is diminished when extreme outliers are present in the sample. Therefore, the assumption of rolling-window F-MB is still too strong and vulnerable. Further, the rolling-window F-MB is subject to a common problem of 2-step estimations, which is that the second step estimation is dependant on the first step results. This approach cannot pass the variability of factor loadings into the second step estimation and, therefore, is insufficient to ensure unbiased estimation of risk prices. The rolling-window approach also views the factor loading as a constant at each time point, causing information carried by the change of factor loading volatilities to be retained within the first step estimation. The time variation of risk prices are thus inflated compared to the true underlying DGP by the rolling window F-MB approach when stochastic volatility is present in factor loadings.
As shown by the innovation of market premium in equation ((ref)), the loading of the capital share factor is expected to be centered at zero, and a strong volatility clustering is expected to be present under rolling-window estimation. Due to heteroskedasticity and the model misspecification problem highlighted by jagannathan1996conditional, risk price estimates should be insignificant but vary dramatically over time.\footnote{An insignificant risk factor in the true equity dynamic might be significant under F-MB estimations.}
To tackle problems in the F-MB procedures, bianchi2017macroeconomic proposes a Bayesian estimation approach, namely the Bayesian time-varying beta with stochastic volatility (B-TVB-SV) model, to consider the SDF and non-arbitrage restriction jointly. Compared to the rolling-window F-MB, this method captures the time variation and variability of factor loadings while maintaining robustness to firm effects.
The B-TVB-SV model for asset return $r_{i,t}$ as a function of risk factor $F_{j,t}$ is:
Factor risk prices $\lambda_{j,t}$ are estimated by:
The B-TVB-SV framework assumes the time-varying betas $\beta_{ij,t}$ and residuals in equation ((ref)) take the following forms:
where $\kappa_{ij,t}$ is the structural break of factor loading $\beta_{ij,t}$, and $\kappa_{iv,t}$ is the structural break of idiosyncratic variance $ln(\sigma^2_{i,t})$. The stochastic terms $\eta_{ij,t}$ and $\upsilon_{i,t}$ follow normal distributions with zero mean and variances $q^2_{ij}$ and $q^2_{iv}$, respectively. A $\kappa_{ij,t}$ equal to one indicates that structural breaks are present in the factor loadings, and $\kappa_{iv,t}$ equal to one indicates that structural breaks are present in the idiosyncratic variance. The advantage of including structural breaks is that the model captures discrete movements of the factor loadings. In equation ((ref)), the innovation of factor loading maintains the random walk properties to retain the shrinkage power of the selected prior to the largest extent. Therefore, the B-TVB-SV approach tackles factor selection automatically.\footnote{Weak priors are used for the distributions of $\beta_{ij,t}$ and $ln(\sigma^2_{i,t})$. Evidence indicates when the number of variable is small (K=5), flat prior works quite well with the sparse specification and performs modest with the dense specification huber2020inducing. The weak prior adopted by V-TVB-SV approach also has shrinkage effects. } Other detailed break and risk price prior specifications and sampling approaches are discussed in the Appendix.
As shown in equations ((ref)) and ((ref)), the model specification of the B-TVB-SV estimation allows volatility change to have structural breaks as well as autocorrelations, incorporating variance effects of the risk factors that are assumed to enter the mean equation. Therefore, the B-TVB-SV approach is a robustness check for the true data dynamics of equity returns: with the model specification of the B-TVB-SV, the risk factor will generate a significant factor loading and risk price estimates if the risk factor enters the mean equation of equity returns. Given that the model misspecification problem is corrected by the B-TVB-SV, the distribution of capital share risk price should be centered at zero at each time $t$ as indicated by equation ((ref)), and the variance of this distribution should change over time, as in equation ((ref)).
As shown by the conditional market return innovation in equation ((ref)), the capital share should be estimated in the variance equation instead of in the mean equation conditionally. Our paper employs a rolling-window multiplicative GARCH approach to estimate the true volatility effect of the capital share factor on equity returns directly. Within each regression window, the asset pricing model estimated by the multiplicative GARCH is as follows:
where $Var(\epsilon_{i,t})=\sigma^2_{i,t}$, and $\sigma^2_{i,t}$ is consistent with the form in equation ((ref)) below. Therefore, the conditional variance is assumed to be correlated with the capital share growth rate.
To test our theoretical predictions in the conditional expectation case, our paper employs the following most general form for the variance equation:
The functional form in equation ((ref)) is motivated by the market return innovation. As indicated by equation ((ref)), the capital share factor is an $O(n^2)$ addend in the variance equation $\sigma^2_{i,t}$. This paper adopts the conditional variance form proposed by judge1988introduction to test the variance equation ((ref)). The capital share factor enters the variance specification as multiplicative heteroskedasticity. Due to the constraint $\sigma^2\geq0$, equation ((ref)) is rewritten as equation ((ref)) for the sake of estimation.
A 60-month window length is selected by this paper for the rolling-window multiplicative GARCH due to the limitation of maximum likelihood convergence. According to the new data dynamics in equations ((ref)) and ((ref)), the coefficient of $log(F^2_{KS,t})$ in the variance equation ((ref)) is expected to be significant over time if equations ((ref)) and ((ref)) hold.
lettau2019capital use quarterly capital share and quarterly portfolio returns converted from monthly data to test capital share growth. In our paper, instead of modifying monthly returns in a relatively ad-hoc manner, we interpolate the capital share using a reasonable indicator to reduce information loss.
Measurement error leads to biased estimation of CAPMs lettau2019capital.\footnote{During the data collection process, the filtering approach introduces measurement error problem.} Long-term capital share growth is adopted to partial out the measurement error effect. In the test of the capital share factor, lettau2019capital compares 1,4,8,12 and 16-quarter capital share growth to tackle measurement error problems. The 4-quarter capital share growth is found to have higher pricing power. \smallbreak Capital share is calculated as $1-Labour~share$. Labour share data is the nonfarm sector labor share, which is identical to that used by lettau2019capital and gomme2004measuring. Data for constructing capital share from FRED, the monthly capital share is obtained by the Chow-lin interpolation.\footnote{See the Appendix.} \smallbreak The original capital share factor (quarterly) constructed by lettau2019capital is:
In equation ((ref)) $F_{KS}$ can be decomposed as the capital share growth rate plus a constant 1, indicating that the factor is partially correlated with the intercept. Therefore, the estimated capital share factor loading is higher due to the partial effect taken from the constant. Additionally, the estimated distribution of capital share factor loading tends to be wider due to a higher estimated variance. Finally, from the perspective of the B-TVB-SV, the break probabilities of the capital share factor is not easily identified if the factor is correlated with the constant. To avoid these problems, we use a 12-month capital share growth rate as a risk factor and test its pricing power. The monthly capital share factor tested in this paper is constructed as:
According to the unconditional expectation of equity returns in equation ((ref)), capital share variability ($E(F^2_{KS})$) enters the unconditional mean equation of equity returns. The $E(F^2_{KS})$ risk factor is constructed based upon the AR(1) innovation process of $F_{KS}$ as in lettau2019capital:
where $e^{KS}_{t+1}$ captures unexpected shocks in capital share growth. The magnitude of the estimate of $\rho^{KS}$ is 0.947, which is statistically significant at 5% level. We obtain the capital share variability factor using the capital share factor constructed by lettau2019capital to avoid measurement error problems.
The innovations of capital share growth and variability are plotted in Figure (ref), and the descriptive statistics of the capital share factor and the capital share variability factor are reported in Table (ref) in the Appendix.
In our paper, the capital share factor and the variability are tested on different groups of portfolio returns. The portfolio groups we test include 25 size/BM, 10 long-term reverse (REV), 25 size/INV, and 25 size/OP sorted portfolio returns. The descriptive statistics of benchmark portfolio returns are reported in Appendix. For the multiplicative GARCH estimation, this paper takes cross-sectional averages of size/BM, REV, size/INV, and size/OP sorted portfolio returns respectively to mimic different market portfolios. All portfolio data are monthly data from the Kenneth R. French Data Library. The time span is January 1964 to August 2018.
To focus on testing the pricing performance of the capital share factor, our paper estimates a parsimonious capital share factor model which only contains a constant and the capital share factor. A preview of equity portfolios is shown in Figure (ref), which plots the monthly average returns on the y-axis and the portfolio capital share betas on the x-axis. Due to the higher variation in monthly data, the $R^2$ estimates are generally lower for each portfolio class compared to the quarterly data estimates by lettau2019capital. In addition, the $R^2$ estimated by REV sorted portfolios is 0.26 in Figure (ref). All other $R^2$ estimated from monthly data deviate modestly from their quarterly counterparts. According to the distribution of points in the scatter plots of Figure (ref), the model fit is high and the capital share factor has substantial explanatory power for expected returns. However, the regression lines for the portfolios deviate from 1, which indicates a potential presence of heteroskedasticity or non-linearity.
The F-MB bootstrap we use is identical to that in lettau2019capital. We therefore carry out 10000 simulations for the bootstrap process. Table (ref) reports the risk prices estimated by the capital share factor model. In this table, all of the lower bootstrap interval bounds are above zero for capital share risk prices ($F_{KS}$), indicating the risk price estimates are all statistically significant at least at the 5% level. For the bootstrap interval of $R^2$ estimates, the lower bound of $R^2$ for REV portfolios is 0.000, while for other portfolios are all above 0.300. Therefore, for REV portfolios, the low $R^2$ in panel B explains the insignificant capital share premium in panel A: instead of low level correlation, the high variation of correlation between portfolio returns and capital share diminishes the pricing power.
To conclude, results derived by the parsimonious unconditional capital share factor model using monthly data are consistent with the results derived by lettau2019capital using quarterly data. The capital share risk prices are significant and positive for all equity characteristic portfolios, indicating that the capital share factor has strong pricing power. Due to different return dynamics from quarterly data, monthly returns generate lower or insignificant $\bar{R}^2$ estimates for all equity portfolios. Therefore, the cross-sectional results of the capital share risk price might vary over time dramatically, and the the increase in the frequency of the data also increases the probabilities of outliers and the variance of risk price estimates. However, the diminishing pricing power of the capital share factor in higher frequency data also indicates that this factor might be correlated with the volatility of equity returns or a potential nonlinearities in the equity return DGP.
In this section, we return to estimates of the factor loadings using a rolling-window regression in the first step of the F-MB procedure. Risk prices are estimated in the same manner as the static F-MB but within each window. Table (ref) reports the rolling-window estimates of the parsimonious capital share factor model. As shown in this table, the capital share risk prices are insignificant for most equity portfolios, and all signs of risk prices are negative. The negative and insignificant risk prices show that the cross-sectional results in the second step deviate dramatically from static results (positive and significant) when estimating the first step using shorter regression windows.
Figure (ref) plots the 12-month rolling-window estimated factor loadings of the parsimonious capital share model, and the portfolio returns estimated are size/BM sorted portfolios. As the figure shows, the factor loadings have small jumps in levels but big structural breaks in volatilities under conditional estimation. The overall level of factor loadings is centered at zero. Figure (ref) also shows a strong volatility clustering pattern in factor loadings, which further enhances the possibility that the SDF in equation ((ref)) and the factor model estimated might be misspecified, in the sense that the capital share factor does not enter the mean equation if we account for the time evolution of risk prices.
Figure (ref) plots the capital share risk prices estimated by the single factor model. This figure shows that, the time variation of risk prices is very high across the sample, and the level of risk prices witnesses frequent structural breaks. In the first step of the rolling-window F-MB estimation, the factor loadings only capture the effects caused by level changes and not the effects caused by volatility changes. In the second step estimation, the factor loadings at each time are treated as a constant, leading to a more volatile risk price series over the time when volatility varies across windows.
Overall, the rolling-window F-MB estimates are consistent with the theoretical model in equations ((ref)) and equation ((ref)) in that the capital share factor loadings are centered at zero with strong volatility clustering. However, this analysis cannot rule out the potential impact of large outliers on risk price estimates due to the very short window length used. The results derived by the rolling-window F-MB procedure support accounting for structural breaks and stochastic volatility for further robustness.
The Bayesian time-varying beta with stochastic volatility (B-TVB-SV) approach by bianchi2017macroeconomic tackles the volatility clustering of the capital share factor loadings found by the rolling-window F-MB approach.\footnote{The prior specification of factor loading allows volatility clustering and frequent structural breaks.} the B-TVB-SV risk price estimates are more robust to outliers than those from the rolling-window F-MB estimation.
The B-TVB-SV uses 2000 burn-ins and 10000 iterations of the Markov chain Monte Carlo (MCMC) as in bianchi2017macroeconomic, with a parsimonious capital share factor model. Following bianchi2017macroeconomic, to robustify structural break estimates, this paper demeans all risk factors within both the training and the estimation samples to cancel out all potential bias caused by multicollinearity between the constant and the risk factors. The demeaned factors will not affect the results estimated by the B-TVB-SV since all level movements and moment conditions are retained in the sample. All Bayesian estimates passed the Geweke (geweke1991evaluating) convergence diagnostic.
As specified by the B-TVB-SV model, $\kappa_{ij,t}$ is a binary variable that equals 0 or 1. Therefore, the estimated time-average break probabilities can be viewed as a structural break test (structural breaks exist when the break probability estimates are non-zero). Figure (ref) plots the time-average break probabilities calculated by averaging all estimated $\kappa_{ij,t}$ in equation ((ref)). To save space, this paper does not plot the time-average break probability for each portfolio. The average break probabilities of capital share are around 0.427 among the four equity portfolio classes. According to the time evolution of factor loadings stated in equation ((ref)), and due to high expected value of $\kappa_{ij,t}$, $\beta_{ij,t}$ follows a jump process with frequent structural breaks over time. This finding is close to the rolling-window F-MB results reported in the previous section, and also justifies the model specification stated in equation ((ref)).
For a further robustness check of the model specification, we use plots of the capital share factor loadings in Figure (ref) and the capital factor risk price in Figure (ref). From Figure (ref), the capital share factor loadings estimated by the Bayesian method reinforces the rolling-window estimation results in that structural breaks are present both in the factor loadings and volatility. The capital share factor loadings for all portfolios are around zero. In Figure (ref), the distribution of the capital share risk price is centered at zero. The mean effect of capital share is only occasionally significant, as shown by several non-zero risk price estimates.
The capital share risk prices estimated by the Bayesian method are reported in Table (ref). The risk prices in this table are estimated conditional on levels and the volatilities of both the factor loadings and the portfolio returns. In this table, the risk prices of the capital share factor are insignificant for all portfolios. With the Bayesian model specification, the factor loadings are shrunk toward zero by the weak prior when the risk factor has little effect on the level of true equity return dynamics. Therefore, capital share risk prices are insignificant when the capital share factor enters the mean of returns, even after ruling out the potential influence of outliers and stochastic volatility. Given a robust empirical evidence obtained by the Bayesian estimation, we conclude that the capital share factor does not enter the conditional stochastic discount factor of equation ((ref)) and the mean equation of the conditional equity premium of equation ((ref)).
The capital share factor affects equity return volatility under conditional expectations, according to equations ((ref)) and ((ref)). The results obtained by the rolling-window F-MB and the Bayesian estimation methods rule out an impact of the capital share factor on the mean of the equity premium. We now conduct a rolling-window Multiplicative GARCH estimation as a direct test of the conditional innovation of equity return in equation ((ref)). The rolling-window multiplicative GARCH specification tested in our paper is consistent with equations ((ref)) and ((ref)).
The estimates for capital share are plotted in Figure (ref). It shows that, in the variance equation, the capital share factor is always significant at the 5% level. Compared to the Bayesian estimates reported in the previous section, the magnitude of coefficient is higher and more stable over the time horizon during which the capital share factor is insignificant in the mean equation (see Figure (ref)). Therefore, the capital share factor has strong a impact on the variance equation, and this variance effect dominates the mean effect under conditional estimations. This empirical evidence justifies the conditional innovation of market return in equation ((ref)).
In our model, the capital share variability is a risk factor under unconditional expectations. The unconditional tests of this factor's pricing power are conducted in the same manner as earlier tests of the capital share factor. We first plot a preview of equity portfolios in Figure (ref). In this figure, although the average of $R^2$ is lower, the $R^2$ estimates across equity portfolios are more stable than those of Figure (ref). Also, the slope of the regression line estimated by the capital share variability factor is closer to 1 than estimated by $F_{KS}$. Therefore, the OLS results of the capital share variability factor are robust to heteroskedasticity or nonlinearity problems.
We also estimate the capital share variability risk price using the F-MB bootstrap technique. The risk price estimates are reported in Table (ref). In this table, panel A reports the single factor model that only includes a constant and capital share variability as the risk factor, and panel B reports a two factor model that includes $F_{KS}$ and the capital share variability factor for comparison. In panel A, for all equity returns, capital share variability risk prices are significant for all equity returns. The $\bar{R^2}$ estimates are stable across different portfolios and, overall, are higher than those estimated by the single capital share factor model in Table (ref). Note that for REV sorted portfolios the $\bar{R^2}$ estimate is insignificant in Table (ref), while all $\bar{R^2}$ estimates are significant in Table (ref). In panel B, the $\bar{R^2}$ estimates are of similar magnitude as those in panel A. Therefore, multicolinearity might be present in the two factor model. In panel B, $F_{KS}$ is strongly dominated by the capital share variability factor. Following the inclusion of the capital share variability factor, the magnitude of capital share risk price decreases for all portfolios and becomes insignificant for REV sorted portfolios. The magnitude of capital share variability risk price also decreases following the inclusion of the capital share factor due to the colinearity between the capital share factor and its high factor volatility. However, as shown by significant capital share variability risk prices in panel B, the partial effect of the capital share variability factor remains significant in the two factor model. According to the risk price estimates in Table (ref) and Table (ref), we conclude that, under unconditional estimation, capital share variability is a stronger risk factor than capital share growth.
In summary, the factor betas in Figure (ref) and the risk price estimates in Table (ref) empirically justify the data dynamics of an unconditional equity premium (see equation ((ref))) that is positively correlated with the capital share variability factor.
Inspired by the work of lettau2019capital in which U.S. asset prices are explained by capital share risks in an unconditional expected return-risk factor regression, we further investigate the role of capital share risks theoretically. Our paper develops a theoretical model of capital share risks and proposes capital share variability as an unconditional risk factor.
Following bansal2004risks, our paper finds consumption growth volatility operates through capital share growth based upon the recursive preference utility framework developed by epstein1989substitution. Under conditional expectations, capital share growth is found to affect the innovation of market returns but is absent from the mean equation of the equity premium. Under unconditional expectations, the capital share variability is a priced risk factor.
We first employ the Fama MacBeth bootstrap technique used by lettau2019capital for unconditional estimations. The conditional estimations carried out in this paper include the rolling-window F-MB suggested by lewellen2006conditional, the B-TVB-SV estimation proposed by bianchi2017macroeconomic to test the impact of capital share factor on the conditional mean equation of equity returns, and a rolling-window multiplicative GARCH model to test the same impact but on the variance equation.
The empirical evidence is in line with the theoretical model developed in Section 3 of our paper. Under unconditional estimations, capital share growth is found to explain the equity return dynamics. Under the rolling-window F-MB, a strong volatility clustering is found in the capital share factor loading. The capital share risk price is found insignificant but exhibits dramatic fluctuations. Under the B-TVB-SV estimation, high structural break probabilities justify the time variation of the capital share factor loadings, and the robust capital share risk price is found insignificant in the mean equation. Significant rolling-window multiplicative GARCH estimates explain the failure of the capital share factor in the mean equation of conditional equity returns: the capital share factor shows a strong multiplicative heteroskedasticity impact on the variance equation of equity return dynamics. Accordingly, we propose a capital share variability factor and test this new factor unconditionally using the F-MB bootstrap technique. We find this factor dominates the capital share factor. Therefore, the impact of the capital share factor on return volatility is the main source of its conditional pricing power. Under unconditional expectations, capital share variability is a strong unconditional risk factor that captures the long-run movements in market volatility.
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