Higher-Order Systematic Comoments and Asset Pricing: New Evidence

The Financial Review 44 (2009) 345--369

Higher-Order Systematic Comomentsand Asset Pricing: New Evidence

Duong NguyenUniversity of Massachusetts Dartmouth

Tribhuvan N. Puri∗University of Massachusetts Dartmouth

Abstract

We provide evidence supporting Rubinstein’s (1973) model that if returns are not nor-mal, measuring risk requires more than just measuring covariance. Higher-order systematiccomoments should be important to risk-averse investors who are concerned about the extremeoutcomes of their investments. Our paper shows that the Fama-French factors [SMB (returnon small stocks less the return on big stocks), HML (return on high book-to-market stocksless the return on low book-to-market stocks)] as well as the momentum and market liquidityfactors can be explained by the higher-order systematic comoments, and it lends support to thetraditional covariance risk-based theory without having to resort to behavior assumptions.

Keywords: higher-order comoments, Fama-French, momentum, market liquidity factors

JEL Classification: G12

∗Corresponding author: Department of Accounting and Finance, Charlton College of Business, Universityof Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747; Phone: 508-999-8759;Fax: 1-508-999-8646; E-mail: [email protected].

We thank Arnold Cowan (the editor) and the two anonymous referees for their many helpful suggestionsand comments while the paper was in preparation. All the remaining errors and omissions, however, areours. D. Nguyen acknowledges the financial support of a summer research grant from the Charlton Collegeof Business.

C© 2009, The Eastern Finance Association 345

346 D. Nguyen and T. N. Puri/The Financial Review 44 (2009) 345–369

1. Introduction

In utility-maximizing general-equilibrium asset pricing models, returns on riskyassets are determined by their covariance with the state variables that represent thefundamental sources of risk in the economy. When changes in state variables adverselyaffect security returns and investors’ wealth, marginal utility of wealth increases forrisk-averse investors. In equilibrium, investors must be compensated by additionalrewards commensurate with the risks they undertake. In the standard capital-assetpricing model (CAPM), the market summarizes presumably all sources of risk. Cross-sectional returns in the CAPM depend on their covariance with a well-diversifiedmarket portfolio. However, Rubinstein (1973) shows that if returns do not followa normal distribution, measuring risk requires more than covariance. Higher-ordercomoments should be an important consideration for risk-averse investors who areconcerned about the extreme outcomes of their investments. In this paper, we seek toexplain the effect of Fama-French, momentum, and market liquidity factors in assetpricing within Rubinstein’s (1973) model. Our findings suggest that these factorscould be proxies for a set of higher-order comoment factors. In particular, we findthat in the cross-sectional analysis, the factors become insignificant in the presenceof a set of ten or 15 systematic comoments for all portfolios sorted by size, book-to-market, momentum, and liquidity. The results are consistent in both subperiodsas well as in several other robustness checks. Our time series analysis yields similarconclusions. The time series regression intercepts (i.e., pricing errors) tend to zerowhen additional comoment factors are included, suggesting that the higher-ordersystematic comoments should be relevant in pricing assets.

Recent research shows that aggregate market liquidity is a fundamental sourceof risk in financial markets. Studies by Chordia, Roll and Subrahmanyam (2000),Hasbrouck and Seppi (2001), and Huberman and Halka (2001) provide evidence ofthe existence of commonality across stocks in liquidity fluctuations. Their findingshave initiated a new research hypothesis: if liquidity shocks are nondiversifiable andhave a varying impact across individual securities, then the more sensitive a stock’sreturn is to such shocks, the greater its expected return should be. This hypothesisis supported by Pastor and Stambaugh (2003). They develop a measure of aggre-gate liquidity, based on daily price reversals, and show that stocks that are moresensitive to the market liquidity factor command a higher required rate-of-returnthan less sensitive stocks. Acharya and Pedersen (2005), Chan and Faff (2005), andSadka (2006) also provide evidence of premiums on systematic liquidity risk (mea-sured as return covariation with particular measures of aggregate liquidity shocks).Thus, it is generally accepted that market liquidity is a priced state variable. How-ever, how much the liquidity factor affects asset pricing is still in debate due to thefact that true liquidity is unobservable. One of our contributions is to show thatliquidity risk is a possible aspect of market risk, which may not be captured inthe traditional CAPM. Using a higher-order parameter-preference model, a gener-alization of the two-moment CAPM, we find that the Pastor–Stambaugh measure of

D. Nguyen and T. N. Puri/The Financial Review 44 (2009) 345–369 347

liquidity becomes insignificant in the presence of higher-order comoments with themarket.

Apart from aggregate market liquidity, empirical research also reports that somenonmarket factors, such as size, book-to-market ratios, and momentum can explaincross-sectional variations in returns. For example, Fama and French (1993, 1996)show that the size factor, SMB (return on small stocks less the return on big stocks),as well as the book-to-market factor, HML (return on high book-to-market stocks lessthe return on low book-to-market stocks), are important in explaining cross-sectionalstock returns. In addition, Jegadeesh and Titman (1993, 2001) analyze the pricemomentum effect in individual stocks. They show that stocks that do well relative tothe market over the last three–12 months tend to perform well in the next few months,and stocks that perform poorly continue to perform poorly. Fama and French (1996)and Jagadeesh and Titman (2001) assert that the momentum effect is one of the mostserious problems with asset pricing. The momentum effect is distinctly different thanthe value effect captured by stock characteristics and is explained by neither theFama-French three-factor model nor the CAPM. Carhart (1997) adds the momentumfactor, MOM (the difference in returns of diversified portfolios of short-term winnersand losers), to the Fama-French three-factor model and finds that the momentumfactor is significantly important in explaining stock returns.

While there is a consensus that the nonmarket factors (SMB, HML, and MOM)are significant in explaining stock returns, there is an ongoing debate about what eco-nomic mechanisms drive these factors to influence stock returns. The debate focuseson two competing types of theories: risk-based and nonrisk-based. The proponentsof the risk-based theories suggest that although SMB, HML, and MOM factors arenot state variables in the conventional sense, they reflect unidentified state variablesthat produce nondiversifiable risks (covariances). These risks are not captured bythe market risk premium, hence, they are priced separately from market betas. Forexample, Fama and French (1993, 1996) view the SMB and HML proxies for certaindistress factors that are not captured in the CAPM. Lettau and Ludvigson (2001) showthat SMB and HML capture common variations in returns because they seem to berelated to variations in a consumption-based risk premium that changes over time.Chung, Johnson and Schill (2006) argue that the Fama-French factors represent thepricing of higher-order comoments. Similarly, Conrad and Kaul (1998) argue thatmomentum profit is a result of cross-sectional variability in expected stock returns.Chordia and Shivakumar (2002) suggest that profits from momentum strategies canbe explained by a set of lagged macroeconomic variables that are related to businesscycles. According to the risk-based explanations, small cap stocks, value stocks andwinner stocks have relatively high average returns because they are risky; they havehigh sensitivity to the fundamental risk factors that are being measured by SMB,HML, and MOM.

Proponents of the nonrisk-based theories, on the other hand, suggest that humanbehavior that deviates from the rational expectation paradigm drives these factors andinfluences stock prices under market imperfections that limit arbitrage. For example,


Lakonishok, Shleifer and Vishny (1994) suggest that the book-to-market factor is theresult of investor bias in earnings-growth extrapolation. Daniel and Titman (1997)contend that it is characteristics not covariances that generate return dispersion. Theyargue that high book-to-market stocks have high returns due to some other reason(possibly overreaction) so the high returns have nothing to do with systematic risk. Intheir opinion, it is the characteristics (size and book-to-market) rather than the covari-ances (sensitivities to SMB and HML) that are associated with high returns. Jegadeeshand Titman (1993) and Arena, Haggard and Yan (2008) suggest that individual stockmomentum can be driven by investor underreaction to firm-specific information.Daniel, Hirshleifer and Subrahmanyam (1998) and Hong and Stein (1999) attributethe momentum anomaly to investor cognitive biases.

In this paper, we provide an alternative explanation for SMB, HML, and MOMwithin the traditional covariance risk-based theory without having to resort to behaviorassumptions. We argue that when returns are non-normal, not only SMB and HML butMOM may proxy for higher-order comoment market factors. Our motivation comesfrom the theoretical model of Rubinstein (1973) and empirical evidence of Chung,Johnson and Schill (2006). Rubinstein (1973) demonstrates that a risk measure, whendistributions are non-normal, requires not only measuring covariance with the marketbut also the higher-order comoments (coskewness, cokurtosis, and so on) with themarket.1 His model suggests that only market risk (comoment) factors should matterto investors. Chung, Johnson and Schill (2006) test Rubinstein’s assertion and find thatwhile each comoment individually is unable to explain returns, a set of comomentstaken together can do so. They find that a set of systematic comoments of the thirdto the tenth orders substantially lowers the level of significance of the Fama-Frenchfactors (SMB and HML). Therefore, they conclude that the SMB and HML factorsare simply proxies for higher-order systematic comoments.

We argue that most covariance risk-based models fail to account for momentumsince they do not consider the fact that a return is not normal; therefore, higher-ordercomoments which measure extreme outcomes of an investment should be considered.

There is also a large literature that examines the role of higher-order comoments,such as coskewness and cokurtosis, in explaining stock returns.2 The idea is thatif the returns are not normal (skewed or leptokurtic), investors are also concernedabout portfolio skewness and kurtosis. If investors’ preferences contain skewness

1 If the return is normal, the first two moments (i.e., mean and variance) alone are sufficient to explain thedistribution. As a result, investors should not care about higher-order moments. However, there is ampleevidence that suggests otherwise (e.g., Fama, 1965; Arditti, 1971; Singleton and Wingender, 1986; andmore recently, Chung, Johnson and Schill, 2006). This implies that investors may care a great deal aboutthe extreme outcomes of their investments.

2 See, for example, Kraus and Litzenberger (1976), Friend and Westerfield (1980), Sears and Wei (1985),Harvey and Siddque (2000), Chen, Hong and Stein (2001), and Hung, Shackleton and Xu (2004). Inparticular, Harvey and Siddque (2000) find that coskewness accounts for part of the explanatory powerof the Fama-French size and book-to-market factors and that coskewness can explain part of the return tomomentum strategies.


and kurtosis, each stock’s contribution to systematic skewness (coskewness) andkurtosis (cokurtosis) may determine the stock’s attractiveness and hence require riskpremiums.

One might argue that most return distributions can be reasonably characterized bythe first four moments, therefore, resorting to further higher moments does not seemappealing. However, as Rubinstein (1973) and Chung, Johnson and Schill (2006) pointout, risk-averse investors have a great deal of concern for extreme outcomes on theirinvestments. Variance, skewness, and kurtosis might give some information aboutthe tail of the outcome distribution. However, only a set of higher-order comomentscan fully specify the tail of the distribution. A precise specification of the right tailvia higher moments is required to measure contingent returns in the events, suchas lotteries, out-of-the-money options, and extreme geopolitical shocks that couldpossibly dry up the market liquidity. Risk-averse investors, therefore, must concernthemselves with the accurate description of the tail.3

Our data exhibit non-normality, and given the evidence that there are factorsother than just the Fama-French factors that influence the cross-section of returns, itis worthwhile to examine whether these factors may be accounted for by higher-ordercomoments. Our intuition is that risk premiums generated by higher-order comomentfactors may account for identified and as yet unidentified sources of risk. The liquidityfactor proposed by Pastor and Stambaugh (2003) still awaits attestation. So far, fewstudies incorporate a liquidity risk factor into an asset pricing model, and those thatdo have limited success in explaining cross-sectional variations in asset returns. Evenless is known about whether liquidity risk can be captured by some attributes ofmarket risk not explained by the CAPM.

Our study is also motivated by a preliminary estimation in which we regress thefactor loadings of size, book-to-market, momentum, and liquidity factors on a set often higher-order systematic comoments and find that the systematic comoment canexplain the factors with a high R2 (0.85–0.89 for size factor, 0.73–0.75 for book-to-market factor, 0.61–0.90 for momentum factor, and 0.50–0.67 for liquidity factor).This evidence suggests that there might be a relation between the common factors andsystematic comoments, as well as an economic relation within a traditional risk-basedcovariance theory.

3 The following example further justifies the importance of higher moments beyond the fourth moment. Aninvestor attempts to optimize a portfolio made of two independent assets: A and B. The payoff distributionon A consists of a loss of $1 (999 times in 1,000) and a gain of $999 (1 time in 1,000). The distribution ofpayoff on B consists of a gain of $1 (999 times in 1,000) and a loss of $999 (1 time in 1,000). Rationally, anyinvestor would choose A because of small downside risk and large upside gain. Therefore, if asset returnsare independent, the optimal portfolio should consist of 100% A. However, based on the mean-variancemodel, the optimal portfolio should be 50% A and 50% B since this is the minimum variance portfolio(both assets have the same mean and variance). Based on the four-moment model, the optimal portfoliois still 50% A and 50% B since it produces minimum variance and kurtosis. The conclusion based on themean-variance-skewness-kurtosis only might not reflect the small downside risk and the probability toearn large upside return (extreme event). Hence, we should consider the higher-order moments to solvethe problem.


For the cross-sectional analysis using the Fama and MacBeth (1973) procedurefor portfolios sorted by size, book-to-market, momentum, and liquidity for 1970–2005, we find that adding a set of systematic comoments of orders three to ten orhigher reduces the explanatory power of SMB, HML, momentum, and the Pastor–Stambaugh market liquidity factors to insignificant for portfolios sorted in variousways. Also, we do not find the similar results when adding a set of standard momentsof orders three to ten or higher. To check the stability of the results, we separatelyexamine 1970–1987 and 1988–2005. The results still hold in both subperiods. Wealso find that the results with a set of ten systematic comoments are very similar tothose with 15 comoments or higher. This is done with the objective of examiningwhether a set of comoments higher than ten would more precisely characterize thereturn distributions. However, the empirical evidence suggests that a set of commentsof orders three to ten is sufficient to capture the extreme outcomes of the investment.

As a check for the robustness of the Fama and MacBeth (1973) procedure, weconduct a time series analysis using a multivariate test of Gibbons, Ross and Shanken(GRS) (1989) and find that the GRS statistics, which test whether the pricing errorsfrom time series regressions are jointly equal to zero, consistently decrease whenmore systematic comoment factors are added to the model and become insignificantwhen enough comoment factors are included. This suggests that the pricing errortends to zero when more comoment factors are added to the model.

Our findings lend support to Rubinstein’s (1973) model in that higher-ordersystematic comoments should be considered in asset pricing. We also provide anotherrisk-based explanation that the common factors SMB, HML, momentum, and liquidityfactors represent higher-order market comoment factors. In particular, we find thatthe Pastor and Stambaugh (2003) liquidity risk is subsumed within the higher-ordermarket comoments. This evidence lends support to the notion that the empiricalknowledge of nonmarket factors in the context of asset pricing models may be of lessvalue because all such factors may be proxies for a set of higher-order comomentswith a well-diversified market portfolio.

2. Methods

2.1. Portfolio formation

We sort all ordinary domestic common stocks (CRSP codes ten and 11) tradedon three stock exchanges (NYSE, Amex, and Nasdaq) into 50 portfolios based onsize, book-to-market, momentum, and liquidity. In particular for size portfolios, atthe end of each calendar year in 1965–2005, all stocks are ranked based on theirmarket capitalization and then sorted into 50 portfolios containing an equal numberof stocks. For book-to-market portfolios, all stocks are ranked by their beginning-of-period book-to-market ratios and then divided into 50 portfolios of equal size.The momentum portfolios are constructed following the procedure described in the


website of Kenneth French.4 At the end of each month t, all stocks are sorted into 50portfolios of equal size based on their prior compound return for months t−two tot−12. For the liquidity portfolio, we use the measure of Pastor and Stambaugh (2003)as an individual firm’s liquidity measure.5 Each year we perform the following linearregression to obtain the annual liquidity measure for each stock:

rei,d+1,t = θi,t + φi,t ri,d,t + γi,t sign

(re

i,d,t

) × vi,d,t + εi,d+1,t , d = 1, . . . , D, (1)

where r i,d,t is the return of stock i on day d in year t; rei,d+1,t = r i,d,t − r y,d,t where

r y,d,t is the return on the CRSP value-weighted market return on day d in year t; andvi,d,t is the dollar volume for stock i on day d in year t. The liquidity measure for stocki in year t is the estimate of γ i,t in the above Equation (1). We compute the liquiditymeasure for a stock in a given year only if there are more than 30 observations toestimate Equation (1) (D > 30). In each year, we form all stocks into 50 liquidityportfolios based on their annual liquidity estimates γ i,t .

Using portfolios as constructed earlier, we compute equally weighted monthlyreturns for each of the 50 portfolios. We subtract the 30-day Treasury bill yield toobtain the excess portfolio return. Once we have constructed the portfolios, we employboth time series and cross-sectional tests to examine the relation between the commonfactors and higher-order systematic comoments.

2.2. Cross-sectional test

We apply the two-step Fama and MacBeth (1973) procedure to our empiricalasset pricing tests. In the first step, we estimate the five-factor model that includesFama and French (1993) momentum, and Pastor and Stambaugh (2003) factors:

r ( j, t) = a0 + armrf b( j, t) + asmbs( j, t) + ahmlh( j, t)

+ amomm( j, t) + aliql( j, t) + e( j, t), (2)

where r ( j , t) is the excess return of portfolio j in month t and b( j , t), s( j , t),h( j , t), m( j , t), and l( j , t) are factor loadings for excess return of portfolio j on factors(Rm-Rf ), SMB and HML, momentum (MOM), and Pastor and Stambaugh (2003)market liquidity (LIQ), respectively, in month t.6

4 Available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french.

5 There are various measures for liquidity that are used in the literature since liquidity has many dimensions.Existing measures typically focus on one dimension. For example, the bid-ask spread measure in Amihudand Mendelson (1986) relates to the transaction cost dimension. The turnover measure of Datar, Naik andRadcliffe (1998) relates to the trading quantity dimension. The measure in Pastor and Stambaugh (2003)employs the concept of price impact to capture the price reaction to trading volume. Liu (2006) uses ameasure to capture the trading speed dimension. It is hard to say one measure is better than another sinceeach measure focuses on one dimension of liquidity. We use the Pastor and Stambaugh’s (2003) measureto construct the liquidity portfolio to be consistent with the market liquidity factor we use in the model.

6 We obtain the data on these factors from Wharton Research Data Services.


For each month t, the factor loadings are computed by regressing portfolio returnsover the last five years on the market factor (Rm-Rf ), SMB, HML, MOM, and LIQ,respectively. The result is a time series for each factor loading from 1970 to 2005(we lose the observations on the first five years in the original sample to estimatethe factor loadings). Once the factor loadings are computed, for each month t weperform cross-sectional regressions of the period portfolio returns on these loadingsas in Equation (2). Repeating this process for all months in 1970–2005, we have 432sets of coefficient estimates. Following Fama and MacBeth (1973) we average theseestimates to get the average coefficients.

In the second step, we examine whether the above factor loadings are still sig-nificant when a set of systematic comoments is added to the model. In particular,for each month t, we perform cross-sectional regressions of excess portfolio returnson the loadings of SMB, HML, MOM, LIQ, and on the systematic comoments asfollows:

r ( j, t) = a0 + asmbs( j, t) + ahmlh( j, t) + amomm( j, t) + aliql( j, t)

+n∑

i=2

ai b(i, j, t) + e( j, t), (3)

where b(i , j , t) is the ith systematic comoment of portfolio j in month t.7 We computethe systematic comoments in month t using the past 60 months of portfolio returnsas follows:

b(i, j, t)

=

60∑τ=1

[r ( j, t − τ ) − 1

60

60∑k=1

r ( j, t−k)

] [r (m, t−τ )− 1

60

60∑k=1

r (m, t − k)

]i−1

60∑τ=1

[r (m, t − τ ) − 1

60

60∑k=1

r (m, t − k)

]i , (4)

where r (m, t) is the return of the CRSP value-weighted index. We compute thesystematic comoments up to the 10th order as in Chung, Johnson and Schill (2006).We also experiment with a set of systematic comoments up to the 15th order to seewhether the findings are robust to the higher-order of comoments or whether thefindings are only chance results.

Since one might argue that including any set of variables would be able to reducethe significant levels of the factors, we also include a set of standard moments (notsystematic comoments) as follows:

7 We include the loading on the market risk premium factor in the set of systematic comoment since theloading is the second systematic comoment.


r ( j, t) = a0 + armr f b( j, t) + asmbs( j, t) + ahmlh( j, t) + amomm( j, t)

+ aliql( j, t) +n∑

i=3

ai m(i, j, t) + e( j, t). (5)

We also use the past 60 months of portfolio returns to compute m(i , j , t) as follows:

m(i, j, t) = 1

60

60∑τ=1

[r ( j, t − τ )]i . (6)

where m(i , j , t) is the standard moment of order i of portfolio j in month t.The two-step Fama and MacBeth (1973) procedure has two major drawbacks.

First is the errors in variables problem because betas are estimated in the first-stageregression and are subsequently used as independent variables in the second-stageregression. Second is the serial correlation of return residuals. As a check for robust-ness, we use Shanken’s (1992) correction to adjust for estimation errors in betas. Theadjusted covariance matrix is calculated as follows:

Adj. Var(a) = V[1 + a′(Z )−1a

] + Z , (7)

where V is the k factor × k factor covariance matrix of mean coefficient estimates andZ is the k × k covariance matrix of monthly risk factors. The respective risk factors arethe SMB, HML, MOM, LIQ, and market risk premium (RMRF) for the covariancefactor and RMRF raised to the (i − 1) power for the higher-order comoment factors.

The Fama and MacBeth (1973) method is designed to account for cross-correlation between residuals. However, it assumes the residuals are uncorrelatedover time, which may not be true. Therefore, the procedure is not robust to serialcorrelation (see Cochrane, 2001, for more detail). We use Newey and West’s (1987)method to account for serial correlation in the error term.

2.3. Time series test

The drawbacks of the cross-sectional test of Fama and MacBeth (1973) arerectified in the time series regression. It is well known that asset pricing modelscan also be evaluated by examining the intercepts from the time series regressionsof portfolio excess returns on the factors. If the regression intercepts, which are thepricing errors, are jointly equal to zero, the model is valid. Another advantage ofthe time series approach is that the errors in variables problem in the Fama andMacBeth (1973) approach can be avoided since the estimation of risk premiums is nolonger necessary, and the implication of the asset pricing theory can be tested by thehypothesis that all the intercepts are jointly equal to zero. To test this hypothesis, weuse the test developed by Gibbons, Ross and Shanken (1989). The GRS statistics canbe used to examine the relation between common factors and higher-order systematic


comoments. We argue that if the common factors represent higher-order comoments,the magnitude of the GRS statistics should decrease when comoment factors are addedto the model.

To apply the procedure, we first estimate the time series regression of the excessreturns on the 50 portfolios (sorted by size, book-to-market, momentum, and liquidity)on the five-factor model using ordinary least squares:

r (i, t) = αi + βi (Rmt − Rft) + δi SMBt + γi HMLt + ηi MOMt + ψi LIQt + eit , (8)

where r (i , t) is the excess return on portfolio i in month t; (Rmt − Rft), SMBt, andHMLt are the three Fama and French (1993) factors related to market premium, firmsize, and the book-to-market ratio; MOMt is the momentum factor; and LIQt is thePastor and Stambaugh (2003) liquidity factor in month t. We then compute the GRSstatistics to test whether the 50 intercepts from these time series regressions are jointlyequal to zero as follows.

Let N be the number of time series observations, L be number of portfolios, Kbe the number of regression parameters including the constant term, and X be theobservation matrix. Then, the GRS test statistic is:(

A′−1∑

A

)N − K − L + 1

L ∗ (N − K ) ∗ ω1,1, (9)

where A is the column vector of the regression parameters,∑

is the variance–covariance matrix of the residuals from the regression, and ω1,1 is the diagonal ele-ment of (X ′ X )−1. Under the null hypothesis that the regression constants are zero,this statistic has an F-distribution with L and (N − K − L + 1) degrees of freedom.

The next step is to add higher-order comoment factors to Equation (8) asfollows:

r (i, t) = αi + βi (Rmt − Rft) + δi SMBt + γi HMLt

+ ηi MOMt + ψi LIQt +K∑

j=3

ηi, j (Rmt − Rft)j−1 + eit, (10)

where (Rmt − Rft) j−1 is the higher-order j comoment factor. For example, Rmt − Rft

is the covariance factor, (Rmt − Rft)2 is the coskewness factor, (Rmt − Rft)3 is thecokurtosis factor, and so on.

When adding each comoment factor, we compute the corresponding GRS statis-tics. Our argument is that if higher-order comoments are relevant in explaining stockreturns and if higher-order comoments can subsume the effects of factors SMB, HML,MOM, and LIQ, then adding these comoments reduces the significance level of theGRS statistics since the intercepts (pricing errors) tend to approach zero.

3. Results

Table 1 provides the summary statistics on return distributions of the portfo-lios sorted by size, book-to-market, momentum, and liquidity. We use three different


Table 1

Summary statistics of portfolio return

The table reports the summary statistics for portfolio returns (size, book-to-market, and momentum) underanalysis. Size portfolios are constructed by sorting stocks into 50 equal-size portfolios based on theirprevious year-end market capitalization. Book-to-market portfolios are constructed by sorting stocks into50 equal-size portfolios based on their previous year-end book-to-market ratios. Momentum portfoliosare constructed by sorting stocks into 50 equal-size portfolios based on their prior compound return frommonth t−two to t−12. Liquidity portfolios are constructed by sorting stocks into 50 equal-size portfoliosbases on their Pastor and Stambaugh’s (2003) liquidity level estimates. All NYSE–Amex–Nasdaq ordinarycommon stocks from 1965 to 2005 are used in computation. Kolmogorov–Smirnov, Cramer-von Mises,Anderson–Darling statistics are used to test normality of portfolio returns.

Size Book-to-market Momentum Liquidityportfolios portfolios portfolios portfolios

Number of portfolio-period 21,600 21,600 21,600 21,600observations

Mean 0.0079 0.0083 0.0082 0.0076Variance 0.0045 0.0046 0.0047 0.0045Skewness 0.0474 0.0517 1.0992 0.4088Kurtosis 3.0200 3.1312 16.8590 4.9550Kolmogorov–Smirnov 0.0345∗∗∗ 0.0371∗∗∗ 0.0639∗∗∗ 0.0405∗∗∗Cramer-von Mises 10.9780∗∗∗ 11.1920∗∗∗ 40.6470∗∗∗ 14.5321∗∗∗Anderson–Darling 79.9370∗∗∗ 80.8740∗∗∗ 259.6540∗∗∗ 104.4443∗∗∗

∗∗∗ indicates statistical significance at 0.01 level.

statistics, Kolmogorov–Smirnov, Cramer-von Mises, and Anderson–Darling, to testthe normality of the portfolio returns. In all cases, the normality is stronglyrejected.

Table 2 reports the correlations among the factor loadings, SMB, HML, MOM,and LIQ. As can be seen from the table, SMB are generally strongly correlated toHML (0.39, 0.35, −0.05, and 0.59 for size, book-to-market, momentum, and liquidityportfolios, respectively). MOM and LIQ generally have low correlations with SMBand HML. For example, the correlation between LIQ and SMB to HML is −0.02and 0.02, respectively, for size portfolios; −0.01 and 0.07, respectively, for book-to-market portfolios; −0.28 and 0.09, respectively, for momentum portfolios; and −0.18and 0.01, respectively, for liquidity portfolios. The correlation between MOM andLIQ is also low (−0.01, 0.01, 0.17, and 0.07 for size, book-to-market, momentum,and liquidity portfolios, respectively). This indicates that MOM and LIQ are separateeffects from each other, as well as from SMB and HML, and may be priced separately.

3.1. Cross-sectional results

We report the two-step Fama and MacBeth (1973) procedure applied to thefive-factor model (the second systematic comoment, SMB, HML, MOM, and LIQ)in Table 3. For each month, factor loadings s, h, m, and l are computed by regress-ing portfolio returns on SMB, HML, MOM, and LIQ, respectively. Then, in eachmonth, portfolio returns are regressed on these loadings (also including the second


Table 2

Correlation among factor loadings SMB, HML, MOM, and LIQ

The table reports the correlation among factor loadings SMB, HML, LIQ, and MOM for portfolio sortedby size, book-to-market, momentum, and liquidity. Size portfolios are constructed by sorting stocks into 50equal-size portfolios based on their previous year-end market capitalization. Book-to-market portfolios areconstructed by sorting stocks into 50 equal-size portfolios based on their previous year-end book-to-marketratios. Momentum portfolios are constructed by sorting stocks into 50 equal-size portfolios based on theirprior compound return from month t−two to t−12. Liquidity portfolios are constructed by sorting stocksinto 50 equal-size portfolios bases on their Pastor and Stambaugh’s (2003) liquidity level estimates. AllNYSE–Amex–Nasdaq ordinary common stocks from 1965 to 2005 are used in computation.

SMB HML MOM LIQ

Panel A: Size portfolio

SMB 1.00HML 0.39 1.00MOM −0.17 0.07 1.00LIQ −0.02 0.02 −0.01 1.00

Panel B: Book-to-market portfolio

SMB 1.00HML 0.35 1.00MOM −0.20 0.06 1.00LIQ −0.01 0.07 0.01 1.00

Panel C: Momentum portfolio

SMB 1.00HML −0.05 1.00MOM −0.30 −0.09 1.00LIQ −0.28 0.09 0.17 1.00

Panel D: Liquidity portfolio

SMB 1.00HML 0.59 1.00MOM −0.20 −0.02 1.00LIQ −0.18 −0.01 0.07 1.00

systematic comoment, which is the loading on the market factor) to get the Fama andMacBeth (1973) coefficients. As can be seen from Table 3, at least two out of fourcoefficients are significant for all portfolios. The coefficients of HML and MOM, hand m, respectively, are significant in all cases, while the coefficient of market liq-uidity factor LIQ is significant in the case of book-to-market and liquidity portfolios.We compute the F-statistics to test whether the coefficients on SMB, HML, MOM,and LIQ are jointly different from zero. In all sorting criteria, the F-statistics rejectthe null hypothesis that all the coefficients are jointly equal to zero, suggesting atleast one of the factors is significant in explaining cross-sectional stock returns.

The main focus of our analysis is to examine how the significant levels of SMB,HML, MOM, and LIQ change when the systematic comoments are added to themodel. Table 4, Panel A, reports the results for size-sorted portfolios. We find that the


Tabl

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0.03

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0041

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00(8

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∗(2

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(−5.

6000

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(5.5

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∗∗∗

(1.2

900)

[13.

3100

]∗∗∗

Boo

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ket

0.03

550.

0026

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00(9

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∗(1

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3.82

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∗∗(3

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∗(3

.140

0)∗∗

∗[7

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∗M

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0.00

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50.

0101

−0.0

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0.43

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0)(−

0.28

00)

(−2.

1800

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(4.1

300)

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0.13

00)

[6.0

300]

∗∗∗

Liq

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0127

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400.

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00)

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∗∗(2

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700]

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Table 4

Fama and MacBeth (1973) regressions for systematic comoments and common factors

This table presents the Fama and MacBeth (1973) regression estimates of the common factor loadingswhen adding higher-order systematic comoments. Size portfolios are constructed by sorting stocks into50 equal-size portfolios based on their previous year-end market capitalization. Book-to-market portfoliosare constructed by sorting stocks into 50 equal-size portfolios based on their previous year-end book-to-market ratios. Momentum portfolios are constructed by sorting stocks into 50 equal-size portfolios basedon their prior compound return from month t−two to t−12. Liquidity portfolios are constructed by sortingstocks into 50 equal-size portfolios bases on their Pastor and Stambaugh’s (2003) liquidity level estimates.All NYSE–Amex–Nasdaq ordinary common stocks from 1965 to 2005 are used in computation. In eachmonth, portfolio returns are regressed on the factor loadings b, s, h, m, l, and the respective number ofsystematic comoments. These factor loadings are computed by regressing portfolio returns using the pastfive year data on the market premium, SMB, HML, MOM, and LIQ factors, respectively. The systematiccomoments are estimated using the same rolling five-year portfolio return with the market return. SMB andHML are Fama and French (1993) common factors. MOM represents the return on a portfolio of winnerstocks less the return on a portfolio of loser stocks (based on their prior compound return from montht−two to t−12). LIQ is the Pastor and Stambaugh (2003) market liquidity factor. The mean coefficientestimates across the sample period are reported with their t-statistics. The F-statistics test the hypothesisthat the s, h, m, and l estimates are jointly equal to zero.

Systematic Mean adj. R2

comoments s h m l [F-statistics]

Panel A: Size portfolios

2nd to 3rd 0.0030 −0.0123 0.0077 −0.0076 0.4600(1.5000) (−5.3600)∗∗∗ (3.0400)∗∗∗ (−1.5200) [11.0400]∗∗∗

2nd to 4th −0.0007 −0.0075 0.0059 −0.0116 0.4700(−0.3100) (−2.8800)∗∗ (2.0200)∗∗ (−2.1500)∗∗ [5.6300]∗∗∗

2nd to 5th −0.0018 −0.0089 0.0075 −0.0129 0.4800(−0.7500) (−2.9000)∗∗ (2.4200)∗∗ (−2.3300)∗∗ [6.4600]∗∗∗

2nd to 6th 0.0020 −0.0121 0.0088 −0.0071 0.4800(0.7300) (−3.3100)∗∗∗ (2.3700)∗∗ (−1.1500) [6.1100]∗∗∗

2nd to 7th 0.0010 −0.0098 0.0095 −0.0072 0.4900(0.3300) (−2.1800)∗∗ (2.3900)∗∗ (−1.0500) [4.5800]∗∗∗

2nd to 8th 0.0030 −0.0135 0.0116 −0.0025 0.4900(0.7400) (−2.6600)∗∗ (2.4600)∗∗ (−0.3100) [5.3900]∗∗∗

2nd to 9th 0.0070 −0.0156 0.0012 0.0034 0.4900(1.5100) (−2.5000)∗∗ (0.2100) (0.3900) [2.6700]∗∗

2nd to 10th 0.0080 −0.0121 −0.0078 0.0045 0.4900(1.3900) (−1.7700) (−1.1600) (0.4700) [1.3800]

2nd to 15th 0.0135 −0.0090 −0.0032 0.0106 0.4500(1.6500) (−0.9800) (−0.3600) (0.8900) [0.9500]

(continued)

significance levels of the coefficients on SMB, HML, MOM, and LIQ successivelydiminish as we add more systematic comoments. Eventually, the factors becomeinsignificant when a set of ten comoments are included.8 Trying a set of 15 comomentsdoes not change the outcome. More interestingly, the F-statistic decreases when

8 The loading on the LIQ factor remains insignificant throughout almost all cases.


Table 4 (continued)




Panel B: Book-to-market portfolios

2nd to 3rd 0.0015 −0.0077 0.0054 0.0043 0.4500(0.8600) (−3.5800)∗∗∗ (2.2200)∗∗ (0.9700) [4.0500]∗∗∗

2nd to 4th 0.0022 −0.0084 0.0068 0.0059 0.4500(0.9900) (−3.2000)∗∗∗ (2.3000)∗∗ (1.1200) [3.9900]∗∗∗

2nd to 5th 0.0006 −0.0071 0.0080 0.0031 0.4600(0.2600) (−2.2400)∗∗ (2.6100)∗∗∗ (0.5700) [3.2100]∗∗

2nd to 6th 0.0027 −0.0074 0.0084 0.0037 0.4600(0.9700) (−2.0400)∗∗ (2.3100)∗∗ (0.6300) [2.4400]∗∗

2nd to 7th 0.0043 −0.0093 0.0083 0.0095 0.4600(1.4600) (−2.0900)∗∗ (1.9800)∗∗ (1.4300) [2.0400]∗

2nd to 8th 0.0052 −0.0087 0.0074 0.0102 0.4700(1.3900) (−1.7900)∗ (1.6500) (1.2900) [1.5200]

2nd to 9th 0.0067 −0.0084 −0.0030 0.0120 0.4700(1.5500) (−1.3600) (−0.5500) (1.4100) [0.7700]

2nd to 10th 0.0025 −0.0015 −0.0079 0.0058 0.4700(0.5200) (−0.2200) (−1.2800) (0.6400) [0.5700]

2nd to 15th 0.0031 0.0013 0.0101 0.0020 0.4200(0.4300) (0.1300) (1.1400) (0.1700) [0.5200]

Panel C: Momentum portfolios

2nd to 3rd −0.0022 −0.0038 0.0099 −0.0014 0.4400(−0.9300) (−1.7900)∗ (3.8200)∗∗∗ (−0.2700) [4.9300]∗∗∗

2nd to 4th −0.0044 −0.0044 0.0061 −0.0045 0.4400(−1.6500) (−1.7500)∗ (1.8900)∗ (−0.7900) [3.2300]∗∗

2nd to 5th −0.0028 −0.0059 0.0042 −0.0013 0.4500(−0.9500) (−1.6000) (1.0900) (−0.2100) [1.6700]

2nd to 6th −0.0065 −0.0008 0.0063 −0.0055 0.4500(−1.9200) (−0.1900) (1.3000) (−0.8500) [1.9200]

2nd to 7th −0.0023 −0.0091 0.0052 0.0036 0.4500(−0.6600) (−1.5400) (0.9400) (0.4700) [1.7600]

2nd to 8th −0.0001 −0.0079 0.0054 0.0062 0.4600(−0.0100) (−1.2900) (0.9500) (0.7100) [0.9300]

2nd to 9th −0.0016 −0.0091 0.0054 0.0045 0.4600(−0.3500) (−1.1100) (0.8300) (0.4800) [0.9900]

2nd to 10th −0.0032 −0.0056 0.0085 0.0013 0.4600(−0.5100) (−0.6300) (1.0800) (0.1200) [0.8100]

2nd to 15th −0.0033 −0.0063 0.0115 0.0005 0.4200(−0.4200) (−0.6600) (1.3400) (0.0400) [0.9400]

(continued)


Table 4 (continued)




Panel D: Liquidity portfolios

2nd to 3rd 0.0010 −0.0044 0.0058 0.0036 0.3690(0.5300) (−2.2800)∗∗ (2.3400)∗∗ (0.7300) [2.3400]∗

2nd to 4th 0.0010 −0.0052 0.0045 0.0043 0.3770(0.4600) (−2.2800)∗∗ (1.6700)∗ (0.8000) [2.1000]∗

2nd to 5th −0.0003 −0.0057 0.0054 0.0018 0.3840(−0.1200) (−2.1300)∗∗ (1.7700)∗ (0.3300) [2.2700]∗

2nd to 6th 0.0006 −0.0061 0.0075 0.0035 0.3900(0.2000) (−1.7400)∗ (1.9700)∗∗ (0.5900) [1.8900]

2nd to 7th 0.0030 −0.0097 0.0080 0.0125 0.3960(0.9700) (−2.0200)∗∗ (1.8700)∗ (1.7200) [2.0100]∗

2nd to 8th −0.0012 −0.0072 0.0077 0.0040 0.4000(−0.3300) (−1.3800) (1.6700)∗ (0.5200) [1.6400]

2nd to 9th −0.0017 −0.0066 0.0089 0.0045 0.4010(−0.4100) (−1.1200) (1.6500)∗ (0.5400) [1.4100]

2nd to 10th −0.0002 −0.0122 0.0072 0.0069 0.4020(−0.0400) (−1.8800) (1.2100) (0.7700) [1.7900]

2nd to 15th −0.0045 −0.0044 0.0123 −0.0010 0.3410(−0.6700) (−0.5800) (1.5400) (−0.1000) [0.3500]

∗∗∗, ∗∗, ∗ indicate statistical significance at 0.01, 0.05 and 0.1 level, respectively.

more systematic comoments are added and becomes insignificant when a set of tenor 15 comoments is included. This implies that if a sufficient number of comomentsis considered, the factors become insignificant in explaining cross-sectional stockreturns. Table 4, Panels B, C, and D, report the results for book-to-market, momentum,and liquidity sorted portfolios, respectively. The findings are very similar to those inPanel A.

One may argue that any sufficiently large set of variables would reduce thesignificance levels of the factors. This is not the case when we add standard moments(not systematic comoments) to the model as in Equation (5). Standard moments ofvarious orders are computed as per Equation (6). The results are in Table 5. In allcases (for size, book-to-market, momentum, and liquidity portfolios), whether we adda set of ten or15 standard moments, the explanatory powers of factors SMB, HML,MOM, and LIQ remain almost unchanged compared to the original levels when nostandard moment is included. The F-statistics are significant, and about the same, inall cases. The findings imply that standard moments do not lower the significance ofcommon factors, whereas the systematic comoments do.

We also observe that some factors (e.g., LIQ in size and momentum portfolios),that are insignificant before adding higher comoments, remain insignificant even after


Table 5

Fama and MacBeth (1973) regressions for standard moments and common factors

This table presents the Fama and MacBeth (1973) regression estimates of the common factor loadingswhen adding a set of standard moments up to order ten or 15. Size portfolios are constructed by sortingstocks into 50 equal-size portfolios based on their previous year-end market capitalization. Book-to-marketportfolios are constructed by sorting stocks into 50 equal-size portfolios based on their previous year-endbook-to-market ratios. Momentum portfolios are constructed by sorting stocks into 50 equal-size portfoliosbased on their prior compound return from month t−two to t−12. Liquidity portfolios are constructed bysorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh’s (2003) liquidity levelestimates. All NYSE–Amex–Nasdaq ordinary common stocks from 1965 to 2005 are used in computation.In each month, portfolio returns are regressed on the factor loadings b, s, h, m, l, and the respective numberof standard moments. These factor loadings are computed by regressing portfolio returns using the pastfive year data on the market premium, SMB, HML, MOM, and LIQ factors, respectively. The standardmoments are estimated using the same rolling five-year portfolio return with the market return. SMB andHML are Fama and French (1993) common factors. MOM represents the return on a portfolio of winnerstocks less the return on a portfolio of loser stocks (based on their prior compound return from montht−two to t−12). LIQ is the Pastor and Stambaugh (2003) market liquidity factor. The mean coefficientestimates across the sample period are reported with their t-statistics. The F-statistics test the hypothesisthat the s, h, m, and l estimates are jointly equal to zero.

Standard Mean adj. R2

moments s h m l [F-statistics]


3rd to 10th 0.0171 −0.0170 0.0082 0.0134 0.5500(5.9500)∗∗∗ (−6.6300)∗∗∗ (2.5700)∗∗ (2.5000)∗∗ [13.7500]∗∗∗

3rd to 15th 0.0160 −0.0156 0.0068 0.0113 0.5400(5.7800)∗∗∗ (−5.8900)∗∗∗ (2.1400)∗∗ (2.1100)∗∗ [13.3900]∗∗∗


3rd to 10th 0.0122 −0.0134 0.0078 0.0113 0.5200(4.7600)∗∗∗ (−5.8200)∗∗∗ (2.7700)∗∗∗ (2.0400)∗∗ [11.6400]∗∗∗

3rd to 15th 0.0130 −0.0131 0.0062 0.0130 0.5100(4.7000)∗∗∗ (−5.6300)∗∗∗ (2.0500)∗∗∗ (2.2100)∗∗ [10.3700]∗∗∗


3rd to 10th 0.0031 −0.0054 0.0092 0.0023 0.5300(1.0300) (−2.6700)∗∗∗ (2.8900)∗∗∗ (0.4800) [3.5700]∗∗∗

3rd to 15th 0.0025 −0.0048 0.0084 −0.0003 0.5000(0.6900) (−2.1100)∗∗ (2.4500)∗∗ (−0.0700) [2.6500]∗∗


3rd to 10th 0.0092 −0.0071 0.0051 0.0130 0.4530(2.8600)∗∗ (−2.3100)∗∗∗ (1.5400) (2.2900)∗∗ [2.9600]∗∗

3rd to 15th 0.0093 −0.0071 0.0069 0.0129 0.4460(2.8800) (−2.3200)∗∗ (2.0600) (2.1400) [3.5100]∗∗∗

∗∗∗ and ∗∗ indicate statistical significance at 0.01 and 0.05 level, respectively.


Table 6

Gibbons, Ross and Shanken (1989) test

This table reports Gibbons, Ross and Shanken (1989) statistics examining whether the intercepts (pricingerrors) from time series regressions are jointly equal to zero for size, book-to-market, momentum, andliquidity portfolios when adding higher-order comoment factors. Size portfolios are constructed by sortingstocks into 50 equal-size portfolios based on their previous year-end market capitalization. Book-to-marketportfolios are constructed by sorting stocks into 50 equal-size portfolios based on their previous year-endbook-to-market ratios. Momentum portfolios are constructed by sorting stocks into 50 equal-size portfoliosbased on their prior compound return from month t−two to t−12. Liquidity portfolios are constructedby sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh’s (2003) liquiditylevel estimates. All NYSE–Amex–Nasdaq ordinary common stocks from 1965 to 2005 are used in thecomputation.

Systematic Size Book-to-market Momentum Liquiditycomoments portfolio portfolio portfolio portfolio

Original 12.37∗∗∗ 10.69∗∗∗ 2.75∗∗ 3.45∗∗∗2nd to 3rd 8.45∗∗∗ 7.10∗∗∗ 2.46∗∗ 2.53∗∗2nd to 4th 8.27∗∗∗ 6.94∗∗∗ 2.68∗∗ 2.49∗∗2nd to 5th 6.61∗∗∗ 5.49∗∗∗ 2.10∗∗ 1.98∗∗2nd to 6th 6.11∗∗∗ 5.19∗∗∗ 1.73∗ 2.01∗∗2nd to 7th 5.03∗∗∗ 4.28∗∗∗ 1.46 1.612nd to 8th 4.23∗∗∗ 3.77∗∗∗ 1.56 1.512nd to 9th 4.00∗∗∗ 2.94∗∗∗ 1.55 1.192nd to 10th 3.69∗∗∗ 2.93∗∗∗ 1.55 1.182nd to 15th 1.60 1.28 0.87 0.62


adding higher systematic comoments or higher standard moments. This means that aset of systematic comoments affects only those factors that are significant and doesnot affect the ones that are not significant before adding more variables. If addingmore variables causes imprecision in estimation, this pattern would be less likely. Thisobservation is additional evidence that our results are not driven by simply adding aset of variables.

3.2. Time series results

Our main concern with the two-step Fama and MacBeth (1973) procedure isthat it might not be robust (see Lewellen, Nagel and Shanken, 2007; Shanken andZhou, 2007; Petersen, 2009). Therefore, we conduct a time series test as a robustnesscheck. In Table 6, we report the GRS statistics used to test the hypothesis that theintercepts from time series regressions are jointly equal to zero for portfolios sortedby size, book-to-market, momentum, and liquidity. The main idea of the time seriestest is that if the intercepts (i.e., pricing errors) from the time series regressions ofportfolio returns on the five-factor model are jointly equal to zero (the GRS statisticsare not significant), the common factors are sufficient to explain stock returns. On theother hand, if the GRS statistics are significantly different from zero, then the model


cannot explain stock returns. In our analysis, we add higher-order comoment factorsto the model and compute the corresponding GRS statistics. If the statistics decreaseconsistently when more comoment factors are added and become insignificant whenenough comoment factors are included, then higher-order comoments are relevant inasset pricing. It also implies that common factors may be proxies for the higher-ordercomoments.

The results in Table 6 support our argument. When the asset pricing model in-cludes only five factors (market risk premium, SMB, HML, MOM, and LIQ), theGRS statistics are 12.37, 10.69, 2.75, and 3.45 for size, book-to-market, momentum,and liquidity portfolios, respectively, which are strongly and significantly differentfrom zero. This suggests that these factors do not sufficiently explain stock returns.As we start adding the higher-order comoment factors, the GRS statistics consistentlydecrease for all portfolios and become insignificant when a set of ten or 15 como-ment factors is added. The results are consistent with those in the cross-sectionalanalysis.

3.3. Other robustness checks

In this section, we present results of other robustness checks for the Fama andMacBeth (1973) procedure. Chung, Johnson and Schill (2006) consider differentreturn horizons: daily, weekly, monthly, quarterly, and semi-annually. However, sincethe Pastor and Stambaugh market-liquidity factor is constructed using monthly dataonly, the data on this factor for other higher frequencies is not available. Thus, tocheck robustness of the results, we divide the sample into two subperiods: (1) 1970–1987 and (2) 1988–2005. The results are in Tables 7 and 8. Since the results with aset of ten systematic comoments are very similar to those with a set of 15 systematiccomoments, we report only those with the ten comoments.

Table 7 shows that for size portfolios, when the model includes only the secondsystematic comoment (covariance), the coefficients of the factor loadings on SMB,HML, MOM, and LIQ are all significant. In particular, the t-statistics of s, h, m, and lare 1.80, −6.55, 5.21, 3.42, respectively. The F-statistic for the joint significance ofthese coefficients is 16.15, which is significant at 1% or below. However, when a set often systematic comoments is included, the t-statistics of s, h, m, and l are 0.55, −0.22,−0.53, and 1.28, respectively, which are insignificant. The F-statistic drops to only1.21, which is also insignificant. The results with book-to-market portfolios are verysimilar to those with size portfolios. For the momentum portfolios, the coefficientof the momentum factor, m, is significant at the original level (t-statistic = 3.20),but becomes insignificant with the addition of a set of comoments of the second totenth orders. The F-statistic also decreases from 3.32 (significant at the 5% level) to0.83 (insignificant). For the liquidity portfolios, the coefficients of HML and MOMfactors are significant at the 5% level, but all become insignificant when the set ofhigher-order comoments is included in the model.


Table 7

Fama and MacBeth (1973) regressions for systematic comoments, standard moments and common factorsin 1970–1987

This table presents Fama and MacBeth (1973) regression estimates of the common factor loadings in 1970–1987 when adding a set of systematic comoments or a set of standard moments of order three to ten. Sizeportfolios are constructed by sorting stocks into 50 equal-size portfolios based on their previous year-end marketcapitalization. Book-to-market portfolios are constructed by sorting stocks into 50 equal-size portfolios based ontheir previous year-end book-to-market ratios. Momentum portfolios are constructed by sorting stocks into 50equal-size portfolios based on their prior compound return from month t−two to t−12. Liquidity portfolios areconstructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh’s (2003) liquiditylevel estimates. In each month, portfolio returns are regressed on the factor loadings b, s, h, m, l, and the respectivenumber of systematic comoments or standard moments. These factor loadings are computed by regressing portfolioreturns using the past five year data on the market premium, SMB, HML, MOM, and LIQ factors, respectively.The systematic comoments and standard moments are estimated using the same rolling five-year portfolio returnwith the market return. SMB and HML are Fama and French (1993) common factors. MOM represents the returnon a portfolio of winner stocks less the return on a portfolio of loser stocks. LIQ is the Pastor and Stambaugh(2003) market liquidity factor. The mean coefficient estimates across the sample period are reported with theirt-statistics. The F-statistics test the hypothesis that the s, h, m, and l estimates are jointly equal to zero. In eachportfolio category, the original indicates the model including only the second systematic comoment (covariance),and the factor loadings on SMB, HML, MOM, and LIQ.




Original 0.0038 −0.0179 0.0158 0.0172 0.4700(1.8000)∗ (−6.5500)∗∗∗ (5.2100)∗∗∗ [3.4200]∗∗∗ [16.1500]∗∗∗

2nd to 10th 0.0057 −0.0018 −0.0055 0.0176 0.4900comoments (0.5500) (−0.2200) (−0.5300) (1.2800) [1.2100]

3rd to 10th 0.0134 −0.0157 0.0110 0.0182 0.5400standard moments (3.3800)∗∗∗ (−5.3600)∗∗∗ (2.5600)∗∗ (3.3600)∗∗∗ [9.0500]∗∗∗


Original 0.0019 −0.0114 0.0071 0.0183 0.4400(0.9000) (−4.8600)∗∗∗ (2.4100)∗∗ (4.3900)∗∗∗ [10.3000]∗∗∗

2nd to 10th 0.0004 0.0048 −0.0048 0.0107 0.4700comoments (0.0400) (0.7000) (−0.5600) (0.8500) (1.5000)

3rd to 10th 0.0131 −0.0124 0.0107 0.0183 0.5000standard moments (3.3800)∗∗∗ (−4.4100)∗∗∗ (2.5900)∗∗ (3.4100)∗∗∗ [8.3900]∗∗∗


Original −0.0006 −0.0022 0.0097 −0.0030 0.3800(−0.2400) (−0.9900) (3.2000)∗∗∗ (−0.7700) [3.3200]∗∗

2nd to 10th −0.0090 0.0062 0.0156 −0.0199 0.4100comoments (−0.8000) (0.6800) (1.3800) (−1.3000) (0.8300)

3rd to 10th 0.0038 −0.0071 0.0118 0.0010 0.4600standard moments (0.5600) (−1.9900)∗∗ (2.1100)∗∗ (0.1300) [3.3200]∗∗


Original −0.0011 −0.0068 0.0065 0.0015 0.3810(−0.5400) (−3.1700)∗∗∗ (2.1100)∗∗ (0.3900) [3.2500]∗∗

2nd to 10th −0.0070 −0.0110 0.0118 −0.0021 0.4200comoments (−0.7600) (−1.4300) (1.2900) (−0.1600) [1.7100]

3rd to 10th 0.0111 −0.0069 0.0107 0.0083 0.4610standard moments (2.3800)∗∗ (−2.3300)∗∗ (2.4100)∗∗ (1.3600) [3.4000]∗∗∗



The results with standard moments in Table 7 are also consistent with thosefor the full sample. The addition of a set of standard moments does not reduce theexplanatory power of the factors for all the size, book-to-market, momentum, andliquidity portfolios. For the momentum portfolios, the coefficients of s and l are notsignificant. This, however, is not because of adding standard moments since they areinsignificant even before we include the set of standard moments. The F-statisticsremain almost unchanged after adding the set. This implies the standard moments donot affect the explanatory power of the factors.

Table 8 presents the results for the second subperiod, 1988–2005. The findingsare generally consistent with those in the first subperiod as well as in the wholesample. Only in the case of size portfolio, adding systematic comoments does notreduce substantially the significance levels of all the factors. SMB and HML remainsignificant and the F-statistics do not decrease substantially. However, in other cases,the results are in line with previous findings.

The Fama and MacBeth (1973) method may be biased because of the errors-in-variables problem. The right-hand side variables in the second-pass cross-sectionalregression are estimates from the first-pass time series regression. We use Shanken’s(1992) adjustment to recalculate all the t-statistics in Table 3. We show results con-sistent with Chung, Johnson and Schill (2006). The adjusted t-statistics are similarwhen only the third comoment is added, but drop substantially and approach zerowhen higher-order comoments beyond the third order are included in the model.The evidence is consistent for all portfolios: size, book-to-market, momentum, andliquidity. Chung, Johnson and Schill (2006) suggest that because the higher-orderright-hand side variables are created, the Shanken (1992) adjustment biases the testresults too much against the factor loadings, causing the t-statistics to approach zero;hence, the adjustment appears to be inappropriate for our studies.

We use Newey–West’s procedure to recalculate the t-statistics adjusted for serialcorrelation in the return residuals that is not accounted in the Fama and MacBeth(1973) method. The results are similar for the Newey–West procedure and the Famaand MacBeth (1973) method. This implies that the serial correlation problem doesnot influence the results.

Our computation so far is based on centered systematic comoments. We alsoreport the results based on noncentered systematic comoments:

b(i, j, t) =

60∑τ=1

[r ( j, t − τ )][r (m, t − τ )]i−1

60∑τ=1

[r (m, t − τ )]i

, (11)

where b(i , j , t) is the ith noncentered systematic comoment of portfolio j in month t,and r (m, t) is the return of the CRSP value-weighted index.


Table 8

Fama and MacBeth (1973) regressions for systematic comoments, standard moments and common factorsin 1988–2005

This table presents Fama and MacBeth (1973) regression estimates of the common factor loadings in 1988–2005 when adding a set of systematic comoments or a set of standard moments of order three to ten. Sizeportfolios are constructed by sorting stocks into 50 equal-size portfolios based on their previous year-end marketcapitalization. Book-to-market portfolios are constructed by sorting stocks into 50 equal-size portfolios based ontheir previous year-end book-to-market ratios. Momentum portfolios are constructed by sorting stocks into 50equal-size portfolios based on their prior compound return from month t−two to t−12. Liquidity portfolios areconstructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh’s (2003) liquiditylevel estimates In each month, portfolio returns are regressed on the factor loadings b, s, h, m, l, and the respectivenumber of systematic comoments or standard moments. These factor loadings are computed by regressing portfolioreturns using the past five year data on the market premium, SMB, HML, MOM, and LIQ factors, respectively.The systematic comoments and standard moments are estimated using the same rolling five-year portfolio returnwith the market return. SMB and HML are Fama and French (1993) common factors. MOM represents the returnon a portfolio of winner stocks less the return on a portfolio of loser stocks. LIQ is the Pastor and Stambaugh(2003) market liquidity factor. The mean coefficient estimates across the sample period are reported with theirt-statistics. The F-statistics test the hypothesis that the s, h, m, and l estimates are jointly equal to zero. In eachportfolio category, the original indicates the model including only the second systematic comoment (covariance),and the factor loadings on SMB, HML, MOM, and LIQ.




Original 0.0045 −0.0076 0.0100 −0.0073 0.4500(1.7000)∗ (−2.1000)∗∗ (2.8300)∗∗∗ (−1.2700) [3.9800]∗∗∗

2nd to 10th 0.0103 −0.0224 −0.0101 −0.0085 0.4800comoments (2.1200)∗∗ (−2.0600)∗∗ (−1.1700) (−0.6400) [3.3200]∗∗∗

3rd to 10th 0.0129 −0.0151 0.0015 −0.0054 0.5500standard moments (4.1100)∗∗∗ (−3.7500)∗∗∗ (0.3700) (−0.6700) [7.8100]∗∗∗


Original 0.0032 −0.0037 0.0082 0.0037 0.4300(1.2300) (−1.1700) (2.3900)∗∗ (0.6600) [1.7800]

2nd to 10th 0.0048 −0.0078 −0.0110 0.0008 0.4700comoments (0.9500) (−0.6800) (−1.2200) (0.0600) [0.5000]

3rd to 10th 0.0113 −0.0144 0.0049 0.0044 0.5200standard moments (3.3500)∗∗∗ (−3.9500)∗∗∗ (1.2800) (0.4500) [5.7500]∗∗∗


Original −0.0004 −0.0069 0.0106 0.0018 0.4900(−0.1500) (−1.9500)∗ (2.7300)∗∗∗ (0.2300) [3.7500]∗∗∗

2nd to 10th 0.0024 −0.0175 0.0014 0.0226 0.5100comoments (0.4000) (−1.1500) (0.1200) (1.4900) [0.9800]

3rd to 10th −0.0026 −0.0044 0.0114 −0.0040 0.6000standard moments (−0.8000) (−1.4200) (2.6300)∗∗∗ (−0.5000) [2.4800]∗∗


Original −0.0009 −0.0032 0.0025 −0.0054 0.3450(−0.4200) (−1.1000) (0.7800) (−0.9800) [1.0500]

2nd to 10th 0.0066 −0.0133 0.0026 0.0160 0.3820comoments (1.3900) (−1.2800) (0.3400) (1.2700) [0.6300]

3rd to 10th 0.0020 −0.0037 0.0040 0.0094 0.4370standard moments (0.5300) (−0.7900) (0.8600) (1.1000) [0.7900]



The findings are also qualitatively similar.9 Overall, our robustness checks areconsistent with the previous findings: adding a set of systematic comoments lowerssubstantially the significance of SMB, HML, MOM, and LIQ.10

4. Conclusion

In this paper, we provide evidence to support Rubinstein’s (1973) theoreticalmodel that if returns do not follow normal distribution, measuring risk requires morethan just measuring covariance. Higher-order systematic comoments should be im-portant to risk-averse investors who are concerned about the extreme outcomes of theirinvestments. We show that not only the Fama-French factors (SMB and HML), butalso momentum and liquidity factors, can be explained by higher-order comoments.

In cross-sectional analysis and by all sorting criteria (size, book-to-market, mo-mentum, and liquidity), we find that adding a set of ten or 15 systematic comomentsreduces substantially the significance levels of the factors (SMB, HML, MOM, andLIQ) and causes them to become insignificant in most cases. One might argue that theresults are driven by imprecision in estimation due to adding more independent vari-ables; we show that it is not the case. We perform a similar analysis with a set of ten or15 standard moments and find that the explanatory powers of the factors remain thesame after including the set of standard moments. Also, in some cases, some factorsremain consistently insignificant before and after adding the set of variables. Thus, itdoes not appear that our results are being driven by simply adding more explanatoryvariables. Our cross-sectional results are also consistent in both subperiods and inseveral other robustness checks.

In the time series analysis, we find that the GRS statistics, which test whetherthe pricing errors from time series regressions are jointly equal to zero, consistentlydecrease when more comoment factors are added to the model and become insignif-icant when enough factors are included. This suggests that the pricing error tends tozero when more comoment factors are added to the model, hence providing additionalevidence that higher-order systematic comoments should be relevant in pricing assets.

We also find that the results with a set of ten systematic comoments arevery similar to those with 15 comoments or higher. This suggests that when more

9 The results for Shanken’s (1992) correction for errors-in-variables, Newey and West’s (1987) adjustmentfor serial correlation, and computation based on noncentered systematic comoments are not reported indetail; however, they are available from the authors.

10 We also use Hansen’s (1982) Generalized Method of Moments (GMM) approach to tackle the errors-in-variables problem and serial correlation in the return residuals as advocated by Cochrane (2001). Whenthe number of parameters to estimate is small, we obtain the risk premiums similar to those obtainedby the Fama and MacBeth (1973) procedure. However, when we add comoment factors, the numbers ofparameters to estimate becomes very large, in most cases, the numerical solutions do not converge, hence,we are unable to estimate the risk premiums of each factor loading in the presence of a set of comomentfactors. For this reason, we do not pursue the GMM approach in our study.


comoments are included, the return distributions would be characterized more pre-cisely; however, empirically, a set of ten comoments is sufficient to capture theextreme outcomes of the investment.

Our findings provide another explanation for the nonmarket factors SMB, HML,and MOM within the traditional covariance risk-based theory without having to resortto behavioral assumptions since these factors may be proxies for higher-order como-ment factors which are state variables. We also find that the Pastor and Stambaugh(2003) liquidity risk might be captured by these comoment risks. The practical impli-cation is that in a well-diversified portfolio, the idiosyncratic moments (e.g., standarddeviation, skewness, kurtosis, etc.) are eliminated, investors only earn compensa-tion for exposure to systematic comoments (e.g., covariance, coskewness, cokurtosis,etc.), and these systematic comoments should be priced when investors consider theirinvestments.

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Higher-Order Systematic Comoments and Asset Pricing: New Evidence

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