This is a repository copy of Can Stochastic Discount Factor Models Explain the Cross Section of Equity Returns?. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/94216/ Version: Accepted Version Article: Abhakorn, Pongrapeeporn, Smith, Peter Nigel orcid.org/0000-0003-2786-7192 and Wickens, Mike (2016) Can Stochastic Discount Factor Models Explain the Cross Section of Equity Returns? Review of Financial Economics. pp. 56-68. ISSN 1058-3300 https://doi.org/10.1016/j.rfe.2016.01.001 Reuse This article is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) licence. This licence only allows you to download this work and share it with others as long as you credit the authors, but you can’t change the article in any way or use it commercially. More information and the full terms of the licence here: https://creativecommons.org/licenses/ Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Can Stochastic Discount Factor Models Explain the Cross Section of Equity Returns? Pongrapeeporn Abhakorna Peter N. Smith Fiscal Policy Office Department of Economics and Related Studies Ministry of Finance of Thailand University of York, UK Phayatai Road, Thailand, 10400 YO10 5DD E-mail: peter.smith@york.ac.uk E-mail: pongrapeeporn.a@mof.go.th Tel.: +66851234466 Fax: +6626183374 Michael R. Wickens Cardiff Business School and Department of Economics and Related Studies University of York, UK YO10 5DD E-mail: mike.wickens@york.ac.uk Abstract We propose a multivariate test based on no-arbitrage conditions under the stochastic discount factor approach, which compares cross-sectional variation in equity returns to the cross-sectional variation in their conditional covariance with the discount factors. Using the multivariate generalized heteroskedasticity in mean model to estimate the 25 portfolios formed on size and book-to-market ratio, together each with its own arbitrage condition, we find that the no-arbitrage test rejects the consumptionbased capital asset pricing model (C-CAPM). Although the conditional covariances of returns with consumption exhibit negative variation across size, they do not vary across the book-to-market ratio. Thus, the C-CAPM can capture size effect, but not value effect. Allowing the coefficients on the consumption covariances to be different largely improves the fit of the C-CAPM, however. The value effect appears to be associated with book-to-market ratio as well as size. Book-to-market ratio separately does not generate information about average returns that cannot be explained by the C-CAPM. One possible explanation for this extra dimension of risk is the investment growth prospect of firms. Low book-to-market ratio firms may be expected to have higher rates of growth while small firms may also be expected to behave similarly. JEL Classification: G12, G14, C32, E44 Keywords: Risk Premium; Equity Return; Stochastic Discount Factor; No-arbitrage Condition a Corresponding author. The views expressed in the paper are those of the authors and do not necessarily represent those of the Fiscal Policy Office. 1. Introduction Size and value effects have long been recognized as “anomalies” both in the capital asset pricing model (CAPM) literature (summarized in Fama and French, 2006 and 2008), and in the consumption-based CAPM (C-CAPM) framework with power utility (see Cochrane, 2008). This paper tests whether stochastic discount factor (SDF) models that satisfy no-arbitrage restrictions can explain the behavior of a cross section of returns on 25 portfolios sorted by firm size and their book-to-market ratio ratio (the 25 Fama-French portfolios). We examine whether portfolios of stocks have different returns due to different conditional covariances between the returns and the relevant discount factors, or because the coefficients of the discount factors vary by portfolio characteristic. This provides a test of no-arbitrage as finding either effect would imply that no-arbitrage does not hold. Instead of modelling separate no-arbitrage conditions for the returns on the 25 Fama-French portfolios, we model them simultaneously employing an SDF framework. We use a multivariate generalized autoregressive conditional heteroskasticity in mean model (MGM) as in Smith and Wickens (2002). This methodology is in contrast to most of the time-series econometric models of equity returns in the literature, which are univariate models and do not include conditional covariances (see for example Ludvigson, 2012). Smith, Sorensen, and Wickens (2008) used the approach adopted in this paper, examining various SDF models, including the standard C-CAPM, to generate models involving macroeconomic variables. Abhakorn, Smith, and Wickens (2013) estimate the MGM for the standard C-CAPM for each of the 25 Fama-French portfolios, and find that the fit of the model is significantly improved by the inclusion of the firm book-to-market value ratio (HML) factor. This paper extends their analysis by estimating the all 25 Fama-French portfolio returns simultaneously and testing for each asset-pricing model whether the conditional covariances of these returns with the relevant discount factors can adequately explain the excess returns of these portfolios. We find that C-CAPM is rejected by the no-arbitrage test. The model can explain the size effect, as the conditional covariance of consumption with firm size is negative, but not the value effect, as the conditional covariance of consumption with 2 the book-to-market ratio does not vary as required across the book-to-market quintiles. We find that the value effect tends to be slightly lower for portfolios in the highest book-to-market quintile - indicating that a lower risk premium - than for portfolios with the lowest book-to-market quintiles. Allowing the coefficients on the conditional covariances to vary across the portfolios improves fit markedly. As CCAPM restricts them to be the same, this too is an indication of the failure of the model. The paper is set out as follows. In Section 2, we briefly review the relevant literature on CAPM and C-CAPM. In Section 3 we describe our theoretical framework for asset pricing and in Section 4 we explain our econometric methodology. In Section 5, we report our empirical results. Section 6 summarizes the findings of this paper. 2. Some relevant literature on CAPM and C-CAPM Evidence that the accounting variables firm size and the book-to-market ratio would be significant if included in the standard CAPM in addition to the market return was first presented by Fama and French (1993 and 2008). This cast doubt on the empirical validity of the CAPM as it suggested that additional pricing factors to the market return were required to successfully explain the cross-section of stock returns. This raises the question of whether such anomalies would also be significant in alternative models to CAPM such as C-CAPM which takes into account the intertemporal nature of the investor optimization problem. Cochrane (2008) found that size and value effects are not significant in C-CAPM More recently, however, a number of studies have attempted to explain the crosssection of equity returns using modified versions of C-CAPM that included either different or additional factors. Lettau and Ludvigson (2001) use the ratio of aggregate consumption to wealth as a conditioning variable in C-CAPM in order to better capture variations in expected returns over time. An alternative way to overcome the slowness of the consumption adjustment process was suggested by Parker and Julliard (2005) who measured the risk premium by its covariance with consumption growth cumulated over many quarters after the return period, see also Jagannathan and Wang (2007). Yogo (2006) proposed a two-factor model that includes nondurable and 3 durable consumption growth. He found that the size and value effects are due to small and value stocks having higher durable consumption betas than large and growth stocks. Savov (2011) suggested the use of household garbage production as a proxy for consumption; as all forms of consumption produce waste, garbage growth should be informative about rates of consumption growth. These modified versions of C-CAPM seem to explain the cross-section of equity returns equally well to the Fama and French three-factor model (Fama and French, 1993). In this paper, rather than asserting that there are alternative or missing factors in C-CAPM, we exploit implications of C-CAPM that are ignored in the papers discussed above while keeping close to the ideas of Fama and French. In particular, we include the two additional factors of Fama and French, and do so using C-CAPM instead of CAPM, Thus we explore the validity of the model but in a multivariate no-arbitrage framework by estimating the 25 Fama-French portfolios simultaneously. It appears from the results of Abhakorn, Smith, and Wickens (2013) that in order to capture the value effect using C-CAPM it is necessary to include both firm size and the book-to-market ratio as when including them individually C-CAPM cannot explain small growth portfolios. They find that HML helps explain the 25 Fama-French portfolios across size quintiles as well as across book-to-market ratio quintiles, and suggest that HML may be associated with the investment growth prospects of firm. This could be the reason why the investment-based asset pricing models of Brennan, Wang, and Xia (2004) and Li, Vassalou, and Xing (2006) are able to explain well the cross-section of equity returns but traditional CAPM is not able to (e.g. Fama and French, 1992 and 2006 and Lewellen and Nagel, 2006). This suggests that consumption contains information about these firm characteristics that is not available through market return. 3. Theoretical Framework 3.1 Stochastic Discount Factor Representations of Asset Pricing Models The basic no-arbitrage pricing equation for a risky asset defines a relationship between the Stochastic Discount Factor (SDF) M t 1 and the risky return, Rt 1 . 1  Et [ M t 1Rt 1 ] 4 (1) where M t 1 is the real stochastic discount factor for period t  1 and for equity, the rate of return in real terms is Rt 1  ( Pt 1  Dt 1 ) / Pt , where Dt 1 are real dividend payments assumed to be made at the start of period t  1 and Pt is the real price of equity (see Cochrane, 2008). If the logarithms of M t 1 , Rt 1 and the risk free rate ( mt 1 , rt 1 , rt f ) are jointly normally distributed, then (1) implies that the expected excess real return on equity is given by 1 Et (rt 1  rt f )  Vt (rt 1 )  Covt (mt 1 , rt 1 ) . 2 (2) where the right-hand side is the risk premium and the variance term is the Jensen effect. Equation (2) can also be expressed in terms of nominal returns. If it 1 is the nominal return on equity, it f is the nominal risk-free rate, Pt c is the consumer price index, and inflation is 1   t 1  Pt c1 / Pt c , the pricing equation (1) becomes 1  Et  M t 1 ( Pt c / Pt c1 )(1  it 1 )  . The no-arbitrage condition for nominal returns is: 1 Et (it 1  it f )  Vt (it 1 )  Covt (mt 1 , it 1 )  Covt ( t 1 , it 1 ) . 2 (3) Comparing (3) and (2), the no-arbitrage condition for the nominal return involves one additional term in the conditional covariance of returns with inflation. More generally, if mt can be represented as a linear function of n  1 factors zi ,t i  1,..., n  1 so that mt   in11i zi ,t , then a general representation of (3) is Et (it 1  it f )   0Vt (it 1 )   in1i Covt ( zi ,t 1 , it 1 ) , (4) where zn ,t   t . The differences between many asset pricing models are in their stochastic discount factor, zi ,t 1 , and the restrictions imposed on the coefficients. We consider three pricing models that are special cases of equation (4): (a) C-CAPM with power utility  The discount factor in this case is M t 1    C t 1 / Ct  where the consumer./investor has utility function, U (Ct )  (Ct1  1) /(1   ) over real expenditure Ct with   constant coefficient of relative risk aversion (CRRA). The nominal no-arbitrage condition in this case is: 5 1 Et (it 1  it f )  Vt (it 1 )   Covt ( ln Ct 1 , it 1 )  Covt ( t 1 , it 1 ) , 2 (5) where  ln Ct 1  Ct 1 / Ct is the growth rate of consumption. C-CAPM with power utility implies that excess returns of different portfolios of equities differ due to their conditional covariance with consumption with a common CRRA. (b) CAPM The CAPM implies that the expected return of an asset must be linearly related to the covariance of its return with the return on the market portfolio through Et ( rt 1  rt f )   t Covt ( rt m1 , rt 1 ) where  t  Et ( rt m1  rt f ) / Vt ( rt m1 ) is the market price of risk and can be interpreted as the CRRA (Merton, 1980). There is no Jensen effect because log-normality is not assumed. The corresponding no-arbitrage condition for nominal returns is Et (it 1  it f )   t Covt (itm1 , it 1 ) (6) where itm1 is the nominal return on the market portfolio. (c) General Stochastic Discount Factor Models General SDF models are based on macroeconomic factors and particular versions of the multifactor model in (4) which also allow the factors to have unrestricted coefficients. We consider general SDF model with up to three macroeconomic factors. Smith, Sorensen, and Wickens (2008) suggest the use of factors that are associated with the business cycle and inflation. They argue that as financial institutions, such as pension funds, are the main holders of equity and act on behalf of investors and often focus on short-term business cycle considerations rather than on longer term performance associated with the utility of their investors. The authors, therefore, use output growth as an additional source of risk to consumption and inflation, but without seeking to give the model a general equilibrium interpretation. 3.2 Testing the Discount Factor Models In all of the SDF models previously discussed, the discount factors are functions of aggregate variables, and thus it is possible to hold the properties of the discount factors constant as one individual asset is compared to another. As the risk premium is represented by the conditional covariance of the returns with the discount factor, we can compare cross-sectional average returns with cross-sectional variation in their 6 conditional covariances with the factors. The implication is that the coefficients on these conditional covariances should be the same across the cross-section of equity returns and stocks have different returns because they have different conditional covariances with the relevant factors. Our estimation method allows these covariances also to vary through time. This relation provides testable restrictions on no-arbitrage conditions, and therefore, it can be interpreted as a no-arbitrage test. Table 1 provides a summary of restrictions for each asset pricing models implied by its no-arbitrage condition. C-CAPM with power utility and nominal returns (M1) implies that the CRRA is constant and should be the same across the cross-section of expected returns for no arbitrage opportunities in the market. M1 is the restricted version of C-CAPM. On the other hand, allowing the coefficients on the conditional covariances of returns with consumption to be different generates an unrestricted version of C-CAPM (M4). The double-sorted, 25 size and book-tomarket equity ratio portfolios generate two more versions of C-CAPM with power utility: restricted book-to-market model (M2) and restricted size model (M3). M2 allows portfolios with different size groups to have different coefficients on the consumption covariances, while M3 allows the coefficients for portfolios with different book-to-market-equity ratio groups to be different. Similarly, these restrictions of C-CAPM are applied to the CAPM, where the market price of risk is expected to be the same across assets, (M5-M8). In addition, the restricted and unrestricted general SDF models, based on two (consumption growth and inflation) and three macroeconomic variables (consumption growth, inflation and industrial production growth), are given by M9-M12. In sum, all the above asset pricing models can be represented as restricted versions of the SDF model, Et (itsb1  it f )  0,sbVt (itsb1 )  1,sbCovt (ct 1 , itsb1 )   2,sbCovt ( t 1 , itsb1 )  3,sbCovt (qt 1 , itsb1 )   4,sbCovt (itm1 , itsb1 ) (7) where s and b indicate size and book-to-market ratio groups that the characteristics portfolios belong to, respectively, and qt is industrial production. The different asset models can be obtained by placing different restrictions on the  i , s, and b. 7 Table 1 Restrictions on the No-arbitrage Condition s and b indicate size and book-to-market groups for the characteristics portfolios. The numbers are in ascending order of magnitude. The smallest size is denoted by s = 1 while the lowest book-to-market ratio is represented by b = 1.  denotes constant coefficient of relative risk aversion (CRRA). i represents a coefficient for each conditional covariance in Equation 8. 0 Models M1: C-CAPM with power utility and nominal return  M2: Restricted book-to-market C-CAPM  M3: Restricted size C-CAPM  M4: Unrestricted C-CAPM  1 2 1 2 1 2 1 2 1 2 3 4  1 0 0 1,s 1 0 0 1,b 1 0 0 1,sb 1 0 0 M5: CAPM 0 0 0 0 M6: Restricted book-to-market CAPM 0 0 0 0   4,s M7: Restricted size CAPM 0 0 0 0  4,b M8: Unrestricted CAPM 0 0 0 0  4,sb 1 2 1  2 1  2 1  2 1 2 0 0 1,sb  2,sb 0 0 1 2 3 0 1,sb 2,sb  3,sb 0  M9: Restricted two-factor SDF model M10: Unrestricted two-factor SDF model M11: Restricted three-factor SDF model M12: Unrestricted three-factor SDF model 8 4. Econometric Framework We follow the same econometric approach here as in Smith and Wickens (2002), and Smith, Sorensen, and Wickens (2008, 2010), and Abhakorn, Smith, and Wickens (2013) by using the multivariate generalized autoregressive conditional heteroskedasticity in mean model (MGM) to estimate the joint distribution of the excess return on equity with macroeconomic factors in such a way that the return satisfies the no-arbitrage condition under the SDF framework. This approach is achieved by including conditional covariances of the excess equity returns and the macroeconomic factors in the mean of the asset pricing equations and constraining the coefficients on these time-varying, conditional covariances according to the noarbitrage condition implied by each asset-pricing model. Let xt 1  (r1,t 1  rt f ,..., ri ,t 1  rt f , ct 1 ,  t 1 , qt 1 ) ' which contains n variables and i returns, as several portfolios are estimated at the same time. This specification is an extension of the MGM in Smith and Wickens (2002). Consumption, inflation, and industrial productions are included, as they give rise to the discount factors in the SDF model, M1-M12 in Table 1, through their conditional covariances with the excess returns. Additional macroeconomic variables can be included in this vector if they improve the estimate of the joint distribution. The MGM model can then be written as x t+1 = + xt + g t+1 + t+1 , where t+1 | It ~ N (0, Ht+1 ) , gt+1  vech(Ht+1 ) . where, is a n  n matrix of coefficients in the is a n  1 vector of constant, vector autoregressive (VAR) part (included to obtain better representation of the error terms), is a n  n matrix of coefficients of in-mean component, t+1 is an n  1 vector of errors, and i  number of equity returns. The vech operator converts the lower triangle of a symmetric matrix into a vector. The error term, t+1 , is conditionally normally distributed with mean zero and with conditional covariance matrix Ht+1 . The first i rows of the model is restricted to satisfy the no-arbitrage condition as follows: 1) the first i rows of must be zero; 2) the first i rows of depends on then specification of each asset pricing model defined in Table 1; 3) the 9 i+1 to i+3 rows of are all zero; and 4) the first i elements of is zero. A likelihood ratio test is used to provide test statistics for the restrictions implied by the no-arbitrage condition in M1-M12 as given in Table 1. While the MGM model is convenient, it is heavily parameterized, which can create numerical problems in finding the maximum of the likelihood function due to the likelihood of being relatively flat, and hence uninformative. Therefore, to complete the model parameterization for the conditional covariance matrix Ht+1 with the view toward restricting the number of coefficients being estimated, the specification of the conditional covariance matrix is chosen to be the vector diagonal model with variance targeting (Ding and Engle, 2001), which can be written as follows, H t+1  H 0 (ii / - aa / - bb / )  aa /  ( / t t )  bb /  H t where  denotes Hadamard product, H 0 is the observed sample covariance matrix, and a and b are n  1 vectors. The number of parameters to be estimated reduces to 2n . This model is particularly attractive when we estimate several excess returns simultaneously, each with its own arbitrage condition. In addition, the zero restrictions on the coefficients for excess returns in the VAR part of the macroeconomic variables are imposed to further reduce the number of parameters in the MGM model. Estimating the restricted and unrestricted C-CAPM (M1 and M4) for the 25 portfolios sorted by size and book-to-market ratio involve 69 and 93 parameters, respectively, while for the CAPM (M5 and M8) involving 53 and 78 parameters; we need to include only the market return, instead of the macroeconomic factors, in the joint distribution. We are unable to estimate M10 and M12 for the 25 portfolios, as doing so involves estimating too many parameters for our sample size (118 and 143 parameters, respectively); hence we include the two data sets of the 10 portfolios formed for size and book-to-market ratio separately to test for these general two- and three-factor SDF models, in addition to using these one-sorted portfolios to contrast the estimation results with the two-characteristics-sorted portfolios. 10 5. Estimation Results 5.1 Data The data are monthly from 1960.2 to 2004.11 for the U.S. (538 observations). The return on the market portfolio is the value-weighted return on all stocks. The return on a risk-free asset is the one-month Treasury bill rate. Table 2 shows the summary statistics for the 25 value-weighted portfolios, which are the intersections of 5 portfolios formed on size and 5 portfolios formed on the ratio of book-to-market ratio. There are also two sets of ten portfolios sorted by size and book-to-market ratio separately (Table 3). All of the return variables are obtained from Kenneth French’s websiteb. Real non-durable growth consumption is from the Federal Reserve Bank of St. Louis. CPI inflation and the volume index of industrial production are both from Datastream. b http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html 11 Table 2 Summary Statistics: 25 Size and Book-to-Market Portfolios The table presents descriptive statistics for the excess returns on the 25 portfolios formed as the intersections of the five size and book-to-market ratio groups. Data and full definition of the returns can be found on Kenneth French’s webpage. The returns are monthly value-weighted from 1960.2 to 2004.11, 538 observations. t-stat is the test statistics for zero mean hypothesis.  ( xt , x t-i ) represents the autocorrelation coefficients over the time interval i month (s). Book-to-Market Equity Quintiles Size Quintiles Low Small 2 3 4 Big -0.07 0.10 0.18 0.34 0.30 Small 2 3 4 Big -0.53 -0.70 -0.65 -0.49 -0.46 Small 2 3 4 Big 72.7 50.7 44.8 46.9 44.2 Small 2 3 4 Big 37 173 413 1068 9511 Small 2 3 4 Big 0.65 0.94 1.71 3.72 36.21 Small 2 3 4 Big 0.20 0.16 0.12 0.11 0.06 Small 2 3 4 Big 0.02 0.02 0.02 0.02 -0.03 2 3 4 Mean 0.54 0.66 0.90 0.47 0.72 0.82 0.58 0.57 0.73 0.38 0.63 0.75 0.39 0.46 0.47 Skewness -0.46 -0.60 -0.59 -0.89 -0.92 -0.81 -0.99 -0.95 -0.59 -0.96 -0.75 -0.32 -0.62 -0.53 -0.15 Normality 110.0 111.1 144.2 90.1 104.7 107.4 94.7 79.9 84.4 112.3 98.4 46.5 62.0 92.6 27.5 Average firm size 39 38 34 175 177 176 421 421 424 1063 1070 1079 7119 6166 5052 Average percent of market value 0.44 0.43 0.46 0.69 0.69 0.63 1.27 1.18 1.00 2.79 2.38 1.98 16.87 11.29 7.43  ( x t , x t 1 ) 0.18 0.20 0.20 0.16 0.17 0.16 0.15 0.16 0.16 0.13 0.11 0.08 0.04 0.00 -0.02  ( xt , xt  6 ) 0.03 0.03 0.02 0.01 0.02 0.02 0.01 0.02 -0.01 0.01 -0.03 -0.03 -0.06 -0.04 -0.06 High Low 0.97 0.89 0.83 0.70 0.49 8.20 7.48 6.86 6.04 4.80 -0.58 -0.76 -0.80 -0.52 -0.36 2.72 2.34 2.07 1.99 1.89 137.7 117.5 124.4 73.3 23.8 -0.18 0.29 0.60 1.29 1.45 26 172 431 1075 4643 0.28 0.28 0.27 0.27 0.26 0.56 0.48 0.71 1.31 4.17 492 152 115 97 106 0.24 0.15 0.14 0.07 0.06 -0.06 -0.07 -0.05 -0.04 0.03 -0.01 -0.01 -0.01 -0.03 0.02 0.00 -0.03 -0.03 -0.03 0.05 12 2 3 4 Standard deviation 6.98 5.97 5.56 6.07 5.36 5.14 5.44 4.92 4.75 5.15 4.83 4.61 4.54 4.29 4.19 Excess Kurtosis 3.38 3.72 4.35 4.03 4.56 4.23 4.52 3.85 3.12 4.93 3.86 1.82 2.60 3.18 1.23 t-statistics for zero mean 1.78 2.56 3.74 1.81 3.13 3.72 2.48 2.68 3.58 1.73 3.02 3.77 2.00 2.50 2.62 Average book-to-market ratio 0.57 0.78 1.03 0.54 0.76 1.005 0.54 0.75 1.004 0.55 0.75 1.03 0.53 0.75 1.004 Average number of firms 312 315 376 110 109 99 84 78 66 73 62 51 66 51 41  ( xt , xt  3 ) -0.09 -0.05 -0.04 -0.05 -0.05 -0.05 -0.01 -0.05 -0.02 -0.04 -0.02 0.01 -0.01 -0.02 0.02  ( x t , x t 12 ) 0.02 0.06 0.08 0.03 0.05 0.08 0.03 0.02 0.04 0.00 0.03 0.06 0.01 0.02 0.02 High 5.85 5.73 5.36 5.35 4.78 4.20 4.32 4.63 2.72 1.17 3.84 3.60 3.59 3.05 2.37 1.85 1.70 1.66 1.70 1.50 603 77 46 34 25 -0.04 -0.05 -0.04 -0.04 -0.01 0.13 0.10 0.08 0.06 0.02 Table 3 Summary Statistics: 10 Industry Portfolios and Explanatory Variables The table presents descriptive statistics for the returns on the 10 portfolios and explanatory variables. The returns are monthly value-weighted from 1960.2 to 2004.11, 538 observations. Data and full definition of the 10 portfolios can be found on Kenneth French’s webpage. im,t+1 and itf are the returns on the market portfolios and one-month Treasury bill rate respectively. Consumption growth, inflation, and industrial production growth are represented by ct+1, t+1, and qt+1 respectively. Std. Dev is the standard deviation. t-stat is the t-statistic for zero mean hypothesis. t-stat is the test statistics for zero mean hypothesis.  ( xt , x t-i ) represents the autocorrelation coefficients over the time interval i month(s). BM denotes book-to-market equity ratio. Firm size, book-to-market equity ratio, percent of the market, and number of firms are in average terms. Mean Std. Dev. Skewness Excess Kurtosis Normality t-stat Firm Size % of Market No. of firms  ( x t , x t 1 )  ( xt , xt  3 )  ( xt , xt  6 )  ( x t , x t 12 ) Mean Std. Dev. Skewness Excess Kurtosis Normality t-stat Firm Size BM % of Market No. of firms  ( x t , x t 1 )  ( xt , xt  3 )  ( xt , xt  6 )  ( x t , x t 12 ) Mean Small 1.04 6.32 -0.53 3.23 94.89 3.81 21 1.47 2123 0.24 2 0.98 6.26 -0.64 3.52 96.69 3.64 78 1.37 523 0.17 3 1.03 5.99 -0.78 3.22 72.24 3.98 139 1.62 347 0.16 4 0.98 5.80 -0.86 3.56 77.02 3.91 219 2.00 272 0.16 Size Deciles 5 6 1.01 0.92 5.55 5.25 -0.85 -0.84 3.42 3.20 73.66 68.42 4.23 4.08 338 512 2.59 3.33 229 194 0.14 0.01 7 0.98 5.11 -0.71 3.33 81.48 4.45 803 4.69 175 0.12 8 0.95 4.98 -0.64 2.49 56.85 4.41 1346 7.39 164 0.09 9 0.91 4.54 -0.56 2.45 52.81 4.64 2597 13.23 152 0.08 Large 0.79 4.26 -0.52 2.13 50.40 4.29 12780 62.31 146 0.01 -0.05 -0.07 -0.07 -0.06 -0.05 -0.04 -0.03 -0.04 -0.03 0.03 0.01 0.02 0.01 0.02 0.02 0.01 0.00 0.00 -0.02 -0.03 0.08 0.04 0.03 0.03 0.01 0.01 0.01 0.01 0.01 0.05 Book-to-market Deciles 5 6 7 0.93 0.99 1.05 4.35 4.36 4.26 -0.70 -0.67 -0.17 4.24 3.54 1.80 121.46 93.76 50.24 4.96 5.27 5.71 723 576 527 0.71 0.82 0.94 7.79 6.20 5.76 308 307 312 0.05 0.03 0.04 8 1.09 4.26 -0.28 2.11 60.74 5.93 433 1.10 4.88 322 0.05 9 1.12 4.64 -0.41 2.11 54.66 5.59 349 1.35 4.29 351 0.09 High 1.21 5.33 -0.37 3.26 112.13 5.25 177 2.03 2.68 433 0.12 Low 0.67 5.24 -0.42 1.67 38.09 2.91 1582 0.20 32.80 592 0.09 2 0.85 4.77 -0.69 2.76 63.18 4.14 1178 0.37 15.27 370 0.07 3 0.87 4.73 -0.81 3.88 92.12 4.28 981 0.49 11.46 333 0.07 4 0.86 4.66 -0.66 3.12 78.47 4.28 813 0.60 8.88 312 0.08 0.02 -0.01 -0.02 -0.03 -0.04 0.01 0.02 -0.01 -0.03 -0.03 -0.02 -0.01 -0.04 -0.04 -0.03 0.00 -0.06 -0.03 -0.02 -0.02 0.02 0.02 0.01 0.01 0.02 0.02 0.01 0.07 0.06 0.07 im , t  1 it f  c t 1   t 1 0.94 0.46 0.23 0.35 Explanatory variables  q t 1 Std. Dev. 4.41 0.23 0.73 0.30 0.75 Skewness -0.46 1.041 -0.04 0.99 -0.62 Excess Kurtosis Normality  ( x t , x t 1 )  ( xt , xt  3 )  ( xt , xt  6 )  ( x t , x t 12 ) im , t  1 0.25 1.90 1.70 1.37 1.68 2.98 44.85 98.95 33.56 82.25 75.70 0.06 0.95 -0.36 0.64 0.36 0.00 0.90 0.14 0.53 0.27 -0.02 0.84 0.01 0.52 0.09 0.02 0.72 -0.07 0.44 -0.04 13 it f Correlation it f  c t 1   t 1 -0.04 1.00  c t 1 0.15 -0.09 1.00   t 1  q t 1 -0.14 0.54 -0.20 1.00 -0.03 -0.16 0.14 -0.10  q t 1 1.00 The descriptive statistics for the excess returns for the 25 portfolios in Table 2 are similar to those in Fama and French (1993 and 2006) for the periods 1963-1991 and 1963-2004 respectively, thus indicating the value effect and relatively weak size effect. This relatively weak size effect is also seen in Table 3 where one-characteristic sorted portfolios are considered. In general, all excess returns and macroeconomic variables appear to have negative skewness, excess kurtosis, and non-normality, except for the risk-free rate and inflation, which display positive skewness and show persistent volatility. 5.2 Estimates 5.2.1 C-CAPM A full set of model estimates with their restricted versions for C-CAPM with power utility and nominal returns is reported in Table 4. A likelihood ratio test is used to examine the hypothesis implied by each restricted model against the unrestricted model, M4. For M1, the conditional covariance of returns with consumption is highly significant, but the size of the coefficient, 83.25, implies an implausibly large CRRA, which is a common feature of consumption-based models (Campbell, 2002, Cochrane, 2008, Yogo (2006), Smith, Sorensen, and Wickens, 2008, 2010) except for C-CAPM with garbage growth of Savov (2011). For M2, all five consumption coefficients are significant, and the likelihood ratio rejects the hypothesis that portfolios within different book-to-market equity ratio quintiles have the same coefficient at any conventional levels. The test statistic for the restrictions against the unrestricted model M4 is close to that for M1, implying that the differences in the coefficients across size have little weight on the behavior of the estimated returns. On the other hand, the likelihood ratio test marginally fails to reject M3 ( p  value  0.0513); restricting the consumption coefficients for portfolios within the same size quintiles to be the same does not exclude significant information about the excess returns. In other words, size has no, or a relatively weak, relation to the consumption coefficient. In fact, the coefficients in M2 for each size quintile look very similar, while those in M3 for each book-to-market equity ratio quintiles increase substantially from the lowest to the highest book-to-market quintiles. In addition, the 14 consumption covariance coefficient for the lowest book-to-market equity ratio quintiles in M3 is not significantly different from zero. M4 has 23 coefficients that are significant at conventional significance levels. All coefficients range widely from 47.79 to 247.14. Looking down each column, there is no clear pattern in the values of the coefficients across the size quintiles, whilst when looking across each row; the coefficients tend to rise as the book-to-market ratio increases. Like the insignificance of the consumption coefficient on the lowest bookto-market quintile in M3, the remaining 2 coefficients for the lowest book-to-market quintile and the first 2 smallest size quintiles are not significantly different from zero. The insignificance of the coefficients for the lowest book-to-market quintiles of the 25 portfolios is consistent with the evidence from other empirical asset pricing studies on the 25 portfolios (Fama and French, 2008, Lettau and Lugvigson, 2001, Parker and Julliard, 2005, Yogo, 2006, and Savov, 2011) where the pricing models they propose also have difficulty explaining the portfolios in the smallest size and lowest book-tomarket quintiles (small growth portfolio). This inability may be due to limits to arbitrage from short-sale constraints for these portfolios, and thus frictionless equilibrium models, including C-CAPM, cannot explain the returns on these small growth portfolios (Yogo, 2006). Figure 1 shows a scatter plot of average actual and estimated excess returns in M1 to M4 for the 25 portfolios. If the pricing model fits the data well, the points should all lie on a 45-degree line. In Figure 1(d), M4 appears to best explain the excess return on these portfolios and is more or less as good as the modified versions of C-CAPM and the Fama and French three-factor model. The differences in the estimated risk premium and actual excess return ranges from 0.01% to 0.18% per month, which is lower than those in M1-M4. Figures 1(a) and 1(b) show that the estimated risk premia from M1 and M2 are similar, implying that imposing the restrictions on size quintiles does not affect the behavior of risk premia. On the other hand, allowing the consumption coefficients to be different, as in M3, improves the performance of the model sharply, except for the 5 portfolios in the lowest book-to-market ratio quintiles. This observation suggests 15 that book-to-market equity ratio seems to have additional information about the average excess returns that is not captured by C-CAPM. Table 4 Estimates of C-CAPM The table presents the estimates of the C-CAPM (M1-M4): 1960.2-2004.11, 538 observations.  denotes the coefficient relative risk aversion and  i represents a coefficient for each conditional covariance in Equation 8. t ( ) and t(i ) are their corresponding t–statistics respectively. The pricing models (M1-M4) are tested against each other using the log-likelihood ratio test. 2log represents the likelihood ratio statistic. The corresponding p-value at 5% significance level is denoted by p-value.  1s t ( 1 s ) 2 lo g p  va lu e  1b t ( 1b ) 2 lo g p  va lu e Size Quintiles Small 2 3 4 Big  1s t ( 1 s )  1b t ( 1b )  t ( ) 83.25 4.11 Small 93.83 4.13 87.27 0.0000 Low 11.49 0.40 31.31 0.0513 Low Panel A: 25 Size and Book-to-Market Portfolios Panel A1: Restricted C-CAPM (M1) 2 lo g p  va lu e 89.30 0.0000 Panel A2: Restricted Book-to-Market C-CAPM (M2) Size Quintiles 2 3 4 Big 81.83 80.50 80.57 87.62 3.89 3.73 3.58 2.62 Panel A3: Restricted Size C-CAPM (M3) Book-to-Market Quintiles 2 3 4 High 62.46 100.51 130.99 128.32 2.76 4.31 5.53 5.64 2 3  sb 145.27 155.03 146.79 146.63 190.97 47.79 66.32 93.06 108.03 139.19 106.31 104.24 119.64 129.48 171.24  81.96 t ( ) 2 lo g 3.29 16.71 Small 141.11 3.88 2 121.66 3.68 Panel A4: Unrestricted C-CAPM (M4) Book-to-Market Quintiles 4 High Low 2 183.46 167.67 0.90 2.55 171.43 164.63 1.46 2.76 177.21 176.68 1.86 3.55 169.83 142.16 2.21 2.82 175.10 247.14 2.16 2.75 Panel B: 10 Size Portfolios Panel B1: Restricted C-CAPM (M1)  t ( ) 7.93 5.96 Low 215.24 4.18 2 235.78 5.42 3 249.82 5.51 2 lo g 4 High 3.32 4.07 3.73 3.97 3.22 4.47 4.70 4.65 4.44 3.35 4.66 4.61 4.45 3.83 3.43 8 153.18 4.31 9 176.75 4.44 Large 181.85 4.01 8 279.32 7.82 9 254.01 6.99 High 250.89 7.33 p  va lu e 0.0534 Panel B2: Unrestricted C-CAPM (M4) Size Deciles 3 4 5 6 7 133.98 124.19 133.63 143.53 137.74 4.06 3.96 4.32 4.26 4.32 Panel C: 10 Book-to-Market Portfolios Panel C1: Restricted C-CAPM (M1) 261.78 3 t ( sb ) p  va lu e 0.7439 Panel C2: Unrestricted C-CAPM (M4) Book-to-Market Deciles 4 5 6 7 252.10 267.28 270.21 272.45 5.88 5.89 6.72 7.44 16 Figure 1 Cross-Sectional Fit: C-CAPM for the 25 Size and Book-to-Market Portfolios The figure plots average actual versus predicted excess returns (% per month) for the 25 size and bookto-market portfolios. The estimated models are (a) M1, (b) M2, and (c) M3, and (d) M4. The average excess returns are adjusted for the Jensen effect. (b) M2: Restricted Book-to-Market C-CAPM 1.4 1.2 1.2 1.0 1.0 Predicted Risk Premium Predicted Risk Premium (a) M1: Restricted C-CAPM 1.4 0.8 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.0 1.4 0.2 0.4 Average Excess Return 1.2 1.2 1.0 1.0 0.8 0.6 0.2 0.2 0.6 0.8 1.2 1.4 1.2 1.4 0.6 0.4 0.4 1.0 0.8 0.4 0.2 0.8 (d) M4: Unrestricted C-CAPM 1.4 Predicted Risk Premium Predicted Risk Premium (c) M3: Restricted Size C-CAPM 1.4 0.0 0.0 0.6 Average Excess Return 1.0 1.2 0.0 0.0 1.4 Average Excess Return 0.2 0.4 0.6 0.8 1.0 Average Excess Return We estimate M1 and M4 for the two one-characteristic sorted portfolios to investigate whether these provide similar information to the double-sorted portfolios. Panels B and C in Table 4 show that C-CAPM with power utility performs better, as neither of the M1 is rejected, suggesting that sorting stocks according to size and book-to-market ratio may more accurately distinguish stocks. For the 10 size portfolios, all coefficients on the consumption covariances in M4 are significant, and the model fits the data better than M1 (Figure 2). This is also true for the 10 book-tomarket ratio portfolios (Figure 3). In addition, the consumption coefficient for the portfolio in the lowest book-to-market quintile is highly significant, while those for 17 the small growth portfolios in the 25 portfolios are not. The descriptive statistics in Table 2 show that the average book-to-market ratios for these two portfolios are similar, while their average firm sizes are very different. Firms in the smallest size and the lowest book-to-market quintiles seem to be much smaller than other firms in the lowest book-to-market quintiles. Therefore, additional information that is not captured by C-CAPM may be associated with both book-to-market equity ratio as well as size. One possible explanation is that this extra dimension of risk arises from the investment growth prospect of firms. Abhakorn, Smith, and Wickens (2013) find that, in the C-CAPM framework, the mimicking return factor related to book-tomarket ratio (HML) can explain the 25 Fama-French portfolios across size quintiles as well as across book-to-market ratio quintiles. In this regard, they assert that HML may represent risk associated with the investment growth prospects of firm as low book-tomarket ratio firms may be expected to have higher rates of growth while small firms may also be expected to behave similarly. This interpretation of the extra dimension of risk is also consistent with Brennan, Wang, and Xia (2004) and Li, Vassalou, and Xing (2006). These two studies propose asset pricing models based on investment related factors that can explain the cross-section of equity returns well. Figure 2 Cross-Sectional Fit: C-CAPM for the 10 Size Portfolios The figure plots average actual versus predicted excess returns (% per month) for the 10 size portfolios. The estimated models are (a) M1 and (b) M4. The average excess returns are adjusted for the Jensen effect. (b) M4: Unrestricted C-CAPM 1.2 1.2 1.0 1.0 Predicted Risk Premium Predicted Risk Premium (a) M1: Restricted C-CAPM 0.8 0.6 0.4 0.2 0.2 0.8 0.6 0.4 0.4 0.6 0.8 1.0 0.2 0.2 1.2 Average Excess Return 0.4 0.6 0.8 Average Excess Return 18 1.0 1.2 Figure 3 Cross-Sectional Fit: C-CAPM for the 10 Book-to-Market Ratio Portfolios The figure plots average actual versus predicted excess returns (% per month) for the 10 book-tomarket ratio portfolios. The estimated models are (a) M1 and (b) M4. The average excess returns are adjusted for the Jensen effect. (b) M4: Unrestricted C-CAPM 1.4 1.3 1.3 1.2 1.2 Predicted Risk Premium Predicted Risk Premium (a) M1: Restricted C-CAPM 1.4 1.1 1.0 1.1 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.7 0.7 1.4 Average Excess Return 0.8 0.9 1.0 1.1 1.2 1.3 1.4 Average Excess Return 5.2.2 CAPM Table 5 reports the estimation results for all versions of CAPM. The market price of risk in M5 is 2.77 with a t-statistics of 2.94 and is lower than those reported in earlier related studies for the U.S. market (Harvey (1989) and Ng (1991)). Comparing M6 to M2, where the coefficients for the first three size quintiles are relatively less significant, suggests that CAPM cannot price relatively small portfolios well and that there is more information in these portfolio returns related to size left unexplained by CAPM than by C-CAPM discussed above. This inability to price small portfolios has nothing to do with the book-to-market ratio. Moreover, as in M3, the coefficient on the market return for the portfolios in the lowest book-to-market ratio in M7 is not significant. 19 Table 5 Estimates of the CAPM The table presents the estimates of the CAPM (M5-M8): 1960.2-2004.11, 538 observations.  denotes the market price of risk and  i represents a coefficient for each conditional covariance in Equation 8. t ( ) and t(i ) are their corresponding t–statistics respectively. The pricing models (M5-M8) are tested against each other using the log-likelihood ratio test. 2log represents the likelihood ratio statistic. The corresponding p-value at 5% significance level is denoted by p-value. Panel A: 25 Size and Book-to-Market Portfolios  2.77  4s t ( 4 s ) 2 lo g p  va lu e  4b t ( 4 b ) 2 lo g p  va lu e Size Quintiles Small 2 3 4 Big Small 1.70 1.56 110.22 0.0000 Low 0.93 0.93 43.04 0.0020 Low 0.02 0.76 1.30 1.94 1.93 t ( ) Panel A1: Restricted CAPM (M5) 2 lo g p  va lu e 2.94 153.72 0.0000 Panel A2: Restricted Book-to-Market CAPM (M6) Size Quintiles 2 3 4 Big 1.69 1.74 2.03 2.61 1.68 1.80 2.11 2.79 Panel A3: Restricted Size CAPM (M7) Book-to-Market Quintiles 2 3 4 High 2.54 3.51 4.57 4.73 2.68 3.75 4.79 4.89 2 2.20 2.30 3.05 2.26 2.22 3  4 sb 3.02 3.55 3.14 3.59 2.70 Panel A4: Unrestricted CAPM (M8) Book-to-Market Quintiles 4 High Low 2 3 4 High 2.59 3.40 3.14 3.53 2.53 3.78 4.10 4.26 4.04 2.91 3.93 4.02 3.92 3.27 2.63 8 3.26 3.80 9 3.23 3.77 Large 2.45 2.88 8 6.96 7.97 9 6.18 6.86 High 6.24 7.04 t ( 4 sb ) 4.37 4.36 4.37 4.16 3.20 4.51 4.34 4.26 3.57 3.21 0.02 0.67 1.15 1.74 1.81 1.75 2.16 3.00 2.26 2.15 Panel B: 10 Size Portfolios  4s t ( 4 s ) Panel B1: Restricted CAPM (M5)  -0.47 t ( ) 2 lo g -0.34 811.30 Small 3.57 3.46 2 3.07 3.27 3 3.41 3.74 p  va lu e 0.0000 Panel B2: Unrestricted CAPM (M8) Size deciles 4 5 6 7 3.13 3.44 3.27 3.29 3.44 3.93 3.69 3.82 Panel C: 10 Book-to-market Portfolios  4b t ( 4 b ) Panel C1: Restricted CAPM (M5)  1.98 t ( ) 2 lo g 1.79 404.85 Low 3.14 3.19 2 4.29 4.73 3 4.68 5.39 p  va lu e 0.0000 Panel C2: Unrestricted CAPM (M8) Book-to-market Deciles 4 5 6 7 4.76 5.50 5.73 6.64 5.51 6.06 6.58 7.76 20 M5, M6, and M7 are rejected relative to the unrestricted model M8 based on their likelihood ratio statistics of 153.72, 110.22, and 43.04 respectively. The likelihood ratio statistics for CAPM are all larger than those for C-CAPM. As in C-CAPM, allowing the coefficients on conditional covariances of market return with individual excess return to be different offers extra information about the cross-section of the equity returns. As can be seen from Figure 4, M8 can explain the variation in the cross section well while M5 is not able to explain both size and value effects as in previous studies (e.g. Fama and French , 1992 and 2006 and Lewellen and Nagel, 2006). In M8, 20 coefficients on conditional covariances of returns with the market return, including one for the market return (the coefficient on the variance), are more than 2 standard errors different from zero, while the other 3 coefficients are significant at the 10% significance level. The coefficients for the first 3 size and lowest book-to-market quintiles are insignificant at any conventional level. These coefficients range from 0.02 to 4.51, exhibiting a clear positive relation with book-to-market ratio. However, any relation between the coefficients and size can be seen only from the portfolios in the last two book-to-market quintiles. Figure 4 Cross-Sectional Fit: CAPM for the 25 Size and Book-to-Market Portfolios The figure plots average actual versus predicted excess returns (% per month) for the 25 size and bookto-market portfolios. The estimated models are (a) M5, (b) M6, and (c) M7, and (d) M8. The average excess returns are adjusted for the Jensen effect. (b) Restricted Book-to-Market CAPM: M6 0.9 0.9 0.7 0.7 Predicted Risk Premium Predicted Risk Premium (a) Restricted CAPM: M5 0.5 0.3 0.1 -0.1 -0.1 0.5 0.3 0.1 0.1 0.3 0.5 0.7 -0.1 -0.1 0.9 Average Excess Return 0.1 0.3 0.5 Average Excess Return 21 0.7 0.9 (d) Unrestricted CAPM: M8 0.9 0.9 0.7 0.7 Predicted Risk Premium Predicted Risk Premium (c) Restricted Size CAPM: M7 0.5 0.3 0.1 -0.1 -0.1 0.5 0.3 0.1 0.1 0.3 0.5 0.7 -0.1 -0.1 0.9 Average Excess Return 0.1 0.3 0.5 0.7 0.9 Average Excess Return M5 for the two sets of 10 portfolios does not perform well since the market price of risk is of relatively low significance and in the case of the 10 size portfolios, the market price of risk has the wrong sign. Therefore, the likelihood ratio test rejects M5 for both 10 portfolios. M8 fits the data better than M5 (Figures 5 and 6), but its coefficients exhibit a negative relation with size and a negative relation with book-tomarket ratio for both 10 portfolios as in the case of the 25 portfolios. On the other hand, the relation between the consumption coefficients and firm characteristics in CCAPM can be seen only in the case of the 25 portfolios. Thus, C-CAPM can explain size effect, but it has difficulty explaining the value effects; this exposure to the value premium appears to be associated with both the book-to-market ratio and, to some extent, with size. 22 Figure 5 Cross-Sectional Fit: CAPM for the 10 Size Portfolios The figure plots average actual versus predicted excess returns (% per month) for the 10 size portfolios. The estimated models are (a) M5 and (b) M8. The average excess returns are adjusted for the Jensen effect. (b) Unrestricted CAPM: M8 1.0 1.0 0.8 0.8 Predicted Risk Premium Predicted Risk Premium (a) Restricted CAPM: M5 0.6 0.4 0.2 0.0 -0.2 -0.2 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 -0.2 -0.2 1.0 0.0 0.2 Average Excess Return 0.4 0.6 0.8 1.0 Average Excess Return Figure 6 Cross-Sectional Fit: CAPM for the 10 Book-to-Market Portfolios The figure plots average actual versus predicted excess returns (% per month) for the 10 book-tomarket ratio portfolios. The estimated models are (a) M5 and (b) M8. The average excess returns are adjusted for the Jensen effect. (a) Restricted CAPM: M5 1.3 1.1 Predicted Risk Premium Predicted Risk Premium 1.1 0.9 0.7 0.5 0.3 0.3 (b) Unrestricted CAPM: M8 1.3 0.9 0.7 0.5 0.5 0.7 0.9 1.1 0.3 0.3 1.3 Average Excess Return 0.5 0.7 0.9 1.1 1.3 Average Excess Return 5.2.3 C-CAPM and CAPM We further investigate the ability of C-CAPM (M1) and CAPM (M4) by comparing the behavior of the conditional covariances in both models, as given in Table 6 below, with the average excess returns of the 25 portfolios shown in Table 2. The 23 consumption covariance obtained from the estimation of C-CAPM decreases, as we move down the column, indicating a negative relation with size. Thus, C-CAPM can capture the size effect. However, in the lowest book-to-market quintiles, average excess returns increase as the size of the portfolios grows larger. This result is consistent with the results in Table 4 where the coefficients for these portfolios in M3 and M4 are not significant from zero. Table 6 Conditional Covariances of the Returns on the 25 Portfolios in M1 and M5 The table shows the average conditional covariances of the returns with consumption and market returns as implied by the C-CAPM and CAPM, respectively. Each conditional covariance is given in percent per month and estimated by the multivariate GARCH in the mean. Book-to-market ratio Quintile Size Quintile Small 2 3 4 Big Small 2 3 4 Big Low 2 3 4 High Panel A: Conditional covariance of consumption 0.32218 0.28800 0.31130 0.22016 0.27699 0.24066 0.23827 0.23237 0.17594 0.20899 0.19671 0.17586 0.16247 0.15597 0.17918 -0.00131 -0.00166 -0.00138 0.00593 0.00496 0.00304 0.00001 -0.00133 -0.00127 -0.00040 Panel B: Conditional covariance of market return 0.32143 0.35714 0.25829 0.25272 0.28785 0.24282 0.22963 0.21088 0.24303 0.17777 0.23056 0.18894 0.18875 0.17934 0.16103 0.18551 0.21742 0.20223 0.15693 0.20226 0.27074 0.24804 0.19669 0.16809 0.16478 C-CAPM appears to miss the value premium completely by not producing dispersion in the consumption covariance across the book-to-market quintiles. In fact, the consumption covariances for the 5 portfolios in the highest book-to-market quintile seem to be slightly lower than those in the lowest book-to-market quintiles, indicating lower risk premium is implied by C-CAPM. On the other hand, the condition covariances of the returns with the market returns in CAPM, in addition to having a similar behavior across book-to-market quintiles as consumption covariances, appear not to be able to capture the size effect as well. The dispersion of the market covariances is not big enough to explain the differences in the excess returns across size, confirming our previous results where the coefficients for the first two size quintiles in M6 were not highly significant. 24 We add a constant term in M1-M8 for the estimation of the 25 portfolios to measure variation in excess returns that was left unexplained in each model. In general, we expect the constant term in C-CAPM to be of more significance than in those in CAPM because pricing asset returns with market return is expected to be more precise than using aggregate consumption data. However, Table 7 shows that for the magnitude of the constant terms in CAPM, 1.98 is larger than that for C-CAPM at 0.86. This larger magnitude of CAPM is present in every restricted version. The magnitude of the constant is also larger than in Fama and French (1993) with a constant for the CAPM being 0.04 to 0.57 (in absolute terms). Table 7 Constant term The table presents the estimates for the constant term, in all versions of the C-CAPM and CAPM, M1M8 in Table 1, for the 25 portfolios formed based on size and book-to-market ratio. The number in the parentless is the t-statistic associated with each constant term. Constant Constant M1 0.84 (5.36) M5 1.98 (9.13) Panel A: C-CAPM M2 1.14 (6.39) Panel B: CAPM M6 1.75 (6.79) M3 0.52 (2.75) M4 0.87 (1.75) M7 0.56 (1.85) M8 0.94 (1.70) The information about the cross-section of equity returns left unexplained in C-CAPM seems to be less than that in the CAPM. Moving from M1 to M4 decreases the significance of the constant terms (except for moving from M1 to M2), suggesting that allowing coefficients of conditional covariances within the same book-to-market ratio to be different is more important than allowing the coefficients to be different across size quintiles; the magnitude and level of significance of the constant terms reduces more when moving from M2 to M3 than when moving from M1 to M2. This argument is also true for CAPM when moving from M5 to M8. 5.2.4 General SDF Models Table 8 reports the estimates of the general two- and three-factor SDF models based on consumption, inflation, and industrial production. We are unable to estimate M10 and M12 for the 25 portfolios due to the high parameterization of the MGM. As in Smith, Sorensen, and Wickens (2008), we find that industrial production plays no role 25 in evaluating asset returns, but inflation is significant. The coefficient on the conditional covariance of inflation for the 10 book-to-market portfolios is positive because the contribution to the risk premium by consumption is higher than it is for actual excess return. The estimation of M9 and M11 for both 10 portfolios shows that the restrictions they provide on M10 and M12 cannot be rejected by the likelihood ratio test, implying that the coefficients for conditional covariance of consumption and inflation with the returns are similar across size and book-to-market deciles. However, M9 does not explain the data better than C-CAPM and the CAPM (Figure 7). Table 8 Estimates for Restricted Macro SDF Models The table presents the estimates for the restricted general SDF models (M9 and M11): 1960.2-2004.11, 538 observations.  i represents a coefficient for each conditional covariance in Equation 8. t(i ) is its corresponding t–statistics respectively. The pricing models (M9 and M11) are tested against their respective unrestricted alternatives (M10 and M12) using the log-likelihood ratio test. 2log represents the likelihood ratio statistic. The corresponding p-value at 5% significance level is denoted by p-value. Model M9 M11 M9 M11 M9 M11 1 2 3 Panel A: 25 Size and Book-to-Market Portfolios 72.93 -115.72 (3.40) (-1.74) 72.82 -116.47 1.42 (3.40) (-1.73) (0.06) Panel B: 10 size portfolios 59.29 -130.87 (2.18) (-2.01) 59.62 -137.76 16.84 (2.17) (-2.08) (0.55) Panel C: 10 Book-to-market portfolios 307.12 147.29 (8.00) (1.92) 305.17 141.17 8.58 (7.87) (1.82) (0.35) 26 2 lo g p  va lu e 23.66 0.1666 33.43 0.1832 23.27 0.1805 32.33 0.2201 Figure 7 Cross-Sectional Fit: Two-Factor SDF Model for the 25 Portfolios The figure plots average actual versus predicted excess returns (% per month) for the 25 size and bookto-market ratio portfolios. The estimated model is M9. The average excess returns are adjusted for the Jensen effect. M9: Restricted Two-Factor SDF Model 1.4 1.2 Predicted Risk Premium 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Average Excess Return 6. Conclusion This paper examines the behavior of the cross-section of equity returns based on the no-arbitrage condition derived from the Stochastic Discount Factor approach to asset pricing. We test whether the conditional covariances of the equity returns across portfolios formed on size and book-to-market ratio with discount factors in each asset pricing model can sufficiently explain the excess returns in these portfolios. Our results indicate that the no-arbitrage test rejects C-CAPM as the model can explain the size effects, but not the value effect. Although the consumption covariances exhibit a negative relation with size, but they do not vary with the book-to-market ratio. This behavior explains why the likelihood ratio test indicates that the coefficients for the consumption covariances are not similar across book-to-market ratios. Allowing the coefficients on the conditional covariances with consumption to be different across portfolios generally improves the fit of C-CAPM. Even without adding any factor to the model, the performance of the resulting unrestricted CCAPM is comparable to the modified version of C-CAPM of Lettau and Ludvigson (2001), Parker and Julliard (2005), Yogo (2006) and Savov (2011). The unexplained variation in excess returns is less than for unrestricted C-CAPM as the significance of the constant term is lower than that in standard C-CAPM. Unrestricted C-CAPM does 27 not explain the small growth portfolios well, but this phenomenon is common to most asset pricing models. Our results confirm the findings of Abhakorn, Smith, and Wickens (2013) that both firm size and the book-to-market ratio need to be included in the model in order discover a value effect. Firm size or the book-to-market ratio on their own does not generate information about average returns that improves on C-CAPM. This requires the double sorting of stocks according to size and the book-to-market ratio for the 25 Fama-French portfolios. This finding suggests that there is an additional dimension of risk left unexplained by C-CAPM or by an SDF model with only one of these factors . This extra dimension of risk seems to be associated with both (small) size and a (low) book-to-market ratio. A possible explanation for this extra dimension of risk is the investment growth prospect of firms, see Abhakorn, Smith, and Wickens (2013), and could be the reason that the investment-based asset pricing models of Brennan, Wang, and Xia (2004) and Li, Vassalou, and Xing (2006) are able to explain the crosssection of equity returns. Our results indicate that C-CAPM with size and the book-to-market ratio as additional factors contains information about cross-section average returns that is not captured by CAPM in previous studies by, for example, Fama and French (1992 and 2006) and Lewellen and Nagel (2006). 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