Journal of Banking & Finance 42 (2014) 11–22

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Journal of Banking & Finance

journal homepage: www.elsevier .com/locate / jbf

Human capital, household capital and asset returns q

http://dx.doi.org/10.1016/j.jbankfin.2014.01.0280378-4266/� 2014 Elsevier B.V. All rights reserved.

q We thank the editor and the referee for their comments and suggestions. Wealso thank seminar participants of Xiamen University and Southwestern Universityof Finance and Economics for discussions. Ren’s research is supported by theNatural Science Foundation of Fujian Province of China (2011J01384), the Funda-mental Research Funds for the Central Universities (2013221022) and the NaturalScience Foundation of China (71301135, 71203189, 71131008). Yuan’s research issupported by the Natural Science Foundation of China (71301135, 71203189).⇑ Corresponding author. Tel.: +86 592 218 6025; fax: +86 592 218 7708.

E-mail address: renxmu@gmail.com (Y. Ren).

Yu Ren a,⇑, Yufei Yuan a, Yang Zhang b

a The Wang Yanan Institute for Studies in Economics, Xiamen University, Fujian 361005, Chinab Department of Economics, Cornell University, Ithaca, NY 14853, USA

a r t i c l e i n f o

Article history:Received 31 August 2013Accepted 18 January 2014Available online 30 January 2014

JEL classification:E21E44D12

Keywords:Human capitalHousehold capitalConsumption-wealth ratioAsset returns

a b s t r a c t

Sousa (2010a) shows that the residuals from the common trend among consumption, financial wealth,housing wealth and human capital, cday, can predict quarterly stock market returns better than cay fromLettau and Ludvigson (2001), which considers aggregate wealth instead. In this paper, we use a moreappropriate proxy of human capital, which alleviates the potential correlation between the residualsand the regressors and makes the estimation more precise. In addition, we extend housing wealth tohousehold capital by taking durable goods into consideration. The new predictor is proposed accordingly.Empirically, we find that our predictor is superior to the other alternatives.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

The predictability of asset returns using macroeconomic vari-ables is one of the most important research areas in finance. Manypredictors have been intensively studied. More recently, a lot ofeconomically motivated predictors have been proposed, for exam-ple, the ratio of housing wealth to human capital (Lustig and vanNieuwerburgh, 2005), the composition risk (Piazzesi et al., 2007;Yogo, 2006), the trend deviation of the long-run relationship be-tween nondurable consumption, non-asset income, wealth andthe relative price of durables to nondurables (Fernandez-Corugedoet al., 2007), the residuals of the trend relationship between hous-ing wealth and labor income (Sousa and wealth, 2010b), as well asthe ratio of asset wealth to human capital (Sousa, 2012a,b,c).

Of all the predictors in the literature, the transitory deviationfrom the common trend in consumption, asset wealth and humancapital (Lettau and Ludvigson, 2001), cay, is one of the most

successful. Economic intuition is that investors who want tosmooth their consumption adjust their current consumption ifthey expect transitory movements in their asset wealth causedby variations in expected returns. When the expected return rises,a forward-looking investor increases his current consumption.Conversely, when the expected return declines, he decreases it.Sousa (2010a) argues that some components of asset wealth havedifferent characteristics and that it is appropriate to disaggregatethem from asset wealth. Using US and UK data, he shows thatthe residuals from the common trend among consumption, finan-cial wealth, housing wealth and human capital, cday, can predictquarterly stock market returns better than cay proposed by Lettauand Ludvigson (2001). Moreover, Afonso and Sousa (2011) findthat cay and cday are not market-restricted as they can also predictstock returns in other OECD countries.

The construction of cay or cday involves human capital. Unob-servable human capital plays important role in recent asset pricingmodels, for example, Julliard (2004) and Wei (2005). However,how to proxy it has not been paid enough attention. The first con-tribution of this paper is that we improve the prediction abilities ofcay and cday by addressing this proxy issue properly.

From a microeconomic perspective, economists have proposedvariables such as labor inputs with various adjustments (Denison,1967), adult literacy rates and school enrollment ratios (Azariadisand Drazen, 1990; Romer, 1990), and, the most popular, averageyears of schooling (Islam, 1995; O’Neill, 1995; Barro, 2001) to

12 Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22

specify human capital stocks. From a macroeconomic perspective,human capital is usually defined as the present value of futurelabor income and is measured in the aggregate (Auerbach et al.,1992; Auerbach et al., 1994). As Macklem (1997) mentions, themacro or aggregate approach has two important advantages: first,it facilitates our understanding of the joint statistical properties ofshocks in income and interest rates; second, at the macro level, thedata requirements are much less onerous, making this approacheasily applicable to different countries.

Both Lettau and Ludvigson (2001) and Sousa (2010a) take themacro approach and substitute human capital (logarithmic value)with a linear function of current labor income (logarithmic value).Although this substitution is supported by economic theory anddata, it is not appropriate to use it to construct cay or cday as bothcay and cday are obtained using the ‘‘dynamic least squares’’ (DLS)regression proposed by Stock and Watson (1993). The DLS specifi-cation adds leads and lags of the first difference of the right-handside variables to a standard ‘‘ordinary least squares’’ (OLS) regres-sion to eliminate the effects of regressor endogeneity on the distri-bution of the least squares estimator. However, if human capital issubstituted by a linear function of current labor income, it causes acorrelation between the residuals of the regression and the leadsand the lags of the first difference components. This correlationjeopardizes the good finite-sample properties of the DLS estima-tors. In order to eliminate it, we follow Macklem (1997) using aMarkov chain to calculate the sum of the expected present valueof labor incomes, and treat this as a proxy for human capital. Thisproduces better estimators.

The second contribution of this paper is a closer examination ofthe importance of wealth composition, as first emphasized bySousa (2010a). Sousa (2010a) disaggregates aggregate wealth intofinancial wealth, human capital and housing wealth, and finds asuperior predictor of financial asset returns over cay. Similar tohousing wealth, durable goods (such as clothing and furniture) alsohave these special characteristics unlike financial wealth. They aredifferent from financial wealth with respect to liquidity, utilityfrom ownership rights, and the different distributions across in-come groups, among others. Many researchers have examinedthese differences, for instance, Hess (1973), Mankiw (1982), Gross-man and Laroque (1990), Caballero (1993) and Hong (1996). More-over, the value of durable goods is increasing rapidly. Recently, itaccounts for around 7% of aggregate wealth. Therefore, we definethe sum of durable goods and housing wealth as household capitaland disaggregate them from aggregate wealth.

So, we use the expected present value of labor incomes as aproxy for human capital, and estimate the transitory deviationfrom the common trend in consumption, financial wealth, humancapital and household capital. We define this transitory deviationas a new predictor, cadh. cadh should outperform cay and cday be-cause the parameters are estimated more precisely and durablegoods are taken into consideration in cointegrating.

Empirically, we collect US quarterly data from 1952 to 2011,and split it into two subsamples. The first is from the first quarterof 1952 to the fourth quarter of 1976; the second is from the firstquarter of 1977 to the fourth quarter of 2011. The reason for doingthis is that the cointegrating vectors among consumption, financialwealth, human capital and household capital are different for thesetwo subsamples. The difference of the cointegrating vectorsreflects the change in the long-run elasticities of consumption withrespect to financial wealth, household capital, and human capital.Specifically, the elasticities with respect to financial wealth andhuman capital increase and decrease respectively, while theelasticity with respect to household capital remains relativelyunchanged.

Finally, we compare the predictive power of cadh; cay and cday.We find that in the first subsample, our predictor can explain at

most 12% variation over the next 8 quarters for in-sample forecast-ing while cay and cday explain, at most, 7% and 9% variation,respectively. In the second subsample, the numbers increase to31%, 26% and 27%, respectively. Moreover, we show that the supe-riority of our predictors is due to both good measure of human cap-ital and usage of household capital. For out-of-sample forecasting,all three predictors improve the mean squared error (MSE) com-pared with the constant return model, and the improvements aresignificant. While, our predictor is the best in terms of MSE.

The rest of this paper is organized as follows: Section 2 de-scribes our estimation model; Section 3 reports the empirical anal-ysis; and Section 4 concludes.

2. A new measure of the consumption-aggregate wealth ratio

As shown by Lettau and Ludvigson (2001), the budget con-straint of a consumer in a representative agent economy is

Wtþ1 ¼ ð1þ Rw;tþ1ÞðWt � CtÞ;

where Wt denotes aggregate wealth at time t;Ct denotes consump-tion at time t and Rw;tþ1 is the return on aggregate wealth betweenperiod t and period t þ 1.

Campbell and Mankiw (1989) show that when the consump-tion-wealth ratio is stationary, the budget constraint can beapproximated by a first-order Taylor expansion:

Dwtþ1 � kþ rw;tþ1 þ ð1� 1=/wÞðct �wtÞ;

where /w is the steady-state ratio of new investment to totalwealth, ðW � CÞ=W , and k is a constant that plays no role in theanalysis. Solving this difference equation forward and imposing thatlimi!1 /i

wðctþi �wtþiÞ ¼ 0, the log consumption-wealth ratio can bewritten as

ct �wt ¼X1i¼1

/iwðrw;tþi � DctþiÞ: ð1Þ

Taking the conditional expectation on both sides of Eq. (1), weobtain

ct �wt ¼ Et

X1i¼1

/iwðrw;tþi � DctþiÞ; ð2Þ

where Et is the expectation operator conditional on informationavailable at time t.

Following Sousa (2010a), we decompose aggregate wealth as

Wt ¼ Ft þ Dt þ Ht ; ð3Þ

where Ft is financial wealth and Ht is human capital, as in Sousa(2010a). However, we define Dt as household capital, the sum ofhousing wealth and durable goods. Durable goods, such as carsand furniture, can provide a service flow for several years. Hencethey constitute part of aggregate wealth. Quantitatively, durablegoods account for 7% of aggregate wealth in 2011, according tothe Bureau of Economic Analysis. Qualitatively, durable goods aresimilar to housing in liquidity, utility from ownership and incomedistribution. Hence we put them together and denote them house-hold capital.

Eq. (3) can be approximated as

wt � af f t þ addt þ ð1� af � adÞht; ð4Þ

where af and ad equal, respectively, the steady-state share of finan-cial wealth holdings in total wealth, F=W , and the steady-stateshare of household capital holdings in total wealth, D=W .

The return to aggregate wealth can be decomposed into threecomponents:

1þ Rw;t � af ð1þ Rf ;tÞ þ adð1þ Rd;tÞ þ ð1� af � adÞð1þ Rh;tÞ: ð5Þ

Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22 13

Campbell (1996) shows that Eq. (5) can be transformed into anapproximation equation for log returns taking the form

rw;t � af rf ;t þ adrd;t þ ð1� af � adÞrh;t: ð6Þ

Substituting Eq. (4) and Eq. (6) into the ex-ante budget constraint,Eq. (2), gives

ct � af f t � addt � ð1� af � adÞht

¼ Et

X1i¼1

/iwðaf rf ;tþi þ adrd;tþi þ ð1� af � adÞrh;tþi � DctþiÞ: ð7Þ

Eq. (7) shows that the trend deviation term will be a good proxy formarket expectations of future financial returns, rf ;tþi, as long as ex-pected future returns on household capital, rd;tþi, returns on humancapital, rh;tþi, and consumption growth, Dctþi, are not too volatile. Toobtain the cointegrating vector, we follow Stock and Watson (1993)to run the DLS regression as

ct ¼ lþ bf f t þ bddt þ bhht þXk

i¼�k

bf ;iDft�i þXk

i¼�k

bd;iDdt�i

þXk

i¼�k

bh;iDht�i þ et: ð8Þ

We need to assume that et is uncorrelated with Dft�i; Ddt�i andDht�i (i ¼ �k; � � � ; k) in order to take full advantage of the DLSregression.

Lettau and Ludvigson (2001) substitute unobservable humancapital with a linear function of labor income in Eq. (8). However,by its definition in macroeconomics, human capital is the sum ofthe discounted values of current and future labor income net oftaxes. Here we denote Lt as labor income, s as the average tax ratecorresponding to labor income, and hence Xt ¼ Ltð1� sÞ denotesnet labor income. Let xt denote the growth rate of net labor incomeand rt denote the discount rate of labor income corresponding tohuman capital. Then, human capital can be expressed as

Ht ¼ Ltð1� sÞ 1þ Et

X1i¼1

Pij¼1

1þ xtþj

1þ rtþj

� �" #( )

� Ltð1� sÞð1þ CtÞ; ð9Þ

where Ct represents the expectation of the term in the inner squarebrackets. We call it the cumulative growth factor. By taking logs onboth sides of Eq. (9), we obtain

ht ’ c0 þ lt þ Ct ; ð10Þ

where ht is the log value of human capital Ht , and lt is the log valueof labor income Lt . Eq. (10) provides support for the literaturesubstituting ht with lt , especially when labor income is very persis-tent and similar to the I (1) process. However, this substitution alsogenerates the serious problem of finite-sample estimation bias dueto Ct . Ct is a function of the current (and expected future) discountrates rt and the growth rates of net labor income xt . Both rt and xt

are endogenous variables of the aggregate economy and are corre-lated with the growth rates of labor income, financial wealth andhousehold capital. This correlation weakens the desirable propertiesof our estimation.

The problem with Eq. (9) is that the cumulative growth factor Ct

is not directly observable. An ambitious approach to measure Ct

would be to develop a fully articulated macro model and theninvoke the rational expectations hypothesis. However, sinceunderstanding the economic mechanism of Ct is not the main pur-

pose of this paper, following Macklem (1997), we assume that 1þxtþj

1þrtþj

follows a Markov process. Given this assumption, there exists afunction F such that Ct ¼ Fð1þxt

1þrtÞ. Because human capital is not ob-

servable, we cannot estimate F directly. A more modest approach

is to estimate a bivariate VAR to characterize the joint distributionof the growth rate of labor income net of tax and the real interestrate. The estimated bivariate VAR is then approximated as a dis-crete-valued finite-state Markov chain (Tauchen, 1986), and thisapproximated system is used to compute the expected discountedvalue of future net income growth.

We follow Macklem (1997) and use VAR (1) to predict xtþ1 andrtþ1. Due to the computational constraints associated with the fi-nite-state Markov chain approximation, the VAR is restricted tobe of the first order. As Macklem (1997) argues, this constraintdoes not appear to be too serious, since the first-order model cap-tures most of the predictive content of past income growth andreal interest rates. A bivariate VAR (1) model to predict xtþ1 andrtþ1 can be expressed as:

xtþ1 ¼ a1 þ h11xt þ h12rt þ �1;tþ1;

rtþ1 ¼ a2 þ h21xt þ h22rt þ �2;tþ1:ð11Þ

Then, we obtain the proxy for human capital based on Eq. (9). Formore technical details, please refer to the Appendix A.

Therefore, besides including durable goods, our decompositiondiffers from those of Lettau and Ludvigson (2001) and Sousa(2010a) by the selection of the proxy for human capital. Theyapproximate unobservable human capital by a linear function oflabor income, whereas, we treat the present value of labor incomein all periods as human capital. This proxy can alleviate the poten-tial correlation between the residuals and the regressors at allleads and lags. Hence, the estimators obtained by DLS are consis-tent and have good finite-sample properties.

We will denote the trend deviation term in our approach ascadh and the analogues in Lettau and Ludvigson (2001) and Sousa(2010a) as cay and cday, respectively.

We need to emphasize several issues before presenting theempirical results. First, there is a vast amount of literature discuss-ing whether it is appropriate to obtain the predictor by regressingthe model as specified in Lettau and Ludvigson (2001). Hahn andLee (2001) find that there are substantial changes in the cointe-grating parameter estimates over the sample period. Brennanand Xia (2005) show that the predictive power of cay arises mainlyfrom a look-ahead bias. Lettau and Ludvigson (2005) explain whythe critique in Brennan and Xia (2005) is misplaced. Koop et al.(2008) investigate the robustness of the results and argue for theuse of Bayesian model averaging. Corte et al. (2010) provide someevidence for the predictability of cay being related to structurebreak. Our paper circumvents this debate, and improve the pre-dictability of cay based on cday of Sousa (2010a). We do not implythat our predictor would not suffer from the same problems. How-ever, improving the predictability of cay is the focus of our papersince it makes sense to investigate those problems of the predictorwhen the predictor has better predictability. Therefore, we followLettau and Ludvigson (2001) and Sousa (2010a) in setting ourregression model.

Second, our regression model uses a simple but effective meth-od to obtain human capital, and easily alleviates the correlation be-tween the regressors and the residuals. In earlier work, Campbell(1996), Shiller (1993), Jagannathan and Wang (1996), Alvarezand Jermann (2004), Lustig and van Nieuwerburgh (2008), and Pal-acios (2011) make particular and very different assumptions to ob-tain human capital. Since measuring human capital is not the mainpurpose of this paper, we do not claim that our proxy is better thanthe alternatives in other studies.

Last, durable goods play an increasing important role in recentempirical studies. Yogo (2006) shows that durable consumption iscrucial to understanding the equity premium, followed by Lustigand Verdelhan (2007), Phalippou (2007), Gomes et al. (2009), and

14 Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22

Yang (2011). Our regression model sheds some light on the impor-tance of durable goods in predictability regression models.

3. Empirical results

3.1. Data

We collect US quarterly data from the first quarter of 1952 tothe fourth quarter of 2011. Consumption, income, tax, population,and the PCE deflator are collected from the Bureau of EconomicAnalysis, NIPA tables. The sources for financial wealth, housingwealth, durable goods and three-month Treasury bill rates arethe Board of Governors of the Federal Reserve System. The financialdata are collected from the Center for Research in Security Prices(CRSP) database. All of the nominal variables are transformed intoreal terms by dividing by the PCE deflator (based on 2005). All thedata are seasonally adjusted at annual rates, measured in billionsof dollars, in per capita terms and expressed in logarithmic form.We lag the wealth data once, so that the observation of wealth int corresponds to the value at the beginning of the period t þ 1. Adetailed data description is in next section. Table 1 summarizesthe statistical properties of the data.

Consumption in this model includes not only nondurable goodsand services (excluding clothing and shoes) but also the serviceflows of household capital composed of durable goods and housingwealth. However, the latter are not observable because residentialinvestment and expenditures on durable goods are only replace-ments and additions to the stock of household capital. Therefore,to measure consumption, we follow the standard methods appliedby Galí (1990), among others. We assume that total consumption isa stationary multiple of nondurable goods and services. So aggre-gate consumption satisfies

Ct ¼ UtCn;t ;

where Ct is aggregate consumption and Cn;t is the consumption ofnondurables and services. Hence, ct ¼ /t þ cn;t where /t ¼ logðUtÞ.We further assume that /t ¼ �/þ mt and mt is subject to i.i.d. normaldistribution. Under this assumption, we can replace ct with cn;t

without changing the linear parameters of Eq. (8).

3.2. Data details

In this section, we describe the data in more details.

� Consumption: Consumption data is the sum of nondurablegoods and services, excluding shoes and clothing. Series com-prises the period 1947Q1–2012Q1. The data source is the USDepartment of Commerce, Bureau of Economic Analysis, NIPATable 2.3.5. The data are transformed to be quarterly, seasonallyadjusted at annual rate, in billions of dollars (2005 prices), percapita and in logarithmic form.

Table 1Statistics properties.

Mean Standard deviation Notation

Treasury bill rate 0.004 0.006 rConsumption 9.73 0.33 cAsset wealth 11.43 0.44 mFinancial wealth 10.99 0.37 fHousing wealth 9.92 0.43 hsHousehold capital 10.39 0.34 dLabor income 9.57 0.27 lLabor income growth rate 0.005 0.01 x

This table reports the mean and the standard deviation of the quarterly data col-lected from 1952Q1 to 2011Q4.

� After-tax labor income: After-tax labor income is defined asthe sum of wage and salary disbursements (line 3), personalcurrent transfer receipts (line 16) and employer contributionsfor employee pension and insurance funds (line 7) minus per-sonal contributions for government social insurance (line 25),employer contributions for government social insurance (line8) and taxes. Taxes are defined as: [(wage and salary disburse-ments (line 3))/(wage and salary disbursements (line 3)+ proprietor’s income with inventory valuation and capital con-sumption adjustments (line 9) + rental income of persons withcapital consumption adjustment (line 12) + personal dividendincome (line 15) + personal interest income (line 14))]⁄

(personal current taxes (line 26)). Series comprises the period1947Q1–2012Q1. The data source is the Bureau of EconomicAnalysis, NIPA Table 2.1. The data are transformed to be quar-terly, seasonally adjusted at annual rate, in billions of dollars(2005 prices), per capita and in logarithmic form.� Human capital: Human capital is defined as the expected pres-

ent value of current and future after-tax labor income. SeeAppendix A for technical details. The data are transformed tobe quarterly, seasonally adjusted at annual rate, in billions ofdollars (2005 prices), per capita and in logarithmic form.� Asset wealth: Asset wealth is defined as the net worth of

households and nonprofit organizations. Series comprises theperiod 1945–2012. The quarterly data are collected from1952Q1. The data source is the Board of Governors of the Fed-eral Reserve System, Flow of Funds Accounts, Table B.100(Series FL152090005.Q). The data are transformed to be quar-terly, seasonally adjusted at annual rate, in billions of dollars(2005 prices), per capita and in logarithmic form.� Financial wealth: Financial wealth is defined as total financial

assets (Series FL154090005.Q) minus total liabilities of house-holds and nonprofit organizations (Series FL154190005.Q).Series comprises the period 1945–2012. The quarterly dataare collected from 1952Q1. The data source is the Board ofGovernors of the Federal Reserve System, Flow of FundsAccounts, Table B.100. The data are transformed to be quarterly,seasonally adjusted at annual rate, in billions of dollars (2005prices), per capita and in logarithmic form.� Housing wealth: Housing wealth is defined as households,

owner-occupied real estate including vacant land and mobilehomes at market value (Series FL155035015.Q) minus house-holds and nonprofit organizations, home mortgages liability(Series FL153165105.Q). Series comprises the period 1945–2012. The quarterly data are collected since 1952Q1. The datasource is the Board of Governors of the Federal Reserve System,Flow of Funds Accounts, Table B.100. The data are transformedto be quarterly, seasonally adjusted at annual rate, in billions ofdollars (2005 prices), per capita and in logarithmic form.� Durable goods: Durable goods are defined as households and

nonprofit organizations’ consumer durable goods (current costbasis, series FL155111005.Q). Series comprises the period1945–2012. The quarterly data are collected since 1952Q1.The data source is the Board of Governors of the Federal ReserveSystem, Flow of Funds Accounts, Table B.100. Household capitalis defined as the sum of housing wealth and durable goods. Thedata are transformed to be quarterly, seasonally adjusted atannual rate, in billions of dollars (2005 prices), per capita andin logarithmic form.� Population: Population series comprises the period 1947Q1–

2012Q1. The data source is the US Department of Commerce,Bureau of Economic Analysis, NIPA Table 2.3.5, Line 40.� Price deflator: Price deflator is the price index for personal con-

sumption expenditure (2005 = 100). Series comprises the period1947Q1–2012Q1. The data source is the US Department of Com-merce, Bureau of Economic Analysis, NIPA Table 2.3.4, Line 1.

1952Q1 1962Q1 1972Q1 1982Q1 1992Q1 2002Q1 2012Q18

9

10

11

12

13

14

15

16

Time

Hum

an C

aptia

l V.S

. Lab

or In

com

e

labor incomehuman capital

Fig. 1. Human capital and labor income.

Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22 15

� Interest rate: Interest rate (risk-free rate) is defined as the 3-month US Treasury bills rate. We take the arithmetic averageof three consecutive months in each quarter to transform theoriginal monthly data to quarterly. Series comprises the periodJanuary 1934–May 2012. The data source is H.15 Publication ofthe Board of Governors of the Federal Reserve System. The dataare transformed to be real values by taking the differencebetween the nominal values and the inflation rates.� Asset returns: Asset returns are the total returns on the S&P

Composite Index or the CRSP NYSE-AMEX-NASDAQ value-weighted index from the CRSP database. The data are trans-formed to be real values by taking the difference between thenominal values and the inflation rates.

3.3. Calculation of human capital

After we collect the data, we calculate human capital in the firststep. Based on Eq. (9), we need to predict the future labor incomegrowth rate, xtþ1; xtþ2,. . ., and future treasury rate, rtþ1; rtþ2,. . .,according to the VAR (1) model described by Eq. (11). Eq. (11)states that:

xtþ1 ¼ a1 þ h11xt þ h12rt þ �1;tþ1;

rtþ1 ¼ a2 þ h21xt þ h22rt þ �2;tþ1:

The estimation results of the VAR (1) model are summarized inTable 2.

Table 2 shows that the estimated h22 is significantly positive,which implies that the real interest rate exhibits positive persis-tence. This is also consistent with the finding in Macklem (1997).Our h12 estimator is insignificant, which means that the interestrates do not significantly affect the wage growth rate of the nextperiod. This can be explained by the driving force of interest rates.If positive technological shocks drive up interest rates, then thegrowth rate of wages could also increase. However, if high interestrates are associated with an inactive economy, the growth rate ofthe wage is dampened. Since both scenarios occasionally occur inthe US, the data do not display a clear relation between interestrates and wage growth rates.

We choose 25�25 points to approximate the labor incomegrowth rates and the three-month Treasury bill rates. The residualsin the VAR are assumed to follow a normal distribution, such thatwe are able to calculate the Markovian transition matrix. Then, weobtain human capital according to Eq. (9). The results are plotted inFig. 1. In order to show the difference between human capital andlabor income, we also plot labor income in Fig. 1.

We run a linear regression

ht ¼ a0 þ a1lt þ zt ;

where zt is the error term and can be regarded as the estimator ofCt . We find the estimated a1 ¼ 0:999 with a confidence interval

Table 2Estimation results.

a1 h11 h12 R2

Estimator 0.005 �0.07 0:12 0.10t-Stat (6.19) (�2.09) (0.99)

a2 h21 h22

Estimator 0:001 �0:07 0:78 0.62t-Stat (3.82) (�2.88) (18.30)

This table reports the VAR estimation results for the growth rate of labor incomeand the discount rate specified by Eq. (11). Significant coefficients at the 5% level arehighlighted in bold.

½0:995;1:004� at the 95% level. This result is not surprising givenour construction of ht .

The important point is the significant correlation relationship ofthe residuals, zt , with the regressors in Eq. (8). This correlation rela-tionship displays finite-sample estimation bias by using lt insteadof ht in the regression specified by Eq. (8). Table 3 reports the cor-relation coefficients. We observe that zt is correlated with all of theregressors except lt . To take full advantage of DLS, zt must beuncorrelated with Dlt�i;Dft�i and Ddt�i. Therefore, these correla-tions will lead to unsatisfied properties of the estimators evenwhen DLS is employed in the estimation.

3.4. Trend relationship and structure break

If consumption, financial wealth, household capital and humancapital are cointegrated, we use DLS to estimate the cointegratingrelationship. The model we study is

cn;t ¼ lþ bf f t þ bddt þ bhht þXk

i¼�k

bf ;iDft�i þXk

i¼�k

bd;iDdt�i

þXk

i¼�k

bh;iDht�i þ et ; ð12Þ

where parameters bf ;bd and bh represent, respectively, the long-runelasticities of consumption with respect to financial wealth, house-hold capital, and human capital. D denotes the first-order differenceoperator, l is a constant, and et is the error term.

Table 3Correlation coefficients.

lt Dlt Dlt�1 Dltþ1

zt 3.63�10�11 �0.10 0.003 �0.04

ft Dft Dft�1 Dftþ1

zt �0.05 �0.09 �0.09 �0.03

dt Ddt Ddt�1 Ddtþ1

zt �0.14 �0.19 �0.11 �0.04

This table reports the correlation coefficients between zt and the regressors in Eq.(8).

Table 4Phillips–Perron unit root test.

Variable c m f hs d l h cay cday cadh

Sample: 1952Q1–1976Q4t-Statistic 0.64 �1.00 �1.26 0.15 0.61 0.09 0.19 �3.12 �3.11 �3.16p-Value 0.99 0.73 0.62 0.97 0.99 0.96 0.97 0.03 0.03 0.03

Sample: 1977Q1–2011Q4t-Statistic �2.18 �1.40 �0.82 �1.88 �2.31 �0.07 0.15 �2.93 �3.31 �3.67p-Value 0.22 0.56 0.81 0.35 0.17 0.95 0.97 0.04 0.02 0.01

This table reports the results of the Phillips–Perron unit root test for the variables and the predictors. The null hypothesis is that there is a unit root in a univariate time serieswith the alternative that the time series is stationary. We test the variables and the predictors for each subsample: 1952Q1–1976Q4 and 1977Q1–2011Q4. c denotesconsumption, m denotes asset wealth as defined in Lettau and Ludvigson (2001), f denotes financial wealth, hs denotes housing wealth as defined in Sousa (2010a), d denoteshousehold capital, l denotes labor income, h denotes human capital, cay denotes the predictor in Lettau and Ludvigson (2001), cday denotes the predictor in Sousa (2010a) andcadh denotes the predictor put forward this paper.

Table 5Estimation results.

cin;t � lþ bif ft þ biddt þ bihht

Sample: 1952Q1–1976Q4b1f b1d b1h

Estimator 0:17 0.07 0:76t-Statistic (7.23) (2.54) (16.41)

Sample: 1977Q1–2011Q4b2f b2d b2h

Estimator 0:24 0:08 0:66t-Statistic (14.73) (4.66) (23.07)

This table summarizes the cointegrating vectors for the subsamples of 1952Q1–1976Q4 and 1977Q1–2011Q4. Significant coefficients at the 5% level are high-lighted in bold.

16 Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22

We split the whole sample into two subsamples. The first isfrom 1952Q1 to 1976Q4 and the second is from 1977Q1 to2011Q4. Then we have

c1n;t ¼ l1 þ b1f f1;t þ b1dd1;t þ b1hh1;t þXk

i¼�k

b1f ;iDf1;t�i

þXk

i¼�k

b1d;iDd1;t�i þXk

i¼�k

b1h;iDh1;t�i þ e1;t; ð13Þ

and

c2n;t ¼ l2 þ b2f f2;t þ b2dd2;t þ b2hh2;t þXk

i¼�k

b2f ;iDf2;t�i

þXk

i¼�k

b2d;iDd2;t�i þXk

i¼�k

b2h;iDh2;t�i þ e2;t: ð14Þ

Eqs. (13) and (14) are set up for the first subsample and the secondsubsample, respectively. Variables have subscripts ‘1’ and ‘2’ toindicate which subsample they belong to. The reason for this splitis that the cointegrating vectors among financial wealth, householdcapital and human capital are different for these two subsamples.1

In order to justify this, we test parameter instability using the Waldstatistic for DLS estimators in Stock and Watson (1993). Specifically,we test the null hypothesis that

H0 : b1f ¼ b2f ; b1d ¼ b2d; b1h ¼ b2h ð15Þ

against the alternative that at least one of the equalities is violated.The relevant Wald statistic is 212.4062 with a p-value of 0, whichstrongly rejects the null.2

Next, we assess the cointegrating relationships by using themethodology of Engle and Granger (1987). We apply the Phil-lips–Perron unit test to every variable. The results are reported inTable 4. We see that consumption, asset wealth, financial wealth,housing wealth, household capital, labor income and human capi-tal contain a unit root for each subsample. If we use them to con-struct cay; cday and cadh, then these three variables are stationary.This confirms the cointegrating relationships for the two subsam-ples as claimed in Lettau and Ludvigson (2001) and Sousa(2010a) and this paper.

3.5. Estimation results

We run the regressions for Eqs. (13) and (14) for the two subs-amples.3 The results are summarized in Table 5.

1 The cointegrating vectors of cay and cday change over these two subsamples aswell.

2 The details of the testing procedure are in the Appendix B.3 We use k ¼ 1. The results are not sensitive to the choice of k.

Table 5 reveals that the estimators of the parameters are consis-tent with our expectations. The sum of our estimators forb1f ; b1d; b1h or b2f ; b2d; b2h is approximately equal to 1, which is con-sistent with the model setup according to Eq. (7). bih and bif revealthe relative shares of human capital and financial wealth in theproduction function. For human capital, the shares are

b1hb1hþb1f

¼ 82% for the first subsample and b2hb2hþb2f

¼ 73% for the sec-

ond subsample. As for financial wealth, the shares are 18% and27%, respectively. These are consistent with the commonly usedvalues of the Cobb–Douglas production function in macroeconom-ics. Moreover, as explained by Sousa (2010a), bid reflects the mar-ginal propensity to consume out of housing wealth. The literature,including Gale et al. (1999) and Carroll (2004), estimates this num-ber to be somewhere between 0.04 and 0.09. Both the estimators ofSousa (2010a) and our estimator are consistent with this.

If we compare the estimators for the two subsamples, we findthat the elasticity of consumption with respect to financial wealthincreases from the first subsample to the second, along with adecreasing elasticity to human capital. This is corroborated bythe changes in personal income composition of the U.S. Accordingto the Bureau of Economic Analysis, from 1952 to 1976, laborincome4 constituted 72% of personal income and the dividends offinancial wealth5 were 10%; while, from 1977 to 2011, these twonumbers become 68% and 16%, respectively. So, from 1977 personalincome became more sensitive to financial wealth and less so to hu-man capital. Therefore, the elasticities of consumption to financialwealth and human capital increase and decrease, respectively.

4 Labor income is equivalent to the compensation of employees in Table 2.1published by the Bureau of Economic Analysis.

5 We regard the dividends of financial wealth as personal income receipts on assetsas published in Table 2.1 of the Bureau of Economic Analysis.

Table 6Estimation results of cointegrated VAR.

Dependent 1952Q1–1976Q4 1977Q1–2011Q4

Variables Dct Dft Ddt Dht Dct Dft Ddt Dht

Dct�1 0:29 0.89 0.27 0:52 0:34 0.99 0.14 0.21(2.48) (1.48) (1.18) (2.13) (3.99) (1.24) (0.56) (0.63)

Dft�1 0.003 0.07 0.04 0.03 0.01 0:21 0:07 �0:06(0.14) (0.67) (0.99) (0.74) (1.16) (2.37) (2.31) (�1.76)

Ddt�1 �0.002 0.17 0.11 0.14 0:08 0.21 0:79 0.10(�0.04) (0.59) (1.03) (1.26) (4.25) (1.18) (13.73) (1.40)

Dht�1 0.055 0.22 0:18 �0.06 0:06 0.33 0.10 �0:18(1.09) (0.85) (1.80) (�0.57) (2.68) (1.55) (1.46) (�2.03)

cadht�1 0.014 0:45 �0.06 0:20 0.005 0:36 0.0004 0:13(0.35) (2.22) (�0.83) (2.47) (0.37) (2.73) (0.01) (2.31)

R2 0.08 0.07 0.11 0.12 0.39 0.10 0.70 0.08

This table reports the estimated coefficients from a cointegrated VAR (1) of the row variable on the column variable for the two subsamples. t-Statistics appear in parentheses.Significant coefficients at the 5% level are highlighted in bold.

Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22 17

From Table 5, we can compute the transitory deviation from thecommon trend in consumption and wealth by cadht ¼ cn;t � 0:17f t

�0:07dt � 0:76ht for the first subsample and cadht ¼ cn;t � 0:24f t

�0:08dt � 0:66ht for the second. To explore whether deviationsfrom these trends are better described as transitory movementsin financial wealth and/or household capital, or as transitory move-ments in consumption and human capital, we use the cointegratedVAR method, regressing the log difference in consumption, finan-cial wealth, household capital and human capital on their own lagsand the lagged value of the estimated trend deviation, cadht�1. Theresults are reported in Table 6. The data has several interestingproperties. First, in both subsamples, consumption growth is pre-dictable by its lag values as also noted by the empirical literature,including Campbell and Mankiw (1989), Lettau and Ludvigson(2001) and Sousa (2010a). This consumption persistence is alsoconsistent with the theoretical literature on habit formation. Theassumption of habit formation implies a slow and gradual adjust-ment of consumption over time.6

Second, the growth rate of household capital, Ddt�1, positivelyand significantly predicts the growth rate of consumption, Dct , inthe second subsample while insignificant in the first. These resultsare consistent with the empirical findings in the literature. Elliott(1980) and Peek (1983) analyze data from 1950 to the mid 1970sand find that fluctuations in household capital are not significantlyfollowed by fluctuations of consumption. While Carroll et al.(2006) and Campbell and Cocco (2007), on the contrary, find thatgrowing household capital has stimulated consumption since themid 1970s. Iacoviello (2004) and Iacoviello (2005) explains thisstimulation effect as relaxed financial constraints brought aboutby an increase in house prices.

Third, household capital also positively predicts itself in the sec-ond subsample. We can understand this by observing the changesin house prices, since housing wealth makes up the main portion ofhousehold capital. The growth rate of house prices is highly persis-tent, as demonstrated by empirical studies, such as Case and Shiller(1989), Hosios and Pesando (1991) and Itô and Hirono (1993). Thetheoretical model also provides several explanations for this per-sistence. For example, the papers by Diàz and Jerez (2010) andHead et al. (2011) apply the frictions of search and match in thehousing market to explain the persistence of house price growth.But this persistence is very weak in the first subsample, as shownby the insignificant coefficient of Ddt�1 in the equation of Ddt .

6 See Chapman (1998) and Fuhrer (2000).

Finally, in the equation of financial wealth growth, the esti-mated coefficients for cadht�1 are large and significant for thetwo subsamples. This shows that the proxy cadht�1 strongly pre-dicts financial wealth growth in both subsamples and henceimplies that the deviations of consumption from its shared trendwith financial wealth, household capital and human capital revealan important transitory variation in the holdings of financialwealth. Therefore, when consumption deviates from its sharedtrend with wealth, financial wealth is forecasted to adjust, whichactually reflects the expectation of rational agents for futurereturns of financial wealth.

3.6. Forecasting quarterly asset returns

As shown by Lettau and Ludvigson (2001) and Sousa (2010a),both cay and cday can predict excess returns of stock markets. Fur-thermore, cday performs better than cay because of the special fea-tures of housing wealth. Here, we follow this path by evaluatingthe prediction performance of our cadh and comparing it withcay and cday. In addition, to highlight the importance of bothhousehold capital and human capital, we explore the role playedby each component.7

We run a simple linear regression for the two subsamples:

rtþ1 þ rtþ2 þ � � � þ rtþH ¼ aþ bxt þ et; ð16Þ

where rtþi is the log excess return, xt is the predictor, H is the fore-casting horizon and a is a constant. The predictor is either cay; cdayor cadh. We estimate the coefficient b and observe the adjusted R2 ofthe regression.

If the predictor has prediction power, the estimated coefficientb should be significant. Furthermore, performance is evaluated bythe fitness of the regression, indicated by the adjusted R2. For eachsubsample, we use the three predictors into the regression one byone, and summarize the results in Table 7. It is clear that when theforecast horizon is 1, none of the three predictors reveals any obvi-ous evidence of predictability. When the horizon spans between 2and 8 quarters, all three predictors demonstrate a strong ability topredict excess returns. If we compare the adjusted R2 for each sam-ple, our cadh delivers the highest of all for the forecasting horizonswe consider, followed by Sousa’s cday. If this comparison is madeacross the subsamples, we observe that predictability is moreprominent from 1977Q1 to 2011Q4. This may be due to the Great

7 The results reported in the following sections are based on the real returnsmeasured by the S&P Composite Index. We also use the returns calculated from theCRSP value-weighted index, and the results are quite similar.

Table 7Long-run forecasting.

Regressor Forecast horizon H

1952Q1–1976Q4 1977Q1–2011Q4

1 2 4 6 8 1 2 4 6 8

Panel A: Our predictorcadht 0:93 3:31 5:64 6:57 6:66 0:70 2:32 3:92 5:27 6:61t-Stat 2.06 2.84 3.45 2.87 2.79 2.68 3.38 3.89 4.47 4.94R2 0.02 0.05 0.12 0.12 0.11 0.03 0.07 0.15 0.23 0.31

Panel B: Lettau and Ludvigson (2001)cayt 0.76 2.93 5:50 6:26 6:79 0:79 2:35 4:03 5:64 7:22t-Stat 1.04 1.74 2.73 2.25 2.29 2.81 3.03 3.47 4.13 5.02R2 0.00 0.02 0.07 0.07 0.07 0.03 0.05 0.12 0.18 0.26

Panel C: Sousa (2010a)cdayt 0.93 3:32 5:80 6:51 6:79 0:85 2:53 4:20 5:76 7:23t-Stat 1.58 2.30 2.83 2.34 2.29 2.93 3.22 3.66 4.17 4.58R2 0.01 0.04 0.10 0.10 0.09 0.03 0.06 0.13 0.20 0.27

This table summarizes the estimation results of the forecasting regression for different forecast horizons. cadht ; cayt and cdayt are the predictors of our paper, Lettau andLudvigson (2001) and Sousa (2010a), respectively. The first row in each panel is the estimator of the coefficient for the corresponding predictor, followed by the t-statistic andadjusted-R2 in the next two rows. Significant coefficients at the 5% level are highlighted in bold.

18 Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22

Moderation in the mid-1980s, when the volatilities of major eco-nomic variables such as real GDP growth, industrial production,monthly payroll employment and the unemployment rate beganto decline.

In addition, Tables 7 shows that both the coefficient estimatesand the R2 statistics increase monotonically over the forecast per-iod. This is in contrast with the findings of Lettau and Ludvigson(2001) and Sousa (2010a). This result might be due to the fact thatthe sample sizes over which the different forecasting regressionsare being carried are not the same.

3.7. Role of household capital and human capital

The importance of both household capital and human capitalshould be emphasized. We construct another two new predictors,which are derived from cday in Sousa (2010a) by updating only onecomponent. First, we still use housing wealth, as Sousa (2010a)does, but we update the proxy of human capital as we argue in Sec-tion 3.2, and denote the new predictor cahh. Next, we return tocday in Sousa (2010a) and replace his housing wealth with thesum of housing wealth plus durable wealth, but continue using la-bor income in the regression. We denote this predictor cady. If bothhousehold capital and human capital play important roles in pre-diction, then these two predictors should have predictive powersomewhere between our cadh and Sousa’s cday. Table 8 reports

Table 8Role of human capital and household capital.

Regressor Forecast horizon H

1952Q1–1976Q4

1 2 4 6 8

Panel D: Role of human capitalcahht 0:95 3:38 5:71 6:58 6:t-Stat 2.06 2.83 3.43 2.82 2.R2 0.02 0.05 0.12 0.12 0.

Panel E: Role of Household Capitalcadyt 0.87 3:14 5:58 6:34 6:t-Stat 1.49 2.19 2.80 2.35 2.R2 0.01 0.04 0.09 0.09 0.

This table summarizes the estimation results of the forecasting regression for differencomponent of cday in Sousa (2010a). First, we still use housing wealth, as Sousa (2010a) da predictor called cahh. Next, we return to cday in Sousa (2010a), and replace his housinlabor income in the regression. By these means, we obtain another predictor, called cadypredictor, followed by the t-statistic and adjusted-R2 in the next two rows. Significant c

the results, which are consistent with our conjecture. For each sub-sample, the values of adjusted R2, when the regressor is cahh orcady, are smaller than the ones in Table 7 Panel A and higher thanthe ones in Table 7 Panel C.

However, we need to emphasize that the reason for replacing la-bor income with human capital is the potential strong correlationbetween the residuals and the regressors. If there is no such concernor the correlation is tiny, labor income is certainly a good proxy forhuman capital in regressions. Replacing it with estimated humancapital may not improve the model performance. Actually, in theliterature, when human capital serves as one variable in a VARmodel, it is quite common to replace it with labor income, see, forexample, Campbell (1996), Palacios-Huerta (2003), and Lustig andvan Nieuwerburgh (2008). When human capital is approximatedby ht ¼ c0 þ lt þ Ct ;Ct contains the net return to human capital,rh;t . Although Campbell (1996), Shiller (1993), and Jagannathanand Wang (1996) have different specifications for rh;t , none of themcorrelates rh;t with past information. So, when the predictors areconstructed by regressing only on the lagged variables, their pre-dictability will not be improved even if labor income is replacedby human capital. We illustrate this by studying one example.

Sousa (2012a), Sousa (2012b), Sousa (2012c) shows that thedeviations of the cointegrating relationship between asset wealthand labor income, denoted by wy, can have predictive power forfuture stock returns. Specifically, he estimates the VECM:

1977Q1–2011Q4

1 2 4 6 8

55 0:69 2:28 3:82 5:12 6:3968 2.69 3.35 3.81 4.38 4.8110 0.03 0.07 0.15 0.22 0.30

77 0:86 2:57 4:31 5:93 7:4737 2.92 3.24 3.72 4.24 4.6909 0.03 0.06 0.13 0.21 0.28

t forecast horizons. cahht and cadyt are the predictors obtained by updating oneoes, but we update the proxy of human capital as we argue in this paper, to constructg wealth with the sum of housing wealth plus durable wealth, but continue using

. The first row in each panel is the estimator of the coefficient for the correspondingoefficients at the 5% level are highlighted in bold.

Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22 19

Dmt

Dlt

� �¼ a½mt þ-lt þ ht þ v� þ

XK

k¼1

DkDmt�k

Dlt�k

� �þ et ; ð17Þ

and the predictor

wyt ¼ mt þ -lt þ ht þ v: ð18Þ

We use our data to obtain wyt and construct wht by replacing laborincome with our human capital estimators. The performance of thepredictive regression is summarized in Table 9. We see that for thefirst subsample, wht is superior to wyt . But for the second subsam-ple, the results are opposite. The main reason is that the differencebetween human capital and labor income contains informationabout future labor income and is more likely to correlate with thelabor income growth rate of current and future periods. We observethis in Table 3. As wyt is obtained by a VECM, where there are onlylags of the labor income growth rate, the estimators are consistentas long as the residuals are white noise (Lütkepohl, 2005). There-fore, there is no advantage to replace labor income with human cap-ital in this case.

3.8. Out-of-sample forecasts

We study the out-of-sample forecasting results by fixing thecointegrating vector. The starting points for the out-of-sampleforecast are the first quarter of 1967 and the first quarter of 2002

Table 9wy versus wh.

Regressor Forecast horizon H

1952Q1–1976Q4

1 2 4 6 8

wht �0:58 �1:69 �2:56 �3:37 �3:t-Stat �3.26 �3.58 �4.31 �5.20 �5.R2 0.10 0.15 0.26 0.34 0.35

wyt �0:67 �1:93 �2:89 �3:75 �3:t-Stat �3.29 �3.58 �4.32 �4.98 �4.R2 0.10 0.15 0.24 0.31 0.30

This table compares the in-sample forecast estimation for the predictors wy and wh. Asincome, which measures the deviation from the cointegration relationship: wyt ¼ mt þestimated in this paper. Significant coefficients at the 5% level are highlighted in bold.

Table 10Out-of-sample forecasting.

Regressor Forecast horizon H

1952Q1–1976Q4

1 2 4 6 8

Panel A: Theil’s Ucadht 0.94 0.89 0.77 0.77 0.8cayt 0.98 0.96 0.85 0.84 0.8cdayt 0.94 0.91 0.79 0.80 0.8

Panel B: MSE-Fcadht 2:36 4:53 11:01 10:66 7:7cayt 0.83 1:61 6:51 6:59 6:0cdayt 2:42 3:85 9:94 8:68 6:5

Panel C: ENC-NEWcadht 1:68 3:41 7:39 6:64 4:3cayt 0:97 1:89 5:07 4:41 3:5cdayt 1:77 3:05 6:82 5:47 3:7

This table reports the results of forecast comparisons of an unrestricted model, which inwith a restricted model, which just includes a constant. The Theil’s U statistic is the ratiotest the null hypothesis that the restricted model has an MSE equal to that of the unrestENC-NEW statistic provides a test of the null hypothesis that the restricted model’sunrestricted contains some information that could improve the restricted model’s forecasfor the out-of-sample forecast are the first quarter of 1967 and the first quarter of 2002, fopreceding the forecast period. Significant statistics at the 5% level are highlighted in bol

for two subsamples, respectively. The model is recursively esti-mated until the quarter immediately preceding the forecast period.For each predictor, we consider two models, the restricted and theunrestricted. The unrestricted model is the same as Eq. (16), andthe restricted model is that of constant returns. We collect theMSEs for the two models, and calculate Theil’s U, the MSE-F statis-tic (McCracken, 2007) and the ENC-NEW statistic (Clark andMcCracken, 2001). Theil’s U is the ratio of the MSE of the unre-stricted model over the MSE of the restricted one. The MSE-F statis-tic is used to test the null hypothesis that the restricted andunrestricted models have equal MSEs with the alternative that therestricted model has a higher MSE. The ENC-NEW statistic is usedto test the null hypothesis that the restricted model encompassesthe unrestricted model; the alternative is that the unrestricted mod-el contains information that could be used to improve the restrictedmodel’s forecast. The results are reported in Table 10.

The first panel of Table 10 displays Theil’s U for the two subs-amples. We see that all the values are less than 1. That means nomatter which predictor is used in the unrestricted model, it alwaysdecreases the MSE of the restricted model. This is consistent withthe empirical findings of Lettau and Ludvigson (2001) and Sousa(2010a). Since the restricted model is the same for the differentpredictors, we can compare the Theil’s U to evaluate predictionperformance for the predictors. Our predictor, cadht delivers thesmallest MSE, followed by cdayt , for all forecast horizons except

1977Q1–2011Q4

1 2 4 6 8

64 -0.10 �0.32 �0:62 �0:93 �1:1235 �1.41 �1.57 �2.12 �2.45 �2.55

0.00 0.01 0.05 0.10 0.12

91 �0:18 �0:54 �0:93 �1:36 �1:5868 �2.15 �2.30 �2.76 �3.27 �3.13

0.02 0.04 0.08 0.15 0.17

in Sousa (2012a) and Sousa (2012b), wy is defined as the ratio of wealth to labor-lt þ ht þ v. wh is constructed by replacing labor income with human capital as

1977Q1–2011Q4

1 2 4 6 8

1 0.94 0.86 0.74 0.71 0.645 0.97 0.93 0.83 0.72 0.603 0.95 0.91 0.80 0.71 0.64

4 2:44 6:30 12:67 14:30 18:132 1:37 2:81 7:77 13:83 21:757 2:13 3:94 9:27 14:36 18:72

9 1:59 3:94 8:17 9:93 13:005 1:04 1:91 4:95 8:95 14:517 1:32 2:35 5:45 8:62 11:56

cludes a constant and a predictor (cayt ; cdayt or cadht) as the explanatory variables,of the MSEs for the unrestricted and restricted models. The MSE-F statistic is used toricted model with the alternative that the unrestricted model has a lower MSE. Theforecast ‘‘encompasses’’ all the unrestricted model with the alternative that thet. We obtain cayt ; cdayt or cadht by fixing the cointegrated vector. The starting pointsr two subsamples. The model is recursively estimated until the quarter immediatelyd.

20 Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22

the last one in the second subsample. Panel B and Panel C reportthat the MSE-F and ENC-NEW statistics are significant in all casesexcept for the short forecast horizon for cayt . The significantMSE-F and ENC-NEW statistics reveal that the difference betweenthe MSEs of the unrestricted and the restricted model is significantand can be interpreted as there being more information containedin the unrestricted model.

4. Conclusion

In this paper, we improve the prediction power of cay in Lettauand Ludvigson (2001) and of cday in Sousa (2010a). Similar to thesetwo papers, we use budget constraint to derive the cointegratingrelationship between consumption and the different componentsof wealth, and then apply the trend deviation denoted by cadh topredict stock returns. Our predictor, cadh, has superior predictionpower over cay and cday in two respects. First, as we use an appro-priate proxy for human capital, we avoid potential endogeneity inthe DLS regression. Specifically, we use a VAR (1) model to forecastthe growth rate of labor income and discount rates, and we sumthe current values of all future labor income. This sum is regardedas current human capital. Second, we further explore the impor-tance of wealth composition. We generalize housing wealth as de-fined in Sousa (2010a) to be household capital, which includesboth housing wealth and durable goods. This is because durablegoods are quantitatively an important part of aggregate wealth,and they are similar to housing in terms of their liquidity, utilityfrom ownership and income distribution.

Empirically, we find that there is a structural change in the databetween 1952–1976 and 1977–2011. First, the cointegrating vec-tors among consumption, financial wealth, household capital andhuman capital are different for each subsample. This difference re-flects a change in the long-run elasticities of consumption with re-spect to financial wealth, household capital, and human capital,which is corroborated by a change in personal incomecompositions.

Second, the relationship between the growth rates of householdcapital and consumption is different. In the first subsample, thegrowth rates of household capital are not significantly associatedwith the future growth rates of consumption. In the second sub-sample, the growth rates of household capital are significantlyand positively related to the future growth rates of consumption.These two phenomena are consistent with the empirical literaturestudying the data before and after the mid-1970s. Sousa (2010a)verifies the growth rate relationship and attributes it to the persis-tence of household assets. We also observe that the growth rates ofhousehold capital have significant positive autocorrelation in thesecond subsample but not in the first one, interestingly consistentwith the discussion of Sousa (2010a). There is currently no discus-sion of any such structural break in US data in the literature.

We compare the in-sample and out-of-sample forecasting per-formances of our predictor, cadh, with cay and cday for the twosubsamples. In both subsamples, cadh outperforms the other two.This supports the importance of our human capital proxy in theregression and durable goods in wealth composition. Moreover,the predictive abilities of the three predictors tend to be strongerpost 1977. A possible explanation is that the Great Moderationmade investors’ expectations more precise.

Appendix A. Calculation of human capital

As defined in macroeconomics, human capital is the sum of thediscounted value of current and future labor income, net of taxes.Here, we denote Lt as labor income, At as the tax correspondingto the labor income, and hence Xt ¼ Lt � At denotes net labor in-

come. Let xt denote the growth rate of net labor income and rt de-note the discount rate. Human capital can be expressed as

Ht ¼ Xt 1þ Et

X1i¼1

Pij¼1

1þ xtþj

1þ rtþj

� �" #( )� Xtð1þ CtÞ; ð19Þ

where Ct represents the expectation of the term in the inner squarebrackets. We call it the cumulative growth factor.

We use a bivariate VAR (1) model to describe the motions of xt

and rt .

xtþ1 ¼ a1 þ h11xt þ h12rt þ �1;tþ1;

rtþ1 ¼ a2 þ h21xt þ h22rt þ �2;tþ1;

which implies the unconditional distribution of

ðxt; rtÞ Na1

a2

� �;R

� �and R ¼ r2

1 r12

r21 r22

� �.

We approximate this VAR process by a discrete-value finite-state Markov chain. We primarily use the method proposed byTauchen (1986) with some modification. By doing so, we can com-pute Ct as the weighted summation over all of the possible out-comes instead of an intractable integral.

We use grids of Nx and Nr to approximate the continuous-val-ued series x and r, respectively, and hence the state space of thediscrete system is Nx Nr � N. The dynamics of the system are de-scribed by an N N matrix of transition probabilities P.

Let x ¼ fxigNxi¼1 denote the state space of the corresponding dis-

crete-valued Markov chain for x and r ¼ frjgNr

j¼1 for r. We truncatethe top 2.5% and the bottom 2.5% tails of the empirical distribu-tions of x and r to eliminate the extreme values. Letxmin; xmax; rmin; rmax denote the minimal and maximal values of thesample after the truncation and denote d1 ¼ xmax�xmin

Nx�1 and

d2 ¼ rmax�rminNr�1 . Then, we can define x and r as

xi ¼ xmin þ ði� 1Þd1

rj ¼ rmin þ ðj� 1Þd2:

We obtain the approximated series ~xt and ~rt for xt and rt , respec-tively, by substituting the observations ðxt ; rtÞ at each period t withthe closest points in x and r. Assume the number of sample periodsis T. Then, we obtain

~xt ¼ xarg min

i2f1;2;...;Nxgjxt�xi j

~rt ¼ rarg min

j2f1;2;...;Nrgjrt�rj j

for t ¼ 1;2; . . . ; T.We use the new sample f~xt ;~rtgT

t¼1 to estimate the VAR modeldefined by

~xtþ1 ¼ ~a1 þ ~h11~xt þ ~h12~rt þ ~�1;tþ1

~rtþ1 ¼ ~a2 þ ~h21~xt þ ~h22~rt þ ~�2;tþ1:

Denote the variance–covariance matrix of the errors

ð~�1;tþ1; ~�2;tþ1Þ as ~R ¼ ~r21 ~r12

~r21 ~r22

� �. Next, we compute the transitional

matrix P. Suppose the current state is ði; jÞ (i 2 f1;2; . . . ;Nxg andj 2 f1;2; . . . ;Nrg). The probability that the current state transitionsinto ði0; j0Þ (i 2 f1;2; . . . ;Nxg and j 2 f1;2; . . . ;Nrg) in the next periodcan be defined as

Probði0; j0ji; jÞ ¼ Prob Fð~xi0 ;~rj0 Þ < Gð~xi;~riÞ þ~�1;tþ1

~�2;tþ1

" #< Fð~xi0 ;~rj0 Þ

!

where Gð~xi;~riÞ ¼~a1 þ ~h11~xi þ ~h12~rj

~a2 þ ~h21~xi þ ~h22~rj

� �. Fð~xi0 ;~rj0 Þ and Fð~xi0 ;~rj0 Þ satisfy

Y. Ren et al. / Journal of Banking & Finance 42 (2014) 11–22 21

Fð~xi0 ;~rj0 Þ ¼

½�1�1� if i0 ¼ 1 and j0 ¼ 1

�1~rj0 � k2

2

" #if i0 ¼ 1 and 1 < j0 6 Nr

~xi0 � k12

�1

" #if 1 < i0 6 Nx and j0 ¼ 1

~xi0 � k12

~rj0 � k22

" #if 1 < i0 6 Nx and 1 < j0 6 Nr

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

and

Fð~xi0 ;~rj0 Þ ¼

11

� �if i0 ¼ Nx and j0 ¼ Nr

1~rj0 þ k2

2

" #if i0 ¼ Nx and 1 6 j0 < Nr

~xi0 þ k12

1

" #if 1 6 i0 6 Nx and j0 ¼ Nr

~xi0 þ k12

~rj0 þ k22

" #if 1 6 i0 < Nx and 1 6 j0 < Nr

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

Given that ð~�1;tþ1; ~�2;tþ1Þ Nð0; ~RÞ, we obtain the transitionprobability Probði0; j0ji; jÞ and hence the whole transition matrix, P.

Now we can calculate Ct . From Eq. (19), we can write Ct as

Ct ¼ Etðqtþ1 þ qtþ1qtþ2 þ . . .Þ

where qtþl¼1þxtþl1þrtþl

. Based on this approximation procedure, we obtainthe approximated qi;j ¼ 1þxi

1þrjand ~qt is defined in a similar way to ~xt

and ~rt . We have

Eð~qtþ1j~qt ¼ qi;jÞ ¼XNx

i0¼1

XNr

j0¼1

pði0; j0; i; jÞqi0;j0

Eð~qtþ1~qtþ2j~qt ¼ qi;jÞ ¼XNx

i00¼1

XNr

j00¼1

XNx

i0¼1

XNr

j0¼1

pði00; j00; i0; j0Þpði0; j0; i; jÞqi0 ;j0 qi00 ;j00

Let QNx�Nr be the matrix of q s in the system with Qði; jÞ ¼ qi;j anddefine ~Q as the vector obtained by stacking the columns of Q Nx�Nr

one on top of each other. Let C be a vector of all the cumulativegrowth factors in the discrete system. In addition, let X be a matrixwith all its rows being ?QT, where T is the transpose operator.Macklem (1994) and Macklem (1997) provided the closed-formsolution for the vector of cumulative growth factor C

C ¼X1a¼1

ðP �XÞal ¼ ½ðI � P �XÞ�1 � I�l

where � denotes element-by-element multiplication, P denotes thetransition matrix, l is a vector of ones, and I is the identity matrix.

However, the convergence condition usually does not hold andis sensitive to the setting of N1;N2 as well as the truncating methodused. In other words, the inverse of ðI � P �XÞ does not necessarilyexist, and thus, Macklem’s formula may not always be true.

In this paper, we only take the summation up to 200 periods.This assumption is well motivated because the average life-spanin the US is between 75–80 years. If we assume employment be-gins between 25–30 years old, the working life is less than50 years. Thus we use the expected summation of discounted netlabor income for the next 50 years to measure the human capital.

Appendix B. Structure test

In order to test the null hypothesis

H0 : b1f ¼ b2f ; b1d ¼ b2d; b1h ¼ b2h ð20Þ

of Eqs. (13) and (14), we construct

~c ¼

c1n;1

..

.

c1n;t1

c2n;1

..

.

c2n;t2

0BBBBBBBBBB@

1CCCCCCCCCCA

~f 1 ¼

f1;1

..

.

f1;t1

0...

0

0BBBBBBBBBB@

1CCCCCCCCCCA

~f 2 ¼

0...

0f2;1

..

.

f2;t2

0BBBBBBBBBB@

1CCCCCCCCCCA: ð21Þ

~d1;~d2;

~h1 and ~h2 can be created similarly to ~f 1;~f 2.

Then, we run the DLS regression

~c ¼ ~lþ b1f~f 1 þ b2f

~f 2 þ b1d~d1 þ b2d

~d2 þ b1h~h1 þ b2h

~h2

þXk

i¼�k

b1f ;iD~f 1;t�i þ � � � þXk

i¼�k

b2h;iD~h2;t�i þ et: ð22Þ

After that, we use the Wald statistic of Stock and Watson (1993) totest the null.

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