Asset Prices and Capital Share Risks: Theory and Evidence
AA SSET P RICES AND C APITAL S HARE R ISKS : T
HEORY AND E VIDENCE
Joseph P. Byrne ∗ Boulis M. Ibrahim † Xiaoyu Zong ‡ th May 2020 A BSTRACT
An asset pricing model using long-run capital share growth risk has recently beenfound to successfully explain U.S. stock returns. Our paper adopts a recursivepreference utility framework to derive an heterogeneous asset pricing model withcapital share risks.While modeling capital share risks, we account for the elevatedconsumption volatility of high income stockholders. Capital risks have strongvolatility effects in our recursive asset pricing model. Empirical evidence is presentedin which capital share growth is also a source of risk for stock return volatility. Weuncover contrasting unconditional and conditional asset pricing evidence for capitalshare risks.
Keywords:
Asset Pricing, Capital Share, Recursive Preference, ConsumptionGrowth, Bayesian Methods.
JEL Classification Codes:
C21, C30, E25, G11, G12. ∗ Edinburgh Business School, School of Social Sciences. Heriot-Watt University. Email: [email protected]. † Edinburgh Business School, School of Social Sciences. Heriot-Watt University. Email: [email protected]. ‡ Correspondence author: Xiaoyu Zong. Edinburgh Business School, School of Social Sciences. Heriot-Watt University. Email:[email protected]. a r X i v : . [ ec on . E M ] J un Introduction
Leading asset pricing theories frequently assume a single representative agent when seeking tomodel expected returns. For instance, Breeden’s (1979) consumption based approach adoptsa representative agent, allowing aggregate consumption growth to systematically price returns.However, stock returns are considerably more volatile than aggregate consumption growth. Thisempirical observation is a cornerstone of the equity premium puzzle, see Mehra and Prescott (1985)and Breeden et al. (2014). When endeavoring to explain the failure of consumption-based assetpricing models, one can relax the homogeneous agent assumption (see Constantinides and Duffie(1996), Chabi-Yo et al. (2014) and Lettau et al. (2019)). Observed differences in stock marketparticipation, resources and/or preferences justify heterogeneous agent asset pricing models. Indeed,Campbell et al. (1993) emphasize a focus upon heterogeneous agent asset price model, since it isshareholder consumption that matter for stock returns.Important recent work by Lettau et al. (2019) adopts heterogeneous agents and proposes a capitalshare risk factor that maps shocks to the assets of high income stock holders’ assets and consumptionto explain U.S. returns. Capital risks account for limited stock market participation and proxy theconcentration of wealth. Lettau et al. (2019) present evidence that a capital risk factor explainsexpected returns, and empirically dominate aggregate consumption growth and the Fama and French(1993) factors. In developing a general equilibrium model with limited stock market participationand inequality, Toda and Walsh (2019) highlight that rising wealth holdings of the richest onepercent predict excess stock returns. Their asset pricing model allows for heterogeneity in riskaversion or beliefs. See also theoretical asset pricing models accounting for heterogeneous beliefsunder recursive preferences by Boroviˇcka (2020).While explaining excess equity returns has been the focus of much academic research, elevated stockreturn variability has also been considered by homogeneous agent models. Seeking to resolve assetpricing puzzles associated with standard models, Bansal and Yaron (2004) develop a consumptionrepresentative agent model with long-run risks based upon the recursive preference utility framework2f Epstein and Zin (1989). The asset pricing puzzles highlighted by Bansal and Yaron (2004) includehigh 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. Goingbeyond the homogeneous framework developed by Bansal and Yaron (2004), our theoretical workgeneralizes the heterogeneous agent assumption of Lettau et al. (2019) and allows us to reconsidertheir empirical evidence, accounting in particular for time-varying risk prices and equity returnvolatility.Our asset pricing model has multiple economic agents: in particular, high and low incomestockholders with different consumption patterns. Using United States wealth distribution datafrom Saez and Zucman (2016), Lettau et al. (2019) identify that high and low income stockholders’consumption behaviour responds differently to capital share growth. Given that high incomestockholders consume primarily out of their wealth, capital share growth can also explain theelevated consumption variability of the richest cohorts. Therefore, our theory captures heterogeneityin the volatility of stockholder consumption growth, and this drives the relationship between capitalshare growth and equity returns. The impact of consumption volatility of wealthy stockholders onthe whole market is captured by capital share change.One novelty of our model is that the same risk factor is analysed separately using conditional andunconditional expectations. Inspired by Campbell and Cochrane (2000), the capital share risk factorin our model is linked to stock returns under both unconditional and conditional expectations. Thecontrasting impact of the capital share factor on equity returns under conditional and unconditionalsettings also serves as a potential explanation of the weakness of the consumption-based CAPMunder conditional estimation (Campbell and Cochrane, 2000). Capital share growth is priced onlyunder unconditional expectations, as found by Lettau et al. (2019), and capital share variability isproposed in our paper to capture long-run market volatility. Therefore, our framework posits thatcapital risks enter the equity return variance equation under conditional estimation, while they enterthe mean equation under unconditional estimation.Motivated by our theoretical framework in which capital shares impact not only the mean but alsothe variance of return dynamics, we test the pricing power of capital risks in a more general settingthan that of Lettau et al. (2019). To avoid firm effects, we first use bootstrapped cross-sectional3egressions to estimate the level and volatility effects of capital share risks in asset prices. Thecapital share risk price is adopted as a benchmark for pricing power in this case. To then test theconditional equity premium dynamics, we investigate the conditional capital share risk prices usingthe rolling-window Fama-MacBeth procedure used by Lewellen and Nagel (2006) and the Bayesianasset price estimation from Bianchi et al. (2017). Additionally, our paper estimates the impact ofcapital share growth on return volatility using rolling-window multiplicative GARCH. Finally, wetest the capital risk variability as the long-run risk factor in the mean equation for U.S. stock returnsusing a bootstrap procedure, as we do for testing the capital share factor. This alternative capitalvariability factor empirically dominates the standard capital share growth factor of Lettau et al.(2019).The paper is structured as follows: Section 2 reviews standard asset pricing models and the capitalshare risk factor proposed by Lettau et al. (2019). Section 3 presents a theoretical asset pricing modelwith recursive preferences and heterogeneous agents, in which consumption volatility operatesthrough capital share risk factors. Section 4 sets out empirical methodologies for estimating overallpricing power, and the level and volatility effects of the capital share factor. Section 5 presents thedata and Section 6 presents evidence on the impact of capital share risks. Section 7 proposes andtests the capital share variability factor. Finally, section 8 concludes on the capital share variabilityand 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 thefuture cash flow of an asset, and the price of an asset can be computed by the expectation of totalfuture cash flows discounted by that discount factor.An asset pricing model can be seen as a special case of the following relationship: E t ( M t +1 r t +1 ) = 0 (1)4here M t +1 denotes the discount factor, r t +1 denotes asset excess returns and E t is the conditionalexpectation given information at time t . The form of the SDF relies heavily on the assumptionsmade by different CAPMs (Cochrane, 2009). Equation (1) is operationalized by agents’ expectationformation and their utility or preferences function(s).The consumption-based capital asset pricing model (CCAPM) developed by Breeden (1979) statesthat, with a representative agent assumption, the SDF is based on the marginal rate of substitutionover aggregate household consumption. The CCAPM assumes the SDF is equal to the time-discountfactor ( δ ) multiplied by the ratio of the marginal utility of aggregate consumption tomorrow U (cid:48) ( C t +1 ) and today U (cid:48) ( C t ) as shown in equation (2), where U denotes the utility function of the representativeagent, and C t denotes consumption at time t . M t +1 = δ U (cid:48) ( C t +1 ) U (cid:48) ( C t ) (2)With an homogeneous agent and power utility, expected returns can be priced by aggregate con-sumption growth. Agents are however more reasonably considered to be heterogeneous, due toimperfect risk-sharing, concentrated wealth and limited stock holder participation. And the CCAPMmay not perform well empirically, see Breeden et al. (2014). According to Lettau et al. (2019), forthe wealthiest households, relative to the least wealthy, aggregate consumption volatility multipliedby 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, andtheir consumption comes almost entirely from labor income. Therefore, aggregate consumptiongrowth also fails to capture redistribution risk. Capital share growth better reflects the consumptionof stockholders, while accounting for stockholder heterogeneity and redistributive shocks (seeGreenwald et al. (2014) and Lettau et al. (2019)). The income shares, and therefore consumptionpatterns, of wealthy capital owners is well represented by the capital share.Lettau et al. (2019) then proposes a linear approximation of the asset price SDF using the capitalshare factor: M t +1 = a + b ( C t +1 C t −
1) + b ( KS t +1 KS t −
1) + µ t +1 (3) Wealth re-distributions between stockholders and workers. KS t denotes capital share, a is related to the time-discountfactor (i.e. α = 1 + ln ( δ ) ), while model parameters b and b are related to the risk aversion ofconsumers. Evidence presented by Lettau et al. (2019) indicates that capital share growth explainsU.S. asset prices and restricting aggregate consumption growth to have no effect (i.e. b = 0 ) is areasonable assumption. Approximation error is denoted by µ and explains other factors.The model proposed by Lettau et al. (2019) does not allow for time variation of the capital shareparameters. Empirically risk factor loadings however may vary over time, and conditional assetpricing models can be justified theoretically (see Jensen (1968), Jagannathan and Wang (1996)and Lewellen and Nagel (2006)). For instance, the static CCAPM fails to capture the effect oftime-varying investment opportunities (Lettau and Ludvigson, 2001). The non-zero unconditionalprice anomalies do not necessarily indicate non-zero conditional alphas, given time-varying factorloadings that are correlated with the equity premium or market volatility (Lewellen and Nagel,2006). Our paper generalises the role of capital share by testing both unconditional and conditionalapproaches. Capital share can be motivated as an asset pricing risk factor in a stylized economy with heteroge-neous agents, see Lettau et al. (2019). In this economy, capital investors own the entire corporatesector, while workers do not participant in the stock market. Capital share is assumed to play a rolethrough its influence on stockholders’ consumption. Therefore, in this case, the stochastic discountfactor 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 (2019) assumption that capital shareonly impacts the top wealth distribution of stockholders to derive a more general case. Here capitalshare growth is assumed to influence both the high and the low income stockholder groups, since6he stock market and the wealth weighted participation rates are not identical (see Lettau et al.(2019)). Our model assumes three income groups, which are high income stockholders, low incomestockholders and labour workers. Labour workers are assumed to be absent from the stock marketas in Lettau et al. (2019). We assume for simplicity a constant relative share of high versus lowincome stockholders. 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 lowincome stockholders into a recursive asset pricing model.Over an infinite horizon, the CCAPM in equation (2) is nested as a special case in asset returnmodels derived from a recursive preference utility framework proposed by Epstein and Zin (1989).This recursive framework permits risk attitudes to be disentangled from the degree of intertemporalsubstitutability, and addresses the importance of consumption uncertainty in asset pricing (Epsteinand Zin, 1989). Epstein and Zin (1989) assume agents are homogeneous in the market, and ourcapital share model relaxes this assumption. In our model, the homogeneous assumptions for therecursive framework are satisfied within each stockholder group. Going beyond the homogeneous agent model of Bansal and Yaron (2004), the stockholder con-sumption growth G t +1 is underpinned by the particular consumption patterns of different investors. According to Lettau et al. (2019), the consumption growth rate of the high income stockholders ismore volatile than that of those who derive income from wages. Intuitively, the top of the wealthdistribution 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 capitalshare growth rate is strongly and positively correlated with the consumption growth rate of thehigh income group, while strongly negative correlated with that of the low income group (Lettauet al., 2019). Accordingly, we define ¯ G t as the weighted average of consumption growth of labour Individual can move between income groups, but the population sizes to move into and out from each income group are thesame. In our paper, G nt denotes the consumption growth of agent n calculated from G nt = C nt +1 /C nt , where C nt is the consumption attime t of agent n . ¯ G t ≈ r p ( w H G Ht + w L G Lt ) + (1 − r p ) G Wt (4)where r p is the stock market participation rate, and G Wt is the consumption of labour workerswithout stocks. The aggregate consumption growth ¯ G is independent from how the income groupsare defined. Our heterogeneous model focuses upon the high ( G Ht ) and low ( G Lt ) income stockholderconsumption growth. The former departs from the aggregate consumption growth ( ¯ G t ) based uponthe capital share and excess volatility, as follows: G Ht = ¯ G t f HKS,t (1 + ξ t ) (5) G Lt = ¯ G t f LKS,t (6)where the stochastic term ξ t ∼ N i.i.d (0 , Σ) captures the excess volatility of the consumption growthrate of the high income group compared to the low income group. The consumption growthvolatility shocks of the high income stockholders are absorbed by the labour workers, and theaggregate consumption growth ¯ G t remains independent from such shocks according to equation(4). The ξ t term therefore defines the variance of the consumption distribution of the economy.Based on the data correlations identified by Lettau et al. (2019), we formulate f HKS,t as a monotonicincreasing function, and f LKS,t as a monotonic decreasing function, of the capital share growth rate.The volatility of G Ht is bounded due to limited resources and productivity growth. If the volatilityof G Lt equals zero, the volatility of aggregate consumption growth ¯ G t must also be zero when f LKS,t is non-zero.Since labour workers do not participate in the stock market, our model only focuses on the partialequilibrium of stockholders. The average stockholder consumption growth G St can be approximatedby the weighted average of high and low income stockholder groups: G St ≈ w H G Ht + w L G Lt (7) The stock market participation rate is about 50% (Lettau et al., 2019). We assumes that the low income stockholders have theaggregate consumption growth volatility. The consumption growth volatility of the low income stockholders is, therefore, higher thanthat of labour workers and lower than that of high income stockholders. Excess volatility of high income stockholder consumptiongrowth is absorbed by labour workers and does not affect ¯ G t . Our model does not make an autocorrelation assumption for ξ t to avoid possible explosive growth. Σ denotes a constant variancefor ξ t . G St = ¯ G t [ w H f HKS,t (1 + ξ t ) + w L f LKS,t ] (8)Given evidence from Saez and Zucman (2016), Lettau et al. (2019) assume that the stockholderconsumption equals to the product of the aggregate consumption and capital share. Inspiredby Lettau et al. (2019), we assume that high (low) income stockholders’ consumption growthis positively (negatively) related to the capital share growth, such that f HKS,t = 1 + F KS,t and f LKS,t = 1 − F KS,t . Therefore, the consumption of each stockholder group contains a persistentcomponent as in Bansal and Yaron (2004). Thus, stockholder consumption growth in equation (7) can be written as: G St = ¯ G t [ w H (1 + F KS,t ) + w L (1 − F KS,t )] + ¯ G t [ w H (1 + F KS,t )] ξ t (9)Any percentage at the top can be used to illustrate how the concentration of wealth affects theintensive margin of the stock market (Lettau et al., 2019). We assume that stockholder wealth andpopulation wealth in the economy are drawn from the same distribution, to solve the population sizesof each stockholder income group. Our model therefore approximates the high income stockholderpopulation weight w H by the stock market participation rate ( w H ≈ r p ), and low income stockholderpopulation weight w L by labour worker population weight in the economy ( w H ≈ − r p ). Giventhe average stock market participation rate is close to 50% over time (Lettau and Ludvigson, 2001),we can simply assume w H = w L for simplicity of the calculation. We define capital share growth as capital share factor F KS,t to be consistent with the notation used by Lettau et al. (2019). The empirical evidence presented by Lettau et al. (2019) points out that bottom 90% wealth distribution is strongly negativelycorrelated with capital share growth. Using Saez and Zucman (2016) data, Lettau et al. (2019) Table 2 identifies that capital sharehas 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 andopposite (OLS coefficient = 1.20, t-statistic = 7.34). We assume two stockholder groups and each group satisfies assumptions of Bansal and Yaron (2004) independently. In equation (9), w H ∈ [0 , by definition. About 95% of F KS falls in the range between -4% and 4% (within two standarddeviations) as shown by the empirical results in Table A9. The bounded volatility of C Ht also implies a bounded excess volatility ξ t .Therefore, the ¯ G t [ w H (1 + F KS,t )] ξ t term is bounded and the stockholder consumption does not witness an explosive growth in thismodel. The wealth-weighted participation rate is lower than the aggregate participation rate, regardless which quantile of wealthdistribution is selected as a benchmark. F KS,t and the factthat w H − w L = 0 , the positive impact of capital share growth on the high income group and thenegative impact of capital share growth cancel one another out. At time t , taking conditional expec-tations of (9) and, therefore, setting the stochastic term to zero, we have the expected stockholderconsumption growth as: E t [ G St ] = E t [ ¯ G t ( w H (1 + F KS,t ) + w L (1 − F KS,t ))]= ¯ G t [1 + E t ( F KS,t )( w H − w L )]= ¯ G t (10)Notice that in equation (10), when the dynamics of low and high income stockholder consumptiongrowth f LKS,t and f HKS,t have different functional forms, their mutual effect on the level of expectedstockholder consumption growth can be tested can be tested by having the capital share factor inthe excess return equation. In contrast, equation (9) contains the ¯ G t [ w H (1 + F KS,t )] ξ t term, whichindicates the stockholder consumption growth volatility operates through capital share growth.The magnitude of aggregate stockholder consumption growth volatility also associates with thepopulation size of the high income group w H , and the excess volatility ξ t . Equations (9) and (10)are consistent with the empirical findings of Lettau et al. (2019) in that the consumption growth ofthe top wealth distribution is more volatile than that of the rest of the population, but the expectedconsumption growth rate of the richest individuals is at around the same level as that of the wholeeconomy.To model stockholder consumption growth, we further assume the aggregate consumption growthrate g t contains the persistent expected growth rate component x t proved by Bansal and Yaron(2004) to define the centre of the consumption growth distribution for the economy. We define theaggregate consumption growth rate g t +1 = log ¯ G t +1 = µ + x t + ση t +1 , and x t is the predictableterm following Bansal and Yaron (2004). According to the dynamics of stockholder consumptiongrowth described by equation (9) and the aggregate consumption growth ¯ G , the stockholder ¯ G t +1 = 1 + g t +1 g t +1 can be written as the following function: g t +1 = µ + x t + w H (1 + F KS,t +1 ) ξ t +1 + ση t +1 (11)Therefore, the time-varying volatility of stockholder consumption growth rate σ t +1 is defined as w H (1 + F KS,t +1 ) ξ t +1 + ση t +1 .The intuition behind the consumption growth rate in equation (11) is as follows. According to Saezand Zucman (2016), 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 (10). However, when the capital share growth is positive, the wealth increase ofindividuals in the high income stockholders is higher than in the low income stockholders, asaddressed by Gabaix et al. (2016). The consumption of the top of the wealth distribution willhave 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 Boguth andKuehn (2013). In the theoretical motivation of their volatility risk factor, the consumption growthis assumed to switch between high and low volatility states. In our model, instead of assuming aMarkov switching process based upon changing beliefs, volatility is explicitly modeled using acapital share factor. In Boguth and Kuehn (2013), consumption growth volatility σ t is assumed tobe a time varying function of high ( σ H ) and low ( σ L ) volatility states: ˆ σ t = b t σ H + (1 − b t ) σ L = b t ( σ H − σ L ) + σ L (12)In our model, according to the innovation of consumption growth in equation (11), the volatility ofconsumption growth is assumed to be: ˆ σ t = w H (1 + F KS,t +1 ) ξ t + ση t (13) The w H (1 + F KS,t +1 ) ξ t +1 term is relatively small given the range of w H and F KS,t +1 , and the definition of ξ t +1 . Therefore,we use a Taylor approximation here. This statement does not conflict with assumed high and low income stockholder consumption growth in equations (5) and (6):the aggregate consumption ¯ G t is also increased when capital share increases, as they are positively correlated. σ H − σ L ) and σ L in equation (12) areexplained in our model as excess volatility of the high to the low income stockholders ( ξ t ) and ση t +1 in equation (13) respectively. Therefore, in equation (13), the counterpart of belief b t is w H (1 + F KS,t +1 ) , see equation (12). In line with the assumption by Lettau et al. (2019), the labour workers do not influence equityprices and, consequently, they are independent from the stock market and their participation is notmodeled. Equity returns are linear functions of the equity premium according to the Capital AssetPricing Model (Cochrane, 2009). To solve the relationship between equity returns and capital sharegrowth, our paper extends the model of Bansal and Yaron (2004) to derive the equity premiumexplicitly. The system stated in our paper is a hybrid system of the constant volatility case (CaseI) and the time-varying volatility case (Case II) of Bansal and Yaron (2004). The volatility ofthe log stockholder consumption growth contains both a constant element σ and a time-varyingpart w H (1 + F KS,t +1 ) ξ t +1 . The dividend growth volatility is correlated with consumption growthvolatility, as suggested by Bansal and Yaron (2004). Therefore, σ d,t +1 is assumed to be partiallycorrelated with both F KS,t +1 ξ t +1 and σ in our model. In our model, the stock market is driven by a persistent growth component ( x t +1 ), capital share andstochastic high income volatility shocks ( ξ t +1 ) based upon equation (11) as follows: x t +1 = ρx t + φ e σe t +1 g t +1 = µ + x t + w H (1 + F KS,t +1 ) ξ t +1 + ση t +1 g d,t +1 = µ d + φx t + φ d σ d,t +1 u t +1 (14) e t +1 , u t +1 , η t +1 ∼ N i.i.d. (0 , ξ t ∼ N i.i.d. (0 , Σ) where the g d,t +1 is the log dividend growth rate, and ρ is the persistence of the expected growth rateprocess. Parameters µ and µ d are the constant component of g t +1 and g d,t +1 , respectively. φ e > The specification of σ d,t +1 also relaxes the setting by Bansal and Yaron (2004) Case II, in which g t +1 and g d,t +1 are cointergated,to be consistent with empirical literature (Campbell and Cochrane, 1999). φ d > allow for parameter calibration. The parameter φ can be interpreted as the leverage ratioon expected consumption growth, see Bansal and Yaron (2004) and Abel (1999). The stochasticerror terms e t +1 , u t +1 , and η t +1 are independent from each other (Bansal and Yaron, 2004). σ is aconstant which captures the volatility of x t +1 and g t +1 . The innovation of g d,t +1 which is foundto be more volatile than g t +1 (Campbell, 1999) is tackled by φ d . Our model therefore formalisesuncertainty in terms of the impact of high income consumption variability, rather than a genericuncertainty as set out by Bansal and Yaron (2004).Based upon the recursive preference utility function, the asset pricing restrictions for gross return R i,t +1 satisfy E t [ δ θ G − θψ t +1 R − (1 − θ ) a,t +1 R i,t +1 ] = 1 (15)where θ = (1 − γ ) / (1 − ψ ) . In equation (15), G t +1 denotes the aggregate consumption growthrate, and R a,t +1 is the gross return on an asset that generates dividends that cover the aggregatestockholder consumption. < δ < is the time discount factor, γ ≥ is the risk-aversionparameter, and ψ ≥ is the intertemporal elasticity of substitution (IES).Given the asset pricing constraint in equation (15), the intertemporal marginal rate of substitution(IMRS): m t +1 = θlogδ − θψ g t +1 + ( θ − r a,t +1 (16)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 Campbell and Shiller (1988b) to derive thefunctional form of the equity premium. The innovation of log gross consumption r a,t +1 and logmarket return r m,t +1 are assumed to follow: r a,t +1 = κ + κ z t +1 − z t + g t +1 (17) r m,t +1 = κ ,m + κ ,m z m,t +1 − z m,t + g d,t +1 (18) Bansal and Yaron (2004) Case II adds time varying volatility and fluctuating economic uncertainty into their model through ageneral error term. Our model does not assume a stochastic innovation of σ in order to isolate the volatility effect generalized by theintroduction of the capital share factor. z t is the log price-consumption ratio ( log ( P t C t ) ) and z m,t is the log price-dividend ratio( log ( P t D t ) ). Therefore, z t and z m,t are assumed to satisfy z t = A + A x t + A ,t ξ t and z m,t = A m, + A m, x t + A ,m,t ξ t . The relevant state variables in solving for the equilibrium are x t and ξ t .We modify the functional form of the log price-consumption and log price-dividend ratios assumedby Bansal and Yaron (2004) to include the time-varying part of stockholder consumption growthvolatility. In addition, we need the dynamics of capital share growth to solve the equity premium. Accordingto Lettau et al. (2019), the capital share growth follows an AR(1) process: F KS,t +1 = ρ KS F KS,t + e KSt +1 (19)where e KSt +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 areexogenous 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 (16)-(19). We first solve the parameters of the persistent consumption growth x t and excess volatility ξ t on price-consumption and price-dividend ratios, which track expected risk prices (Campbell andCochrane, 2000). In our model, the resulting A and A ,m are identical to Bansal and Yaron (2004).The sensitivity of the price-consumption (and price-dividend) ratio to the excess volatility ξ t is D t denotes the dividend. A and A m, are constants; A and A m, are parameters of the persistent consumption growth component x t ; A ,t and A ,m,t are parameters of the excess volatility ξ t See Bansal and Yaron (2004) Case II. The constant is not significant according to our AR(1) estimation. Detailed proofs are provided in the Appendix. A ,t (and A ,m,t ) are constants when we hold w H , ρ KS and F KS,t constant: A ,t = 1 − ψ − κ w H ρ KS F KS,t (20) A ,m,t = θ − − θψ − κ ,m w H − w H ρ KS ψ (1 − κ ,m ) F KS,t (21)According to the parameters of excess volatility in equations (20) and (21), given the stochasticnature of ξ t , the capital share growth does not have an impact on the magnitude but affects theuncertainty of the price-consumption and price-dividend ratios. Therefore, due to constant excessvolatility between two adjacent periods, capital share growth does not shift the expected rate ofreturn under short-run (conditional) expectations. However, in the long-run, volatility shocks fail tofeature in expectations and the increased uncertainty of returns generate redistribution risks betweenhigh and low income stockholders.We now set out equity returns conditionally and unconditionally. The difference between these twosettings is due to the difference between conditional and unconditional expectations of ξ t +1 . Condi-tioning on information at t , E t ( ξ t +1 ) = ξ t due to smoothed consumption, while the unconditionalexpectation of ξ t +1 is . We now derive equity premiums under conditional and unconditional expectations, respectively. Conditional on information at time t , all shocks alter agents’ expectations. The conditional innova-tion of the pricing kernel m t +1 is: m t +1 − E t ( m t +1 ) = λ η ση t +1 + λ e σe t +1 + λ ξ,t +1 ξ t +1 (22)The conditional innovation of market return r m,t +1 is: r m,t +1 − E t ( r m,t +1 ) = φ d σ d,t +1 u t +1 + λ m,e σe t +1 + λ m,ξ,t +1 ξ t +1 (23) A ,t and A ,m,t are derived in Appendix. Full details are in the Appendix.
15n equations (22) and (23), λ m,e , λ η and λ e are constants, while λ m,ξ,t +1 and λ ξ,t +1 are functionsof e KSt +1 . Therefore, the conditional pricing kernel innovation in equation (25) is only correlatedto unexpected capital share growth e KSt , but the conditional market return innovation is correlatedwith capital share growth through σ d,t +1 .Following Bansal and Yaron (2004), the continuous equity premium in the presence of time-varyingeconomic uncertainty is E t ( r m,t +1 − r f,t ) = − ( λ m,e λ e − . λ m,e ) σ + 0 . φ d σ d,t +1 + E t ( λ m,ξ,t +1 λ ξ,t +1 − . λ m,ξ,t +1 )= − ( λ m,e λ e − . λ m,e ) σ + 0 . φ d σ d,t +1 (24)At time t , the conditional expectation E t ( ξ t +1 ) = ξ t , so the effect of predictable capital sharegrowth is omitted in equation (24). As shown by equation (24), the conditional equity premium isconstant and has one source of systematic risk that relates to fluctuations in expected consumptiongrowth σ . However, the capital share factor enters the innovation of market return in equation (23).Hence, the excess volatility of the high income group, through capital share growth, is linked to thevariability of equity returns. We estimate the conditional equity premium in equation (24) using theshort-window regression as suggested by Lewellen and Nagel (2006) and the Bayesian approachproposed by Bianchi et al. (2017).Under unconditional expectations, we do not allow unexpected shocks of parameters. The uncondi-tional innovation of the pricing kernel is as follows: m t +1 − E ( m t +1 ) = λ η ση t +1 + λ e σe t +1 + λ uξ,t +1 ξ t +1 (25)The unconditional innovation of market return is: r m,t +1 − E ( r m,t +1 ) = φ d σu t +1 + λ m,e σe t +1 + λ um,ξ,t +1 ξ t +1 (26)Detailed functional forms of the parameters in equations (25) and (26) are in the Appendix. Usingequations (25) and (26), the unconditional equity premium is calculated as: E ( r m,t +1 − r f,t ) = − ( λ m,e λ e − . λ m,e − . φ d ) σ + E [ λ um,ξ,t +1 λ uξ,t +1 − . λ um,ξ,t +1 ) ] (27) See the Appendix. E [ λ ur,ξ,t +1 λ uξ,t +1 − . λ ur,ξ,t +1 ) ] is positively correlated with E ( F KS,t +1 ) . Under uncon-ditional expectations, the equity premium is a function of fluctuations in expected consumptiongrowth σ and capital share variability E ( F KS,t +1 ) .The intuition behind E ( F KS,t +1 ) as an unconditional risk factor is as follows. The variance ofconsumption volatility, captured by e KSt in our model, is very small and gets magnified underunconditional expectations because of the long-lasting nature of the volatility shock (Bansal andYaron, 2004). Intuitively, the ratio of the conditional risk premium to the conditional volatilityof the market portfolio fluctuates with consumption volatility (Bansal and Yaron, 2004). Themaximal Sharpe ratio approximated by volatility of the pricing kernel innovation also varies withconsumption 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, ψ , is justified by the survival analysis of Boroviˇcka (2020) which studies a two-agent model from theperspective of beliefs, where under different belief styles, IES is found to be greater than 1 to ensurethe long-run coexistence of two heterogeneous agents. When ψ > holds, the negative coefficientof capital share growth in parameters A ,t in equation (20) and A ,m,t in equation (21) ensures thatcapital share growth is negatively correlated with the uncertainty in the price-consumption andthe price-dividend ratio. In response to lower expected expected rates of return uncertainty, assetdemand rises to generate positive risk price of the capital share variability in our model. The utilitystudy of Colacito et al. (2018) also highlights that increased macroeconomic volatility increases thestochastic discount factor under the recursive utility framework, thus raises expected returns andgenerates a positive volatility risk price. We estimate the unconditional equity premium in equation(27) using a Fama-MacBeth approach suggested by Lettau et al. (2019) and capital share variabilityas a risk factor. Given our theoretical framework, the capital share variability risk price is expectedto be positive.The model specification of Boguth and Kuehn’s (2013) consumption volatility risk (CVR) factoris a potential explanation of the nonlinear relationship between the equity premium and capital See the Appendix. t , consumption growth volatility has only two latentstates, but ξ t is an unknown stochastic variable and, hence, this is a setup that is consistent with thequadratic relationship between equity returns and capital share growth in our model.To conclude, our theoretical model indicates that: under conditional expectations, the capital sharefactor 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 captureslong-run market volatility. We employ both unconditional and conditional estimation approaches to examine the empiricalimportance of capital share risks and to test the predictions of our model. The unconditionalestimation is a bootstrapped Fama and MacBeth (1973) procedure, which corrects both cross-sectional correlations and the firm effect of equity returns. Conditional estimations include Lewellenand Nagel’s (2006) rolling-window regressions, the Bayesian time-varying beta with stochasticvolatility (B-TVB-SV) estimation from Bianchi et al. (2017), and a rolling-window multiplicativeGARCH. The rolling-window and the B-TVB-SV approaches assume that a risk factor enters themean equation of the stochastic discount factor as shown in equation (3) to test the explanatorypower of capital share growth on the level of equity returns. The rolling-window and the B-TVB-SVestimates are expected to generate statistically insignificant capital share factor loadings accordingto the conditional stochastic discount factor in equation (22) and the conditional equity premiumin equation (24). The rolling-window multiplicative GARCH assumes that the capital share factorinfluences market volatility only, and is used to test the conditional innovation of market return inequation (23). 18 .1 Unconditional Cross Sectional Regressions
The risk price measures the risk-reward relationship between factors and returns. Fama andMacBeth’s (FMB) two-step procedure is widely used in estimating risk prices of factors and intesting 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 effectof 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 ofthis factor will be significant. However, when a risk factor enters the variance equation of the truedynamics, the ordinary least squares (OLS) regression in the first step of the static F-MB approachwill be biased due to heteroskedasticity problems. The risk price estimate of this factor will also besignificant 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 andestimate the unconditional equity premium of the capital share variability factor in equation (27).The F-MB bootstrap is based upon the static F-MB procedure, and can be used to correct bothcross-sectional correlations and firm effects (Lettau et al., 2019) while it constraints the factorloadings 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, Lettau et al. (2019)adopt the non-overlapping block residual bootstrap for both steps of the F-MB procedure. Althoughit is argued that utilizing the overlapping bootstrap is a more robust method, Andrews (2004)compares overlapping and non-overlapping block bootstraps, and reaches the conclusion thatalthough 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 Lettauet al. (2019) for the capital share factor. The optimal length of the bootstrap block should increase as the sample size increases to maintainthe consistency of moments and distribution functions (Horowitz, 1997). In the first step, our sample In the first step time series regression, capital share growth is assumed to follow an AR(1) process to factor in the serialcorrelation. .Therefore, the optimal block-length is ( ) ≈ following Hall et al. (1995).The second step involves 25 portfolio returns, and the optimal block-length is identical to Lettauet al. (2019).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 inferthe functional form of the true dynamics of equity returns. When assuming that capital share growthis a risk factor in the mean equation, the time variation of factor loadings are not captured by simplyestimating one time series regression in the first step of F-MB bootstrap approach, hence the riskprices are estimated unconditionally. Additionally, the non-linearity in the dynamics of equityreturns is omitted, leading to biased F-MB bootstrap estimates of the capital share risk price.According to the theoretical justification in equation (23), the capital share factor explains thevariance of equity returns under conditional expectations. Also, the true unconditional risk factorshould be the capital share variability according to equation (27). In our paper, the monthly F-MBbootstrap estimates are benchmarks of pricing power. The monthly risk price of the capital sharefactor is expected to be significant due to its multicollinearity with capital share variability E ( F KS,t ) in a single capital share factor model. In a two factor model including both the capital share factorand capital share variability, the latter can be expected to dominate according to the unconditionalequity premium in equation (27).
As shown by equation (24), the mean equation of the equity premium is independent from the capitalshare factor. We adopt rolling-window regressions to estimate factor loadings in a conditionalmanner as suggested by Lewellen and Nagel (2006). Our paper estimates the F-MB first stepregression following Lewellen and Nagel (2006). The window length selected in the first step F-MB The bootstrap only estimate January 1974 to August 2018 which is consistent with sample span for B-TVB-SV approach. 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.
20s 12 months. The second step is identical to the original cross-sectional regression of the F-MBapproach. The results of the rolling-window regression serves as a benchmark for the true DGP offactor 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, theregression 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 regressionwindow (Fama and French, 1988). Another function of the rolling-window regression is to serve asa 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 anappropriate approximation for time-varying factor loadings, only conditional on the assumption thatthere are no structural breaks present within each window. The time variations are still not fullycaptured due to the ad-hoc window length selection: robustness of the rolling-window approach isdiminished 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 acommon problem of 2-step estimations, which is that the second step estimation is dependant onthe first step results. This approach cannot pass the variability of factor loadings into the secondstep estimation and, therefore, is insufficient to ensure unbiased estimation of risk prices. Therolling-window approach also views the factor loading as a constant at each time point, causinginformation carried by the change of factor loading volatilities to be retained within the first stepestimation. The time variation of risk prices are thus inflated compared to the true underlying DGPby the rolling window F-MB approach when stochastic volatility is present in factor loadings.As shown by the innovation of market premium in equation (24), the loading of the capital sharefactor is expected to be centered at zero, and a strong volatility clustering is expected to be presentunder rolling-window estimation. Due to heteroskedasticity and the model misspecification problemhighlighted by Jagannathan and Wang (1996), risk price estimates should be insignificant but varydramatically over time. An insignificant risk factor in the true equity dynamic might be significant under F-MB estimations. .2.2 The Bayesian Time-Varying Beta With Stochastic Volatility Model To tackle problems in the F-MB procedures, Bianchi et al. (2017) proposes a Bayesian estimationapproach, namely the Bayesian time-varying beta with stochastic volatility (B-TVB-SV) model, toconsider the SDF and non-arbitrage restriction jointly. Compared to the rolling-window F-MB, thismethod captures the time variation and variability of factor loadings while maintaining robustnessto firm effects.The B-TVB-SV model for asset return r i,t as a function of risk factor F j,t is: r i,t = β i ,t + K (cid:88) j =1 β ij,t F j,t + σ i,t (cid:15) i,t (cid:15) i,t ∼ N (0 , (28)Factor risk prices λ j,t are estimated by: r i,t = λ ,t + K (cid:88) j =1 λ j,t β ij,t + e i,t e i,t ∼ N (0 , τ ) (29)The B-TVB-SV framework assumes the time-varying betas β ij,t and residuals in equation (28) takethe following forms: β ij,t = β ij,t − + κ ij,t η ij,t j = 0 , ..., K (30) ln ( σ i,t ) = ln ( σ i,t − ) + κ iv,t υ i,t i = 0 , ..., N (31)where κ ij,t is the structural break of factor loading β ij,t , and κ iv,t is the structural break of id-iosyncratic variance ln ( σ i,t ) . The stochastic terms η ij,t and υ i,t follow normal distributions withzero mean and variances q ij and q iv , respectively. A κ ij,t equal to one indicates that structuralbreaks are present in the factor loadings, and κ iv,t equal to one indicates that structural breaksare present in the idiosyncratic variance. The advantage of including structural breaks is that themodel captures discrete movements of the factor loadings. In equation (30), the innovation of factorloading maintains the random walk properties to retain the shrinkage power of the selected priorto the largest extent. Therefore, the B-TVB-SV approach tackles factor selection automatically. Weak priors are used for the distributions of β ij,t and ln ( σ 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 (Huber et al., 2020). Theweak prior adopted by V-TVB-SV approach also has shrinkage effects. t as indicated by equation (24), and the variance of this distribution shouldchange over time, as in equation (23). As shown by the conditional market return innovation in equation (23), the capital share should beestimated in the variance equation instead of in the mean equation conditionally. Our paper employsa rolling-window multiplicative GARCH approach to estimate the true volatility effect of the capitalshare factor on equity returns directly. Within each regression window, the asset pricing modelestimated by the multiplicative GARCH is as follows: r i,t = β i ,t + (cid:15) i,t (cid:15) i,t ∼ N (0 , σ i,t ) (32)where V ar ( (cid:15) i,t ) = σ i,t , and σ i,t is consistent with the form in equation (34) below. Therefore, theconditional 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 thefollowing most general form for the variance equation: σ i,t = γ KS F KS,t (33)The functional form in equation (33) is motivated by the market return innovation. As indicated byequation (23), the capital share factor is an O ( n ) addend in the variance equation σ i,t . This paperadopts the conditional variance form proposed by Judge et al. (1988) to test the variance equation2333). The capital share factor enters the variance specification as multiplicative heteroskedasticity.Due to the constraint σ ≥ , equation (33) is rewritten as equation (34) for the sake of estimation. σ i,t = exp [ λ + λ log ( F KS,t )] (34)A 60-month window length is selected by this paper for the rolling-window multiplicative GARCHdue to the limitation of maximum likelihood convergence. According to the new data dynamics inequations (32) and (33), the coefficient of log ( F KS,t ) in the variance equation (34) is expected to besignificant over time if equations (23) and (24) hold. Lettau et al. (2019) use quarterly capital share and quarterly portfolio returns converted frommonthly data to test capital share growth. In our paper, instead of modifying monthly returns in arelatively ad-hoc manner, we interpolate the capital share using a reasonable indicator to reduceinformation loss.
Measurement error leads to biased estimation of CAPMs (Lettau et al., 2019). Long-term capitalshare growth is adopted to partial out the measurement error effect. In the test of the capitalshare factor, Lettau et al. (2019) compares 1,4,8,12 and 16-quarter capital share growth to tacklemeasurement error problems. The 4-quarter capital share growth is found to have higher pricingpower.Capital share is calculated as − Labour share . Labour share data is the nonfarm sector laborshare, which is identical to that used by Lettau et al. (2019) and Gomme and Rupert (2004). Datafor constructing capital share from FRED, the monthly capital share is obtained by the Chow-lininterpolation. During the data collection process, the filtering approach introduces measurement error problem. See the Appendix. F qKS = KS qt +4 KS qt (35)In equation (35) F KS can be decomposed as the capital share growth rate plus a constant 1, indicatingthat the factor is partially correlated with the intercept. Therefore, the estimated capital share factorloading is higher due to the partial effect taken from the constant. Additionally, the estimateddistribution 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 isnot easily identified if the factor is correlated with the constant. To avoid these problems, we use a12-month capital share growth rate as a risk factor and test its pricing power. The monthly capitalshare factor tested in this paper is constructed as: F KS = KS t +12 KS t − (36)According to the unconditional expectation of equity returns in equation (27), capital share variability( E ( F KS ) ) enters the unconditional mean equation of equity returns. The E ( F KS ) risk factor isconstructed based upon the AR(1) innovation process of F KS as in Lettau et al. (2019): F KS,t +1 = ρ KS F KS,t + e KSt +1 (37)where e KSt +1 captures unexpected shocks in capital share growth. The magnitude of the estimate of ρ KS is 0.947, which is statistically significant at 5% level. We obtain the capital share variabilityfactor using the capital share factor constructed by Lettau et al. (2019) to avoid measurement errorproblems.The innovations of capital share growth and variability are plotted in Figure 1, and the descriptivestatistics of the capital share factor and the capital share variability factor are reported in Table A9in the Appendix. In our paper, the capital share factor and the variability are tested on different groups of portfolioreturns. The portfolio groups we test include 25 size/BM, 10 long-term reverse (REV), 25 size/INV,25nd 25 size/OP sorted portfolio returns. The descriptive statistics of benchmark portfolio returns arereported in Appendix. For the multiplicative GARCH estimation, this paper takes cross-sectionalaverages of size/BM, REV, size/INV, and size/OP sorted portfolio returns respectively to mimicdifferent market portfolios. All portfolio data are monthly data from the Kenneth R. French DataLibrary. The time span is January 1964 to August 2018.
To focus on testing the pricing performance of the capital share factor, our paper estimates aparsimonious capital share factor model which only contains a constant and the capital share factor.A preview of equity portfolios is shown in Figure 2, which plots the monthly average returns on they-axis and the portfolio capital share betas on the x-axis. Due to the higher variation in monthly data,the R estimates are generally lower for each portfolio class compared to the quarterly data estimatesby Lettau et al. (2019). In addition, the R estimated by REV sorted portfolios is 0.26 in Figure2. All other R estimated from monthly data deviate modestly from their quarterly counterparts.According to the distribution of points in the scatter plots of Figure 2, the model fit is high and thecapital share factor has substantial explanatory power for expected returns. However, the regressionlines for the portfolios deviate from 1, which indicates a potential presence of heteroskedasticity ornon-linearity.The F-MB bootstrap we use is identical to that in Lettau et al. (2019). We therefore carry out 10000simulations for the bootstrap process. Table 1 reports the risk prices estimated by the capital sharefactor model. In this table, all of the lower bootstrap interval bounds are above zero for capital sharerisk prices ( F KS ), indicating the risk price estimates are all statistically significant at least at the5% level. For the bootstrap interval of R estimates, the lower bound of R for REV portfolios is0.000, while for other portfolios are all above 0.300. Therefore, for REV portfolios, the low R inpanel B explains the insignificant capital share premium in panel A: instead of low level correlation,26 able 1: Expected Return Capital Share Risk Prices Size/BM REV Size/INV Size/OP β F KS ¯ R Notes:
This table reports F-MB bootstrap estimations of risk prices (%) of the capital share factor. Thestochastic discount factor in equation (3) is tested by the single factor model stated in this table: β isthe constant and F KS is the capital share factor constructed as 12-month capital share growth. Portfolioreturns used for estimation are REV, size/BM, size/INV, and size/OP sorted portfolios. Bootstrapped 95%confidence intervals are reported in square brackets. ** denotes the estimate is significant at 5% level. *denotes the estimate is significant at 10% level. The sample spans the period January 1974 to August 2018. the high variation of correlation between portfolio returns and capital share diminishes the pricingpower.To conclude, results derived by the parsimonious unconditional capital share factor model usingmonthly data are consistent with the results derived by Lettau et al. (2019) 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 dynamicsfrom quarterly data, monthly returns generate lower or insignificant ¯ R estimates for all equityportfolios. Therefore, the cross-sectional results of the capital share risk price might vary over timedramatically, and the the increase in the frequency of the data also increases the probabilities ofoutliers and the variance of risk price estimates. However, the diminishing pricing power of thecapital share factor in higher frequency data also indicates that this factor might be correlated withthe volatility of equity returns or a potential nonlinearities in the equity return DGP.27 .2 Conditional Cross Sectional Regressions Table 2: Expected Return Capital Share Beta Rolling Regressions
Size/BM REV Size/INV Size/OP β F KS -0.190* -0.166 -0.093 -0.017(0.058) (0.191) (0.926) (0.867) R Notes:
This table reports risk prices (%) of the capital share factor. Conditional equity premium in equation(24) is tested by including capital share factor F KS , which is the 12-month capital share growth, in themean equation. In this table, statically insignificant F KS rules out the possibility that the capital sharefactor is priced under conditional expectations. Portfolio returns used for estimation are REV, size/BM,size/INV, and size/OP sorted portfolios. The model estimated is a single capital share factor model, where β is the constant and F KS,t is the capital share factor. the P-values are reported in parentheses belowestimates. ** and * denote significance at the 5% and 10% levels, respectively. Sample spans the periodJanuary 1974 to August 2018.
In this section, we return to estimates of the factor loadings using a rolling-window regression inthe first step of the F-MB procedure. Risk prices are estimated in the same manner as the staticF-MB but within each window. Table 2 reports the rolling-window estimates of the parsimoniouscapital share factor model. As shown in this table, the capital share risk prices are insignificantfor most equity portfolios, and all signs of risk prices are negative. The negative and insignificantrisk prices show that the cross-sectional results in the second step deviate dramatically from staticresults (positive and significant) when estimating the first step using shorter regression windows.Figure 3 plots the 12-month rolling-window estimated factor loadings of the parsimonious capitalshare model, and the portfolio returns estimated are size/BM sorted portfolios. As the figureshows, the factor loadings have small jumps in levels but big structural breaks in volatilities underconditional estimation. The overall level of factor loadings is centered at zero. Figure 3 also shows28 strong volatility clustering pattern in factor loadings, which further enhances the possibility thatthe SDF in equation (3) and the factor model estimated might be misspecified, in the sense that thecapital share factor does not enter the mean equation if we account for the time evolution of riskprices.Figure 4 plots the capital share risk prices estimated by the single factor model. This figure showsthat, the time variation of risk prices is very high across the sample, and the level of risk priceswitnesses frequent structural breaks. In the first step of the rolling-window F-MB estimation, thefactor loadings only capture the effects caused by level changes and not the effects caused byvolatility changes. In the second step estimation, the factor loadings at each time are treated asa constant, leading to a more volatile risk price series over the time when volatility varies acrosswindows.Overall, the rolling-window F-MB estimates are consistent with the theoretical model in equations(23) and equation (24) in that the capital share factor loadings are centered at zero with strongvolatility clustering. However, this analysis cannot rule out the potential impact of large outlierson risk price estimates due to the very short window length used. The results derived by therolling-window F-MB procedure support accounting for structural breaks and stochastic volatilityfor further robustness.
The Bayesian time-varying beta with stochastic volatility (B-TVB-SV) approach by Bianchi et al.(2017) tackles the volatility clustering of the capital share factor loadings found by the rolling-window F-MB approach. the B-TVB-SV risk price estimates are more robust to outliers thanthose 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 Bianchi et al. (2017), with a parsimonious capital share factor model. Following Bianchiet al. (2017), to robustify structural break estimates, this paper demeans all risk factors within boththe training and the estimation samples to cancel out all potential bias caused by multicollinearity The prior specification of factor loading allows volatility clustering and frequent structural breaks. κ 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 (structuralbreaks exist when the break probability estimates are non-zero). Figure 5 plots the time-averagebreak probabilities calculated by averaging all estimated κ ij,t in equation (30). To save space,this paper does not plot the time-average break probability for each portfolio. The average breakprobabilities of capital share are around 0.427 among the four equity portfolio classes. Accordingto the time evolution of factor loadings stated in equation (30), and due to high expected value of κ ij,t , β ij,t follows a jump process with frequent structural breaks over time. This finding is closeto the rolling-window F-MB results reported in the previous section, and also justifies the modelspecification stated in equation (30).For a further robustness check of the model specification, we use plots of the capital share factorloadings in Figure 6 and the capital factor risk price in Figure 7. From Figure 6, the capital sharefactor loadings estimated by the Bayesian method reinforces the rolling-window estimation resultsin that structural breaks are present both in the factor loadings and volatility. The capital sharefactor loadings for all portfolios are around zero. In Figure 7, the distribution of the capital sharerisk price is centered at zero. The mean effect of capital share is only occasionally significant, asshown by several non-zero risk price estimates.The capital share risk prices estimated by the Bayesian method are reported in Table 3. The riskprices in this table are estimated conditional on levels and the volatilities of both the factor loadingsand the portfolio returns. In this table, the risk prices of the capital share factor are insignificant forall portfolios. With the Bayesian model specification, the factor loadings are shrunk toward zeroby 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 meanof returns, even after ruling out the potential influence of outliers and stochastic volatility. Givena robust empirical evidence obtained by the Bayesian estimation, we conclude that the capital30 able 3: Expect Return-Capital Share Beta Bayesian Regression Average Std.err t-stat p-value 2.5% 50% 97.5%Panel A: size/BM sorted portfolios β F KS -0.017 0.197 -0.085 0.932 -7.296 -0.019 7.784Panel B: REV sorted portfolios β F KS β F KS -0.054 0.157 -0.344 0.731 -6.505 -0.015 8.066Panel D: size/OP sorted portfolio β F KS Notes:
This table reports Bayesian time-varying beta with stochastic volatility proposed by Bianchi et al.(2017). The conditional equity premium in equation (24) is tested by including capital share factor F KS ,which is the 12-month capital share growth, in the mean equation. Estimates in this table are robustto time variation and volatility clustering of factor loadings. Risk prices (%) in panels A, B, C and Dare estimated by a single capital share factor model using size/BM, REV, size/INV, and size/OP sortedportfolios, respectively. The 2.5%, 50%, and 97.5% quantiles of estimated risk price distribution areincluded in this table. ** and * denote significance at the 5% and 10% levels, respectively. Data usedare monthly from January 1964 to August 2018. The first 10-year data are used as training sample forhyperparameter estimation, and the sample used for estimation spans January 1974 to August 2018. share factor does not enter the conditional stochastic discount factor of equation (22) and the meanequation of the conditional equity premium of equation (24). The capital share factor affects equity return volatility under conditional expectations, accordingto equations (23) and (24). The results obtained by the rolling-window F-MB and the Bayesianestimation methods rule out an impact of the capital share factor on the mean of the equity31remium. We now conduct a rolling-window Multiplicative GARCH estimation as a direct testof the conditional innovation of equity return in equation (23). The rolling-window multiplicativeGARCH specification tested in our paper is consistent with equations (32) and (33).The estimates for capital share are plotted in Figure 8. It shows that, in the variance equation,the capital share factor is always significant at the 5% level. Compared to the Bayesian estimatesreported in the previous section, the magnitude of coefficient is higher and more stable over the timehorizon during which the capital share factor is insignificant in the mean equation (see Figure 7).Therefore, the capital share factor has strong a impact on the variance equation, and this varianceeffect dominates the mean effect under conditional estimations. This empirical evidence justifiesthe conditional innovation of market return in equation (23).
In our model, the capital share variability is a risk factor under unconditional expectations. Theunconditional tests of this factor’s pricing power are conducted in the same manner as earlier testsof the capital share factor. We first plot a preview of equity portfolios in Figure 9. In this figure,although the average of R is lower, the R estimates across equity portfolios are more stable thanthose of Figure 2. Also, the slope of the regression line estimated by the capital share variabilityfactor is closer to 1 than estimated by F KS . Therefore, the OLS results of the capital share variabilityfactor are robust to heteroskedasticity or nonlinearity problems.We also estimate the capital share variability risk price using the F-MB bootstrap technique. Therisk price estimates are reported in Table 4. In this table, panel A reports the single factor modelthat only includes a constant and capital share variability as the risk factor, and panel B reports atwo factor model that includes F KS and the capital share variability factor for comparison. In panelA, for all equity returns, capital share variability risk prices are significant for all equity returns. The ¯ R estimates are stable across different portfolios and, overall, are higher than those estimated bythe single capital share factor model in Table 1. Note that for REV sorted portfolios the ¯ R estimateis insignificant in Table 1, while all ¯ R estimates are significant in Table 4. In panel B, the ¯ R F KS is strongly dominated by the capital share variabilityfactor. Following the inclusion of the capital share variability factor, the magnitude of capital sharerisk price decreases for all portfolios and becomes insignificant for REV sorted portfolios. Themagnitude of capital share variability risk price also decreases following the inclusion of the capitalshare 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 ofthe capital share variability factor remains significant in the two factor model. According to the riskprice estimates in Table 1 and Table 4, we conclude that, under unconditional estimation, capitalshare variability is a stronger risk factor than capital share growth.In summary, the factor betas in Figure 9 and the risk price estimates in Table 4 empirically justify thedata dynamics of an unconditional equity premium (see equation (27)) that is positively correlatedwith the capital share variability factor. Inspired by the work of Lettau et al. (2019) in which U.S. asset prices are explained by capital sharerisks in an unconditional expected return-risk factor regression, we further investigate the role ofcapital share risks theoretically. Our paper develops a theoretical model of capital share risks andproposes capital share variability as an unconditional risk factor.Following Bansal and Yaron (2004), our paper finds consumption growth volatility operates throughcapital share growth based upon the recursive preference utility framework developed by Epstein andZin (1989). Under conditional expectations, capital share growth is found to affect the innovation ofmarket returns but is absent from the mean equation of the equity premium. Under unconditionalexpectations, the capital share variability is a priced risk factor.We first employ the Fama MacBeth bootstrap technique used by Lettau et al. (2019) for unconditionalestimations. The conditional estimations carried out in this paper include the rolling-window F-MBsuggested by Lewellen and Nagel (2006), the B-TVB-SV estimation proposed by Bianchi et al.33 able 4: Capital Share Variability as a Risk Factor
Size/BM REV Size/INV Size/OPPanel A: capital share variability α E ( F KS ) ¯ R α F KS E ( F KS ) ¯ R Notes:
This table reports F-MB bootstrap estimations of risk prices (%) of capital share variability. Capitalshare variability is an unconditional risk factor according to the unconditional equity premium in equation(27). In this table, portfolio returns used for estimation are REV, size/BM, size/INV, and size/OP sortedportfolios. Bootstrapped 95% confidence intervals are reported in square brackets. ** denotes the estimateis significant at 5% level. * denotes the estimate is significant at 10% level. Sample spans the periodJanuary 1974 to August 2018. (2017) 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 varianceequation.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 factor34oading. The capital share risk price is found insignificant but exhibits dramatic fluctuations. Underthe B-TVB-SV estimation, high structural break probabilities justify the time variation of thecapital share factor loadings, and the robust capital share risk price is found insignificant in themean equation. Significant rolling-window multiplicative GARCH estimates explain the failureof the capital share factor in the mean equation of conditional equity returns: the capital sharefactor shows a strong multiplicative heteroskedasticity impact on the variance equation of equityreturn dynamics. Accordingly, we propose a capital share variability factor and test this new factorunconditionally using the F-MB bootstrap technique. We find this factor dominates the capital sharefactor. Therefore, the impact of the capital share factor on return volatility is the main source of itsconditional pricing power. Under unconditional expectations, capital share variability is a strongunconditional risk factor that captures the long-run movements in market volatility.35 eferences
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Review of FinancialStudies . hhz121. 39 igure 1: Capital share growth and variability (%).
The sample spans January 1964 toAugust 2018. i g u re : C a p i t a l Sh a re B e t a s . T h i s p l o t d e p i c t s t h e b e t a s c on s t r u c t e dby t h e F - M B r e g r e ss i ono f a v e r a g e po r t f o li o r e t u r n s on ca p it a l s h a r e b e t a . T h e po r t f o li o s e s ti m a t e d i n c l ud e R E V , s i ze / B M , s i ze / I NV a nd s i ze / O P s o r t e dpo r t f o li o s o r u s i ng a ll e qu iti e s t og e t h e r . R e s ti m a t e s o f eac h r e g r e ss i on a r e r e po r t e d i n t h e g r a ph . T h e s a m p l e s p a n s t h e p e r i od J a nu a r y1974 t o A ugu s t . i g u re : - m o n t h r o lli n g - w i nd o w e s t i m a t i o n o f c a p i t a l s h a re f a c t o r l oa d i n g s , s i n g l e f a c t o r m o d e l . T h e f ac t o r l o a d i ng s a r ee s ti m a t e du s i ng m on t h l y s i ze / B M s o r t e dpo r t f o li o r e t u r n s a nd12 - m on t h w i ndo w l e ng t h . T h e % c on fi d e n ce i n t e r v a l s a r e p l o tt e du s i ngd a s h e d li n e . S a m p l e s p a n sJ a nu a r y1974 t o A ugu s t . igure 4: Rolling-window capital share factor risk price (%). Following Fama and MacBeth(1973) and Lewellen and Nagel (2006), the factor loadings are estimated using monthly size/BMsorted portfolio returns and 12-month window length. The 95% confidence intervals are plottedusing dashed line. The sample spans January 1974 to August 2018. i g u re : B - T V B - S V av er ag e b re a kp r o b a b ili t i e s o ff a c t o r l oa d i n g s , s i n g l ec a p i t a l s h a re f a c t o r m o d e l . T h e b r ea kp r ob a b iliti e s a r ee s ti m a t e du s i ng25 s i ze / B M s o r t e dpo r t f o li o s . A v e r a g e p r ob a b iliti e s r e po r t e d a r e t h e ti m e - a v e r a g e f o r eac hpo r t f o li o s . T h e s a m p l e s p a n sJ a nu a r y1964 t o A ugu s t . T h e fi r s t - y ea r d a t a i n t h e s a m p l e i s u s e d f o r t r a i n i ng , a nd t h e s a m p l ee s ti m a t e d c ov e r sJ a nu a r y1974 t o A ugu s t . i g u re : B - T V B - S V c a p i t a l s h a re f a c t o r l oa d i n g s . F ac t o r l o a d i ng s a r ee s ti m a t e dby t h e s i ng l e ca p it a l s h a r e f ac t o r m od e l u s i ng m on t h l y s i ze / B M s o r t e dpo r t f o li o r e t u r n s . T h e % c on fi d e n ce i n t e r v a l s a r e p l o tt e du s i ngd a s h e d li n e . T h e s a m p l e s p a n sJ a nu a r y1964 t o A ugu s t . T h e fi r s t - y ea r d a t a i n t h e s a m p l e i s u s e d f o r t r a i n i ng , a nd t h e s a m p l ee s ti m a t e d c ov e r sJ a nu a r y1974 t o A ugu s t . igure 7: Bayesian capital share risk price (%). This figure plots risk prices estimated by thesingle capital share factor model using monthly size/BM sorted portfolio returns. The 95% confidenceintervals are plotted using dashed lines. The sample spans January 1964 to August 2018. The first10-year data in the sample is used for training, and the sample estimated covers January 1974 toAugust 2018. igure 8: 60 month rolling-window multiplicative GARCH estimates (%). This figure showsestimates for testing conditional market return innovation in equation (23) in which the capital sharefactor enters the variance equation of equity returns. The coefficient of capital share factor is estimatedusing monthly average returns of size/BM sorted portfolios. The 95% confidence intervals are plottedusing dashed lines. The sample spans January 1974 to August 2018. i g u re : C a p i t a l s h a re va r i a b ili t y b e t a s . T h i s p l o t d e p i c t s t h e b e t a s c on s t r u c t e dby t h e F - M B r e g r e ss i on o f a v e r a g e po r t f o li o r e t u r n s on ca p it a l s h a r e v a r i a b ilit yb e t a . T h e po r t f o li o s e s ti m a t e d i n c l ud e R E V , s i ze / B M , s i ze / I NV a nd s i ze / O P s o r t e dpo r t f o li o s o r u s i ng a ll e qu iti e s t og e t h e r . R e s ti m a t e s o f eac h r e g r e ss i on a r e r e po r t e d i n t h e g r a ph . T h e s a m p l e s p a n s t h e p e r i od J a nu a r y1974 t o A ugu s t . ppendix This appendix is not for publication and describes the Bayesian time varying beta with stochasticvolatility (B-TVB-SV) specification, the detailed theoretical induction of our model, the constructionof the dataset, and data basic statistics and estimations.
A The B-TVB-SV model specification
Bianchi et al. (2017) assumes the structural breaks are independent both across portfolio returns andover time. Equation (A.1) defines the structural break probabilities:
P r [ κ ij,t = 1] = π ij i = 1 , ..., NP r [ κ iv,t = 1] = π iv j = 0 , ..., K (A.1)The probabilities π ij and π iv are sampled using a uninformative prior to retain the robustness ofestimations. The priors are assumed to follow beta distributions: π ij ∼ Beta ( a ij , b ij ) i = 1 , ..., Nπ iv ∼ Beta ( a iv , b iv ) j = 0 , ..., K (A.2)The structural break estimation in Bianchi et al. (2017) uses an efficient generation of mixingvariables developed by Gerlach et al. (2000). In modeling intervention in dynamic mixture models,this sampling approach allows the state matrix to be singular and, hence, estimations are allowedto depend on unknown parameters. The breaks innovations κ ij,t in equation (30) are assumed tobe conditional on the residual variance matrix ( Σ ), the break probability matrix of σ ( K σ ), thesimulated model parameter θ , excess returns R , and factors F . In equation (31), κ iv,t is assumedto follow a similar innovation process to κ ij,t . The conditional variance parameters of the size ofthe structural breaks are assumed to follow an inverted Gamma-2 distribution, of which the shapeparameter is linked to the scale parameter (Bianchi et al., 2017).The prior of the second step risk prices is a mixture of 10 random normal distributions. Priors ofthese normal distributions are proposed by Omori et al. (2007). The risk price prior is as follows: λ ∼ M N ( λ, V ) (A.3)1he prior of τ in equation (29) follows a inverse Gamma-2 distribution with shape parameter ¯ ψ and scale parameter Ψ , where Ψ = Ψ + ( r − βλ ) (cid:48) ( r − βλ ) (A.4)The risk prices are sampled conditional on the price error matrix r − βλ linking the time-seriesregression in equation (28) and the second-step cross-sectional regression in equation (29). There-fore, although the risk prices are estimated in a similar manner to the F-MB procedure within eachiteration, the estimated standard deviations of risk prices are robust when a firm effect is present inportfolio returns. B Theoretical Framework
We derive the impact of high income shareholder excess volatility on the price-consumption ratio(see equation (20) in the main text) and the price-dividend ratio (see equation (21) in the main text).Also, the conditional and unconditional innovation of the pricing kernel (equations (22) and (25)),the conditional and unconditional innovation of equity returns (equations (23) and (26)). The equitypremium with conditional and unconditional expectations (equations (24) and (27)) in the main textare derived in this section.With Epstein and Zin (1989) recursive preferences, the asset pricing restrictions for gross return R i,t +1 satisfy E t [ δ θ G − θψ t +1 R − (1 − θ ) a,t +1 R i,t +1 ] = 1 (B.1)where θ = (1 − γ ) / (1 − ψ ) . In equation (15), G t +1 denotes the aggregate consumption growth rate,and R a,t +1 denotes the gross return on an asset that generates dividends that cover the aggregateshareholder consumption. < δ < is the time discount factor, γ ≥ is the risk-aversionparameter, and ψ ≥ is the intertemporal elasticity of substitution (IES).2ur system equation is: x t +1 = ρx t + φ e σe t +1 g t +1 = µ + x t + w H (1 + F KS,t +1 ) ξ t +1 + ση t +1 g d,t +1 = µ d + φx t + φ d σ d,t +1 u t +1 (B.2) e t +1 , u t +1 , η t +1 ∼ N i.i.d. (0 , ξ t ∼ N (0 , Σ) According to Bansal and Yaron (2004), dividend growth volatility is correlated with consumptiongrowth volatility. Thus, σ d,t +1 is partially correlated with F KS,t +1 ξ t +1 .The IMRS is m t +1 = θlogδ − θψ g t +1 + ( θ − r a,t +1 (B.3)Consumption return follows: r a,t +1 = κ + κ z t +1 − z t + g t +1 (B.4)where z t = A + A x t + A ,t ξ t (B.5)Following Bansal and Yaron (2004), assuming r a,t +1 = r i,t +1 , IMRS in equation (B.3) indicates: logδ − ψ g t +1 + r a,t +1 = 0 (B.6)Substituting equations (B.2), (B.4) and (B.5) into equation (B.6), we get: logδ + (1 − ψ )( µ + x t + w H (1 + E t ( F KS,t +1 )) E t ( ξ t +1 ) + ση t +1 )+ κ + κ ( A + A ρx t + A φ e σ + E t ( A ,t +1 ) E t ( ξ t +1 )) − ( A + A x t + A ,t ξ t ) = 0 (B.7)To ensure equation (B.7) holds, the following must hold: (1 − ψ ) x t + κ ρA x t − A x t = 0 (B.8) (1 − ψ ) w H E t ( F KS,t +1 ) E t ( ξ t +1 ) + κ E t ( A ,t +1 ) E t ( ξ t +1 ) − A ,t ξ t = 0 (B.9)Notice that although the long-run expectation of ξ t +1 is zero, this term is relatively stable between t and t + 1 . Our model assumes the existence of an r ξ such that ξ t − r ξ < ξ t +1 < ξ t + r ξ due3o smoothed consumption of each income group. r ξ is a very small number which allows ξ t +1 to deviate from ξ t while ruling out explosive growth. Therefore, E t ( ξ t +1 ) ≈ ξ t . Our model alsoassumes E t ( A ,t +1 ) = A ,t due to the following relationship derived from equation (B.9): E t ( A ,t +1 ) = (1 − ψ ) w H ρ KS F KS,t κ + A ,t (B.10)Assume that the value of A ,t equals A , at t = 0 , it is easy to solve that: A ,t = (1 − ψ ) w H [ κ ρ KS ( κ − ρ KS ) ( F KS, − F KS,t κ t ) + A , κ t ] (B.11)According to Bansal and Yaron (2004) and Campbell and Shiller (1988a), the magnitude of κ is very close to 1. The value of A ,t is bounded by definition, thus the true κ and A , are notconcerns. As shown by equation (B.11), E t ( A ,t +1 ) = (1 − ψ ) w H [ κ ρ KS ( κ − ρ KS ) ( F KS, − ρ KS F KS,t κ t +11 ) + A , κ t +11 ] ≈ (1 − ψ ) w H [ κ ρ KS ( κ − ρ KS ) ( F KS, − F KS,t κ t ) + A , κ t ]= A ,t (B.12)when κ ≈ ρ KS ≈ . Therefore, assuming E t ( A ,t +1 ) = A ,t = E ( A ,t +1 ) is reasonable.Following Lettau et al. (2019), our paper assumes that the capital share growth rate follows anAR(1) process : F KS,t +1 = ρ KS F KS,t + e KSt (B.13)The functional form of A and A ,t can be solved: A = 1 − ψ − κ ρ (B.14) A ,t = 1 − ψ − κ w H ρ KS F KS,t (B.15)Following the same steps used in deriving the consumption premium, our paper further derives theequity premium. Equity returns have the following functional form: r m,t +1 = κ ,m + κ ,m z t +1 − z t + g d,t +1 (B.16) The constant is not significant due to the AR(1) estimation. The magnitude of ρ KS is 0.947. z t = A ,m + A ,m x t + A ,m,t ξ t (B.17)To further derive the equity premium r m,t , our paper invokes the Euler condition E [ exp ( m t +1 + r m,t +1 )] = 1 . The following condition holds: θlogδ − θψ g t +1 + ( θ − r a,t +1 + r m,t +1 = 0 (B.18)To solve A ,m and A ,m,t , substitute equations (B.3), (B.4), (B.16) and (B.17) into equation (B.18),collecting all terms containing x t and ξ t respectively: ( θ − − θψ ) x t + ( θ − κ ρ − A x t + κ ,m A ,m ρx t − A ,m x t φx t = − ψ x t + κ ,m A ,m ρx t − A ,m x t + φx t = 0 (B.19) ( θ − − θψ ) w H (1 + E t ( F KS,t +1 )) E t ( ξ t +1 ) + ( θ − κ E t ( A ,t +1 ) E t ( ξ t +1 ) − A ,t ξ t )+ κ ,m E t ( A ,m,t +1 ) E t ( ξ t +1 ) − A ,m,t ξ t = ( θ − − θψ − ψ ρ KS F KS,t ) w H + κ ,m A ,m,t − A ,m,t = 0 (B.20)The functional form of A ,m and A ,m,t can now be solved as: A ,m = φ − ψ − κ ρ (B.21) A ,m,t = θ − − θψ − κ ,m w H − w H ρ KS ψ (1 − κ ,m ) F KS,t (B.22)
B.0.1 Conditional on information set at time t
The conditional innovation of consumption return is: r a,t +1 − E t ( r a,t +1 ) = ση t +1 + κ A φ e σe t +1 + [ w H (1 + F KS,t +1 ) + κ A ,t +1 ] ξ t +1 − E t [ w H (1 + F KS,t +1 ) + κ A ,t +1 ] ξ t +1 = ση t +1 + λ r,e σe t +1 + λ r,ξ,t +1 ξ t +1 (B.23)5he conditional innovation of the pricing kernel is: m t +1 − E t ( m t +1 ) =( θ − − θψ ) ση t +1 + ( θ − κ A φ e ) σe t +1 + ( θ − κ ( A ,t +1 − E t ( A ,t +1 ))] ξ t +1 = λ η ση t +1 + λ e σe t +1 + λ ξ,t +1 ξ t +1 (B.24)In equations (B.23) and (B.24), the parameters are as follows: λ r,e = κ − ψ − κ ρ φ e (B.25) λ r,ξ,t +1 = ( w H + κ − ψ − κ w H ρ KS ) e KSt +1 (B.26) λ η = θ − − θψ (B.27) λ e = ( θ − κ − ψ − κ ρ φ e ) (B.28) λ ξ,t +1 = ( θ − κ − ψ − κ w H ρ KS ) e KSt +1 (B.29)The conditional consumption premium in the presence of time-varying economic uncertainty is E t ( r a,t +1 − r f,t ) = cov t (( m t +1 − E t ( m t +1 ))( r a,t +1 − E t ( r a,t +1 )) + 0 . V ar t ( r a,t +1 )= − ( λ η + λ r,e λ e − . λ r,e − . σ + E t ( λ r,ξ,t +1 λ ξ,t +1 − . λ ξ,t +1 ) (B.30)The conditional innovation of equity return is: r m,t +1 − E t ( r m,t +1 ) = φ d σ d,t +1 u t +1 + κ ,m A ,m φ e σe t +1 + κ ,m A ,m,t +1 ξ t +1 = φ d σ d,t +1 u t +1 + λ m,e σe t +1 + λ m,ξ,t +1 ξ t +1 (B.31)In equation (B.31), the parameters are as follows: λ m,e = κ ,m φ − ψ − κ ρ (B.32) λ m,ξ,t +1 = κ ,m w H ρ KS ψ (1 − κ ,m ) e KSt +1 (B.33)6he conditional equity premium in the presence of time-varying economic uncertainty is E t ( r m,t +1 − r f,t ) = cov t (( m t +1 − E t ( m t +1 ))( r m,t +1 − E ( r m,t +1 )) + 0 . V ar ( r m,t +1 )= − ( λ m,e λ e − . λ m,e ) σ + 0 . φ d σ d,t +1 + E t ( λ m,ξ,t +1 λ ξ,t +1 − . λ m,ξ,t +1 ) (B.34)where E t ( λ m,ξ,t +1 λ ξ,t +1 − . λ m,ξ,t +1 ) = 0 due to E t ( e KSt +1 ) = 0 ; σ g is close to σ due to verysmall ξ . Therefore, the expected equity premium can be viewed as a constant when the modelonly contains capital share growth as the independent variable. The deviation of equity returns iscorrelated with σ d,t +1 which is a function of F KS,t +1 and ξ r +1 . In our conditional model, F KS,t +1 isa variable that enters the variance equation. B.0.2 Unconditional case
Under unconditional expectations, E ( ξ t ) = 0 . Therefore, the unconditional innovation of consump-tion return is: r a,t +1 − E ( r a,t +1 ) = ση t +1 + κ A φ e σe t +1 + [ w H (1 + F KS,t +1 ) + κ A ,t +1 ] ξ t +1 = ση t +1 + λ r,e σe t +1 + λ ur,ξ,t +1 ξ t +1 (B.35)The unconditional innovation of the pricing kernel is: m t +1 − E ( m t +1 ) =( θ − − θψ ) ση t +1 + ( θ − κ A φ e ) σe t +1 + ( θ − κ A ,t +1 ) ξ t +1 = λ η ση t +1 + λ e σe t +1 + λ uξ,t +1 ξ t +1 (B.36)The unconditional consumption premium in the presence of time-varying economic uncertainty is E t ( r a,t +1 − r f,t ) = cov (( m t +1 − E t ( m t +1 ))( r a,t +1 − E ( r a,t +1 )) + 0 . V ar ( r a,t +1 )= − ( λ η + λ r,e λ e − . λ r,e − . σ + E [ λ ur,ξ,t +1 λ uξ,t +1 − . λ ur,ξ,t +1 ) ] (B.37)7n equations (B.35), (B.36) and (B.37), the parameters are as follows: λ r,e = κ − ψ − κ ρ φ e (B.38) λ ur,ξ,t +1 = w H (1 + F KS,t +1 ) + κ − ψ − κ w H ρ KS F KS,t +1 (B.39) λ η = θ − − θψ (B.40) λ e = ( θ − κ − ψ − κ ρ φ e ) (B.41) λ uξ,t +1 = ( θ − κ − ψ − κ w H ρ KS F KS,t +1 ) (B.42)The unconditional innovation of equity returns is: r m,t +1 − E ( r m,t +1 ) = φ d E ( σ d,t +1 ) u t +1 + κ ,m A ,m φ e σe t +1 + κ ,m A ,m,t +1 ξ t +1 = φ d σu t +1 + λ m,e σe t +1 + λ um,ξ,t +1 ξ t +1 (B.43)The unconditional expectation of E ( σ d,t +1 ) equals to σ due to E ( ξ t ) = 0 . In equations (B.43) and(B.46), the parameters are as follows: λ um,e = κ ,m φ − ψ − κ ρ (B.44) λ um,ξ,t +1 = κ ,m [ θ − − θψ − κ ,m w H − w H ρ KS ψ (1 − κ ,m ) F KS,t +1 ] (B.45)Therefore, the unconditional equity premium in the presence of time-varying economic uncertaintyis E ( r m,t +1 − r f,t ) = cov (( m t +1 − E t ( m t +1 ))( r m,t +1 − E ( r m,t +1 )) + 0 . V ar ( r m,t +1 )= − ( λ m,e λ e − . λ m,e − . φ d ) σ + E [ λ um,ξ,t +1 λ uξ,t +1 − . λ um,ξ,t +1 ) ] (B.46)where E [ λ m,ξ,t +1 λ ξ,t +1 − . λ um,ξ,t +1 ) ] is a function of E ( F KS ) . Given the DGP of capital sharegrowth in equation (B.13), the E ( F KS ) is a predicted value derived by an AR(1) model. In ourunconditional model, E ( F KS ) is a risk factor that enters the mean equation.8 .1 Factor Interpolation This paper estimates the risk exposure and risk premium of the capital share factor in a monthlysetting. However, the highest frequency of capital share data is quarterly. We interpolate capitalshare into monthly data due to the following reasons: 1) to avoid likely information loss whenconverting monthly portfolio returns into quarterly data; 2) to maintain a high degree of freedom inthe training set in Bayesian estimations; 3) to avoid projection errors: in the projection process ofthe capital share factor, the quarterly horizon is more sensitive than the monthly horizon in termsof model missimplification (Lamont, 2001). To convert the factor into monthly frequency, thispaper adopts the Chow-Lin interpolation approach, which is a linear regression based model withautocorrelation in the error term (Chow and Lin, 1971).
B.1.1 Indicator calculation
The commonly used Chow-Lin interpolation (Chow and Lin, 1971) and other alternative interpola-tion approaches (see Fernandez (1981), Litterman (1983), etc.) are all based upon the assumptionthat the monthly observations of interest satisfy a multiple regression relationship with some relatedseries. Accordingly, regression based interpolation methods require related series as indicators tocapture the latent monthly movement out of a quarterly time series.The capital share at time t , denoted by KS t , can be calculated as KS t = 1 − LS t (B.47)under the assumption that all risk sharing across workers and stockholders is imperfect (Lettau et al.,2019). LS t denotes labour share at time t .Table A1 shows the personal income and its disposition. The personal income and the compensationof employees are selected by this paper for indicator construction. An additional assumption ismade to increase the robustness of the indicator, as shown in equation (B.48), which is that the shareof compensation of employees is constantly proportional to the labour share. ES t = γ m LS t (B.48)9 able A1: Personal income and its disposition (FRED, 2019b)Unit: Bil. of $ 2011:12 Percentage 1972:01 PercentagePersonal income Compensation of employees 8,283.50 61% 644.5 72%
Proprietors’ income with inventoryvaluation and capital consumption adjustments 1,286.10 9% 80.2 9%Rental income of persons with capitalconsumption adjustment 508.3 4% 21.1 2%Personal income receipts on assets 2,049.30 15% 122.4 14%Personal current transfer receipts 2,367.10 17% 81 9%Less: Contributions for governmentsocial insurance, domestic 922 7% 50.3 6%Less: Personal current taxes 1,478.80 11% 97.5 11%Equals: Disposable personal income 12,093.60 89% 801.3 89%Less: Personal outlays 11,153.00 82% 694.5 77%Equals: Personal saving 940.5 7% 106.8 12%
Notes:
Personal income is the income obtained from provision of labour, land, and capital used in currentproduction and the net current transfer payments received from business and government. Percentage denotes theproportion of each element in personal income. Data selected are monthly, and covers the period from January1972 to December 2011.
In equation (B.48), ES t denotes the compensation of employees share over personal income, and γ m is a constant.The intuition behind the indicator selection is simple. Labour share is calculated by labour compen-sation divided by national income. Lettau et al. (2019) uses the labour share of national income inthe nonfarm business sector to compute capital share. However, national income is only availablequarterly. Therefore, personal income is the most appropriate proxy for monthly interpolation dueto its relevantly stable relationship with national income. In Table A1, personal income refers to thebroad measure of household income, and the compensation of employees denotes the gross wagespaid to employees within a certain period. Personal income is calculated by national income Labour compensation: compensation of employees in national currency. Here the period is one year. Gomme and Rupert (2004) show that indirect taxes and subsidies are stableover time. Hence, when studying the movement of data, the difference between national incomeand personal income can be ignored, because the difference is mainly caused by indirect tax andsubsidies.The calculation method for ES t is as follows: ES t = Com t P I t (B.49)where Com t denotes the compensation of employees and P I t denotes personal income.To roughly estimate γ m , this paper assumes that γ m and γ q share the same data generation process(DGP). Quarterly compensation share ES q and labour share LS q can be used to calculate quarterly γ q using the following function: γ q = ES q LS q (B.50)Table A2 shows the descriptive statistics of γ q calculated using equation (B.50). The standard Table A2: Descriptive Statistics of γ q Min. 1st Qu. Median Mean 3rd Qu. Max. Std.dev
Notes: γ q is estimated by compensation of employee share in personal income over labourshare (equation B.50). Data is quarterly and covers the sample period 1972:Q1 − γ q is assumed to share the same DGP as γ m . deviation of γ q is 0.020, and the mean and median are close to each other. The dispersion of γ q is low according to the descriptive statistics. Therefore, monthly γ m can be treated as a constantaccording to properties of quarterly γ q . Personal income equals to national income minus corporate profits with inventory valuation and capital consumption adjustments,taxes on production and imports less subsidies, contributions for government social insurance, net interest and miscellaneous paymentson assets, business current transfer payments (net), current surplus of government enterprises, and wage accruals less disbursements,plus personal income receipts on assets and personal current transfer receipts (FRED, 2019a)
Ind t , is calculatedas follows: Ind t = 1 − ES t (B.51)Figures A1 and A2 show the patterns of quarterly capital share factor and indicator, respectively.Although the capital share factor is overall more volatile compared to the indicator, comovementsbetween them can still be found easily by eyeballing the two figures. Figure A1: Capital share (quarterly).
B.1.2 Interpolation of Capital Share
Chow and Lin (1971) proposes an interpolation approach based upon the assumption of a regressionrelationship between the latent monthly time series of interest and indicators. Based upon Chow-Linmethod, Fernandez (1981) and Litterman (1983) approaches introduce unit roots in the error term.12 igure A2: Indicator Dynamics
This paper adopts the Chow-Lin approach for interpolation and also takes potential autocorrelationsin the error term of the target time series into consideration.Therefore, this paper assumes the following relationship holds: KS monthly = β + β ind Ind + µ (B.52)The error term µ has the following form to avoid spurious discontinuities between the last month ofthe previous year and the first month of the next year: µ t = ρ µ t − + (cid:15) t (B.53)where KS monthly denotes the target time series data matrix after interpolation. Ind is the monthlyindicator. µ t is assumed to be an autocorrelated variable as shown in equation (B.53). The covariancematrix of µ is denoted by V . β and β Ind denote the constant and the coefficient of the indicator,respectively. ρ is the coefficient of µ t − and captures the autocorrelation is present in the error term. (cid:15) t is i.i.d. and follows a standard normal distribution.13he generalized least squares estimators are defined as follows in this paper: β Ind = (
Ind (cid:48) V − Ind ) − Ind (cid:48) V − KS monthly (B.54)where V = C ( A (cid:48) A ) − C (cid:48) (B.55)In equation (B.55), A is an auxiliary matrix with the following form ( n equals to the quarterly datalength) to factor in the autocorrelation of the error term: A = (1 − ρ ) . . . − ρ . . . − ρ . . . ... ... . . . . . . ... ... .... . . . . . − ρ − ρ n × n (B.56) C is an n × n matrix with the following form: C = . . . . . .. . . . . . n × n (B.57)Grid search is used in the estimation process of the autocorrelation coefficient ρ . The objectivefunction of grid searches could be the Weighted Least Square or the Log Likelihood Function. Theformats of the objective functions are as follows (Bournay and Laroque, 1979): W LS = µ (cid:48) V − µ (B.58) LL = − n ln (2 π µ (cid:48) V − µn − − log ( | V | ) − n (B.59)To select proper options of the Chow-Lin interpolation, Table A3 shows the information criteriavalues under different settings. According to this table, the first element Chow-Lin interpolation14ith constant and WLS as an objective function has the lowest AIC and BIC. Hence, this paperchooses this Chow-Lin setting to generate artificial monthly capital share data. Table A3: Information Criteria of Different Chow-Lin Settings
Chow-Lin Settings (N=160, n=480, Quarterly to Monthly)
Last Element(opc, rl) WLS LLAIC BIC AIC BIC (0, [ ]) -11.222 -11.183 -11.201 -11.162(1, [ ]) -11.384 -11.327 -11.349 -11.291
First Element(opc, rl) WLS LLAIC BIC AIC BIC (0, [ ]) -11.349 -11.310 -11.329 -11.291(1, [ ]) -11.404 -11.346 -11.373 -11.315
Notes: opc denotes the option related to the constant. When opc equals zero or one, the regressionincludes zero or one constant respectively. rl denotes the innovational parameter. rl = [ ] indicates theautocorrelation parameter ρ is generated by grid search, and the calculation process adopts 100 gridsof ρ ∈ [0 . , . . WLS and LL denotes the objective function for the grid search: Weighted LeastSquare and Log Likelihood Function respectively. The coefficients calculated by the Chow-Lin interpolation are shown in Table A4. The estimatedconstant and the indicator coefficient are both larger than two standard deviations. Although theestimated ρ is close to the upper bound (0.999) of the grid search, since ρ does not go beyond 1,the conditions of partition of residuals still hold (Bournay and Laroque, 1979). Figure A3 plots theinterpolated monthly capital share data. 15 able A4: Chow-Lin coefficients under selected model specification Values Std.dev t-statisticsConstant ( β ) β Ind ρ Notes:
Bold denotes significant or feasible autocorrelation coefficients. β and β Ind are both significant at95% confident level. The estimated autocorrelation coefficient ρ is within the range of ρ ∈ [0 . , . for grid search, indicating no unit roots present in the error term. Figure A3:
Interpolated Capital Share Descriptive Statistics
The descriptive statistics of all portfolio returns and control factors are in Tables (A5) to (A8) below:
Table A5: 10 REV sorted portfolio returns (%)
10 Size/REV sorted portfolios, value-weightedPortfolio/Factor Mean Median Std. dev. Sharpe ratioLoPRIOR 1.000 1.135 7.184 0.139PRIOR2 1.152 1.140 5.674 0.203PRIOR3 1.154 1.375 5.040 0.229PRIOR4 1.039 1.335 4.656 0.223PRIOR5 0.996 1.165 4.422 0.225PRIOR6 0.907 1.240 4.270 0.213PRIOR7 0.892 1.105 4.227 0.211PRIOR8 0.881 1.155 4.375 0.201PRIOR9 0.750 0.855 4.676 0.161HiPRIOR 0.676 0.795 5.403 0.125
Notes:
Data frequency is monthly. Time span of data is from January 1964 to August 2018. able A6: 25 Size/BM sorted portfolio returns (%)
25 Size/BM sorted portfolios, value-weightedPortfolio/Factor Mean Median Std. dev. Sharpe ratioSMALLLoBM 0.681 1.060 7.854 0.087ME1BM2 1.213 1.523 6.849 0.177ME1BM3 1.192 1.254 5.934 0.201ME1BM4 1.407 1.450 5.644 0.249SMALLHiBM 1.491 1.485 5.946 0.251ME2BM1 0.923 1.376 7.094 0.130ME2BM2 1.174 1.456 5.924 0.198ME2BM3 1.273 1.530 5.374 0.237ME2BM4 1.315 1.528 5.197 0.253ME2BM5 1.367 1.788 5.964 0.229ME3BM1 0.920 1.546 6.515 0.141ME3BM2 1.20 1.505 5.383 0.223ME3BM3 1.19 1.486 4.943 0.230ME3BM4 1.268 1.442 4.855 0.261ME3BM5 1.414 1.524 5.587 0.253ME4BM1 1.035 1.157 5.823 0.178ME4BM2 1.018 1.215 5.052 0.201ME4BM3 1.091 1.354 4.906 0.222ME4BM4 1.229 1.420 4.720 0.260ME4BM5 1.210 1.423 5.626 0.215BIGLoBM 0.893 0.998 4.569 0.195ME5BM2 0.915 1.073 4.375 0.209ME5BM3 0.942 1.215 4.231 0.223ME5BM4 0.872 0.995 4.581 0.190BIGHiBM 1.052 1.319 5.326 0.198
Notes:
Data frequency is monthly. Time span of data is from January 1964 to August 2018. able A7: 25 Size/INV sorted portfolio returns (%)
25 Size/INV sorted portfolios, value-weightedPortfolio/Factor Mean Median Std. dev. Sharpe ratioSMALLLoINV 1.353 1.376 7.183 0.188ME1INV2 1.357 1.413 5.599 0.242ME1INV3 1.385 1.631 5.603 0.247ME1INV4 1.266 1.561 5.926 0.214SMALLHiINV 0.783 1.028 7.049 0.111ME2INV1 1.280 1.636 6.298 0.203ME2INV2 1.296 1.586 5.209 0.249ME2INV3 1.315 1.490 5.197 0.253ME2INV4 1.287 1.595 5.633 0.228ME2INV5 0.900 1.235 6.893 0.131ME3INV1 1.263 1.460 5.673 0.223ME3INV2 1.310 1.475 4.775 0.274ME3INV3 1.196 1.383 4.761 0.251ME3INV4 1.206 1.495 5.273 0.229ME3INV5 0.919 1.307 6.441 0.143ME4INV1 1.160 1.455 5.318 0.218ME4INV2 1.127 1.388 4.709 0.239ME4INV3 1.152 1.402 4.620 0.249ME4INV4 1.154 1.269 4.867 0.237ME4INV5 0.972 1.224 6.240 0.156BIGLoINV 1.083 1.125 4.554 0.238ME5INV2 0.937 0.920 3.957 0.237ME5INV3 0.894 1.000 4.066 0.220ME5INV4 0.883 1.045 4.379 0.202BIGHiINV 0.877 1.113 5.390 0.163
Notes:
Data frequency is monthly. Time span of data is from January 1964 to August 2018. able A8: 25 Size/OP sorted portfolio returns (%).
25 Size/OP sorted portfolios, value-weightedPortfolio/Factor Mean Median Std. dev. Sharpe ratioSMALLLoOP 0.955 0.980 7.218 0.132ME1OP2 1.331 1.471 5.791 0.230ME1OP3 1.273 1.581 5.583 0.228ME1OP4 1.357 1.505 5.739 0.237SMALLHiOP 1.240 1.477 6.546 0.190ME2OP1 1.001 1.522 6.944 0.144ME2OP2 1.193 1.633 5.640 0.212ME2OP3 1.209 1.510 5.244 0.230ME2OP4 1.194 1.298 5.509 0.217ME2OP5 1.352 1.698 6.143 0.220ME3OP1 0.948 1.208 6.535 0.145ME3OP2 1.154 1.485 5.091 0.227ME3OP3 1.138 1.384 4.866 0.234ME3OP4 1.146 1.286 5.106 0.225ME3OP5 1.302 1.554 5.753 0.226ME4OP1 0.955 1.077 6.044 0.158ME4OP2 1.087 1.391 5.057 0.215ME4OP3 1.066 1.250 4.720 0.226ME4OP4 1.131 1.293 4.833 0.234ME4OP5 1.200 1.558 5.307 0.226BIGLoOP 0.753 1.051 5.444 0.138ME5OP2 0.753 0.926 4.412 0.171ME5OP3 0.903 1.033 4.325 0.209ME5OP4 0.870 1.127 4.357 0.200BIGHiOP 0.992 1.106 4.273 0.232
Notes:
Data frequency is monthly. Time span of data is from January 1964 to August 2018. able A9: Risk Factors Descriptive Statistics (%) Mean Median Std.dev. Sharpe ratio F KS January 1964 - January 1974 -0.245 -0.502 2.690 -0.091January 1974 - August 2018 0.435 0.195 2.336 0.186January 1964 - August 2018 0.310 0.074 2.416 0.129 E ( F KS ) January 1964 - January 1974 0.065 0.024 0.085 0.764January 1974 - August 2018 0.051 0.017 0.083 0.615January 1964 - August 2018 0.054 0.018 0.084 0.643
Notes: F KS denotes the capital share factor and E ( F KS ) denotes the capital share variability factor. Thetraining sample spans January 1964 to January 1974. The sample used for estimation spans January 1974to August 2018. The full sample spans January 1964 to August 2018.denotes the capital share variability factor. Thetraining sample spans January 1964 to January 1974. The sample used for estimation spans January 1974to August 2018. The full sample spans January 1964 to August 2018.