The intertemporal relation between expected returns …faculty.msb.edu/tgb27/BaliJFE2008.pdf · Journal of Financial Economics 87 (2008) 101–131 The intertemporal relation between

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Journal of Financial Economics 87 (2008) 101–131

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The intertemporal relation between expected returns and risk$

Turan G. Bali

Baruch College, Zicklin School of Business, One Bernard Baruch Way, New York, NY, 10010, USA

Received 12 August 2004; received in revised form 30 January 2007; accepted 2 March 2007

Available online 4 September 2007

Abstract

This paper explores the time-series relation between expected returns and risk for a large cross section of industry and

size/book-to-market portfolios. I use a bivariate generalized autoregressive conditional heteroskedasticity (GARCH)

model to estimate a portfolio’s conditional covariance with the market and then test whether the conditional covariance

predicts time–variation in the portfolio’s expected return. Restricting the slope to be the same across assets, the risk-return

coefficient is highly significant with a risk–aversion coefficient (slope) between one and five. The results are robust to

different portfolio formations, alternative GARCH specifications, additional state variables, and small sample biases.

When conditional covariances are replaced by conditional betas, the risk premium on beta is estimated to be in the range of

3% to 5% per annum and is statistically significant.

r 2007 Elsevier B.V. All rights reserved.

JEL classification: G12; G13; C51

Keywords: ICAPM; Conditional CAPM; Conditional covariance; Risk aversion; Conditional beta; Market risk premium; Intertemporal

hedging demand

1. Introduction

In his seminal paper, Merton (1973) derives an intertemporal capital asset pricing model (ICAPM) that hasformed the basis for much empirical research. The model predicts that an asset’s expected return depends onits covariance with the market portfolio and with state variables that proxy for investment opportunities. Testsof the model have taken two different forms. The first type follows Merton (1980) and French, Schwert, andStambaugh (1987) and explores the time-series relation between the market’s conditional expected return andits conditional variance. The second type, of which tests of the Sharpe and Lintner capital asset pricing model(CAPM) are a special case, focuses on the cross-sectional relation between expected return and risk.

ee front matter r 2007 Elsevier B.V. All rights reserved.

neco.2007.03.002

William Schwert (the editor) an anonymous referee, Gurdip Bakshi, Peter Carr, John Campbell, David Chapman, Ozgur

ne Ferson, Edward Kane, John Merrick, Lin Peng, Pedro Santa-Clara, Robert Schwartz, Jun Wang, Chu Zhang, and

ants at Baruch College, Boston College, and Cornell University for helpful discussions. I also thank Kenneth R. French

rge amount of historical data publicly available in his online data library. I owe a great debt to Liuren Wu for his extremely

ts and suggestions on earlier versions of this paper. Excellent research support was provided by Yi Tang.

ess: [email protected]

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dx.doi.org/10.1016/j.jfineco.2007.03.002

mailto:[email protected]

ARTICLE IN PRESST.G. Bali / Journal of Financial Economics 87 (2008) 101–131102

My study extends time-series tests of the ICAPM to many risky assets. The prediction of Merton (1980) thatexpected returns should be related to conditional risk applies not only to the market portfolio but also toindividual stocks and portfolios: Expected returns for any asset should vary through time with the asset’sconditional covariance with the market portfolio (assuming that hedging demands are not too large). To beinternally consistent, the relation should be the same for all stocks and portfolios, i.e., the predictive slope onthe conditional covariance for any asset should be the representative investor’s relative risk aversion. I exploitthis cross-sectional consistency condition and estimate the common time-series relation across a wide varietyof stock portfolios formed based on industry, size, book-to-market, and beta.

With monthly data from 1926 to 2002, I use a bivariate generalized autoregressive conditionalheteroskedasticity (GARCH) model to estimate the conditional covariance between the excess returns oneach portfolio and the market portfolio. Then, I estimate a system of time-series regressions of portfolios’excess returns on their conditional covariances with the market, while constraining all regressions to have thesame slope coefficient. According to Merton (1973), the common slope coefficient represents the averagerelative risk aversion of market investors. My estimation generates positive and highly significant relative riskaversion coefficients, with magnitudes between one and five. The identified positive risk–return relation isrobust to different portfolio formations, alternative GARCH specifications, additional state variables, andsmall sample biases.

To compare with the literature, I also estimate the risk–return relation using a single series on the marketportfolio and, furthermore, on each of the industry portfolios. I find that the relative risk aversion estimatesfrom these single-series regressions are often statistically insignificant, as found in many earlier studies.Furthermore, the estimates vary greatly from negative to positive as I estimate the relation using differentseries, illustrating the large sample variation in the single-series coefficient estimates. By expanding the studyof the time-series relation to a large cross section, I not only achieve cross-sectional consistency among theportfolios, but also gain statistical power in identifying a significant slope coefficient.

My expansion of the time-series relation to multiple stock portfolios while maintaining cross-sectionalconsistency also moves my paper closer to the conditional CAPM literature, which studies the cross-sectionalimplications of the ICAPM. When I restrict the intercepts and slopes to be the same across portfolios, mysystem of equations becomes closer to the cross-sectional regressions, but with intertemporal stabilityconstraint on the relative risk aversion coefficient. Alternatively, if one replaces the conditional covariancewith the conditional beta, she can regard the system of equations with common intercepts and slopes as cross-sectional regressions with intertemporal stability constraint on the market risk premium. My estimation on theconditional beta formulation also generates highly significant and positive market risk premium estimates, inthe range of 3–5% per annum.

The literature often reports insignificant estimates from either time-series or cross-sectional regressions.I attribute these insignificant estimates to the low statistical power caused by focusing narrowly onthe intertemporal risk–return relation of a single market return series or by focusing on the unconditionalmeasures or rolling window estimates of market risk. I contribute to the literature by expanding the analysisof the intertemporal relation to a large cross section of stock portfolios and by expanding the cross-sectional analysis to each conditional time step based on the bivariate GARCH estimates. Also, res-tricting the intercepts and slopes to be the same across portfolios in the system of equations, I considerboth the intertemporal and cross-sectional implications of the ICAPM. I achieve consistency acrossboth dimensions and gain statistical power by using GARCH-based conditional risk measures and by poolingthe data.

When the investment opportunity is stochastic, investors adjust their investment to hedge against futureshifts in the investment opportunity and achieve intertemporal consumption smoothing. Hence, covariationswith state of the investment opportunity induce additional risk premiums on an asset. I identify a series offinancial and macroeconomic factors and study whether their conditional covariances with portfolio excessreturns induce additional risk premiums. Estimation shows that covariances with the size factor (SMB) ofFama and French (1993) and the Treasury bill rate generate significantly negative slope coefficients. Hence,increases in a portfolio’s covariance with both factors predict a lower excess return on the portfolio. In thecontext of the Merton (1973) ICAPM, the negative slope estimates suggest that an increase in SMB or the risk-free rate predicts a decrease in optimal consumption and hence an unfavorable shift in the investment

ARTICLE IN PRESST.G. Bali / Journal of Financial Economics 87 (2008) 101–131 103

opportunity. Meanwhile, covariances with default spread and aggregate dividend yield generate significantlypositive slope coefficients.

The paper is structured as follows. Section 2 discusses the studies that form the background of my work.Section 3 describes the data set and the estimation methodology. Section 4 presents the estimation results.Section 5 performs robustness analysis. Section 6 concludes.

2. Literature review

The Merton (1973) ICAPM predicts both a time-series and a cross-sectional relation between expectedreturns and risk. My paper focuses on the time-series aspect of the relation and is hence related to a large bodyof empirical literature that attempts to measure the intertemporal risk–return relation on the stock marketportfolio. The findings from the literature are inconclusive. Using different sample periods, different data sets,or different econometric methodologies often leads to different findings on the risk–return trade-off.

Many studies fail to identify a statistically significant intertemporal relation between risk and return of themarket portfolio. French, Schwert, and Stambaugh (1987) find that the coefficient estimate is not significantlydifferent from zero when they use past daily returns to estimate the monthly conditional variance. Goyal andSanta-Clara (2003) and Bali, Cakici, Yan, and Zhang (2005) obtain similar insignificant results using the sameconditional variance estimator but over a longer sample period. Chan, Karolyi, and Stulz (1992) employ abivariate GARCH-in-mean model to estimate the conditional variance. They also fail to obtain a significantcoefficient estimate for the United States. Baillie and DeGennaro (1990) replace the normal distributionassumption in the GARCH-in-mean specification with a fat-tailed t-distribution. Their estimates remaininsignificant. Campbell and Hentchel (1992) use the quadratic GARCH (QGARCH) model of Sentana (1995)to determine the existence of a risk-return trade-off within an asymmetric GARCH-in-mean framework. Theirestimate is positive for one sample period and negative for another sample period, but neither is statisticallysignificant. Glosten, Jagannathan, and Runkle (1993) use monthly data and find a negative but statisticallyinsignificant relation from two asymmetric GARCH-in-mean models. Based on semi-nonparametric densityestimation and Monte Carlo integration, Harrison and Zhang (1999) find a significantly positive risk andreturn relation at one-year horizon, but they do not find a significant relation at shorter holding periods suchas one month.

Several studies even find that the intertemporal relation between risk and return is negative. Examplesinclude Campbell (1987), Breen, Glosten, and Jagannathan (1989), Turner, Startz, and Nelson (1989), Nelson(1991), Glosten, Jagannathan, and Runkle (1993), Whitelaw (1994), and Harvey (2001).

A few studies do provide evidence supporting a positive risk–return relation. French, Schwert, andStambaugh (1987) identify a negative relation between the portfolio return and the unpredictable componentof volatility. They interpret this finding as indirect evidence that the ex ante conditional volatility is positivelyrelated to the ex ante excess return. Furthermore, when they estimate a GARCH-in-mean model with dailydata, they find a positive and statistically significant relation at the daily level. Using the symmetric GARCHmodel of Bollerslev (1986), Chou (1988) finds a significantly positive relation with weekly data. Bollerslev,Engle, and Wooldridge (1988) use a multivariate GARCH-in-mean process to model the conditional meanand the conditional covariance of returns on stocks, bonds, and bills with the excess market return. They finda small but nevertheless significant risk–return relation. Scruggs (1998) includes the long-term governmentbond returns as a second factor of the bivariate EGARCH-in-mean model and finds the partial relationbetween the conditional mean and conditional variance to be positive and significant. Ghysels, Santa-Clara,and Valkanov (2005) introduce a new variance estimator that uses past daily squared returns, and theyconclude that the monthly data are consistent with a positive relation between conditional expected excessreturn and conditional variance. Bali and Peng (2006) examine the intertemporal relation between risk andreturn using high-frequency data. Based on realized, GARCH, implied, and range-based volatility estimators,they find a positive and significant link between the conditional mean and conditional volatility of marketreturns at the daily level.

By emphasizing the cross-sectional consistency of the intertemporal relation, my work is also related to theconditional CAPM literature, the focus of which is to examine whether allowing beta to be time-varying canbetter explain the cross-sectional behavior of excess returns. Important studies include Ang and Chen (2007),


Harvey (1989), Jagannathan and Wang (1996), Ferson and Harvey (1999), Lettau and Ludvigson (2001), andLewellen and Nagel (2006).

3. Data and estimation

The Merton (1973) ICAPM implies the following equilibrium relation between risk and return:

m ¼ ASþ OB, (1)

where m 2 <n denotes the expected excess return on a vector of n risky assets, A reflects the average relativerisk aversion of market investors, S 2 <n denotes the covariance of the excess returns with the marketportfolio, B 2 <k measures the market’s aggregate reaction to shifts in a k-dimensional state vector thatgoverns the stochastic investment opportunity, and O 2 <n�k measures the covariance between excess returnson the n risky assets and the k state variables. For any risky asset or portfolio i, the relation becomes

mi � r ¼ Asim þ oixB, (2)

where sim denotes the covariance between the returns on the risky asset i and the market portfolio m, and oix

denotes a (1� k) row of covariances between the return on risky asset i and the k state variables x.Many empirical studies focus on the time-series implication of the equilibrium relation in Eq. (2) and apply

it narrowly to the market portfolio. Without the hedging demand component ðO ¼ 0Þ, this focus leads to thefollowing risk-return relation:

mm � r ¼ As2m. (3)

When considering stochastic investment opportunity, the literature often implicitly or explicitly projects thecovariance vector oix linearly to the state variables x to obtain the following relation:

mm � r ¼ As2m þ gx. (4)

My work in this article differs from the above literature in two major ways. First, I estimate theintertemporal relation Eq. (2) not on the single series of the market portfolio, but simultaneously on manystock portfolios, and constrain all these portfolios to have the same cross-sectionally consistentproportionality coefficients A and B. Second, I directly estimate the conditional covariances sim and oix

using bivariate GARCH models. I do not make any linear projection assumptions on the state variables.In the Merton (1973) original setup, the two covariance matrices S and O are assumed to be constant.

Nevertheless, the empirical literature has estimated the relation assuming time-varying covariances. I do thesame in this paper. In principle, if the covariances are stochastic, they would represent additional sources ofvariation in the investment opportunity and induce extra intertemporal hedging demand terms.

3.1. Data

I estimate the intertemporal relation in Eq. (2) using a wide variety of stock portfolios. Using portfoliosinstead of individual stocks reduces the workload to a manageable level and reduces the noise in individualstocks. Based on Kenneth French’s online data library, I estimate the intertemporal relation using two broadsets of portfolios. One set is constructed according to different industry groups as in Fama and French (1997).The other is formed based on the quintiles of firm sizes and book-to-market ratios, as described in Fama andFrench (1993). For state variables, I use the firm-size (SMB) and book-to-market (HML) risk factors formedby Fama and French (1993). I also consider commonly used macroeconomic variables such as the relativeTreasury bill rate, default spread, term spread, and dividend-price ratio.

3.1.1. Industry portfolios

Fama and French (1997) assign each NYSE, AMEX, and Nasdaq stock to an industry portfolio at the endof June of each year based on its four-digit standard industrial classification (SIC) code at that time. Then,they compute both the value-weighted and the equal-weighted returns from July 1926 to December 2002. I usethe value-weighted monthly returns following Fama and French (1997).


French’s online data library provides return data on several different industry partitions. The definitions ofthe industries and the corresponding SIC codes are available on his data library. I repeat my analysis using allthese different partitions when possible. The stocks are partitioned into five, ten, 12, 17, 30, 38, and 48industries. Data on five-, ten-, 12-, 17-, and 30-industry portfolios are available monthly from July 1926 toDecember 2002, yielding a total of 918 monthly observations for each series. I use these portfolios for the fullsample analysis. Data on 38-industry portfolios have missing observations in six of the 38 industries. I dropthis partition from my analysis. Data on 48-industry portfolios are fully available after July 1963. I include thispartition when I perform subsample analysis from July 1963 to December 2002, with 474 observations foreach series. Over time, French has revised the industry definitions. I perform a robustness check by repeatingmy analysis on some of the recently revised portfolios. The estimated risk–return relations are similar acrossdifferent portfolio definitions or revisions or both. The results reported in this paper are based on industrydefinitions and portfolio returns that I downloaded from the library in early 2003.

3.1.2. Fama and French size/book-to-market portfolios

Fama and French (1993) form 25 stock portfolios according to the quintiles of the stocks’ marketcapitalization (size, ME) and book-to-market equity ratios (BM). Monthly data from July 1926 to December2002 on these 25 portfolios are available on French’s online data library. To construct the portfolios, in Juneof each year, they rank all NYSE stocks in Center for Research in Security Prices (CRSP) based on thequintiles of ME. Then, they break NYSE, Amex, and Nasdaq stocks into five size groups based on thebreakpoints of the NYSE stock quintiles. They also break all the NYSE, Amex, and Nasdaq stocks into fiveBM groups based on NYSE stock quintiles on BM ratios.

The S5B5 portfolio (biggest size, highest BM) has missing observations from July 1930 to June 1931. I usetwo approaches to deal with these missing observations in my estimation. First, I replace the missing valueswith the average returns on the other four portfolios within the same size quintile (S5B1, S5B2, S5B3, andS5B4). Second, I replace the missing values with the average returns on the other four portfolios within thesame BM quintile (S1B5, S2B5, S3B5, and S4B5). Both approaches yield similar results. I report the resultsbased on the first approach.

3.1.3. Fama and French size and book-to-market risk factors

Fama and French (1993) also form two common risk factors related to firm sizes (SMB) and book-to-market equity ratios (HML). In a series of papers, Fama and French (1993, 1995, 1996) show the importanceof these two factors. To construct these two factors, Fama and French first construct six portfolios accordingto the rankings on ME and BM ratios. In June of each year, they rank all NYSE stocks in CRSP based onME. Then they use the median NYSE size to split NYSE, Amex, and Nasdaq stocks into two groups, smalland big (S and B). They also break NYSE, Amex, and Nasdaq stocks into three BM groups based on thebreakpoints for bottom 30% (Low), middle 40% (Medium), and top 30% (High) of the ranked values of BMfor NYSE stocks. They construct the SMB factor as the difference between the return on the portfolio of smallsize stocks and the return on the portfolio of large size stocks, and the HML factor as the difference betweenthe return on the portfolio of high BM stocks and the return on the portfolio of low BM stocks.

In addition to SMB and HML, Fama and French (1993) use the excess market return as a proxy for themarket factor in stock returns. The excess return on the market portfolio is the value-weighted return on allNYSE, Amex, and Nasdaq stocks (from CRSP) minus the one-month Treasury bill rate (from IbbotsonAssociates). The three factors are also available on French’s data library.

3.1.4. Macroeconomic variables

Several studies find that macroeconomic variables associated with business cycle fluctuations can predict thestock market (see Fama and Schwert, 1977; Keim and Stambaugh, 1986; Chen, Roll, and Ross, 1986;Campbell and Shiller, 1988; Fama and French, 1988, 1989; Schwert, 1989, 1990; Fama, 1990; Campbell, 1991;Ferson and Harvey, 1991; Goyal and Santa-Clara, 2003; Ghysels, Santa-Clara, and Valkanov, 2005; Bali,Cakici, Yan, and Zhang, 2005). The commonly chosen variables include Treasury bill rates, default spreads,term spreads, and dividend-price ratios. I study how variations in these variables predict variations in the


investment opportunity and how incorporating covariances with these variables affects the intertemporal risk-return relation.

I obtain monthly data on the three-month Treasury bill and ten-year Treasury bond yields from CRSP. Thethree-month Treasury bill data are available from July 1926 to December 2002. I follow standard practice inusing the detrended relative rate (RREL), defined as the difference between the three-month T-bill rate and its12-month backward moving average. I construct the term spread (TERM) as the difference between the yieldson the ten-year Treasury bond and the three-month Treasury bill. The ten-year Treasury data are availablefrom May 1941 to December 2002.

I obtain monthly BAA- and AAA-rated corporate bond yields during my sample period from theFederal Reserve statistical release, and I construct the default spread as the yield difference between thetwo rating groups. I also download monthly aggregate dividend–price ratio data over my sample periodfrom Robert Shiller’s website, http://aida.econ.yale.edu/�shiller/. The log dividend-price ratio (DP) is definedas the log difference between last 12-month dividends and the current level of the Standard & Poor’s (S&P)500 index.

3.2. Estimating conditional covariances

I estimate the conditional covariance between excess returns on asset i and the market portfolio m based onthe following bivariate GARCH(1,1) specification:

Ri;tþ1 ¼ ai0 þ ai

1Ri;t þ �i;tþ1, (5)

Rm;tþ1 ¼ am0 þ am

1 Rm;t þ �m;tþ1, (6)

Et½�2i;tþ1� � s2i;tþ1 ¼ gi

0 þ gi1�

2i;t þ gi

2s2i;t, (7)

Et½�2m;tþ1� � s2m;tþ1 ¼ gm

0 þ gm1 �

2m;t þ gm

2 s2m;t, (8)

Et½�i;tþ1�m;tþ1� � sim;tþ1 ¼ gim0 þ gim

1 �i;t�m;t þ gim2 sim;t, (9)

where Ri;tþ1 and Rm;tþ1 denote the time (t+1) excess return on asset i and the market portfolio m over a risk-free rate, respectively, and Et½:� denotes the expectation operator conditional on time t information. TheGARCH specifications do not arise directly from the ICAPM model, but they provide a parsimoniousapproximation of the form of heteroskedasticity typically encountered with financial time-series data. Similarconditional covariance specifications are used in Baillie and Bollerslev (1992), Bollerslev (1990), Bollerslev,Engle, and Wooldridge (1988), Bollerslev and Wooldridge (1992), Ding and Engle (2001), Engle and Kroner(1995), Engle and Mezrich (1996), Engle, Ng, and Wooldridge (1990), and Kroner and Ng (1998).

When considering stochastic investment opportunities governed by a set of state variables, I estimate theconditional covariance between each portfolio i and each state variable x, oix, using an analogous bivariateGARCH specification:


1Ri;t þ �i;tþ1, (10)

xtþ1 ¼ ax0 þ ax

1xt þ �x;tþ1, (11)

Et½�2i;tþ1� � s2i;tþ1 ¼ gi

0 þ gi1�

2i;t þ gi

2s2i;t, (12)

Et½�2x;tþ1� � s2x;tþ1 ¼ gx

0 þ gx1�

2x;t þ gx

2s2x;t, (13)

Et½�i;tþ1�x;tþ1� � oix;tþ1 ¼ gix0 þ gix

1 �i;t�x;t þ gix2 oix;t. (14)

I estimate the conditional covariances of each portfolio with the market portfolio and with each statevariable using the maximum likelihood method. Using �i;t and Vt to denote the bivariate demeaned excess

http://aida.econ.yale.edu/~shiller/

http://aida.econ.yale.edu/~shiller/


return vector and the conditional covariance matrix forecasts,

�t ¼Ri;t � ai

0 � ai1Ri;t�1

Rm;t � am0 � am

1 Rm;t�1

" #or

Ri;t � ai0 � ai

1Ri;t�1

xt � ax0 � ax

1xt�1

" #, (15)

Vt ¼s2i;t sim;t

sim;t s2m;t

" #or

s2i;t oix;t

oix;t s2x;t

" #, (16)

I can write the log-likelihood function as

LðYÞ ¼ �1

2

XN

t¼1

lnð2pÞ þ ln jVtj þ �Tt V�1t �t

� �, (17)

whereY denotes the vector of parameters in the specifications Eqs. (5)–(9) or Eqs. (10)–(14) and N denotes thenumber of monthly observations for each series.

Table 1

Maximum likelihood estimates of the conditional covariance dynamics

Entries report the maximum likelihood estimates of the parameters that govern the dynamics of the conditional covariance between the

excess returns on the market portfolio and the excess returns on each of the 30 industry portfolios (Panel A) and the 25 size/book-to-

market (BM) portfolios (Panel B):

sim;tþ1 ¼ gim0 þ gim

1 �i;t�m;t þ gim2 sim;t.

In the notation for the size/BM portfolios, the firm size increases from S1 to S5, and the BM ratio increases from B1 to B5. The t-statistics

of the parameter estimates are in parentheses. The estimation is based on monthly returns from July 1926 to December 2002.

Panel A. Covariance between excess returns on the market portfolio and on each of the 30 industry portfolios

Industrya gim0 gim

1 gim2

Food 0.00010 0.126 0.830

(5.43) (9.81) (59.07)

Beer 0.00009 0.085 0.875

(4.54) (7.66) (61.13)

Smoke 0.00010 0.107 0.850

(4.07) (9.13) (64.06)

Games 0.00025 0.113 0.824

(6.02) (10.08) (64.84)

Books 0.00042 0.266 0.620

(8.37) (10.88) (40.63)

Hshld 0.00059 0.178 0.535

(9.29) (10.95) (26.55)

Clths 0.00048 0.174 0.658

(9.28) (9.75) (35.27)

Hlth 0.00053 0.152 0.596

(9.88) (9.13) (29.63)

Chems 0.00025 0.106 0.813

(7.80) (7.67) (50.87)

Txtls 0.00016 0.120 0.831

(6.64) (10.04) (67.91)

Cnstr 0.00025 0.093 0.834

(8.89) (7.91) (62.74)

Steel 0.00029 0.074 0.842

(7.18) (5.48) (40.05)

FabPr 0.00048 0.122 0.732

(10.04) (7.23) (34.70)

ARTICLE IN PRESS

Table 1 (continued )

Panel A. Covariance between excess returns on the market portfolio and on each of the 30 industry portfolios

Industrya gim0 gim

1 gim2

ElcEq 0.00019 0.109 0.839

(5.22) (10.12) (55.24)

Autos 0.00029 0.141 0.770

(7.31) (10.39) (49.99)

Carry 0.00024 0.083 0.825

(8.24) (9.25) (69.99)

Mines 0.00012 0.134 0.818

(5.19) (7.52) (51.59)

Coal 0.00008 0.048 0.921

(3.58) (6.57) (77.16)

Oil 0.00010 0.088 0.871

(5.00) (8.18) (72.75)

Util 0.00007 0.100 0.872

(5.10) (12.11) (111.92)

Telcm 0.00008 0.130 0.822

(4.71) (9.17) (66.29)

Servs 0.00004 0.116 0.870

(5.66) (12.18) (111.52)

BusEq 0.00021 0.131 0.800

(6.56) (9.98) (63.55)

Paper 0.00010 0.091 0.871

(4.69) (10.30) (66.38)

Trans 0.00021 0.095 0.840

(5.07) (8.45) (44.74)

Whlsl 0.00010 0.116 0.854

(6.61) (9.17) (77.50)

Rtail 0.00059 0.132 0.596

(18.48) (7.94) (22.55)

Meals 0.00014 0.106 0.846

(4.69) (8.65) (57.18)

Fin 0.00024 0.131 0.793

(6.89) (8.68) (53.34)

Other 0.00007 0.105 0.872

(4.35) (10.57) (93.71)

Panel B. Covariance between excess returns on the market portfolio and on each of the 25 size/BM portfolios

Size/BM gim0 gim

1 gim2

S1B1 0.00015 0.109 0.855

(5.41) (9.06) (59.29)

S1B2 0.00019 0.136 0.831

(6.41) (10.34) (55.87)

S1B3 0.00017 0.103 0.842

(7.76) (8.50) (58.16)

S1B4 0.00012 0.084 0.870

(6.31) (8.59) (64.43)

S1B5 0.00011 0.077 0.875

(5.55) (6.71) (56.22)

S2B1 0.00023 0.136 0.797

(6.30) (9.99) (44.47)

S2B2 0.00015 0.097 0.853

(5.45) (7.65) (48.16)

S2B3 0.00013 0.088 0.862

(7.14) (10.01) (80.37)

S2B4 0.00013 0.094 0.859

(6.32) (7.88) (55.43)

T.G. Bali / Journal of Financial Economics 87 (2008) 101–131108

ARTICLE IN PRESS


Panel B. Covariance between excess returns on the market portfolio and on each of the 25 size/BM portfolios

Size/BM gim0 gim

1 gim2

S2B5 0.00009 0.074 0.896

(5.17) (7.94) (73.40)

S3B1 0.00022 0.196 0.756

(6.92) (13.65) (59.16)

S3B2 0.00021 0.104 0.820

(8.88) (7.42) (50.75)

S3B3 0.00017 0.097 0.836

(7.07) (8.83) (56.47)

S3B4 0.00018 0.134 0.810

(7.11) (10.79) (62.29)

S3B5 0.00017 0.141 0.809

(5.99) (9.96) (50.27)

S4B1 0.00013 0.148 0.819

(5.36) (12.76) (64.89)

S4B2 0.00013 0.117 0.839

(5.79) (10.19) (59.89)

S4B3 0.00016 0.115 0.829

(6.51) (7.58) (53.25)

S4B4 0.00011 0.102 0.857

(5.79) (8.35) (58.08)

S4B5 0.00014 0.131 0.829

(5.91) (13.99) (75.31)

S5B1 0.00016 0.117 0.825

(6.57) (9.58) (66.38)

S5B2 0.00018 0.133 0.896

(6.55) (9.23) (46.22)

S5B3 0.00013 0.109 0.835

(6.01) (8.76) (55.56)

S5B4 0.00012 0.111 0.837

(4.99) (7.98) (44.88)

S5B5 0.00010 0.113 0.852

(4.47) (8.73) (49.95)

aFood: Food Products; Beer: Beer & Liquor; Smoke: Tobacco Products; Games: Recreation; Books: Printing and Publishing; Hshld:

Consumer Goods; Clths: Apparel; Hlth: Healthcare, Medical Equipment, Pharmaceutical Products; Chems: Chemicals; Txtls: Textiles;

Cnstr: Construction and Construction Materials; Steel: Steel Works; FabPr: Fabricated Products and Machinery; ElcEq: Electrical

Equipment; Autos: Automobiles and Trucks; Carry: Aircraft, ships, and railroad equipment; Mines: Precious Metals, Non-Metallic, and

Industrial Metal Mining; Coal: Coal; Util: Utilities; Telcm: Communication; Servs: Personal and Business Services; BusEq: Business

Equipment; Paper: Business Supplies and Shipping Containers; Trans: Transportation; Whlsl: Wholesale; Rtail: Retail; Meals:

Restaraunts, Hotels, Motels; Fin: Banking, Insurance, Real Estate, Trading; Other: Everything Else.

T.G. Bali / Journal of Financial Economics 87 (2008) 101–131 109

Table 1 presents the parameter estimates for the conditional covariance between the excess portfolio returnsand the excess market returns for the 30 industry and 25 size/BM portfolios. The persistence of the conditionalcovariance dynamics on each series is measured by the sum of gim

1 and gim2 . The estimated value of gim

1 þ gim2 is in

the range of 0.713–0.986 for the 30 industry portfolios and 0.923–0.969 for the 25 size/BM portfolios. Hence, Iobserve more cross-sectional variations in the persistence of the conditional covariances among differentindustry portfolios than across the size/BM partitions.

Table 2 reports the sample mean, standard deviation, minimum, maximum, and autocorrelation statistics ofthe estimated conditional covariance series for the whole sample period of July 1926 to December 2002. Forcomparison, I also report the unconditional covariances between the excess portfolio return with the excessmarket return. For both the 30 industry and 25 size/BM portfolios, the time-series averages of the conditionalcovariances are similar to the corresponding unconditional covariance estimates. The standard deviations ofthe conditional covariances are large compared with their means. Hence, it is important to allow the

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Table 2

Summary statistics for the conditional covariance and conditional beta estimates for 1926–2002

Entries report the unconditional covariances (Uncond) between the excess monthly returns on market portfolio and 30 industry (Panel

A) and 25 size/book-to-market (BM) (Panel B) portfolios, and the sample average (Average), standard deviation (Std), minimum (Min),

maximum (Max), and first- and 12th-order monthly autocorrelations (Auto) of the corresponding conditional covariance estimates. For

comparison, the last three columns of the table report the corresponding unconditional beta and the sample average and standard

deviation of the conditional beta, bi;t ¼ sim;t=s2m;t. The sample period is from July 1926 to December 2002.

Panel A. 30 industry portfolios

(Un)conditional covariances (Un)conditional betas

Industrya Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std

Food 0.0023 0.0023 0.0028 �0.0004 0.0270 0.95 0.63 0.76 0.78 0.75

Beer 0.0029 0.0027 0.0035 �0.0010 0.0318 0.97 0.71 0.96 0.94 0.91

Smoke 0.0020 0.0019 0.0022 �0.0006 0.0207 0.95 0.60 0.64 0.68 0.74

Games 0.0042 0.0039 0.0052 0.0012 0.0572 0.95 0.62 1.38 1.28 1.19

Books 0.0033 0.0034 0.0048 0.0003 0.0576 0.81 0.38 1.08 1.17 1.14

Hshld 0.0027 0.0024 0.0023 0.0011 0.0373 0.74 0.29 0.87 0.96 0.89

Clths 0.0031 0.0028 0.0029 0.0007 0.0349 0.82 0.39 1.01 1.07 1.02

Hlth 0.0027 0.0025 0.0019 0.0005 0.0266 0.81 0.38 0.87 0.86 0.79

Chems 0.0031 0.0031 0.0028 0.0012 0.0271 0.94 0.61 1.02 1.08 1.02

Txtls 0.0035 0.0034 0.0043 0.0006 0.0395 0.95 0.59 1.15 1.14 1.10

Cnstr 0.0035 0.0032 0.0031 0.0014 0.0278 0.95 0.64 1.16 1.19 1.22

Steel 0.0040 0.0035 0.0035 0.0015 0.0364 0.94 0.59 1.31 1.36 1.28

FabPr 0.0037 0.0034 0.0031 0.0015 0.0377 0.90 0.53 1.23 1.29 1.20

ElcEq 0.0040 0.0037 0.0045 0.0012 0.0463 0.95 0.64 1.30 1.26 1.17

Autos 0.0037 0.0035 0.0045 0.0008 0.0530 0.90 0.53 1.21 1.20 1.15

Carry 0.0036 0.0031 0.0028 0.0006 0.0271 0.95 0.66 1.18 1.06 0.99

Mines 0.0027 0.0028 0.0033 0.0007 0.0362 0.93 0.57 0.90 0.97 0.94

Coal 0.0022 0.0024 0.0012 0.0006 0.0103 0.95 0.48 0.73 1.06 1.08

Oil 0.0027 0.0026 0.0027 0.0008 0.0214 0.96 0.69 0.87 0.90 0.93

Util 0.0025 0.0024 0.0034 �0.0006 0.0317 0.97 0.74 0.81 0.72 0.70

Telcm 0.0020 0.0018 0.0022 0.0001 0.0239 0.94 0.58 0.66 0.61 0.67

Servs 0.0026 0.0031 0.0021 �0.0009 0.0166 0.90 0.38 0.83 1.25 1.21

BusEq 0.0032 0.0030 0.0030 0.0010 0.0284 0.93 0.60 1.04 1.09 1.01

Paper 0.0029 0.0029 0.0030 0.0009 0.0281 0.96 0.65 0.95 1.02 0.97

Trans 0.0035 0.0031 0.0037 0.0012 0.0450 0.96 0.55 1.16 1.09 1.06

Whlsl 0.0034 0.0035 0.0044 0.0009 0.0405 0.96 0.63 1.10 1.17 1.12

Rtail 0.0029 0.0028 0.0019 0.0008 0.0246 0.80 0.39 0.96 0.88 0.84

Meals 0.0029 0.0028 0.0026 0.0007 0.0241 0.96 0.65 0.96 1.02 1.10

Fin 0.0035 0.0032 0.0043 0.0011 0.0516 0.94 0.61 1.14 1.06 1.08

Other 0.0032 0.0034 0.0042 0.0000 0.0368 0.97 0.67 1.04 1.11 1.14

Panel B. 25 size/BM portfolios


Size/BM Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std

S1B1 0.0050 0.0047 0.0065 0.0012 0.0668 0.95 0.65 1.65 1.50 1.46

S1B2 0.0046 0.0048 0.0082 0.0008 0.0919 0.92 0.52 1.51 1.47 1.39

S1B3 0.0042 0.0038 0.0051 0.0010 0.0489 0.95 0.61 1.38 1.25 1.27

S1B4 0.0040 0.0034 0.0047 0.0009 0.0457 0.96 0.67 1.30 1.12 1.23

S1B5 0.0042 0.0034 0.0045 0.0010 0.0411 0.96 0.67 1.39 1.14 1.19

S2B1 0.0038 0.0036 0.0034 0.0010 0.0302 0.93 0.54 1.24 1.30 1.26

S2B2 0.0038 0.0035 0.0042 0.0010 0.0415 0.96 0.62 1.25 1.20 1.15

S2B3 0.0036 0.0032 0.0042 0.0011 0.0423 0.96 0.63 1.19 1.09 1.19

S2B4 0.0037 0.0033 0.0044 0.0010 0.0416 0.96 0.67 1.21 1.10 1.14

S2B5 0.0041 0.0041 0.0048 0.0008 0.0424 0.97 0.70 1.35 1.20 1.22

S3B1 0.0039 0.0040 0.0056 0.0009 0.0661 0.91 0.50 1.27 1.26 1.21

S3B2 0.0035 0.0031 0.0031 0.0010 0.0294 0.94 0.63 1.13 1.13 1.08


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S3B3 0.0035 0.0031 0.0037 0.0009 0.0355 0.95 0.63 1.13 1.07 1.01

S3B4 0.0034 0.0033 0.0047 0.0010 0.0521 0.95 0.61 1.12 1.09 1.12

S3B5 0.0042 0.0039 0.0064 0.0010 0.0725 0.94 0.59 1.37 1.20 1.17

S4B1 0.0033 0.0034 0.0036 0.0009 0.0345 0.94 0.61 1.08 1.15 1.09

S4B2 0.0033 0.0032 0.0041 0.0010 0.0390 0.95 0.60 1.09 1.06 1.02

S4B3 0.0033 0.0031 0.0040 0.0010 0.0425 0.95 0.63 1.09 1.06 1.04

S4B4 0.0036 0.0033 0.0049 0.0009 0.0513 0.96 0.65 1.17 1.04 1.00

S4B5 0.0044 0.0042 0.0072 0.0008 0.0775 0.95 0.63 1.45 1.23 1.26

S5B1 0.0030 0.0029 0.0029 0.0010 0.0278 0.94 0.62 0.98 1.04 1.06

S5B2 0.0028 0.0027 0.0029 0.0010 0.0314 0.94 0.58 0.92 0.97 0.94

S5B3 0.0030 0.0028 0.0037 0.0009 0.0404 0.96 0.65 0.98 0.92 0.91

S5B4 0.0034 0.0033 0.0050 0.0003 0.0526 0.96 0.65 1.12 0.96 0.94

S5B5 0.0039 0.0036 0.0061 0.0000 0.0641 0.96 0.65 1.26 1.07 1.10









conditional covariances to vary over time when estimating the ICAPM relations. I measure the cross correlationsbetween the different conditional covariance and market portfolio conditional variance estimates, and I find thatthe conditional covariance series show highly positive cross correlations with one another and with theconditional variance of the market portfolio. The full-sample correlation estimates range from 0.85 to 0.99.

Given the conditional covariance estimates for each portfolio ðsim;tÞ and the conditional variance estimatesfor the market portfolio ðs2m;tÞ, I also compute the conditional betas for each portfolio i as bi;t ¼ sim;t=s2m;t. Inthe last three columns of Table 2, I report the sample average and standard deviation of the conditional betaestimates, as well as the unconditional beta estimates for each portfolio, bi ¼ CovðRi;RmÞ=VarðRmÞ. Thesample averages of the conditional betas are close to the magnitudes of the corresponding unconditional betas.

Table 3 presents the same descriptive statistics for the unconditional and conditional covariance and betaestimates over the subsample period of July 1963 to December 2002. Similar to my findings in Table 2, thetime-series averages of the conditional covariances and betas are similar to their corresponding unconditionalestimates for both the 30 industry and 25 size/BM portfolios. The major difference between the whole sampleand subsample estimates is that the volatilities of the conditional covariances and betas are much lower for theCompustat era because the subsample period (1963–2002) does not include the extremely volatile GreatDepression era. As shown in Table 3, the standard deviations of conditional betas are in the range of 0.28–0.42for 30 industry portfolios and 0.26–0.46 for 25 size/BM portfolios. These estimates are similar to the findingsof Lewellen and Nagel (2006) but larger than the rolling window estimates of Fama and French (2006). Icannot compare my estimates with those of Ang and Chen (2007) because they do not report theunconditional standard deviation of their stochastic betas. For the sample period of 1964–2001, Lewellen andNagel (2006) find the standard deviation of monthly betas to be roughly 0.30 for a small-minus-big portfolio,0.25 for a value-minus-growth portfolio, and 0.60 for a winner-minus-loser portfolio. For the sample period of1963–2001, the standard deviation of 60-month rolling window beta estimates of Fama and French (2006) isroughly 0.26 for a small-minus-big portfolio and 0.24 for a value-minus-growth portfolio. Although it is notcomparable with the unconditional standard deviation of my conditional betas, Ang and Chen (2007) find theconditional standard deviation of their stochastic betas in the range of 0.13–0.17 for the value and growthportfolios.

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Table 3

Summary statistics for the conditional covariance and conditional beta estimates for 1963–2002

Entries report the unconditional covariances (Uncond) between the excess monthly returns on market portfolio and 30 industry (Panel

A) and 25 size/book-to-market (BM) (Panel B) portfolios, and the sample average (Average), standard deviation (Std), minimum (Min),

maximum (Max), and first- and 12th-order monthly autocorrelations (Auto) of the corresponding conditional covariance estimates. For

comparison, the last three columns of the table report the corresponding unconditional beta and the sample average and standard

deviation of the conditional beta, bi;t ¼ sim;t=s2m;t. The sample period is from July 1963 to December 2002.


Industrya (Un)conditional covariances (Un)conditional betas

Uncond Average Std Min Max Auto(1) Auto(12) Uncond Average Std

Food 0.0015 0.0016 0.0009 0.0007 0.0069 0.95 0.49 0.76 0.74 0.31

Beer 0.0015 0.0016 0.0011 0.0004 0.0076 0.95 0.47 0.75 0.76 0.30

Smoke 0.0014 0.0014 0.0010 0.0007 0.0077 0.97 0.65 0.70 0.72 0.36

Games 0.0026 0.0027 0.0013 0.0013 0.0113 0.95 0.51 1.30 1.33 0.38

Books 0.0021 0.0021 0.0016 0.0011 0.0187 0.95 0.52 1.03 1.02 0.32

Hshld 0.0019 0.0019 0.0010 0.0011 0.0112 0.95 0.43 0.92 0.92 0.30

Clths 0.0022 0.0023 0.0013 0.0007 0.0150 0.93 0.47 1.10 1.08 0.35

Hlth 0.0018 0.0018 0.0008 0.0005 0.0098 0.93 0.38 0.88 0.90 0.35

Chems 0.0020 0.0020 0.0009 0.0012 0.0091 0.94 0.55 0.98 1.02 0.41

Txtls 0.0020 0.0020 0.0012 0.0006 0.0109 0.96 0.54 0.98 0.98 0.33

Cnstr 0.0023 0.0023 0.0011 0.0011 0.0084 0.95 0.50 1.11 1.12 0.28

Steel 0.0022 0.0023 0.0010 0.0013 0.0087 0.96 0.60 1.09 1.11 0.31

FabPr 0.0024 0.0024 0.0010 0.0016 0.0115 0.96 0.53 1.16 1.18 0.32

ElcEq 0.0025 0.0025 0.0013 0.0012 0.0106 0.94 0.50 1.23 1.21 0.40

Autos 0.0020 0.0020 0.0011 0.0008 0.0118 0.94 0.50 0.99 1.00 0.34

Carry 0.0023 0.0023 0.0009 0.0013 0.0080 0.94 0.47 1.10 1.12 0.33

Mines 0.0018 0.0018 0.0013 0.0007 0.0123 0.90 0.44 0.90 0.90 0.38

Coal 0.0020 0.0020 0.0011 0.0008 0.0081 0.97 0.67 0.98 0.98 0.36

Oil 0.0016 0.0016 0.0009 0.0008 0.0060 0.94 0.45 0.78 0.77 0.34

Util 0.0011 0.0011 0.0008 0.0004 0.0062 0.96 0.55 0.53 0.53 0.27

Telcm 0.0016 0.0016 0.0010 0.0001 0.0057 0.98 0.70 0.79 0.75 0.36

Servs 0.0029 0.0029 0.0017 0.0011 0.0106 0.96 0.60 1.41 1.38 0.37

BusEq 0.0026 0.0025 0.0015 0.0010 0.0101 0.97 0.66 1.26 1.20 0.42

Paper 0.0019 0.0020 0.0010 0.0009 0.0083 0.94 0.46 0.94 0.95 0.28

Trans 0.0023 0.0023 0.0010 0.0012 0.0080 0.95 0.53 1.11 1.14 0.37

Whlsl 0.0023 0.0023 0.0013 0.0011 0.0108 0.95 0.53 1.13 1.13 0.36

Rtail 0.0021 0.0022 0.0009 0.0011 0.0078 0.95 0.49 1.04 1.03 0.30

Meals 0.0024 0.0023 0.0013 0.0007 0.0104 0.96 0.55 1.16 1.13 0.38

Fin 0.0020 0.0021 0.0011 0.0011 0.0085 0.94 0.42 0.99 0.99 0.31

Other 0.0022 0.0022 0.0013 0.0002 0.0095 0.94 0.47 1.06 1.08 0.35




S1B1 0.0029 0.0029 0.0016 0.0011 0.0108 0.95 0.54 1.44 1.39 0.44

S1B2 0.0025 0.0025 0.0016 0.0012 0.0129 0.95 0.55 1.22 1.20 0.38

S1B3 0.0022 0.0021 0.0012 0.0010 0.0085 0.95 0.57 1.07 1.11 0.46

S1B4 0.0020 0.0020 0.0011 0.0010 0.0093 0.95 0.55 0.99 0.98 0.33

S1B5 0.0021 0.0020 0.0012 0.0009 0.0098 0.95 0.55 1.01 0.99 0.34

S2B1 0.0029 0.0029 0.0014 0.0012 0.0106 0.96 0.58 1.44 1.44 0.35

S2B2 0.0024 0.0024 0.0012 0.0013 0.0125 0.96 0.56 1.17 1.17 0.32

S2B3 0.0021 0.0021 0.0012 0.0010 0.0100 0.95 0.54 1.02 1.03 0.32

S2B4 0.0020 0.0020 0.0010 0.0011 0.0078 0.95 0.52 0.96 0.97 0.32

S2B5 0.0021 0.0021 0.0011 0.0011 0.0096 0.94 0.55 1.04 1.07 0.45

S3B1 0.0028 0.0028 0.0019 0.0010 0.0160 0.95 0.52 1.36 1.36 0.35

S3B2 0.0022 0.0022 0.0010 0.0012 0.0091 0.96 0.60 1.10 1.12 0.38


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S3B3 0.0020 0.0020 0.0010 0.0012 0.0087 0.94 0.48 0.97 0.98 0.30

S3B4 0.0018 0.0018 0.0009 0.0011 0.0073 0.96 0.55 0.89 0.91 0.33

S3B5 0.0020 0.0020 0.0011 0.0011 0.0093 0.95 0.51 0.98 1.00 0.33

S4B1 0.0026 0.0026 0.0016 0.0009 0.0113 0.96 0.56 1.26 1.23 0.35

S4B2 0.0022 0.0022 0.0012 0.0009 0.0101 0.96 0.55 1.07 1.07 0.29

S4B3 0.0020 0.0020 0.0010 0.0010 0.0080 0.94 0.46 0.97 0.99 0.31

S4B4 0.0018 0.0019 0.0009 0.0009 0.0073 0.94 0.51 0.90 0.92 0.32

S4B5 0.0020 0.0021 0.0012 0.0008 0.0095 0.95 0.58 0.99 1.01 0.35

S5B1 0.0021 0.0021 0.0011 0.0011 0.0089 0.95 0.45 1.01 1.01 0.27

S5B2 0.0019 0.0020 0.0010 0.0010 0.0093 0.95 0.56 0.95 0.95 0.29

S5B3 0.0017 0.0018 0.0008 0.0010 0.0068 0.95 0.53 0.86 0.86 0.26

S5B4 0.0016 0.0016 0.0009 0.0009 0.0065 0.94 0.46 0.78 0.80 0.30

S5B5 0.0016 0.0016 0.0009 0.0009 0.0071 0.95 0.50 0.79 0.81 0.32









The affinity in magnitudes between the sample averages of the conditional estimates and their unconditionalcounterparts show that my bivariate GARCH specifications generate reasonable conditional variance andcovariance estimates. As a further sensibility check, I compute the value-weighted averages of the conditionalcovariances of all the portfolios within each portfolio partition, using the average firm size of each portfolioreported in Kenneth French’s online data library to determine the weight. Then, I compare the properties ofthe weighted average conditional covariances with the conditional variance of the market portfolio. The toptwo panels of Fig. 1 plot the time series of the value-weighted covariances and the market variances for the 30industry and 25 size/BM portfolios. In each panel, the solid line denotes the weighted averages of theconditional covariances, and the dashed line is the market portfolio conditional variance benchmark. Theweighted-average covariances are all in the same range as the market portfolio variances. The two series ineach panel move closely together. The bottom two panels of Fig. 1 plot the time series of the value-weightedconditional betas (solid lines) and compare them with the market beta (one, dashed lines) benchmark for the30 industry and 25 size/BM portfolios. The value-weighted averages fluctuate around one within a narrowband. The deviations come from temporal mismatches in portfolio weights. The composition of the marketportfolio is updated monthly based on market capitalization. In contrast, the compositions of the industry andsize/BM portfolios are updated annually. Furthermore, when I average the conditional covariances and betasof these portfolios, I use firm size information reported in French’s data library, which is also updatedannually. The weights mismatches from these two layers generate deviations between the weighted-averageconditional betas and the market portfolio beta.

As a further robustness check, I use the GARCH conditional betas in a conditional market-modelregression of Shanken (1990):

Ri;t ¼ a0;i þ a1;ibi;t þ ðb0;i þ b1;ibi;tÞRm;t þ ei;t; i ¼ 1; 2; . . . ; n, (18)

where Ri;t is the return on portfolio i at month t, Rm;t is the return on market portfolio at month t, and bi;t isthe time-varying conditional beta of portfolio i at month t. If the GARCH conditional betas are unbiased thenthe coefficients b0;i and b1;i should be zero and one, respectively. Pagan and Schwert (1990) use a similar test to

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Fig. 1. Value-weighted conditional covariances and conditional betas. In each panel, the solid line denotes the value-weighted average of

the conditional covariance (top panels) and conditional beta (bottom panels) estimates across the different portfolios within the 30

industry portfolio partition (left panels) and the 25 size/book-to-market (BM) portfolio partition (right panels). The dashed line in top

panels denotes the conditional variance of the market portfolio and the dashed line in bottom panels denotes the beta of market portfolio.


compare the relative performance of alternative conditional volatility models in terms of their ability toproduce unbiased forecasts of future monthly volatility. I use the Wald test to determine whether the jointhypothesis of b0;i ¼ 0, b1;i ¼ 1 can be rejected for the 30 industry and 25 size/BM portfolios. As shown inAppendix A, the Wald statistics distributed as Chi-square with two degrees of freedom are smaller than thecritical values, indicating that the conditional betas are unbiased for both the whole sample and subsampleperiods.

I also compute the cross-sectional average of b0;i and b1;i as well as the cross-sectional average of theirstandard errors for 55 portfolios pooled from the 30 industry and 25 size/BM portfolios. For the sampleperiod of July 1963–December 2002, the cross-sectional mean of b0;i is about 0.0019 with the average standarderror of 0.2440 and the cross-sectional mean of b1;i is about 1.0335 with the average standard error of 0.2565.

3.3. Estimating the risk-return relations

Given the conditional covariances, we estimate the intertemporal relation from the following system ofequations,

Ri;tþ1 ¼ Ci þ Asim;tþ1 þ Boix;tþ1 þ ei;tþ1; i ¼ 1; 2; . . . ; n, (19)

where n denotes the number of portfolios and also the number of equations in the estimation. For example, inthe five-industry partition case, I simultaneously estimate n ¼ 5 equations. As I make the industry partitionfiner from five to ten, 12, 17, 30, and 48, I am estimating increasingly larger systems of equations. I constrainthe slope coefficients (A, B) to be the same across all portfolios for cross-sectional consistency. I allow theintercept Ci to differ across different portfolios. Under the null hypothesis of ICAPM, the intercepts should bezero. I use deviations of the intercept estimates from zero as a test against the validity and sufficiency of theICAPM specification.

I estimate the system of equations using a weighted least square method that allows me to place constraintson coefficients across equations. I compute the t-statistics of the parameter estimates accounting forheteroskedasticity and autocorrelation as well as contemporaneous cross-correlations in the errors fromdifferent equations. The estimation methodology can be regarded as an extension of the seemingly unrelatedregression (SUR) method, the details of which are in Appendix B.


4. The risk-return relation

I first report the estimation results on the intertemporal risk-return relation assuming zero intertemporalhedging demand. Then, I discuss the cross-sectional implications from the perspective of the conditionalCAPM. I conclude this section by discussing additional risk premiums induced by conditional covarianceswith various financial and macroeconomic variables.

4.1. The intertemporal risk– return relation

Table 4 reports the common slope estimates and the t-statistics from the following system of equations:

Ri;tþ1 ¼ Ci þ Asim;tþ1 þ ei;tþ1; i ¼ 1; 2; . . . ; n, (20)

under different portfolio partitions and for both the full sample period from July 1926 to December 2002(Panel A) and a more recent subsample from July 1963 to December 2002 (Panel B). The estimates arerelatively stable across different portfolio partitions, between 1.22 and 1.78 during the full sample periodand between 2.14 and 4.16 during the subsample period. The t-statistics show that all estimates arehighly significant. The consistent estimates and high t-statistics across different portfolio partitions andsample periods suggest that the identified positive risk–return relation is not only significant, but also robust.Based on the relative risk aversion interpretation, the magnitudes of the estimates are also economicallysensible.

For comparison, I also do what the traditional literature does and estimate the risk-return relation over thefull sample period using the single series of the market portfolio:

Rm;tþ1 ¼ 0:0038ð1:4162Þ

þ 0:8034ð0:8777Þ

s2m;tþ1 þ em;tþ1, (21)

where I report the t-statistics in parentheses below the coefficient estimates. The slope estimate is lower thanwhat I obtain in Table 4. More important, the t-statistic for the slope estimate is small. The slope estimateis no longer significantly different from zero. To carry this exercise one step further, I also estimate the

Table 4

The intertemporal risk-return relation with cross-sectional consistency

Entries report the common slope estimates and the t-statistics (in parentheses) from the following system of equations:

Ri;tþ1 ¼ Ci þ Asim;tþ1 þ ei;tþ1; i ¼ 1; 2; . . . ; n,

where n denotes the number of portfolio partitions and the number of regression equations in the estimation. Each row reports the

estimates based on one portfolio partition. The t-statistics adjust for heteroskedasticity and autocorrelation for each series and cross-

correlations among the portfolios. BM ¼ book-to-market.

Sample Period July 1926–December 2002 July 1963–December 2002

5 Industry 1.7808 2.1161

(2.4901) (2.1767)

10 Industry 1.7258 2.1425

(2.5128) (2.4274)

12 Industry 1.2195 2.3681

(2.5731) (2.6354)

17 Industry 1.3142 3.4923

(2.5366) (2.5573)

30 Industry 1.3843 3.4937

(2.5207) (2.4961)

48 Industry – 3.8675

(2.5806)

25 Size/BM 1.2512 4.1597

(2.5487) (2.5164)


risk-return relation based on a single portfolio series:

Ri;tþ1 ¼ Ci þ Aisim;tþ1 þ ei;tþ1. (22)

I repeat this exercise for all the industry portfolios and the market portfolio, altogether 75 series. The 75series are the collection of five-, ten-, 12-, 17-, 30-industry portfolios and the market portfolio. The estimationdiffers from Eq. (20) by relaxing the cross-sectional consistency requirement. The single-series estimation cangenerate different relative risk aversion coefficients from different portfolio series and hence does notguarantee consistency across different portfolios. Fig. 2 plots the histogram of the 75 estimates in the left paneland the corresponding t-statistics in the right panel. From the left panel, I observe that the slope estimates arewidely dispersed, from �0.24 to 7.67. The right panel shows that the t-statistics also vary dramatically from�0.28 to 5.18. Of the 75 estimates, only 31 have t-statistics greater than 1.96 in absolute magnitude. Therefore,by narrowly focusing on one series instead of making full use of the cross sections, the estimation loses itsstatistical power. The estimates show both large cross-sectional sample variation and low statisticalsignificance.

The R-squared estimates for each return series in Eq. (20) are between 1% and 5%. Since the equations areforecasting equations on portfolio returns, the low percentages are consistent with market efficiency and are in

Fre

quency

Fre

quency

Estimates for Ai

t-statistics for Ai

15

10

5

0−1

−1 0 1 2 3 4 5 6

0

8

7

6

5

4

3

2

1

0

1 2 3 4 5 6 7 8

Fig. 2. Risk–return coefficient estimates based on single portfolio series. The two panels plot the histograms of the risk–return coefficient

estimates (left panel) and t-statistics (right panel) based on each of the 75 different industry and market portfolio series, which are

collections of the five-, ten-, 12-, 17-, and 30-industry portfolio series and the market portfolio for the sample period of July 1926 to

December 2002.


the same range as those from the single-series forecasting regressions without cross-sectional constraints. Thelow explained percentages also suggest that the fundamental risk–return relation is inherently difficult toidentify. By using a large cross section of return series and constraining them to share the same slopecoefficient, I not only achieve cross-sectional consistency in pricing all portfolios according to the samerisk–return relation, but also gain statistical power in identifying significant, robust, and economically sensiblerisk–return coefficients.

4.2. The cross-sectional implications of the ICAPM

By expanding my analysis of the time-series relation to a large cross section of stock portfolios whilemaintaining cross-sectional consistency, I also move closer to the conditional CAPM literature, which focuseson the cross-sectional relations between excess returns and market beta.

Ignoring intertemporal hedging demands, the ICAPM predicts that the expected excess returns vary bothcross sectionally and over time with their conditional covariances with the market. My results in Section 4.1confirm the time-series prediction of the ICAPM as I identify a positive and highly significant intertemporalrisk–return trade-off. However, the existing literature finds little evidence supporting the cross-sectionalimplication. Many studies find that little cross-sectional relation exists between unconditional betas andexpected excess returns (e.g., Fama and French, 1992; Jagannathan and Wang, 1996). In this subsection, Ireconcile with the literature and highlight the fundamental connection between the time-series and the cross-sectional risk–return relations implied by the ICAPM.

Generically, I can write the risk–return relation in the following two forms in terms of covariance and beta,respectively:

Covariance : Ri;tþ1 ¼ Ci þ Asim;tþ1 þ ei;tþ1;

Beta : Ri;tþ1 ¼ f0;t þ fm;tbim;tþ1 þ ei;tþ1;(23)

both of which can be interpreted as either time-series or cross-sectional relations. According to Merton (1973),the slope estimate from the covariance regression represents the average relative risk aversion of marketinvestors, whereas the slope estimate from the beta regression represents the market risk premium. Section 4.1focuses on the covariance equation and treats it as a time-series relation with time-constant coefficients. Iestimate the time-series relation while constraining the slope coefficients to be the same across all portfolios,but I allow the intercept to vary across different portfolios. In contrast, the unconditional and conditionalCAPM literatures often focus on the beta equation and treat it as a cross-sectional relation across differentstocks or portfolios. Unconditional CAPM assumes that beta is constant over time and hence the same cross-sectional relation holds across all time periods. Conditional CAPM allows beta to be time-varying andcontends that the cross-sectional relation holds period by period.

Fama and MacBeth (1973) estimate the cross-sectional relation period by period and then report thesummary statistics of time series on the intercept and slope estimates. For comparison, I start my analysis byrepeating a similar exercise using the whole-sample estimates of unconditional covariance and beta. First, Iestimate the unconditional covariance and unconditional beta of each portfolio using the entire return dataand then run the cross-sectional regressions for each month from July 1926 to December 2002. In this exercise,the conditional covariances and betas (sim;tþ1 and bim;tþ1) in Eq. (23) are replaced by their unconditionalcounterparts.

In Panel A of Table 5, I report the time-series averages and the Newey and West (1987) t-statistics of theintercept and slope estimates based on the 30 industry and 25 size/BM portfolios. I also repeat the estimationusing the one hundred size/beta portfolios formed as in Jagannathan and Wang (1996). The results are similarto those presented in Table 5. To save space, henceforth I report results only based on the 30 industryportfolios and the 25 size/BM portfolios. The results are similar to those reported in the existing literature: Theaverage slope estimate is not significantly different from zero, and the average intercept is large and highlysignificant. The results are similar whether the cross-sectional regressions are on beta or covariance. Thesefindings have prompted some to claim that unconditional beta does not matter. At an earlier stage of thestudy, I also estimate the unconditional covariance and unconditional beta of each portfolio using 60 months

ARTICLE IN PRESS

Table 5

The cross-sectional implications of the intertemporal capital asset pricing model (ICAPM)

Sample period: July 1926–December 2002

30 Industry portfolios 25 Size/BM portfolios

Intercept Slope Intercept Slope

Panel A. Fama and MacBeth regressions with unconditional covariance and beta estimates

Covariance 0.0104 �0.0566 0.0076 1.1285

(5.7439) (�0.0659) (2.5128) (0.9411)

Beta 0.0104 �0.0002 0.0076 0.0034

(5.7439) (�0.0659) (2.5128) (0.9411)

Panel B. Fama and MacBeth regressions with conditional covariance and beta estimates

Covariance 0.0038 1.3157 0.0042 1.2116

(3.1682) (2.2977) (3.2744) (2.3432)

Beta 0.0038 0.0041 0.0042 0.0037

(3.1682) (2.1706) (3.2744) (2.2054)

Panel C. Cross-sectional relations with intertemporal stability

Covariance 0.0035 1.4326 0.0037 1.3922

(2.9671) (2.4702) (3.0105) (2.5809)

Beta 0.0037 0.0046 0.0040 0.0040

(3.1370) (2.2804) (3.1982) (2.3965)

Sample period: July 1963–December 2002

Panel A. Fama and MacBeth regressions with unconditional covariance and beta estimates

Covariance 0.0101 �0.0641 0.0178 �3.0248

(3.4344) (�0.0352) (4.4060) (�1.3265)

Beta 0.0101 �0.0001 0.0178 �0.0061

(3.4344) (�0.0352) (4.4060) (�1.3265)

Panel B. Fama and MacBeth regressions with conditional covariance and beta estimates

Covariance �0.0014 3.1572 �0.0015 3.6807

(�1.1581) (2.3051) (�1.0106) (2.3682)

Beta �0.0014 0.0065 �0.0015 0.0074

(�1.1581) (2.2140) (�1.0106) (2.2405)

Panel C. Cross-sectional relations with intertemporal stability

Covariance �0.0023 3.5473 �0.0026 4.1905

(�1.0896) (2.4880) (�1.1213) (2.5333)

Beta �0.0024 0.0075 �0.0027 0.0083

(�1.1002) (2.3117) (�1.1966) (2.3779)


of past-return data, as in Fama and French (1992). Then, I run the cross-sectional regressions for each monthfrom July 1931 to December 2002. In this exercise, the conditional covariances and betas are replaced by theirunconditional counterparts based on a five-year rolling window. The qualitative results are similar to thoseobtained from the whole-sample unconditional estimates of market risk. The average slope coefficient is notsignificantly different from zero, whereas the average intercept is highly significant. Similar results are alsoobtained for the subsample period of July 1963–December 2002.

Some of the earlier studies on conditional CAPM, e.g., Jagannathan and Wang (1996), focus on anunconditional implication of the conditional relation by regressing excess returns on the unconditional betaand an unconditional covariance term that accounts for the covariation between the conditional beta and theconditional market risk premium. They find significant slope estimates on the additional covariance term, butthe slope estimate on the unconditional beta remains insignificant.

However, a more direct test of the conditional CAPM is to estimate the conditional relation directly, insteadof estimating an unconditional implication of the conditional relation. Given my GARCH estimates on the


conditional variance and covariances, I can estimate the conditional cross-sectional relations directly as in Eq.(23) month by month and then analyze the sample properties of the estimates following Fama and MacBeth(1973). Panel B of Table 5 reports the time-series averages and Newey and West (1987) t-statistics of themonthly estimates on the intercept and the slope of the conditional relations from July 1926 to December2002. By testing the conditional CAPM conditionally, I obtain positive and significant slope estimate on boththe conditional covariance and the conditional beta. The coefficient estimates on the conditional covarianceare 1.32 (with a t statistic of 2.30) for the 30 industry portfolios and 1.21 (with a t-statistic of 2.34) for the 25size/BM portfolios, similar to my time-series relation estimates in Section 4.1. The coefficient estimates on theconditional betas are 0.41% with a t-statistic of 2.17 for the 30 industry and 0.37% with a t-statistic of 2.21 for25 size/BM portfolios. The corresponding annualized market risk premiums are 4.92% and 4.44%,respectively, both of which are in economically sensible ranges. Therefore, my estimation suggests thatconditional beta matters when estimated conditionally using the bivariate-GARCH model. Another notablepoint in data for sample period July 1926–December 2002 in Table 5 is that the intercepts drop considerably(but remain significantly positive in the full sample) in moving from Panel A to Panel B. For the 25 size/BMportfolios in the full sample, the point estimates of the risk premiums in Panels A and B are similar and thegreater statistical significance in Panel B comes primarily from lower standard errors because there is generallya larger spread in betas in the conditional regressions.

I can go one step further and link the conditional CAPM result tightly to my time-series estimation inSection 4.1. By keeping the intercept and slope in Eq. (23) constant across both time and cross-section, I caninterpret the estimates either as intertemporal relations with cross-sectional consistency or as cross-sectionalrelations with intertemporal stability constraint on the relative risk aversion (for the covariance regression) orthe market risk premium (for the beta regression). Table 5 reports the results from the pooled regression inPanel C. Similar to what I found in Section 4.1, the slope coefficient on the conditional covariance regression isabout 1.4 and highly significant. The slope estimates on the conditional beta regression are also highlysignificant. The monthly estimates of 0.46% and 0.40% correspond to annualized market risk premium of5.52% and 4.8%, respectively.

I have so far discussed the cross-sectional implications of the ICAPM for the whole-sample period of July1926–December 2002. Table 5 also presents results for the subsample period of July 1963–December 2002.Similar to what I found from the whole sample, the slope coefficients on the conditional covariance andconditional beta regressions are statistically significant, whereas the cross-sectional relation between expectedreturns and unconditional measures of market risk is flat.

Overall, my results from both samples indicate that conditional beta and conditional covariance do matter.The expected excess return on a risky portfolio increases with its conditional covariance with the marketportfolio, or equivalently with its conditional beta. The estimated relative risk reversion coefficient and themarket risk premium are not only statistically significant, but also in economically sensible ranges. The ofteninsignificant estimates in the literature can be attributed to the low statistical power caused by focusingnarrowly on either the intertemporal risk–return relation of a single market return series or the unconditionalmeasures of covariance and beta. I contribute to the literature by expanding the analysis of the intertemporalrelation to a large cross section of stock portfolios and by expanding the cross-sectional analysis to eachconditional time step based on bivariate-GARCH estimates. Furthermore, by simultaneously considering theintertemporal and cross-sectional implications of the ICAPM, I not only achieve consistency across both timeand cross section, but I also gain statistical power by using the GARCH conditional risk measures and bypooling data along the two dimensions.

4.3. Risk premiums induced by conditional covariation with SMB and HML

When the investment opportunity is stochastic, investors adjust their investment to hedge against futureshifts in the investment opportunity and achieve intertemporal consumption smoothing. Hence, covariationswith state of the investment opportunity induce additional risk premiums on an asset. In this subsection, I takethe size (SMB) and book-to-market (HML) factors of Fama and French (1993) to describe the state of theinvestment opportunity, and I investigate whether covariations with these two factors induce additional riskpremiums on a stock portfolio. I measure the conditional covariance of each portfolio with these two factors


and estimate the following system of equations:

Ri;tþ1 ¼ Ci þ Asim;tþ1 þ Bsois;tþ1 þ Bhoih;tþ1 þ ei;tþ1; i ¼ 1; 2; . . . ; n, (24)

where ois;tþ1 and oih;tþ1 measure the time-t expected conditional covariance between the time-(t +1) excessreturn on portfolio i and the two risk factors SMB and HML, respectively. I compute the sample mean,standard deviation, minimum, and maximum values of ois;tþ1 and oih;tþ1 for the full sample period of July1926–December 2002. The sample mean of ois;tþ1 is in the range of 0.00033–0.00210, and the standarddeviations range from 0.00031 to 0.00244. The volatilities of ois;tþ1 are large compared with their means. Formost of the 25 size/BM portfolios, both the mean and the standard deviation of oih;tþ1 are small, especially ascompared with the values of ois;tþ1. From the estimates on Bs and Bh, one can learn how investors react to thecovariations with the two state variables. If upward shocks in the two state variables predict favorable shifts inthe investment opportunity, we would expect risk averse investors to invest less in an asset when the asset’scorrelation with the state variables increases. As a consequence, I would expect the excess return on the assetto increase with increased correlation with the state variables and hence a positive estimate for the slopecoefficient B. However, negative estimates for the coefficient B would indicate that upward shocks in the statevariable predict unfavorable shifts in the investment opportunity. Increased correlation with the state variablewould induce a positive intertemporal hedging demand and a negative risk premium.

Table 6 reports the slope estimates and t-statistics of different restricted and unrestricted versions of thesystem of equations in Eq. (24) on the 30 industry and 25 size/BM portfolios. For both portfolio partitionsand in all specifications that include conditional covariances with the SMB factor, the coefficient estimates Bs

on the SMB factor are strongly negative. Thus, an increase in the covariance of an asset return with the SMB

factor predicts a decrease in the asset’s expected excess return for the next period. The negative estimatessuggest that upward movements in the SMB factor predict unfavorable shifts in the investment opportunity.

Campbell (1993, 1996) argues that an unfavorable shift in the investment opportunity can come in the formof heightened market volatility. Based on my negative coefficient estimate, I conjecture that an increase in theSMB factor can potentially predict an increase in future market volatility. To test this conjecture, I computethe correlation between SMB and the changes in the next month’s realized variance on the market portfolioexcess return, (s2m;tþ1 � s2m;t). I compute the realized variance according to the following formula:

s2m;t ¼XDt

d¼1

R2md þ 2

XDt

d¼2

Rm;dRm;d�1, (25)

Table 6

Risk premiums induced by conditional covariation with firm-size (SMB) and book-to-market (HML) factors

Entries report the slope estimates and t-statistics (in parentheses) from the system of equations,

Ri;tþ1 ¼ Ci þ Asim;tþ1 þ Bsois;tþ1 þ Bhoih;tþ1 þ ei;tþ1; i ¼ 1; 2; . . . ; n,

where ois;t and oih;t measure the time-t expected conditional covariance between the excess return on portfolio i and the two risk factors

SMB and HML, respectively. Estimation is based on monthly data from July 1926 to December 2002. BM ¼ book-to-market.

Portfolios A Bs Bh

30 Industry 2.2221 �4.1552 –

(3.3150) (�2.1821)

30 Industry 1.2783 – 2.8201

(2.4836) (0.7392)

30 Industry 2.1342 �4.1137 2.5993

(3.2713) (�2.1723) (1.0004)

25 Size/BM 2.1440 �3.3612 –

(3.1529) (�2.2989)

25 Size/BM 1.2343 – 0.1566

(2.4588) (0.9284)

25 Size/BM 2.1903 �3.3801 �0.3796

(3.3219) (�2.3849) (�0.1362)


where Dt is the number of trading days in month t and Rm;d is the value-weighted market portfolio return onday d. The second term on the right-hand side adjusts for the autocorrelation in daily returns, as in French,Schwert, and Stambaugh (1987) and Goyal and Santa-Clara (2003). The daily data on the CRSP value-weighted index are available from July 2, 1962 to December 31, 2002. I estimate the market portfolio’smonthly realized variance over this sample period and measure its correlation with the SMB factor. Consistentwith my conjecture, the correlation coefficient between SMBt and monthly changes in realized varianceðs2m;tþ1 � s2m;tÞ is positive and at about 13%. I experiment with alternative realized variance measures andobtain similarly positive correlation estimates. Moskowitz (2003) also reports evidence that the SMB factorpredicts future market volatility. Therefore, the negative risk premium on the SMB factor might be partiallyinduced by its positive prediction of future market volatility.

By contrast, the coefficient estimates on the HML factor are not statistically significant, indicating thatinvestors do not react consistently to movements in the HML factor. While not reported, similar results areobtained on the two risk factors from other portfolio partitions.

From the cross-sectional perspective of the conditional CAPM, I can also estimate the risk-return relationfollowing Fama and MacBeth (1973) by first estimating the cross-sectional relation at each month using theconditional covariances:

Ri;tþ1 ¼ Ct þ Atsim;tþ1 þ Bs;tois;tþ1 þ Bh;toih;tþ1 þ ei;tþ1, (26)

and then analyzing the time-series properties of the cross-sectional slope estimates. Table 7 reports the time-series averages and the Newey and West (1987) t-statistics of the slope coefficients from different versions ofthe cross-sectional regressions in Eq. (26). The results are similar to those in Table 6. The coefficient estimateson the covariance with the SMB factor are significantly negative, while the estimates on the covariance withthe HML factor are insignificant. Taken together, the results indicate that the relations between the expectedexcess returns and their conditional covariances with the market, SMB, and HML factors are similar whetherconsidering the relation intertemporally or cross-sectionally.

4.4. Risk premiums induced by conditional covariation with macroeconomic variables

Researchers often choose certain macroeconomic variables to control for shifts in the investmentopportunity. The commonly chosen variables include the short-term Treasury bill rates, default spreads on

Table 7

Fama and MacBeth regressions using conditional covariances with firm-size (SMB) and book-to-market (HML) factors

Entries report the time-series averages and the Newey and West t-statistics (in parentheses) of the slope coefficient estimates from the

following cross-sectional regressions at each month t:

Ri;tþ1 ¼ Ct þ Atsim;tþ1 þ Bs;tois;tþ1 þ Bh;toih;tþ1 þ ei;tþ1,

where ois;t and oih;t measure the time-t expected conditional covariance between the excess return on portfolio i and the two risk factors

SMB and HML, respectively. Estimation is based on monthly data from July 1926 to December 2002. BM ¼ book-to-market.

Portfolios A Bs Bh

30 Industry 2.1314 �2.2620 –

(2.7953) (�2.0705)

30 Industry 1.2630 – 1.7214

(2.3914) (0.9221)

30 Industry 1.9806 �2.1431 1.5317

(2.6586) (�2.0618) (0.8434)

25 Size/BM 2.0851 �2.1164 –

(2.6126) (�2.2320)

25 Size/BM 1.2332 – 0.7921

(2.3530) (0.5865)

25 Size/BM 1.8487 �2.1037 0.2352

(2.4026) (�2.1736) (0.4172)


corporate bonds, term spreads on interest rates, and aggregate log dividend–price ratios. To analyze howthese macroeconomic variables vary with the investment opportunity and whether covariations with theminduce additional risk premiums, I first estimate the conditional covariance of these variables with excessreturns on each portfolio and then analyze how the portfolio excess returns respond to their conditionalcovariance with these macroeconomic variables. In estimating the conditional covariances, I use the detrendedrelative Treasury bill rates, the monthly changes in default spreads, the term spreads, and the log dividend-price ratios, as described in Section 3.1.4. Table 8 reports the estimation results for the 30 industry and the25 size/BM portfolios. For each variable, I first estimate the intertemporal relation based on the longestcommon sample of the available data. Then, I re-estimate the relation using a more recent subsample thatstarts in July 1963.

The estimates in Table 8 reveal several important results. First, incorporating any of these macroeconomicvariables does not dramatically alter the relative risk aversion estimates. In all cases, the relative risk aversionestimates are positive and highly significant. Second, the coefficient on the covariance with the relativeTreasury bill rate is significantly negative, indicating that a short rate increase predicts a downward shift in theoptimal consumption. This negative estimate is consistent with Merton’s conjecture in the original Merton(1973) paper, where he uses the example of stochastic interest rate to illustrate the role of intertemporalhedging demand. It is also consistent with the findings of several empirical studies (e.g., Campbell, 1987) thatthe Treasury bill rate predicts positively future market volatility. However, the coefficient estimates on

Table 8

Risk premiums induced by conditional covariation with macroeconomic variables

Entries report the slope estimates and t-statistics (in parentheses) from the following system of equations using the 30 industry and 25

size/book-to-market (BM) portfolios,

Ri;tþ1 ¼ Ci þ Asim;tþ1 þ Bxoix;tþ1 þ ei;tþ1; i ¼ 1; 2; . . . ; n,

where sim;tþ1 measures the conditional covariance between the excess return on each portfolio ðRi;tþ1Þ and the market portfolio ðRm;tþ1Þ,

and oix measures the conditional covariance of the portfolio return with a macroeconomic variable Xt, which includes the relative

Treasury bill rate (RRELt), monthly changes in the default spread (DEFt), monthly changes in the term spread (TERMt), and monthly

changes in the log dividend-price ratio (DIVt). Estimation is based on both the full sample starting July 1926 (except for TERMt, which

starts in May 1941) and a subsample starting July 1963.

July 1926–December 2002 July 1963–December 2002

Xt A Bx A Bx


RRELt 2.9891 �0.3906 4.0387 �0.5713

(2.6174) (�2.2379) (2.5611) (�2.3585)

DEFt 2.9652 2.5179 2.9019 1.9708

(2.7906) (2.4611) (2.5202) (1.9010)

TERMt 2.7830 0.7906 3.5709 0.2305

(2.5123) (1.2660) (2.6000) (0.9203)

DIVt 3.7211 4.7549 2.3577 2.0152

(2.9899) (2.6946) (2.8010) (2.3894)


RRELt 3.0202 �0.5721 4.1077 �0.7451

(2.5975) (�2.0935) (2.5300) (�2.3509)

DEFt 3.2086 1.5535 3.0042 1.4749

(2.6712) (2.3180) (2.4073) (2.0219)

TERMt 3.3451 0.5001 3.4711 0.4025

(2.4904) (1.6101) (2.6115) (1.2983)

DIVt 3.8712 4.4777 2.9225 2.5461

(2.9365) (2.4586) (2.7826) (2.2000)


covariances with the default spread and dividend-price ratio are significantly positive, indicating that upwardmovements in these variables predict favorable shifts in the investment opportunity.

As a robustness check, I also directly incorporate the lagged macroeconomic variables and the lagged excessreturns to the system of equations. Table 9 presents the parameter estimates and t-statistics for the 30 industryand 25 size/BM portfolios for the common sample period from July 1963 to December 2002. The relative riskaversion estimates range from 3.05 to 3.72 for the 30 industry and from 3.84 to 4.11 for the 25 size/BMportfolios. They all remain highly significant for both portfolio partitions. In regressions with one laggedvariable, I obtain significantly negative coefficient estimate on the relative T-bill rate and significantly positivecoefficient estimates on the default spread, the term spread, the dividend yield, and the lagged marketand portfolio returns. When I include all the lagged variables at the same time, the statistical significanceof the default spread, term spread, lagged market return disappears, but the relative T-bill rate, thedividend yield, and the lagged portfolio return remain significant predictors of the one-month-ahead excessportfolio returns.

Table 9

Directly incorporating lagged macroeconomic variables

Entries report the slope estimates and t-statistics (in parentheses) from the following system of equations:

Ri;tþ1 ¼ Ci þ Asim;tþ1 þ BX t þ ei;tþ1; i ¼ 1; 2; . . . ; n,

where sim;tþ1 measures the conditional covariance between the excess return on each portfolio ðRi;tþ1Þ and the market portfolio ðRm;tþ1Þ,

and Xt denotes lagged macroeconomic variables, which includes the relative Treasury bill rate (RRELt), default spread (DEFt), term

spread (TERMt), log dividend-price ratio (DIVt), lagged excess return on the market portfolio ðRm;tÞ,and lagged excess return on the

corresponding industry or size/book-to-market (BM) portfolio ðRi;tÞ. Data are monthly from July 1963 to December 2002.

sim;tþ1 RRELt DEFt TERMt DIVt Rm;t Ri;t


3.1541 (2.4492) �0.5601 – – – – –

(�2.3800)

3.1367 (2.4784) – 0.9548 – – – –

(2.0258)

3.4504 (2.4512) – – 0.2902 – – –

(2.2611)

3.0785 (2.3951) – – – 0.0068 – –

(2.3180)

3.6980 (2.4524) – – – – 0.0826 –

(2.5620)

3.7191 (2.4658) – – – – – 0.0842

(2.6572)

3.0528 (2.3209) �0.5773 0.0213 �0.1021 0.0097 �0.0421 0.0829

(�2.2600) (0.3561) (�0.9812) (2.0550) (�1.1919) (2.6044)


3.8675 (2.4780) �0.5923 – – – – –

(�2.4292)

3.8392 (2.4296) – 0.8673 – – – –

(2.0402)

4.0891 (2.5341) – – 0.3223 – – –

(2.3204)

3.9284 (2.4880) – – – 0.0081 – –

(2.3402)

4.1145 (2.5341) – – – – 0.1205 –

(2.7919)

4.0866 (2.5102) – – – – – 0.1193

(2.7480)

3.9901 (2.4710) �0.5723 0.0076 �0.0010 0.0110 �0.0171 0.1098

(�2.3804) (0.2621) (�0.1812) (2.3984) (�0.7523) (2.5251)


4.5. Abnormal returns

In estimating the system of time-series relations, I allow the intercept to be different for different portfolios.These intercepts capture the abnormal returns on each portfolio that cannot be explained by the conditionalcovariances with the market portfolio and other factors.

Fig. 3 displays the histograms of the abnormal return estimates (left panel) and t-statistics (right panel) forthe full sample period from 1926 to 2002 on various industry portfolios. The histogram contains the estimateson 74 industry portfolios pooled from the five-, ten-, 12-, 17-, and 30-industry partitions. Hence, someestimates refer to the same industry portfolios, or subsamples of them, in different portfolio partitions. Tosave space, I do not report results on the specifications including covariance with SMB, HML, andmacroeconomic factors.

Out of the 74 portfolios, only seven generate statistically significant estimates for the abnormal returns.Thus, the model explains fairly well the excess returns on different industry portfolios. When I collect theabnormal return estimates on the more recent subsample from 1963 to 2002 on all the 122 industry portfoliosfrom the five-, ten-, 12-, 17-, 30-, and 48-industry partitions, we find that only 21 out of 122 abnormal returnestimates are significantly different from zero. Incorporating conditional covariances with other state variables

Fre

quency

Estimates for monthly abnormal returns, in percent

Fre

quency

t-statistics for abnormal return estimates

14

12

10

8

6

4

2

0

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5

14

12

10

8

6

4

2

0

Fig. 3. Histograms of the estimates and t-statistics of abnormal returns on industry portfolios for the sample period of July

1926–December 2002. The left panel plots the histogram of the abnormal return estimates on different industry portfolio series. The right

panel plots the histogram of the t-statistics of these estimates.


do not dramatically alter the general conclusion. Nevertheless, when I constrain the intercept to be the sameacross all portfolios, the common intercept estimate becomes significantly different from zero, as shown inTable 5. Thus, although the conditional covariance with the market (or conditional beta) predicts positively onportfolio returns, it cannot fully explain all the cross-sectional variations in portfolio returns.

I also analyze the abnormal returns on 25 size/BM portfolios for the full sample period. The unconditionalmean excess returns increase with the BM ratio, especially at small size quintile. The mean excess returnsdecrease with the firm size except for the lowest BM quintile. Out of the 25 portfolios, 23 of them havesignificantly positive mean excess returns at the 95% confidence level. Consistent with Fama and French (1993),I find that small and value stocks have larger unconditional mean excess returns than big and growth stocks.

I also estimate the abnormal returns from my simultaneous equations on the full sample period withoutincluding any additional covariance terms with other state variables. Out of the 25 estimates, only eight arestatistically significant. Thus, I confirm the results from the industry portfolios that the conditional covariancewith the market can explain a significant portion of the cross-sectional variation of the size/BM portfolio returns.Nevertheless, the abnormal returns still show some remaining structures. For example, similar to the pattern inthe unconditional means, the abnormal returns increase with the BM ratios, especially for small firms, indicatingthat the ICAPM model cannot fully explain the cross-sectional variation of the portfolio excess returns.

5. Robustness analysis

To check further the robustness of my findings, I consider variations in the conditional covarianceestimation, and I analyze the effect of small sample bias on my estimates.

5.1. Alternative conditional variance and covariance specifications

Because the conditional variance and covariance of stock market returns are not observable, differentapproaches and specifications used in estimating the conditional variance and covariance could lead todifferent conclusions. In this paper, I use the bivariate GARCH(1,1) specification in Eqs. (5)–(14) to obtainconditional variance and covariance estimates. In this subsection, I investigate whether changing thesespecifications influences my results on the risk-return relation estimates.

The literature has considered two major variations on the conditional variance estimation. One incorporatesexogenous instrumental variables such as short-term interest rates in the conditional variance forecastingequation (e.g., Campbell, 1987). The other is to allow downward return movements and upward returnmovements to have different impacts on the conditional volatility forecasts (e.g., Glosten, Jagannathan, andRunkle, 1993; Ding and Engle, 1993). To investigate whether such variations in the variance forecastingspecification alter my conclusion, I re-estimate the conditional variance and covariance using the followingalternative specification:


1Ri;t þ �i;tþ1,

Rj;tþ1 ¼ aj0 þ aj

1Rj;t þ �j;tþ1,

s2i;tþ1 ¼ gi0 þ gi

1�2i;t þ gi

2s2i;t þ gi

3�2i;tI�i;t þ gi

4rt,

s2j;tþ1 ¼ gj0 þ gj

1�2i;t þ gj

2s2i;t þ gj

3�2j;tI�i;t þ gj

4rt,

sij;tþ1 ¼ rijsi;tþ1sj;tþ1, ð27Þ

where Rj denotes the excess return on the market portfolio in estimating the covariance with the marketportfolio ðsimÞ and denotes the state variable x in estimating the covariance with the state variable ðoixÞ, I�i;t isan indicator function that equals one when �t is negative and zero otherwise, and rt denotes the short-termrisk-free interest rate. The indicator function generates an asymmetric GARCH effect between positive andnegative shocks. In this specification, I assume constant correlation, rij , following Bollerslev (1990).

Because I find that the SMB factor predicts future changes in the market portfolio return volatility, I alsoconsider yet another specification, in which we replace the short rate in Eq. (27) with the SMB factor as apredictor of the conditional variance. I estimate both specifications using the maximum likelihood method.

ARTICLE IN PRESS

Table 10

Intertemporal relation estimates under alternative conditional covariance specifications

Entries report the slope estimates and t-statistics (in parentheses) from the following system of equations using the 30 industry and 25

size/book-to-market (BM) portfolios,

Ri;tþ1 ¼ Ci þ Asim;tþ1 þ Bsois;tþ1 þ ei;tþ1; i ¼ 1; 2; . . . ; n,

where the conditional covariances sim;tþ1 and sis;tþ1 are estimated using different generalized autoregressive conditional heteroskedasticity

(GARCH) specifications. Different rows report estimates on different portfolio partitions, different sample periods, and different

restricted and unrestricted versions of the system of equations. SMB ¼ Firm size factor; BM ¼ book-to-market.

Asymmetric GARCH with short rate Asymmetric GARCH with SMB

Portfolios Sample A Bs A Bs

30-Industry 1926–2002 1.4669 — 1.3119 —

(2.5464) (2.4563)

30-Industry 1926–2002 1.9119 �3.8944 1.8740 �3.1014

(2.6243) (�2.2006) (2.5009) (�2.0998)

30-Industry 1963–2002 4.4480 — 3.2873 —

(2.8755) (2.7995)

30-Industry 1963–2002 5.4494 �5.8529 4.2591 �4.2006

(2.9354) (�2.3197) (2.9112) (�2.2626)

25 Size/BM 1926–2002 2.2568 — 2.1315 —

(2.5860) (2.6021)

25 Size/BM 1926–2002 4.0461 �4.8482 3.2034 �3.2643

(3.0125) (�2.3550) (2.8440) (�2.1783)

25 Size/BM 1963–2002 4.4586 — 4.0416 —

(2.6300) (2.6671)

25 size/BM 1963–2002 5.5185 �5.0519 5.2104 �4.8770

(2.7124) (�2.1975) (2.9015) (�2.1596)


With the new estimates on the variance and covariances, I reestimate the system of equations and analyze howthe alternative covariance specifications affect the risk–return relation estimates.

Table 10 reports the common slope estimates for the system of equations for several representativespecifications, portfolio partitions, and sample periods under the two alternative GARCH specifications forthe conditional covariances. In all cases, the estimates are similar to those obtained from the originalcovariance specification. The relative risk aversion coefficient estimates are positive and statisticallysignificant, and the coefficient estimates on the covariance with the SMB factor are negative and highlysignificant.

5.2. Small sample bias

Stambaugh (1999), Amihud and Hurvich (2004), and Lewellen (2004) point out that a small sample biasexists in time-series predictive regressions when the regression disturbances are correlated with the regressors’innovations. The expectation of the regression disturbance conditional on the future values of regressors nolonger equals zero. My whole sample from July 1926 to December 2002 spans 76.5 years of data generating918 monthly observations for each series. The construction of the system of regression equations amounts to afurther increase in the effective sample size. Hence, I expect the effect of small sample bias to be small.

I perform three exercises to check the significance of the small sample bias. First, I apply the augmentedregression method of Amihud and Hurvich (2004) and estimate the relative risk aversion coefficient for each ofthe 25 size/BM portfolios on the subsample from July 1963 to December 2002. By comparing the rawregression estimates with the bias-corrected values on each portfolio series, I find that the small bias is verysmall. The average bias-corrected estimate across the 25 portfolios is 4.19. The maximum difference betweenthe bias corrected estimate and the corresponding raw estimates is merely 0.39, leading to a maximum bias of


less than 10%. I expect the bias to be even smaller for the whole sample and when I estimate the system ofregression equations.

Second, I estimate the system of equations that restrict the relative risk aversion coefficient to be the sameacross different portfolios. In calculating the small sample bias corrections, I assume that the effective numberof observations is 474 instead of 25� 474. Based on the Amihud and Hurvich (2004) methodology, the bias-corrected estimate for the relative risk aversion coefficient is 4.01 with a bias-corrected t-statistic of 2.41. Thecorresponding estimate without bias correction is 4.16, leading to a bias of less than 4%.

Finally, as in Lewellen (1999), I analyze the small sample bias following the randomization technique ofNelson and Kim (1993) in panel data setting to correct for the small sample bias. First, from the SURestimation of the predictive regression in Eq. (20), I record all the regression residuals from each excess returnseries. Second, I perform a first-order autoregression on the conditional covariances ðsim;tþ1Þ and record theregression residuals on each conditional covariance series. Third, I randomize the residuals on the excessreturns to create pseudo excess return series under the null hypothesis of no predictability (zero slope). In eachsimulation, the residuals from the predictive regression and the autoregressions for the conditional covariancesare randomized simultaneously, hence the correlation that drives the small sample bias is preserved. Thepseudo excess returns are generated as the unconditional mean plus the randomized residual. I also randomizethe residuals on the conditional covariance series to generate pseudo-conditional covariance series underthe estimated AR(1) structure. Fourth, I perform the SUR estimation on the system of equations using thepseudo-excess returns and conditional covariances. I repeat this procedure one thousand times. Because thetrue slope coefficient is assumed to be zero, the mean of the one thousand slope estimates provides an estimateon the bias in the original SUR estimation. The original slope coefficient reported in Table 5 is 1.3843 with at-statistic of 2.5207. The bias-adjusted coefficient is 1.2204 with a one-sided p-value of 0.0107. Small samplebias-adjusted p-value is computed as the percentage of times the simulated t-statistics are greater than thesample t-statistic. Overall, all three tests confirm that small sample bias does not significantly affect myconclusions.

6. Conclusion

Empirical studies of the intertemporal capital asset pricing model diverge into two perpendiculardimensions. Studies that focus on the intertemporal risk–return relation often choose to use merely one returnseries on the market portfolio, ignoring the model’s implication that the same relation between excess returnsand their conditional covariances with the market should hold across all stocks to guarantee cross-sectionalconsistency. However, some of the earlier studies on conditional CAPM deal with an unconditionalimplication of the conditional relation by regressing excess returns on the unconditional beta and anunconditional covariance term that accounts for the covariation between the conditional beta and theconditional market risk premium. What this exercise ignores is that a more direct test of the conditionalrelation is to estimate the conditional relation itself, instead of estimating the unconditional implication of theconditional relation. The narrow focus of both strands of the literature often leads to insignificant estimates onmarket beta or market variance.

In this paper, I propose to study jointly the intertemporal and cross-sectional implications of the ICAPMmodel. I estimate the intertemporal risk–return relation using a large cross section of stock portfolios. By sodoing, I not only guarantee the cross-sectional consistency of the estimated intertemporal relation, but alsogain statistical power by pooling multiple time series together for a joint estimation with common slopecoefficients. The estimated risk-return coefficient, or the average relative risk aversion coefficient according toMerton (1973), is positive, statistically significant, and robust to variations in the portfolio formulationmethods, incorporation of conditional covariances with financial or macroeconomic variables, andconditional covariance specifications. The magnitude of the estimates is also economically sensible, rangingfrom one to five.

From the perspective of the conditional CAPM literature, I can also regard my specification as estimatingthe cross-sectional conditional risk–return relation with an intertemporal stability constraint on the relativerisk aversion coefficient. Alternatively, I can assume intertemporal stability on the market risk premium andreplace the conditional covariance in the regressor with the conditional beta. The estimated market risk


premium is also highly significant and positive. The magnitude is between 3% and 5% per annum, alsoeconomically sensible estimates for the market risk premium.

When investigating the intertemporal hedging demands and the associated risk premiums induced byconditional covariances with different financial and macroeconomic state variables, I find that positivecovariation with the SMB factor and risk-free interest rates induces a negative premium, but positivecovariation with default spread and dividend yield induces a positive premium. Nevertheless, incorporatingthe conditional covariation with any of these state variables does not change the positive risk premiuminduced by conditional covariation with the market portfolio or the conditional beta.

Existing studies often contend that the unconditional beta does not matter in cross-sectional regressions andthat the intertemporal risk–return relation is not significantly positive in time-series regressions. By pooling thetime series and cross section together, I find that the GARCH-based conditional covariance and beta matterand that they generate significant and reasonable risk premiums. I also find that the intertemporal risk–returntrade-off is significantly positive and the relative risk aversion estimates are within a reasonable range. Thesignificant and sensible estimates highlight the added benefits of using conditional measures of risk andsimultaneously maintaining cross-sectional consistency and intertemporal stability in estimating theconditional capital asset pricing models.

Appendix A. Wald test

Table A1 presents the Wald statistics and the corresponding p-values from testing whether the GARCHconditional betas are unbiased based on the following conditional market-model regression:

Ri;t ¼ a0;i þ a1;ibi;t þ ðb0;i þ b1;ibi;tÞRm;t þ ei;t,

Table A1

Industrya July 1926 July 1963 Size/BM July 1926 July 1963

December 2002 December 2002 December 2002 December 2002

Wald p-value Wald p-value Wald p-value Wald p-value

Food 1.77 0.41 1.84 0.40 S1B1 1.92 0.38 1.88 0.39

Beer 1.91 0.38 1.79 0.41 S1B2 1.97 0.37 1.67 0.43

Smoke 1.54 0.46 1.34 0.51 S1B3 1.82 0.40 1.56 0.46

Games 1.24 0.54 1.40 0.50 S1B4 1.31 0.52 1.26 0.53

Books 0.97 0.62 1.19 0.55 S1B5 0.73 0.70 1.77 0.41

Hshld 1.41 0.49 0.36 0.84 S2B1 1.76 0.41 1.83 0.40

Clths 0.83 0.66 0.56 0.76 S2B2 0.83 0.66 1.52 0.47

Hlth 0.41 0.81 0.35 0.84 S2B3 0.34 0.84 0.92 0.63

Chems 0.88 0.64 0.71 0.70 S2B4 0.79 0.67 1.63 0.44

Txtls 0.78 0.68 0.70 0.71 S2B5 1.10 0.58 1.27 0.53

Cnstr 0.57 0.75 0.33 0.85 S3B1 1.59 0.45 1.37 0.50

Steel 1.36 0.51 1.10 0.58 S3B2 1.80 0.41 1.44 0.49

FabPr 1.09 0.58 1.24 0.54 S3B3 0.77 0.68 0.66 0.72

ElcEq 1.35 0.51 1.42 0.49 S3B4 0.92 0.63 1.85 0.40

Autos 1.18 0.56 1.27 0.53 S3B5 1.57 0.46 1.30 0.52

Carry 1.40 0.50 1.11 0.57 S4B1 1.53 0.47 1.21 0.55

Mines 0.76 0.68 0.24 0.88 S4B2 1.92 0.38 0.99 0.61

Coal 1.60 0.45 1.44 0.49 S4B3 1.79 0.41 1.21 0.55

Oil 0.85 0.65 0.63 0.73 S4B4 1.19 0.55 0.36 0.83

Util 1.28 0.53 0.98 0.61 S4B5 1.28 0.53 1.05 0.59

Telcm 0.97 0.61 0.68 0.71 S5B1 0.85 0.65 0.76 0.68

Servs 1.19 0.55 1.08 0.58 S5B2 1.34 0.51 1.34 0.51

BusEq 1.44 0.49 1.40 0.50 S5B3 1.66 0.44 1.76 0.42

Paper 1.74 0.42 1.82 0.40 S5B4 1.31 0.52 1.23 0.54

ARTICLE IN PRESS

Table A1 (continued )

Industrya July 1926 July 1963 Size/BM July 1926 July 1963

December 2002 December 2002 December 2002 December 2002

Wald p-value Wald p-value Wald p-value Wald p-value

Trans 1.22 0.54 1.13 0.57 S5B5 1.26 0.53 0.74 0.69

Whlsl 1.45 0.48 1.67 0.43

Rtail 1.02 0.60 0.92 0.63

Meals 1.22 0.54 1.34 0.51

Fin 0.95 0.62 0.52 0.77

Other 1.00 0.61 0.75 0.69









where Ri;t is the return on portfolio i at month t, Rm;t is the return on market portfolio at month t, and bi;t isthe time-varying conditional beta of portfolio i at month t. If the GARCH conditional betas are unbiased,then the coefficients b0;i and b1;i should be zero and one, respectively. Test results are based on the monthlyseries for the 30 industry and 25 size/BM portfolios over the sample periods July 1926–December 2002 andJuly 1963–December 2002. Wald statistics are distributed as Chi-square with two degrees of freedom.

Appendix B. Estimation of a system of regression equations

Consider a system of n equations, of which the typical ith equation is

yi ¼ X ibi þ ui, (28)

where yi is a N� 1 vector of time-series observations on the ith dependent variable, Xi is a N� ki matrix ofobservations of ki independent variables, bi is a ki� 1 vector of unknown coefficients to be estimated, and uiisa N� 1 vector of random disturbance terms with mean zero. Parks (1967) proposes an estimation procedurethat allows the error term to be both serially and cross-sectionally correlated. In particular, he assumes thatthe elements of the disturbance vector u follow an AR(1) process:

uit ¼ riuit�1 þ �it; rio1, (29)

where �it is serially independently but contemporaneously correlated:

Covð�it�jtÞ ¼ sij ; 8i; j; and Covð�it�jsÞ ¼ 0; for sat (30)

Eq. (28) can then be written as

yi ¼ X ibi þ Pi�i, (31)

with

Pi ¼

ð1� r2i Þ�1=2 0 0 . . . 0

rið1� r2i Þ�1=2 1 0 . . . 0

r2i ð1� r2i Þ�1=2 ri 0 . . . 0

:

:

:

rN�1i ð1� r2i Þ

�1=2 rN�2i rN�3

i . . . 1

26666666666664

37777777777775. (32)


Under this setup, Parks presents a consistent and asymptotically efficient three-step estimation techniquefor the regression coefficients. The first step uses single equation regressions to estimate the parameters ofautoregressive model. The second step uses single equation regressions on transformed equations to estimatethe contemporaneous covariances. Finally, the Aitken estimator is formed using the estimated covariance,

b ¼ ðXTO�1X Þ�1XTO�1y, (33)

where O � E½uuT� denotes the general covariance matrix of the innovation. In my application, I use theaforementioned methodology with the slope coefficients restricted to be the same for all portfolios. Inparticular, I use the same three-step procedure and the same covariance assumptions as in Eqs. (29)–(32) toestimate the covariances and to generate the t-statistics for the parameter estimates.

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