HIGHER ORDER SYSTEMATIC CO-MOMENTS AND ASSET-PRICING: NEW
EVIDENCE
Duong Nguyen*
Tribhuvan N. Puri*
Address for correspondence:
Tribhuvan N. Puri, Professor of Finance
Chair, Department of Accounting and Finance
Charlton College of Business
University of Massachusetts Dartmouth
285 Old Westport Road
North Dartmouth, MA 02747
Voice: 508-999-8759
tpuri@umassd.edu
*University of Massachusetts, Dartmouth
HIGHER ORDER SYSTEMATIC CO-MOMENTS AND ASSET-PRICING: NEW
EVIDENCE
Abstract:
In this paper, we provide evidence supporting Rubinstein (1973)’s theoretical model that
if returns do not follow normal distribution, measuring risk requires more than just measuring
covariance, higher order systematic co-moments should be important to risk averse investors
who are concerned about the extreme outcomes of their investments. Our paper provides a
contribution to the existing literature that not only Fama-French factors (SMB, HML) but also
the momentum factor can be explained by higher order systematic co-moments, hence lends a
support to the traditional covariance risk-based framework without having to resort to behavior
assumptions. We also find that higher order co-moments might subsume the effect of Pastor and
Stambaugh liquidity risk. Our results are consistent in both cross sectional and time series
framework as well as in several robustness checks.
1
I.
Introduction
In utility maximizing general equilibrium asset pricing models, returns on risky assets are
determined by their covariance with the state variables that represent the fundamental sources of
risks in the economy. When changes in state variables adversely affect security returns and
investors’ wealth, marginal utility of wealth increases for risk-averse investors. Hence in
equilibrium investors must be compensated by additional reward commensurate with the risk that
they undertake. In the standard capital asset pricing model (CAPM), which is a single state
variable asset pricing model, the market summarizes presumably all the sources of risk. Crosssectional returns within CAPM depend upon their sensitivity to a well diversified market
portfolio.
Recent researches show that aggregate market liquidity is yet another fundamental source
of risk driving financial markets. Studies by Chordia, Roll, and Subrahmanyam (2000),
Hasbrouck and Seppi (2001), and Huberman and Halka (2001) provide evidence of the existence
of commonality across stocks in liquidity fluctuations. Their findings have initiated a new
research hypothesis that if liquidity shocks are non-diversifiable and have a varying impact
across individual securities, the more sensitive a stock’s return to such shocks, the greater its
expected return should be. This hypothesis has been supported by Pastor and Stambaugh (2003).
They develop a measure of aggregate liquidity, based on daily price reversals, and show that
stock whose returns are more sensitive to market liquidity factor command higher required rate
of return than stocks whose returns are less sensitive to market liquidity factor. Acharya and
Pedersen (2005), Sadka (2006) also provide evidence of premium of systematic liquidity risk
(measured as return covariation with particular measures of aggregate liquidity shocks). Thus it
is generally accepted that market liquidity is a priced state variable, however, to what extent the
2
liquidity factor has important bearing on asset pricing is still in debate. One of our contributions
in this paper is to try to resolve this debate. We argue that liquidity risk might be captured by
certain market risk not captured by the CAPM.
Empirical finance research has also documented that some non-market factors such as
size, book-to-market ratios, and momentum can explain cross-sectional variations in returns. For
example, Fama and French (1993, 1995) show that the size factor, SMB (return on small stocks
less the return on big stocks), and the book-to-market factor, HML (return on high book-tomarket stocks less the return on low book-to-market stocks), are significantly important in
explaining cross sectional of stock returns. Jagadeesh and Titman (1993, 2001) document price
momentum of individual stocks. They show that the stocks that do well relative to the market
over the last three to twelve months tend to do well in the next few months and stocks that do
poorly continue to do poorly. Gundy and Martin (2001), Fama and French (1996) assert that the
momentum effect is one of the most serious problems to asset pricing. This effect is distinct from
the value effect captured by stock characteristics and is explained neither by the Fama-French
three-factor model nor by the CAPM. Carhart (1997) adds a momentum factor MOM (the
difference in returns of diversified portfolios of short-term winner and losers) to the FamaFrench three-factor model and finds that the momentum factor is significantly important in
explaining stock returns.
While there is a consensus that these non-market factors, i.e., SMB, HML, and MOM are
significant in explaining stock returns, there is an ongoing debate about what economic
mechanisms drive these factors. The debate focuses on two competing categories: risk-based
explanations and non-risk based explanations. The proponents of the risk-based explanations
suggest that although SMB, HML, and MOM factors are themselves not state variables in the
3
conventional senses, they reflect unidentified state variables that produce non-diversifiable risks
(covariances) not captured by market returns and are priced separately from market betas. For
example, Fama and French (1993, 1996) view SMB, HML proxy for certain distress factors that
are not captured in the CAPM. Lettau and Ludvigson (2001) show that SMB and HML capture
common variation in returns because they seem to be related to variation in a consumption-based
risk premium that changes over time. Chung, Johnson and Schill (2006) argue that the FamaFrench factors simply proxy for the pricing of higher order co-moments. Conrad and Kaul (1998)
argue that momentum profit is a result of cross sectional variability in expected stock returns.
Chordia and Shivakumar (2002) suggest that profits to momentum strategies can be explained by
a set of lagged maroeconomic variables that are related to business cycle. According to this riskbased explanation, small cap stocks, value stocks, and winner stocks have high average returns
because they are risky – they have high sensitivity to the fundamental risk factors that are being
measured by SMB, HML, and MOM.
Proponents of the non-risk based explanations, on the other hand, suggest that the human
behaviors that deviate from rational expectation theory affect stock prices when market frictions
limit arbitrage drives these factors. For example, Lakonishok, Shleifer and Vishny (1994)
suggest that the book-to-market factor proxies for investor bias in earnings-growth extrapolation.
Daniel and Titman (1997) contend that it is "characteristics, not covariances," that produce return
dispersion. They argue that high book-to-market stocks have high returns due to some other
reason (possibly overreaction), so that the high returns have nothing to do with systematic risk.
In their opinion, it is the characteristics (size and book-to-market) rather than the covariances
(sensitivities to SMB and HML) that are associated with high returns. Jegadeesh and Titman
(1993) suggested that individual stock momentum might be driven by investor underreaction to
4
firm-specific information. Daniel, Hirshleifer, and Subrahmanyam (1998), Hong and Stein
(1999) attribute the momentum anomaly to investor cognitive biases.
In this paper, we attempt to provide another explanation for the non-market factors, SMB,
HML, MOM within the traditional co-variance risk-based framework, without having to resort to
behavior assumptions. We argue that when the returns are not normal, not only SMB and HML
but also MOM proxy for certain market risk factors, not captured by CAPM, measured by a set
of higher order co-moments. Our motivation comes from the theoretical model of Rubinstein
(1973) and empirical evidence in Chung et al (2006). Rubinstein (1973) demonstrates that a risk
measure when distributions are non-normal, requires not only measuring covariance with the
market but higher order co-moments (co-skewness, co-kurtosis, and so on). His model suggests
that only market risk factors (co-moment factors) should matter to investor. Chung et al. (2006)
tests Rubinstein’s assertion and find that, while each co-moment individually is unable to explain
returns, a set of co-moments taken together can do so. They find that a set of systematic comoments of the order 3rd through 10th substantially reduces the level of significance of FamaFrench factors (SMB and HML). One might argue that any set of variables would be able to
reduce the significance levels of Fama-French factors if enough of them are included. To rule out
such a possibility, Chung et al employ a set of standard moments and find that the significance
levels of SMB and HML remain the same in almost all the cases. Therefore, they conclude that
the SMB and HML factors are simply proxies for higher order systematic co-moments.
Chung et al. (2006) provide a useful starting point. We know that price momentum is one
of the most serious challenges to asset pricing as most co-variance risk-based models fail to
account for it (Fama and French 1996 and Jagadeesh and Titman 2001). Much research,
accordingly, has attributed the momentum strategy profit to behavior biases. We argue that most
5
co-variance risk-based models fail to account for momentum since they do not consider the fact
that return is not normal, therefore, higher-order co-moments which measure extreme outcomes
of an investment should be considered.
If the return is normal, the first two moments (i.e., mean and variance) alone are
sufficient to explain the distribution. As a result, investors should not care about higher order
moments. However, there is ample evidence that suggest otherwise (see for example, Fama
1965, Arditti 1971, Singleton and Wingender 1986, and more recently, Chung, Johnson, and
Schill 2006). This implies investors may care a great deal about the extreme outcomes of their
investment.
There is also a large literature examining at the role of higher order co-moments such as
co-skewness and co-kurtosis in explaining stock returns. The basic idea is that if the returns are
not normal (skewed or leptokurtic), investors are also concerned about portfolio skewness and
kurtosis. If investors’ preferences contain skewness and kurtosis, each stock’s contribution to
systematic skewness (coskewness) and kurtosis (cokurtosis) may determine stock’s
attractiveness and hence require risk premiums. Starting from Kraus and Litzenberger (1976),
many studies provide evidence to support the importance of coskewness and cokurtosis (see, for
example, Friend and Westerfield 1980, Sears and Wei 1985, Harvey and Siddique 2000, Chen,
Hong and Stein 2001, Hung, Shackleton, and Xu 2004). More interestingly, Harvey and Siddique
(2000) find that co-skewness accounts for part of the explanatory power of Fama-French size and
book-to-market factors and that co-skewness can explain part of the return to momentum
strategies.
One might argue that most return distributions can be reasonably characterized by the
first four moments, hence resorting to further higher moments does not seem to be appealing.
6
However, as with Rubinstein (1973) and Chung et al (2006), we argue that there is no reason to
stop with the third or fourth moment. Risk-averse investors have a great concern about the
extreme outcomes of their investment or the tail of the return distribution. Variance, skewness,
and kurtosis might give some information about the tail, but a set of higher order co-moments is
capable of characterizing the likelihood of extreme outcomes.
Lotteries and out-of-money options are examples of why investors care about higher
moments. For example, the price of a lottery in the US is $1, the expected value is $0.45, hence
the expected return is -55 percent. However, lottery is still popular since the downside risk is
small and investors have chance to win lottery, which is an example of extreme outcomes.
Similarly, the value of out-of-money options should be zero, but the options still have value
since options limit the downside risk and give the chance to earn upside return. These examples
imply that investors care about the right tail of return distribution, not only skewness and
kurtosis, but also higher order moments.
To justify further the importance of higher moments, consider the following lottery
example: an investor attempts to optimize a portfolio made of two independent assets: “Buy” and
“Sell”. The two assets have the following payoffs:
“Buy” payoff:
•
A $1 loss (999 times in 1000)
•
A $999 gain (1 time in 1000)
“Sell” Payoff:
•
A $1 gain (999 times in 1000)
•
A $999 loss (1 time in 1000)
From the above distribution, any rational investor would choose “Buy” because of small
7
downside risk and large upside gain. Therefore, if asset returns are independent, the optimal
portfolio should consist of 100 percent “Buy”. However, it might not be the case when we
consider this lottery in the mean-variance-skewness-kurtosis framework:
We compute the “Buy” and “Sell” central moments. The results are as follows: 1
•
“Buy” and “Sell” have the same mean (zero) and variance (999),
•
Skewness is 31.57 for “Buy” and -31.57 for “Sell”,
•
Kurtosis is 998 for both “Buy” and “Sell”.
Based on the mean-variance framework, the optimal portfolio should be 50 percent “Buy” and
50 percent “Sell” since this is the minimum variance portfolio (both assets have the same mean
and variance). Based on the four moment framework, the optimal portfolio is still 50 percent
“Buy” and 50 percent “Sell” since it produces minimum variance, kurtosis. The conclusion based
on the mean-variance-skewness-kurtosis only might not reflect the small downside risk and the
probability to earn large upside return (extreme event). As a result, we should consider the total
return distribution to solve the problem, hence the higher order moments should be considered.
Therefore, we argue that it is worthwhile to examine whether the effect of momentum
can be accounted for when higher order co-moments are considered. We also examine whether
the liquidity risk can be related to higher order systematic co-moments. The liquidity factor
proposed by Pastor and Stambaugh still awaits for attestation. So far few studies incorporate a
liquidity risk factor into an asset pricing model, and those that do observe limited success in
explaining cross-sectional variation in asset returns. Even less is known about whether liquidity
risk can be captured by some forms of market risk not explained by the CAPM.
1
The proof is available upon request.
8
Our study is also motivated by a preliminary examination. We regress the factor loadings
of size, book-to-market, momentum, and liquidity factors on a set of ten higher order systematic
co-moments and find that the systematic comoment can explain these loadings with high Rsquares (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). 2 This evidence suggests that there might be a
relationship between the common factors and systematic co-moments, hence may provide an
explanation for economic mechanism behind these factors within a traditional risk-based
covariance framework.
For the cross sectional analyses, using Fama-Macbeth (1973) procedure for portfolios
sorted by size, book-to-market, momentum, and liquidity for the period 1970-2005, we find that
adding a set of systematic co-moments of order 3 through 10 or higher reduces the explanatory
power of SMB, HML, momentum, and Pastor-Stambaugh market liquidity factors to
insignificance in almost all cases. Also, consistent with Chung et al (2006), we do not find the
similar results when adding a set of standard moments of order 3 through 10 or higher. To check
the stability of the results, we divide the sample into two sub-samples: 1970-1987 and 19882005, and perform similar analyses. The results still hold in both sub-periods. We also find that
the results with a set of 10 systematic co-moments are very similar to those with 15 co-moments
or higher. This was done with an objective to verify whether a set of co-moments higher than 10
would more precisely characterize the return distributions. However, the empirical evidence
suggests that a set of comments of order 3 through 10 is sufficient to capture the extreme
outcomes of the investment.
For the time series analysis, we use multivariate test of Gibbon, Ross, and Shanken
(GRS) (1989) and find that the GRS statistics, which test whether the pricing errors from time
2
The detail results are reported in Appendix Table A.1.
9
series regression are jointly equal to zero, consistently decrease when more systematic comoments are added to the model, and become insignificant when enough co-moments are
included. This suggests that the pricing error tends to zero when more co-moments are added to
the model.
Our findings lend support to Rubinstein (1973) in that higher order systematic comoments should be considered in asset pricing. We also provide another risk-based explanation
for SMB, HML, and MOM that these common factors might proxy for higher order co-moment,
which is some certain market risk capturing extreme outcomes of the return distribution and this
risk is not captured in the CAPM. Furthermore, we find that Pastor and Stambaugh liquidity risk
might be subsumed by the effect of these higher order systematic co-moments. This evidence
lends support to the notion that the empirical knowledge of non-market factors in the context of
asset pricing models may be of less value because all such factors might be proxied by a set of
higher order co-moments with a well diversified market portfolio.
This paper is organized as follows. The next section discusses the data and methodology
employed in the paper. Section III presents the empirical findings and section IV concludes the
paper
II.
Data and Methodology
We examine the issue of high order co-moments and common factors in asset pricing
under both cross-sectional and time-series framework. The Fama-MacBeth (1973)’s procedure is
employed for cross sectional analysis while the multivariate test of Gibbon-Ross-Shanken (1989)
is applied for time series analysis. We form stocks into portfolios based on size, book-to-market,
momentum, and liquidity. The detailed procedure is as follows.
10
1. Portfolio formation
We sort all ordinary common stocks (with the CRSP share code = 10 and 11) traded on
three stock exchanges, NYSE, AMEX, and NASDAQ, into 50 portfolios based on size, book-tomarket, and momentum and liquidity. In particular, for size portfolios, at the end of each
calendar year in the period 1965-2005, all stocks are ranked based on their market capitalization
and sorted into 50 portfolios of equal number of 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 along the line with the procedure in Ken
French’s website. 3 In particular, at the end of each month t, all stocks are sorted into 50
portfolios of equal size based on their prior compound return from month t-2 to t-12. For the
liquidity portfolio, we use the liquidity measure of Pastor and Stambaugh (2003). More
specifically, each year we perform the following linear regression to obtain the annual liquidity
measure for each stock
rie,d +1,t = θ i ,t + φi ,t ri ,d ,t + γ i ,t sign(rie,d ,t ) × vi ,d ,t + ε i ,d +1,t
d = 1, ..., D
(1)
Where ri ,d ,t is the return of stock i on day d in year t; rie,d +1,t = ri ,d ,t − ry ,d ,t where ry , d ,t is the return
on the CRSP value-weighted market return on day d in year t; and v i , d ,t is the dollar volume for
stock i on day d in year t. The liquidity measure for stock i in year t is the estimate of γ i,t in the
above regression (1). We compute the liquidity measure for a stock in a given year only if there
are more than 30 observations to estimate (1) (D >30). Then in each year, we form all stocks into
50 liquidity portfolios based on their annual liquidity estimates γ i,t .
Using portfolios constructed above, we compute equally weighted monthly returns for
each of the 50 portfolios. We subtract the 30-day Treasury bill yield to obtain the excess
3
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french
11
portfolio return. Once we have constructed the portfolios, we employ both time series and crosssectional tests to examine the relationship between the common factors and higher order
systematic co-moments.
2. Cross-sectional test
We apply the two-step Fama-Macbeth (1973) procedure to our empirical asset pricing
tests. In particular, first we test the five-factor model that includes Fama-French, momentum, and
Pastor-Stambaugh factors as follows:
r ( j , t ) = a 0 + a rmrf brm ( j , t ) + a smb bs ( j , t ) + a hml bh ( j , t ) + a mom bm ( j , t ) + a liq bl ( j , t ) + e( j , t )
(2)
Where, r ( j , t ) is the excess return of portfolio j in month t and brm ( j , t ) , bs ( j , t ) , bh ( j , t ) ,
bm ( j , t ) , bl ( j , t ) are factor loadings for excess return of portfolio j on factors (Rm-Rf), SMB and
HML, momentum (MOM), and Pastor-Stambaugh market liquidity (LIQ) respectively, in month
t. 4
For each month t, the factor loadings are computed by regressing portfolio returns over
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 first five-year data
in the original sample in order to estimate the factor loadings). Once the factor loadings are
computed, for each month t, we perform cross-sectional regressions of the period portfolio
returns on the factor loadings as in equation (2). Repeating this process for all months in the
period 1970-2005, we have 432 sets of coefficient estimates. Following Fama-MacBeth, we
average these estimates to get the average coefficients.
4
We thank the Wharton Research Database Service (WRDS) for proving us the data on these factors.
12
Next, we examine whether the above factor loadings are still significant when a set of
systematic co-moments is added to the model. In particular, for each month t, we perform crosssectional regressions of excess portfolio returns on the loadings of SMB, HML, MOM, and LIQ,
and on the systematic co-moments as follows:
n
r ( j , t ) = a 0 + a smb bs ( j , t ) + a hml bh ( j , t ) + a mom bm ( j , t ) + aliq bl ( j , t ) + ∑ a i b(i, j , t ) + e( j , t ) ,
(3)
i =2
where b(i, j , t ) is the ith systematic co-moment of portfolio j in month t. 5 We compute the
systematic co-moments in month t using the past 60 months of portfolio returns as follows:
1 60
1 60
⎡
⎤⎡
⎤
(
)
(
)
(
)
τ
τ
r
j
t
r
j
t
k
r
m
t
r (m, t − k )⎥
,
,
,
−
−
−
−
−
∑
∑
∑
⎢
⎥
⎢
60 k =1
60 k =1
⎦⎣
⎦
b(i, j , t ) = τ =1 ⎣
i
60
1 60
⎤
⎡
(
)
−
−
τ
r
m
t
r (m, t − k )⎥
,
∑
∑
⎢
60 k =1
τ =1 ⎣
⎦
60
i −1
,
(4)
where, r (m, t ) is the return of the CRSP value weighted index. We compute the systematic comoments up to the 10th order as in Chung et al (2006). We also experiment a set of systematic
co-moments up to the 15th order to see whether the findings in Chung et al are robust to higher
order of co-moments or the findings are only chance results. Our findings which are described in
the next section show that the two set of systematic co-moments give similar results.
Since one might argue that including any set of variables would be able to reduce the
significant levels of the factors, to address this issue, we also include a set of standard moments
(not systematic co-moments) as follows:
n
r ( j , t ) = a 0 + a rmrf brm ( j , t ) + a smb bs ( j , t ) + a hml bh ( j , t ) + a mom bm ( j , t ) + aliq bl ( j , t ) + ∑ a i m(i, j , t )
i =3
+ e( j , t ) ,
(5)
5
We include the loading on the market risk premium factor in the set of systematic co-moment since the loading is
the 2nd systematic co-moment.
13
where, m(i, j , t ) is the standard moment order i of portfolio j in month t. We also use the past 60
months of portfolio returns to compute m(i, j , t ) as follows
m (i , j , t ) =
1 60
[r ( j, t − τ )]i
∑
60 τ =1
(6)
The two-step Fama-MacBeth procedure has become a standard for estimation and testing
of different asset pricing models. However, this approach has two major drawbacks: (1) error in
variable, because betas are estimated in the first-stage regression and then subsequently used as
independent variables in the second-stage regression, and (2) serial correlation in return residual.
As a check for robustness, we use Shanken (1992)’s correction to adjust for estimation errors in
betas. The adjusted covariance matrix is calculated as follows
[
]
Adj. Var(a) = V 1 + a '( Z ) −1 a + Z
(7)
Where V is the k factor × k factor covariance matrix of mean coefficient estimates and Z is the k
× k covariance matrix of monthly risk factors. The respective risk factors are the SMB, HML,
MOM, LIQ, and market risk premium (RMRF) for the covariance factor, and RMRF raised to
the (i-1) power for the higher-order co-moment factors.
The Fama-MacBeth method is designed in such a way that it accounts for cross
correlation between return residual. However, it assumes the residuals are not correlated over
time, which might not be correct. Therefore, the procedure is not robust to serial correlation (see
Cochrane 2001 for more detail). We use Newey-West (1987)’s method to account for serial
correlation in the error term. 6
6
See Newey-West (1987) for more detail
14
3. Time Series Test
It is well-known that asset pricing models can also be evaluated by examining the
intercepts from the time series regressions of portfolio excess returns on the factors. If the
regression intercepts, which are the pricing errors, are jointly equal to zero, the model is valid.
Another advantage of time series approach is to avoid the error in variable problem in the FamaMacBeth approach since the estimation of risk premium is no longer necessary and the
implication of the asset pricing theory can be tested by the hypothesis that all the intercepts are
jointly equal to zero. To test this hypothesis, we use the test developed by Gibbon, Ross, and
Shanken (1989) (GRS). The GRS statistics can be used to examine the relation between common
factors and higher order systematic co-moments. We argue that if the common factors proxy for
higher order co-moments, the magnitude of GRS statistics should decrease when co-moment
factors are added to the model. The procedure is follows
First, we estimate the time-series regression of the excess returns 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 ) = α i + β i ( Rmt − R ft ) + δ i SMBt + γ i HMLt + η i MOM t + ψ i LIQt + eit
(8)
where r (i, t ) is the excess return on portfolio i in month t, ( Rmt − R ft ), SMBt , HMLt are the
Fama and French (1993) three factors related to market premium, firm size, and the book-tomarket ratio, MOM t is the momentum factor, and LIQt is the Pastor and Stambaugh (2003)
liquidity factor in month t. We then compute the GRS statistics to test whether the 50 intercepts
from these time series regressions are jointly equal to zero as follows:
15
Let N be the number of time series observations, L be number of portfolios, K be the
number of regression parameters including the constant term, and X be the observation matrix.
Then, the GRS test statistic is given by
(A' ∑ A) L *N(−N K− −K L) *+ω1
−1
1,1
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 element of ( X ' X ) . Under the null
−1
hypothesis that the regression constants are zero, this statistic has an F-distribution with L and
(N – K – L + 1) degrees of freedom.
The findings are presented in the next section. Briefly, we find that the GRS statistics
strongly reject the null hypothesis that the time series intercepts are jointly equal to zero.
The next step is we add higher order co-moment factors to (8) as follows
r (i, t ) = α i + β i ( Rmt − R ft ) + δ i SMBt + γ i HMLt + η i MOM t + ψ i LIQt + ∑η i , j (Rmt − R ft )
K
j −1
j =3
+ eit
Where
(R
(9)
mt
− R ft )
j −1
is the higher order j co-moment factor. For example, Rmt − R ft is the
covariance factor, (Rmt − R ft ) is the coskewness factor, (Rmt − R ft ) is the co-kurtosis factor, and
2
3
so on.
When adding each co-moment factor, we compute the corresponding GRS statistics. Our
argument is that if higher order co-moments are relevant in explaining stock return and if higher
order co-moments can subsume the effects of factors SMB, HML, MOM, LIQ, then adding these
co-moments reduces the significance level of the GRS statistics since the intercepts (pricing
errors) tend to approach zero.
16
III.
Empirical findings
Table 1 provides a summary statistics for the distributions of portfolios returns for size-
sorted, book-to-market sorted, and momentum sorted portfolios. We use three statistics, namely
Kolmogorov-Smirnov, Cramer-von Mises, Anderson-Darling, to test the normality of the
portfolio returns. In all cases, the normality is strongly rejected. Since the returns do not follow
normal distribution, the first two moments alone should not be sufficient to characterize the
return distribution hence higher order moments should be considered.
Table 2 reports the correlation among factor loadings, SMB, HML, MOM, and LIQ. As
can be seen from the table, SMB are generally strongly correlated to HML (0.39, 0.35, -0.05, and
0.59 for size, book-to-market, momentum, and liquidity portfolios, respectively). MOM and LIQ
generally have low correlation with SMB and HML. For example, the correlation between LIQ
and SMB, HML is -0.02 and 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 -0.18
and 0.01, respectively for liquidity portfolios. The correlation between MOM and LIQ is also
low (-0.01, 0.01, 0.17, 0.07 for size, book-to-market, momentum, and liquidity portfolios,
respectively). This indicates that MOM and LIQ are separate effect from each other as well as
from SMB and HML and might be priced separately.
A. Cross Sectional Results
We report the two-step Fama-MacBeth (1973)’s procedure applied to the five factor
model (2nd systematic co-moment, SMB, HML, MOM, and LIQ) in table 3. For each month,
factors loadings: s, h, m, l, are computed by regressing portfolio returns on the SMB, HML,
MOM, LIQ, respectively. Then, in each month, portfolio returns are regressed on these factor
17
loadings (also including the 2nd systematic co-moment, which is the loading on the market
factor) to get the Fama-MacBeth coefficients. As can be seen from the table, at least two out of
four factor loadings are significant for all portfolios. The factor loadings on HML and MOM: h
and m, respectively, are significant in all cases, while the factor loading on market liquidity
factor LIQ is significant in the case of book-to-market and liquidity portfolios. We compute the
F-statistics to examine whether the coefficients on SMB, HML, MOM, and LIQ are jointly
different from zero. In all sorting criteria, the F-statistics reject the null hypothesis that all factor
loadings are jointly equal to zero, suggesting at least 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 co-moments are added to the model. Table 4, Panel
A reports the results for size sorted portfolios. We find that the significance levels of factor
loadings on SMB, HML, MOM, and LIQ successively diminish as we add more systematic comoments. Eventually, the factor loadings become insignificant when a set of 10 co-moments is
included. 7 We also experiment with a set of 15 co-moments instead of a set of 10 co-moments.
We find that the results are not different. More interestingly, the magnitude of the F-statistics
that tests the joint significance of coefficients of the factors decreases when more systematic comoments are added and become insignificant when a set of 10 or 15 co-moments is included.
This implies that if a sufficient number of co-moments are considered, the factor loadings
become insignificant in explaining cross sectional stock returns. Panel B, C and D report the
results for book-to-market, momentum, and liquidity sorted portfolios, respectively. The findings
are very similar to those in Panel A.
7
The significant level of loading on LIQ factor basically remains insignificant throughout all cases.
18
Since one may argue that any set of variable would be able to reduce the significance
levels of the factor loadings if a sufficient number of them are included. Such is not the case
when we add standard moments (not systematic co-moments) to the model. Standard moments
are computed as in equation (5). The results are reported in Table 5. In all cases (for size, bookto-market, momentum, and liquidity portfolios), whether we add a set of 10 or15 standard
moments, the explanatory powers of factor loadings on SMB, HML, MOM, and LIQ remain
almost unchanged compared with the original levels when no standard moment is included. The
F-statistics are significant and their magnitudes are similar in all cases. The findings imply that
standard moments do not reduce the significance of common factors, but the systematic comoments do. This is consistent with Rubinstein (1973)’s model in that if return is not normal,
risk averse investors should be concerned about higher order co-moments.
Another noteworthy observation is that some factor loadings (e.g., LIQ), that are
insignificant before adding higher co-moments, remain insignificant even after adding higher
systematic co-moments or higher standard moments. This means that a set of systematic comoments affects only those factors that are significant, and does not affect the ones that are not
significant before adding more variables. A set of standard moments does not reduce the
significance levels of the factors regardless they are initially significant or insignificant. If adding
more variables causes imprecision in estimation, this should not be the case. In some other
context, this observation is additional evidence that our results are not driven by simply adding a
set of variables.
19
B. Time Series Results
Our main concern with the two-step Fama-MacBeth procedure is that it might not be
robust (see Shanken and Zhou 2007, Lewellen, Nagel, and Shanken 2007, Petersen 2007).
Therefore, we conduct time series test as an alternative asset pricing test for robustness check.
We report the GRS statistics testing the hypothesis whether the intercepts from time series
regressions are jointly equal to zero for portfolios sorted by size, book-to-market, momentum,
and liquidity in Table 6. The main idea in the time series test is that if the intercepts (i.e., pricing
errors) from time series regressions of portfolio returns on the five-factor model are jointly equal
to zero, hence the GRS statistics are not significant, the common factors are sufficient to explain
stock returns. On the other hand, if the GRS statistics are significant different from zero, then the
model is not able to explain stock returns. In our analysis, we add higher order co-moment
factors to the model and compute the corresponding GRS statistics. If the statistics decrease
consistently when more co-moment factors are added and become insignificant when enough comoment factors are included, then higher order co-moments should be considered in asset
pricing, common factors may be proxies for these co-moments.
The results in Table 6 support our argument. When the asset pricing model includes only
five factors: market risk premium, SMB, HML, MOM, LIQ, the GRS statistics are 12.37, 10.69,
2.75, and 3.45 for size, book-to-market, momentum, and liquidity portfolios, respectively, which
are strongly significant different from zero. This suggests these factors are not sufficient to
explain stock return. Next, we start adding the higher order co-moment factors, the GRS
statistics consistently decrease for all portfolios and become insignificant when a set of 10 or 15
co-moment factors are added. The results here are consistent with those in the cross-sectional
20
analyses in that the factors SMB, HML, MOM, LIQ might proxy for higher order co-moments
when explaining stock returns.
C. Other robustness checks
In this section, we present results of other robustness checks for the Fama-MacBeth
procedure, namely, (1) test the stability of the results by dividing the sample into two subperiods, (2) correct for error-in-variable problem using Shanken (1992)’s adjustment, and (3)
adjust for serial correlation in return residuals using Newey-West (1987)’s method.
Chung et al (2006) consider different return horizons: daily, weekly, monthly, quarterly,
and semi-annually. However, since the Pastor and Stambaugh market liquidity factor is
constructed using monthly data only, the data on this factor for other higher frequencies is not
available. Thus, in order to check robustness of the results, we divide the sample into two subperiods: (1) 1970-1987 and (2) 1988-2005. The results are reported in table 7 and 8. Since the
results with a set of 10 systematic co-moments are very similar to those with a set of 15
systematic co-moments, we report only those with the 10 co-moments.
Table 7 shows that for size portfolios, when the model includes only the 2nd systematic
co-moment (covariance), the coefficients of the factor loadings on SMB, HML, MOM, LIQ are
all significant. In particular, the t-statistics of s, h, m, l are 1.80, -6.55, 5.21, 3.42, respectively.
The F-statistics for the joint significance of these loadings is 16.15, which is significant at 1
percent or below. However, when a set of 10 systematic co-moments is included, the t-statistics
of s, h, m, l, are 0.55, -0.22, -0.53, 1.28, respectively, which are insignificant. The F-statistics
reduces to only 1.21, which is also insignificant. The results with book-to-market portfolios are
very similar to those with size portfolios. For the momentum portfolios, the loading on
momentum factor, m, is significant at the original level (t-stat = 3.20), but becomes insignificant
21
with the addition of a set of co-moments of the order 2nd to 10th. The F-statistics also reduces
from 3.32 (significant at 5 percent level) to 0.83 (insignificant). For the liquidity portfolios, the
loadings on HML and MOM factors are significant at 5 percent level but all become insignificant
when the set of higher order co-moments is included in the model.
The results with standard moments in panel B, Table 7 are also consistent with those
obtained for the full sample. The addition of a set of standard moment does not reduce the
explanatory power of factor loadings for the size, book-to-market, momentum, and liquidity
portfolios. For the momentum portfolios, the coefficients of s and l are not significant. This,
however, is not because of adding standard moments since they are insignificant even before we
include the set of standard moments. The magnitudes of F-statistics remain almost unchanged
before and after adding the set. This implies the standard moments do not affect the explanatory
power of the factors on stock return.
Table 8 presents the results for the second sub-period: 1988-2005. The findings are
generally consistent with those in the first sub-period as well as in the whole sample. Only in the
case of size portfolio, adding systematic co-moments does not reduce substantially the
significance levels of all factor loadings. SMB and HML remain significant and the magnitude of
F-stat does not reduce substantially. However, in other cases, the results are in line with previous
findings.
The Fama-MacBeth may be biased because of the error-in-variable problem. The righthand-side variables in the second-pass cross sectional regression are estimates from the first-pass
time series regression. We use Shanken (1992)’s adjustment to recalculate all the t-statistics in
Table 4. The results are reported in Appendix table A.2. We document consistent results with
Chung et al (2006). The adjusted t-statistics are similar when only the 3rd comoment is added, but
22
drop substantially and approach zero when higher order co-moments beyond the 3rd are included
in the model. The evidence is consistent for all portfolios: size, book-to-market, momentum, and
liquidity. Chung et al suggest that because of the higher-order right-hand-side variables are
created, the Shanken adjustment biases the test results 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 serial
correlation in the return residuals which is not accounted in the Fama-Macbeth method. The
results are reported in Appendix table A.3. We document similar findings as with the FamaMacBeth method. This implies that the serial correlation problem does not influence the results.
Our computation so far is based on centered systematic co-moments. We also report the
results based on non-centered systematic co-moments. 8 The findings are also qualitatively
similar (see Appendix Table A.4).
Overall, our robustness checks are consistent with the previous findings that adding a set
of systematic co-moments reduces substantially the significance levels of factor loadings: SMB,
HML, MOM, and LIQ.
8
The non-centered systematic co-moments are computed as follows
60
b(i, j , t ) =
[r ( j, t − τ )][r (m, t − τ )]
∑
τ
i −1
=1
60
[r (m, t − τ )]
∑
τ
where
b(i, j , t ) is the ith non-centered systematic co-moment of
i
=1
portfolio j in month t, r (m, t ) is the return of the CRSP value weighted index.
23
IV.
Conclusion
In this paper, we attempt to provide evidence to support Rubinstein (1973)’s theoretical
model that if returns do not follow normal distribution, measuring risk requires more than just
measuring covariance, higher order systematic co-moments should be important to risk averse
investors who are concerned about the extreme outcomes of their investments. We show that not
only the Fama-French factors (SMB and HML), but also momentum and liquidity factors can be
explained by higher order co-moments.
In cross-sectional analyses, for all sorting criteria (size, book-to-market, momentum, and
liquidity), we find that adding a set of 10 or 15 systematic co-moments reduces substantially the
significance levels of the factors: SMB, HML, MOM, and LIQ and causes them to become
insignificant in most cases. One might argue that the results are driven by imprecision in
estimation due to adding more independent variables, we show that it is not the case. We perform
a similar analysis with a set of 10 or 15 standard moments and find that the explanatory powers
of the factors remain the same after including the set of standard moments. Also, in some cases,
some factors remain consistently insignificant before and after adding the set of variables. Thus,
it does not appear that our results are being driven by simply adding more explanatory variables.
Our cross sectional results are also consistent in both sub-periods and several other robustness
checks.
In time-series analysis, we find that the Gibbon-Ross-Shanken statistics, which test
whether the pricing errors from time series regressions are jointly equal to zero, consistently
decrease when more systematic co-moments are added to the model, and become insignificant
when enough co-moments are included. This suggests that the pricing error tends to zero when
24
more co-moments are added to the model, hence providing evidence that higher order systematic
co-moments should be relevant in pricing assets.
We also find that the results with a set of 10 systematic co-moments are very similar to
those with 15 co-moments or higher. This suggests that while the more number of higher comoments are included, the return distributions would be characterized more precisely, but
empirically, a set of 10 order co-moments is sufficient to capture the extreme outcomes of the
investment.
Our findings provide another explanation for the non-market factors, SMB, HML, MOM
within the traditional co-variance risk-based framework, without having to resort to behavior
assumptions since these factors are explained by certain market risk not captured in the CAPM
and measured by a set of higher order systematic co-moments. We also find that the Pastor and
Stambaugh liquidity risk might be captured by these co-moments. The practical implication is
that in a well-diversified portfolio, the idiosyncratic moments (e.g., standard deviation,
skewness, kurtosis, etc.) are eliminated, investors only earn compensation for exposure to
systematic co-moments (e.g., co-variance, co-skewness, co-kurtosis, etc.), and these systematic
co-moments should be priced when investors consider their investments.
25
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29
Table 1: Summary Statistics of Portfolio Return
The table reports the summary statistics for portfolio returns (size, book-to-market, and momentum) under analysis.
Size portfolios are constructed by sorting stocks into 50 equal-size portfolios based on their previous year-end
market capitalization. Book-to-market portfolios are 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 based on their prior compound return from month t-2 to t-12. Liquidity portfolios are
constructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh (2003)’s liquidity
level estimates. All NYSE-AMEX-NASDAQ ordinary common stocks from 1965-2005 are used in computation.
Kolmogorov-Smirnov, Cramer-von Mises, Anderson-Darling statistics are used to test normality of portfolio returns.
***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
Size portfolios
Book-to-market
portfolios
Momentum
portfolios
Liquidity
portfolios
Number of portfolioperiod observation
21600
21600
21600
21600
Mean
0.0079
0.0083
0.0082
0.0076
Variance
0.0045
0.0046
0.0047
0.0045
Skewness
0.0474
0.0517
1.0992
0.4088
Kurtosis
3.0200
3.1312
16.8590
4.9550
KolmogorovSmirnov
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***
30
Table 2: Correlation among factor loadings SMB, HML, LIQ, and MOM
The table reports the correlation among factor loadings SMB, HML, LIQ, and MOM for portfolio sorted by size, bookto-market, momentum, and liquidity. Size portfolios are constructed by sorting stocks into 50 equal-size portfolios
based on their previous year-end market capitalization. Book-to-market portfolios are 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 based on their prior compound return from month t-2 to t12. Liquidity portfolios are constructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and
Stambaugh (2003)’s liquidity level estimates. All NYSE-AMEX-NASDAQ ordinary common stocks from 1965-2005
are used in computation.
Panel A: Size portfolio
SMB
HML
MOM
LIQ
SMB
1.00
HML
0.39
1.00
MOM
-0.17
0.07
1.00
LIQ
-0.02
0.02
-0.01
1.00
HML
MOM
LIQ
Panel B: book-to-market portfolio
SMB
SMB
1.00
HML
0.35
1.00
MOM
-0.20
0.06
1.00
LIQ
-0.01
0.07
0.01
1.00
HML
MOM
LIQ
Panel C: Momentum portfolio
SMB
SMB
1.00
HML
-0.05
1.00
MOM
-0.30
-0.09
1.00
LIQ
-0.28
0.09
0.17
31
1.00
Panel D: Liquidity portfolio
SMB
HML
MOM
SMB
1.00
HML
0.59
1.00
MOM
-0.20
-0.02
1.00
LIQ
-0.18
-0.01
0.07
32
LIQ
1.00
Table 3: Fama-Macbeth Regression Results
This table reports Fama-MacBeth regression estimates for size, book-to-market, and momentum portfolios. Size
portfolios are constructed by sorting stocks into 50 equal-size portfolios based on their previous year-end market
capitalization. Book-to-market portfolios are 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 based on their prior compound return from month t-2 to t-12. Liquidity portfolios are constructed by
sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh (2003)’s liquidity level estimates.
All NYSE-AMEX-NASDAQ ordinary common stocks from 1965-2005 are used in computation. In each month,
portfolio returns are regressed on the factor loadings b, s, h, m, l. These factor loadings are computed by regressing
portfolio returns using the past five year data on the market premium, SMB, HML, MOM, and LIQ factors,
respectively. SMB and HML are Fama-French (1993) common factors, MOM represents the return on a portfolio of
winner stocks less the return on a portfolio of loser stocks (based on their prior compound return from month t-2 to t12). LIQ is the Pastor-Stambaugh market liquidity factor. The mean coefficient estimates across the sample period
are reported with their t-statistics. The F-statistics test the joint significance of the s, h, m, l estimates.
***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
b
s
h
m
l
Mean adj. R2
[F-stat]
Size
0.0356
(8.94)***
0.0041
(2.45)**
-0.0127
(-5.60)***
0.0129
(5.54)***
0.0049
(1.29)
0.46
[13.31]***
Book-tomarket
0.0355
(9.01)***
0.0026
(1.53)
-0.0076
(-3.82)***
0.0077
(3.39)***
0.0110
(3.14)***
0.44
[7.39]***
0.0011
(0.26)
-0.0005
(-0.28)
-0.0045
(-2.18)**
0.0101
(4.13)**
-0.0005
(-0.13)
0.43
[6.03]***
0.01272
(4.10)***
-0.0012
(-0.68)
-0.0046
(-2.74)**
0.0040
(1.96)**
0.0075
(2.04)**
0.342
[3.17]***
Portfolios
Momentum
Liquidity
33
Table 4: Systematic Co-moments and Common Factors
This table reports Fama-MacBeth regression estimates for size, book-to-market, momentum, and liquidity portfolios
when adding systematic higher order co-moments. Size portfolios are constructed by sorting stocks into 50 equal-size
portfolios based on their previous year-end market capitalization. Book-to-market portfolios are 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 based on their prior compound return from
month t-2 to t-12. Liquidity portfolios are constructed by sorting stocks into 50 equal-size portfolios bases on their
Pastor and Stambaugh (2003)’s liquidity level estimates. All NYSE-AMEX-NASDAQ ordinary common stocks from
1965-2005 are used in computation. In each month, portfolio returns are regressed on the factor loadings b, s, h, m, l,
and the respective number of systematic co-moments. These factor loadings are computed by regressing portfolio
returns using the past five year data on the market premium, SMB, HML, MOM, and LIQ factors, respectively. The
systematic co-moments are estimated using the same rolling five-year portfolio return with the market return. SMB
and HML are Fama-French (1993) common factors, MOM represents the return on a portfolio of winner stocks less
the return on a portfolio of loser stocks (based on their prior compound return from month t-2 to t-12). LIQ is the
Pastor-Stambaugh market liquidity factor. The mean coefficient estimates across the sample period are reported with
their t-statistics. The F-statistics test the joint significance of the s, h, m, l estimates.
***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
Panel A: size portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
0.0030
(1.50)
-0.0007
(-0.31)
-0.0018
(-0.75)
0.0020
(0.73)
0.0010
(0.33)
0.0030
(0.74)
0.0070
(1.51)
0.0080
(1.39)
0.0135
(1.65)
-0.0123
(-5.36)***
-0.0075
(-2.88)**
-0.0089
(-2.90)**
-0.0121
(-3.31)***
-0.0098
(-2.18)**
-0.0135
(-2.66)**
-0.0156
(-2.50)**
-0.0121
(-1.77)
-0.0090
(-0.98)
0.0077
(3.04)***
0.0059
(2.02)**
0.0075
(2.42)**
0.0088
(2.37)**
0.0095
(2.39)**
0.0116
(2.46)**
0.0012
(0.21)
-0.0078
(-1.16)
-0.0032
(-0.36)
-0.0076
(-1.52)
-0.0116
(-2.15)**
-0.0129
(-2.33)**
-0.0071
(-1.15)
-0.0072
(-1.05)
-0.0025
(-0.31)
0.0034
(0.39)
0.0045
(0.47)
0.0106
(0.89)
34
Mean adj. R2
[F-stat]
0.46
[11.04]***
0.47
[5.63]***
0.48
[6.46]***
0.48
[6.11]***
0.49
[4.58]***
0.49
[5.39]***
0.49
[2.67]**
0.49
[1.38]
0.45
[0.95]
Panel B: book-to-market portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
Mean adj. R2
[F-stat]
0.0015
(0.86)
0.0022
(0.99)
0.0006
(0.26)
0.0027
(0.97)
0.0043
(1.46)
0.0052
(1.39)
0.0067
(1.55)
0.0025
(0.52)
0.0031
(0.43)
-0.0077
(-3.58)***
-0.0084
(-3.20)***
-0.0071
(-2.24)**
-0.0074
(-2.04)**
-0.0093
(-2.09)**
-0.0087
(-1.79)*
-0.0084
(-1.36)
-0.0015
(-0.22)
0.0013
(0.13)
0.0054
(2.22)**
0.0068
(2.30)**
0.0080
(2.61)***
0.0084
(2.31)**
0.0083
(1.98)**
0.0074
(1.65)
-0.0030
(-0.55)
-0.0079
(-1.28)
0.0101
(1.14)
0.0043
(0.97)
0.0059
(1.12)
0.0031
(0.57)
0.0037
(0.63)
0.0095
(1.43)
0.0102
(1.29)
0.0120
(1.41)
0.0058
(0.64)
0.0020
(0.17)
0.45
[4.05]***
0.45
[3.99]***
0.46
[3.21]**
0.46
[2.44]**
0.46
[2.04]*
0.47
[1.52]
0.47
[0.77]
0.47
[0.57]
0.42
[0.52]
s
h
m
l
Mean adj. R2
[F-stat]
-0.0022
(-0.93)
-0.0044
(-1.65)
-0.0028
(-0.95)
-0.0065
(-1.92)
-0.0023
(-0.66)
-0.0001
(-0.01)
-0.0016
(-0.35)
-0.0032
(-0.51)
-0.0033
(-0.42)
-0.0038
(-1.79)*
-0.0044
(-1.75)*
-0.0059
(-1.60)
-0.0008
(-0.19)
-0.0091
(-1.54)
-0.0079
(-1.29)
-0.0091
(-1.11)
-0.0056
(-0.63)
-0.0063
(-0.66)
0.0099
(3.82)***
0.0061
(1.89)*
0.0042
(1.09)
0.0063
(1.30)
0.0052
(0.94)
0.0054
(0.95)
0.0054
(0.83)
0.0085
(1.08)
0.0115
(1.34)
-0.0014
(-0.27)
-0.0045
(-0.79)
-0.0013
(-0.21)
-0.0055
(-0.85)
0.0036
(0.47)
0.0062
(0.71)
0.0045
(0.48)
0.0013
(0.12)
0.0005
(0.04)
0.44
[4.93]***
0.44
[3.23]**
0.45
[1.67]
0.45
[1.92]
0.45
[1.76]
0.46
[0.93]
0.46
[0.99]
0.46
[0.81]
0.42
[0.94]
Panel C: momentum portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
35
Panel D: liquidity portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
Mean adj. R2
[F-stat]
0.0010
(0.53)
0.0010
(0.46)
-0.0003
(-0.12)
0.0006
(0.20)
0.0030
(0.97)
-0.0012
(-0.33)
-0.0017
(-0.41)
-0.0002
(-0.04)
-0.0045
(-0.67)
-0.0044
(-2.28)**
-0.0052
(-2.28)
-0.0057
(-2.13)
-0.0061
(-1.74)
-0.0097
(-2.02)
-0.0072
(-1.38)
-0.0066
(-1.12)
-0.0122
(-1.88)
-0.0044
(-0.58)
0.0058
(2.34)**
0.0045
(1.67)
0.0054
(1.77)
0.0075
(1.97)
0.0080
(1.87)
0.0077
(1.67)
0.0089
(1.65)
0.0072
(1.21)
0.0123
(1.54)
0.0036
(0.73)
0.0043
(0.80)
0.0018
(0.33)
0.0035
(0.59)
0.0125
(1.72)
0.0040
(0.52)
0.0045
(0.54)
0.0069
(0.77)
-0.0010
(-0.10)
0.369
[2.34]*
0.377
[2.10]*
0.384
[2.27]*
0.390
[1.89]
0.396
[2.01]*
0.400
[1.64]
0.401
[1.41]
0.402
[1.79]
0.341
[0.35]
36
Table 5: Standard Moments and Common Factors
This table reports Fama-MacBeth regression estimates for size, book-to-market, momentum, and liquidity portfolios
when adding a set of standard moment up to order 10 or 15. Size portfolios are constructed by sorting stocks into 50
equal-size portfolios based on their previous year-end market capitalization. Book-to-market portfolios are
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 based on their prior compound
return from month t-2 to t-12. Liquidity portfolios are constructed by sorting stocks into 50 equal-size portfolios
bases on their Pastor and Stambaugh (2003)’s liquidity level estimates. All NYSE-AMEX-NASDAQ ordinary
common stocks from 1965-2005 are used in computation. In each month, portfolio returns are regressed on the factor
loadings b, s, h, m, l, and the respective number of standard moments. These factor loadings are computed by
regressing portfolio returns using the past five year data on the market premium, SMB, HML, MOM, and LIQ
factors, respectively. The standard moments are estimated using the same rolling five-year portfolio return with the
market return. SMB and HML are Fama-French (1993) common factors, MOM represents the return on a portfolio of
winner stocks less the return on a portfolio of loser stocks (based on their prior compound return from month t-2 to t12). LIQ is the Pastor-Stambaugh market liquidity factor. The mean coefficient estimates across the sample period
are reported with their t-statistics. The F-statistics test the joint significance of the s, h, m, l estimates.
***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
s
h
m
l
Mean adj. R2
[F-stat]
3rd to 10th
0.0171
(5.95)***
-0.0170
(-6.63)***
0.0082
(2.57)**
0.0134
(2.50)**
0.55
[13.75]***
3rd to 15th
0.0160
(5.78)***
-0.0156
(-5.89)***
0.0068
(2.14)**
0.0113
(2.11)**
0.54
[13.39]***
Standard
moments
A. Size portfolios
B. Book-to-market portfolios
3rd to 10th
0.0122
(4.76)***
-0.0134
(-5.82)***
0.0078
(2.77)***
0.0113
(2.04)**
0.52
[11.64]***
3rd to 15th
0.0130
(4.70)***
-0.0131
(-5.63)***
0.0062
(2.05)***
0.0130
(2.21)**
0.51
[10.37]***
C. Momentum portfolios
3rd to 10th
0.0031
(1.03)
-0.0054
(-2.67)***
0.0092
(2.89)***
0.0023
(0.48)
0.53
[3.57]***
3rd to 15th
0.0025
(0.69)
-0.0048
(-2.11)**
0.0084
(2.45)**
-0.0003
(-0.07)
0.50
[2.65]**
3rd to 10th
0.0092
(2.86)**
-0.0071
(-2.31)***
0.0051
(1.54)
0.0130
(2.29)**
0.453
[2.96]**
3rd to 15th
0.0093
(2.88)
-0.0071
(-2.32)**
0.0069
(2.06)
0.0129
(2.14)
0.446
[3.51]***
D. Liquidity portfolios
37
Table 6: Gibbon-Ross-Shanken (1989) Test
This table reports Gibbon-Ross-Shanken (1989) statistics examining whether the intercepts (pricing errors) from time
series regressions are jointly equal to zero for size, book-to-market, momentum, and liquidity portfolios when adding
higher order co-moment factors. Size portfolios are constructed by sorting stocks into 50 equal-size portfolios based
on their previous year-end market capitalization. Book-to-market portfolios are 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 based on their prior compound return from month t-2 to t-12. Liquidity
portfolios are constructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh
(2003)’s liquidity level estimates. All NYSE-AMEX-NASDAQ ordinary common stocks from 1965-2005 are used in
computation.
***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
Systematic
Co-moments
original
Size
portfolio
12.37***
Book-to-market
portfolio
10.69***
Momentum
portfolio
2.75**
Liquidity
portfolio
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
1.61
2nd to 8th
4.23***
3.77***
1.56
1.51
2nd to 9th
4.00***
2.94***
1.55
1.19
2nd to 10th
3.69***
2.93***
1.55
1.18
2nd to 15th
1.60
1.28
0.87
0.62
38
Table 7: Sub-period 1970-1987
This table reports Fama-MacBeth regression estimates in the sub-period 1970-1987 for size, book-to-market,
momentum, and liquidity portfolios when adding a set of systematic co-moments or a set of standard moments of
order 3 through 10. Size portfolios are constructed by sorting stocks into 50 equal-size portfolios based on their
previous year-end market capitalization. Book-to-market portfolios are constructed by sorting stocks into 50 equalsize portfolios based on their previous year-end book-to-market ratios. Momentum portfolios are constructed by
sorting stocks into 50 equal-size portfolios based on their prior compound return from month t-2 to t-12. Liquidity
portfolios are constructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh
(2003)’s liquidity level estimates. In each month, portfolio returns are regressed on the factor loadings b, s, h, m, l,
and the respective number of systematic co-moments or standard moments. These factor loadings are computed by
regressing portfolio returns using the past five year data on the market premium, SMB, HML, MOM, and LIQ
factors, respectively. The systematic co-moments and standard moments are estimated using the same rolling fiveyear portfolio return with the market return. SMB and HML are Fama-French (1993) common factors, MOM
represents the return on a portfolio of winner stocks less the return on a portfolio of loser stocks. LIQ is the PastorStambaugh market liquidity factor. The mean coefficient estimates across the sample period are reported with their tstatistics. The F-statistics test the joint significance of the s, h, m, l estimates. In each panel A and B, the original
indicates the model including only the 2nd systematic co-moment (covariance), and the factor loadings on SMB,
HML, MOM, and LIQ.***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
Panel A: Regressions with systematic co-moments
s
h
m
l
Mean adj. R2
[F-stat]
Original
0.0038
(1.80)*
-0.0179
(-6.55)***
0.0158
(5.21)***
0.0172
[3.42]***
0.47
[16.15]***
2nd to 10th
0.0057
(0.55)
-0.0018
(-0.22)
-0.0055
(-0.53)
0.0176
(1.28)
0.49
[1.21]
Systematic
co-moments
A. Size portfolios
B. Book-to-market portfolios
Original
0.0019
(0.90)
-0.0114
(-4.86)***
0.0071
(2.41)**
0.0183
(4.39)***
0.44
[10.30]***
2nd to 10th
0.0004
(0.04)
0.0048
(0.70)
-0.0048
(-0.56)
0.0107
(0.85)
0.47
(1.50)
C. Momentum portfolios
Original
-0.0006
(-0.24)
-0.0022
(-0.99)
0.0097
(3.20)***
-0.0030
(-0.77)
0.38
[3.32]**
2nd to 10th
-0.0090
(-0.80)
0.0062
(0.68)
0.0156
(1.38)
-0.0199
(-1.30)
0.41
(0.83)
Original
-0.0011
(-0.54)
-0.0068
(-3.17)***
0.0065
(2.11)**
0.0015
(0.39)
0.381
[3.25]**
2nd to 10th
-0.0070
(-0.76)
-0.0110
(-1.43)
0.0118
(1.29)
-0.0021
(-0.16)
0.420
[1.71]
D. Liquidity portfolios
39
Panel B: Regressions with standard moments
s
h
m
l
Mean adj. R2
[F-stat]
0.0134
(3.38)***
-0.0157
(-5.36)***
0.0110
(2.56)**
0.0182
(3.36)***
0.54
[9.05]***
-0.0124
(-4.41)***
0.0107
(2.59)**
0.0183
(3.41)***
0.50
[8.39]***
0.0038
(0.56)
-0.0071
(-1.99)**
0.0118
(2.11)**
0.0010
(0.13)
0.46
[3.32]**
0.0111
(2.38)**
-0.0069
(-2.33)**
0.0107
(2.41)**
0.0083
(1.36)
0.461
[3.40]***
Standard
moments
A. Size portfolios
3rd to 10th
B. Book-to-market portfolios
3rd to 10th
0.0131
(3.38)***
C. Momentum portfolios
3rd to 10th
D. Liquidity portfolios
3rd to 10th
40
Table 8: Sub-period 1988-2005
This table reports Fama-MacBeth regression estimates in the sub-period 1988-2005 for size, book-to-market,
momentum, and liquidity portfolios portfolios when adding a set of systematic co-moments or a set of standard
moments of order 3 through 10. Size portfolios are constructed by sorting stocks into 50 equal-size portfolios based
on their previous year-end market capitalization. Book-to-market portfolios are 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 based on their prior compound return from month t-2 to t-12. Liquidity
portfolios are constructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh
(2003)’s liquidity level estimates In each month, portfolio returns are regressed on the factor loadings b, s, h, m, l,
and the respective number of systematic co-moments or standard moments. These factor loadings are computed by
regressing portfolio returns using the past five year data on the market premium, SMB, HML, MOM, and LIQ
factors, respectively. The systematic co-moments and standard moments are estimated using the same rolling fiveyear portfolio return with the market return. SMB and HML are Fama-French (1993) common factors, MOM
represents the return on a portfolio of winner stocks less the return on a portfolio of loser stocks. LIQ is the PastorStambaugh market liquidity factor. The mean coefficient estimates across the sample period are reported with their tstatistics. The F-statistics test the joint significance of the s, h, m, l estimates. In each panel A and B, the original
indicates the model including only the 2nd systematic co-moment (covariance), and the factor loadings on SMB,
HML, MOM, and LIQ.***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
Panel A: Regressions with systematic co-moments
s
h
m
l
Mean adj. R2
[F-stat]
Original
0.0045
(1.70)*
-0.0076
(-2.10)**
0.0100
(2.83)***
-0.0073
(-1.27)
0.45
[3.98]***
2nd to 10th
0.0103
(2.12)**
-0.0224
(-2.06)**
-0.0101
(-1.17)
-0.0085
(-0.64)
0.48
[3.32]***
Systematic
co-moments
A. Size portfolios
B. Book-to-market portfolios
Original
0.0032
(1.23)
-0.0037
(-1.17)
0.0082
(2.39)**
0.0037
(0.66)
0.43
[1.78]
2nd to 10th
0.0048
(0.95)
-0.0078
(-0.68)
-0.0110
(-1.22)
0.0008
(0.06)
0.47
[0.50]
C. Momentum portfolios
Original
-0.0004
(-0.15)
-0.0069
(-1.95)*
0.0106
(2.73)***
0.0018
(0.23)
0.49
[3.75]***
2nd to 10th
0.0024
(0.40)
-0.0175
(-1.15)
0.0014
(0.12)
0.0226
(1.49)
0.51
[0.98]
Original
-0.0009
(-0.42)
-0.0032
(-1.10)
0.0025
(0.78)
-0.0054
(-0.98)
0.345
[1.05]
2nd to 10th
0.0066
(1.39)
-0.0133
(-1.28)
0.0026
(0.34)
0.0160
(1.27)
0.382
[0.63]
D. Liquidity portfolios
41
Panel B: Regressions with standard moments
s
h
m
l
Mean adj. R2
[F-stat]
0.0129
(4.11)***
-0.0151
(-3.75)***
0.0015
(0.37)
-0.0054
(-0.67)
0.55
[7.81]***
-0.0144
(-3.95)***
0.0049
(1.28)
0.0044
(0.45)
0.52
[5.75]***
-0.0026
(-0.80)
-0.0044
(-1.42)
0.0114
(2.63)***
-0.0040
(-0.50)
0.60
[2.48]**
0.0020
(0.53)
-0.0037
(-0.79)
0.0040
(0.86)
0.0094
(1.10)
0.437
[0.79]
Standard
moments
A. Size portfolios
3rd to 10th
B. Book-to-market portfolios
3rd to 10th
0.0113
(3.35)***
C. Momentum portfolios
3rd to 10th
D. Liquidity portfolios
3rd to 10th
42
Appendix
Table A1: R-squares of the regressions of factor loadings (SMB, HML, MOM, LIQ) on a
set of systematic co-moments
This table reports the R-squares from the regressions of factor loadings (SMB, HML, MOM, LIQ) on a set of 10 or
15 systematic comoments for size, book-to-market, and momentum portfolios when adding systematic higher order
co-moments. Size portfolios are constructed by sorting stocks into 50 equal-size portfolios based on their previous
year-end market capitalization. Book-to-market portfolios are 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 based on their prior compound return from month t-2 to t-12. Liquidity portfolios
are constructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh (2003)’s liquidity
level estimates. These factor loadings are computed by regressing portfolio returns using the past five year data on
the market premium, SMB, HML, MOM, and LIQ factors, respectively. The systematic co-moments are estimated
using the same rolling five-year portfolio return with the market return. SMB and HML are Fama-French (1993)
common factors, MOM represents the return on a portfolio of winner stocks less the return on a portfolio of loser
stocks. LIQ is the Pastor-Stambaugh market liquidity factor.
Panel A: Common factors and a set of 10 systematic co-moments
Factors
Size
portfolio
Book-to-market
portfolio
Momentum
portfolio
Liquidity
portfolio
SMB
0.89
0.88
0.85
0.85
HML
0.75
0.73
0.74
0.73
MOM
0.62
0.63
0.90
0.61
LIQ
0.54
0.50
0.67
0.51
Panel B: Common factors and a set of 15 systematic co-moments
Factors
Size
portfolio
Book-to-market
portfolio
Momentum
portfolio
Liquidity
portfolio
SMB
0.91
0.90
0.86
0.87
HML
0.77
0.75
0.76
0.75
MOM
0.66
0.67
0.92
0.65
LIQ
0.58
0.54
0.69
0.55
43
Table A2: Shanken (1992)’s correction for Fama-MacBeth estimates
This table reports Shanken (1992)’s correction for error-in-variable problem in Fama-MacBeth regression estimates
for size, book-to-market, momentum, and liquidity portfolios when adding systematic higher order co-moments. Size
portfolios are constructed by sorting stocks into 50 equal-size portfolios based on their previous year-end market
capitalization. Book-to-market portfolios are 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 based on their prior compound return from month t-2 to t-12. Liquidity portfolios are constructed by
sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh (2003)’s liquidity level estimates.
All NYSE-AMEX-NASDAQ ordinary common stocks from 1965-2002 are used in computation. In each month,
portfolio returns are regressed on the factor loadings b, s, h, m, l, and the respective number of systematic comoments. These factor loadings are computed by regressing portfolio returns using the past five year data on the
market premium, SMB, HML, MOM, and LIQ factors, respectively. The systematic co-moments are estimated using
the same rolling five-year portfolio return with the market return. SMB and HML are Fama-French (1993) common
factors, MOM represents the return on a portfolio of winner stocks less the return on a portfolio of loser stocks (based
on their prior compound return from month t-2 to t-12). LIQ is the Pastor-Stambaugh market liquidity factor.
***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
Panel A: size portfolios
Systematic
Co-moments
Original
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 10th
s
h
m
l
0.0041
(1.50)**
0.0030
(0.79)
-0.0007
(-0.007)
-0.0018
(-0.008)
0.0020
(0.000)
0.0080
(4×10-10)
-0.0127
(-3.77)***
-0.0123
(-2.91)***
-0.0075
(-0.070)
-0.0089
(-0.032)
-0.0121
(-0.000)
-0.0121
(-5×10-10)
0.0129
(3.47)***
0.0077
(1.60)
0.0059
(0.049)
0.0075
(0.026)
0.0088
(0.000)
-0.0078
(-3×10-10)
0.0049
(0.88)
-0.0076
(-0.85)
-0.0116
(-0.052)
-0.0129
(-0.025)
-0.0071
(-0.000)
0.0045
(1.5×10-10)
s
h
m
l
0.0026
(0.95)
0.0015
(0.48)
0.0022
(0.25)
0.0006
(0.030)
0.0027
(0.000)
0.0025
(39×10-10)
-0.0076
(-2.54)**
-0.0077
(-2.08)**
-0.0084
(-0.82)
-0.0071
(-0.261)
-0.0074
(-0.000)
-0.0015
(-17×10-10)
0.0077
(2.14)
0.0054
(1.24)
0.0068
(0.59)
0.0080
(0.304)
0.0084
(0.000)
-0.0079
(-95×10-10)
0.0110
(2.15)*
0.0043
(0.58)
0.0059
(0.29)
0.0031
(0.067)
0.0037
(0.000)
0.0058
(43×10-10)
Panel B: book-to-market portfolios
Systematic
Co-moments
original
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 10th
44
Panel C: momentum portfolios
Systematic
Co-moments
original
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 10th
s
h
m
l
-0.0005
(-0.21)
-0.0022
(-0.72)
-0.0044
(-1.08)
-0.0028
(-0.002)
-0.0065
(-0.000)
-0.0032
(-0.6×10-10)
-0.0045
(-1.74)*
-0.0038
(-1.37)
-0.0044
(-1.15)
-0.0059
(-0.004)
-0.0008
(-0.000)
-0.0056
(-0.8×10-10)
0.0101
(3.12)***
0.0099
(2.83)**
0.0061
(1.23)
0.0042
(0.002)
0.0063
(0.000)
0.0085
(1.3×10-10)
-0.0005
(-0.11)
-0.0014
(-0.23)
-0.0045
(-0.54)
-0.0013
(-0.000)
-0.0055
(-0.000)
0.0013
(1.6×10-10)
s
h
m
l
0.0010
(0.32)
0.0010
(0.02)
-0.0003
(-0.000)
0.0006
(0.000)
-0.0002
(-1.1×10-10)
-0.0044
(-1.41)
-0.0052
(-0.13)
-0.0057
(-0.010)
-0.0061
(-0.002)
-0.0122
(-50×10-10)
0.0058
(1.43)
0.0045
(0.09)
0.0054
(0.008)
0.0075
(0.002)
0.0072
(32×10-10)
0.0036
(0.49)
0.0043
(0.04)
0.0018
(0.001)
0.0035
(0.000)
0.0069
(20×10-10)
Panel D: liquidity portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 10th
45
Table A3: Newey-West adjustment for Fama-MacBeth estimation
This table reports Newey-West (1987)’s adjustment for serial correlation in return residual problem in FamaMacBeth regression estimates for size, book-to-market, momentum, and liquidity portfolios when adding systematic
higher order co-moments. Size portfolios are constructed by sorting stocks into 50 equal-size portfolios based on
their previous year-end market capitalization. Book-to-market portfolios are 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 based on their prior compound return from month t-2 to t-12. Liquidity
portfolios are constructed by sorting stocks into 50 equal-size portfolios bases on their Pastor and Stambaugh
(2003)’s liquidity level estimates. All NYSE-AMEX-NASDAQ ordinary common stocks from 1965-2002 are used in
computation. In each month, portfolio returns are regressed on the factor loadings b, s, h, m, l, and the respective
number of systematic co-moments. These factor loadings are computed by regressing portfolio returns using the past
five year data on the market premium, SMB, HML, MOM, and LIQ factors, respectively. The systematic comoments are estimated using the same rolling five-year portfolio return with the market return. SMB and HML are
Fama-French (1993) common factors, MOM represents the return on a portfolio of winner stocks less the return on a
portfolio of loser stocks (based on their prior compound return from month t-2 to t-12). LIQ is the Pastor-Stambaugh
market liquidity factor.
***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
Panel A: size portfolios
Systematic
Co-moments
original
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
0.0041
(2.23)**
0.0030
(1.37)
-0.0007
(-0.29)
-0.0018
(-0.72)
0.0020
(0.73)
0.0010
(0.34)
0.0030
(0.78)
0.0070
(1.45)
0.0080
(1.38)
0.0135
(1.73)
-0.0127
(-4.35)***
-0.0123
(-4.35)***
-0.0075
(-2.56)**
-0.0089
(-2.72)***
-0.0121
(-3.04)***
-0.0098
(-1.99)**
-0.0135
(-2.59)**
-0.0156
(-2.50)**
-0.0121
(-1.79)
-0.0090
(-0.93)
0.0129
(5.26)***
0.0077
(2.70)***
0.0059
(1.73)*
0.0075
(2.13)**
0.0088
(2.08)**
0.0095
(2.15)**
0.0116
(2.28)**
0.0012
(0.19)
-0.0078
(-1.12)
-0.0032
(-0.34)
0.0049
(1.13)
-0.0076
(-1.28)
-0.0116
(-1.83)*
-0.0129
(-2.09)**
-0.0071
(-1.00)
-0.0072
(-0.95)
-0.0025
(-0.30)
0.0034
(0.37)
0.0045
(0.46)
0.0106
(0.88)
46
Panel B: book-to-market portfolios
Systematic
Co-moments
original
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
0.0026
(1.41)
0.0015
(0.79)
0.0022
(0.92)
0.0006
(0.26)
0.0027
(0.97)
0.0043
(1.56)
0.0052
(1.37)
0.0067
(1.52)
0.0025
(0.52)
0.0031
(0.43)
-0.0076
(-3.27)***
-0.0077
(-3.09)***
-0.0084
(-2.96)***
-0.0071
(-1.98)**
-0.0074
(-1.77)*
-0.0093
(-1.91)*
-0.0087
(-1.60)
-0.0084
(-1.28)
-0.0015
(-0.21)
0.0013
(0.13)
0.0077
(3.24)***
0.0054
(2.18)**
0.0068
(2.20)**
0.0080
(2.70)***
0.0084
(2.20)**
0.0083
(1.93)*
0.0074
(1.56)
-0.0030
(-0.53)
-0.0079
(-1.14)
0.0101
(1.12)
0.0110
(2.92)***
0.0043
(0.93)
0.0059
(1.05)
0.0031
(0.55)
0.0037
(0.60)
0.0095
(1.56)
0.0102
(1.33)
0.0120
(1.46)
0.0058
(0.67)
0.0020
(0.19)
s
h
m
l
-0.0005
(-0.26)
-0.0022
(-0.83)
-0.0044
(-1.44)
-0.0028
(-0.84)
-0.0065
(-1.69)
-0.0023
(-0.57)
-0.0001
(-0.01)
-0.0016
(-0.33)
-0.0032
(-0.45)
-0.0033
(-0.39)
-0.0045
(-2.00)**
-0.0038
(-1.59)
-0.0044
(-1.53)
-0.0059
(-1.44)
-0.0008
(-0.17)
-0.0091
(-1.26)
-0.0079
(-1.10)
-0.0091
(-1.04)
-0.0056
(-0.60)
-0.0063
(-0.61)
0.0101
(4.53)***
0.0099
(4.22)***
0.0061
(2.13)**
0.0042
(1.25)
0.0063
(1.33)
0.0052
(0.96)
0.0054
(1.02)
0.0054
(0.94)
0.0085
(1.20)
0.0115
(1.41)
-0.0005
(-0.13)
-0.0014
(-0.26)
-0.0045
(-0.79)
-0.0013
(-0.21)
-0.0055
(-0.81)
0.0036
(0.45)
0.0062
(0.75)
0.0045
(0.51)
0.0013
(0.13)
0.0005
(0.04)
Panel C: momentum portfolios
Systematic
Co-moments
original
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
47
Panel D: liquidity portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
0.0010
(0.51)
0.0010
(0.43)
-0.0003
(-0.12)
0.0006
(0.19)
0.0030
(0.92)
-0.0012
(-0.33)
-0.0017
(-0.44)
-0.0002
(-0.04)
-0.0045
(-0.72)
-0.0044
(-2.15)**
-0.0052
(-2.14)
-0.0057
(-2.06)
-0.0061
(-1.66)
-0.0097
(-2.00)
-0.0072
(-1.46)
-0.0066
(-1.19)
-0.0122
(-1.96)
-0.0044
(-0.64)
0.0058
(2.34)
0.0045
(1.62)
0.0054
(1.74)
0.0075
(1.97)
0.0080
(1.84)
0.0077
(1.67)
0.0089
(1.73)
0.0072
(1.23)
0.0123
(1.81)
0.0036
(0.77)
0.0043
(0.86)
0.0018
(0.36)
0.0035
(0.61)
0.0125
(1.76)
0.0040
(0.54)
0.0045
(0.58)
0.0069
(0.83)
-0.0010
(-0.12)
48
Table A4: Non-centered Systematic Co-moments and Common Factors
This table reports Fama-MacBeth regression estimates for size, book-to-market, and momentum portfolios when
adding non-centered systematic higher order co-moments. Size portfolios are constructed by sorting stocks into 50
equal-size portfolios based on their previous year-end market capitalization. Book-to-market portfolios are
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 based on their prior compound
return from month t-2 to t-12. Liquidity portfolios are constructed by sorting stocks into 50 equal-size portfolios
bases on their Pastor and Stambaugh (2003)’s liquidity level estimates. All NYSE-AMEX-NASDAQ ordinary
common stocks from 1965-2002 are used in computation. In each month, portfolio returns are regressed on the factor
loadings b, s, h, m, l, and the respective number of systematic co-moments. These factor loadings are computed by
regressing portfolio returns using the past five year data on the market premium, SMB, HML, MOM, and LIQ
factors, respectively. The non-centered systematic co-moments are estimated using the same rolling five-year
portfolio return with the market return without demeaning. SMB and HML are Fama-French (1993) common factors,
MOM represents the return on a portfolio of winner stocks less the return on a portfolio of loser stocks (based on
their prior compound return from month t-2 to t-12). LIQ is the Pastor-Stambaugh market liquidity factor. The mean
coefficient estimates across the sample period are reported with their t-statistics. The F-statistics test the joint
significance of the s, h, m, l estimates.
***, **, * denote significance level at 1 percent, 5 percent, and 10 percent respectively.
Panel A: size portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
0.0096
(5.65)***
0.0056
(2.78)**
0.0019
(0.86)
0.0034
(1.30)
0.0003
(0.09)
-0.0004
(-0.11)
0.0002
(0.05)
0.0008
(0.16)
-0.0006
(-0.07)
-0.0019
(-1.09)
0.0005
(0.23)
-0.0013
(-0.53)
-0.0008
(-0.27)
0.0064
(1.41)
0.0054
(1.06)
0.0066
(1.00)
0.0031
(0.42)
0.0135
(1.51)
0.0076
(3.05)***
0.0057
(2.08)**
0.0072
(2.51)**
0.0063
(1.82)*
0.0074
(1.96)**
0.0082
(1.97)**
0.0058
(1.18)
-0.0012
(-0.20)
0.0026
(0.28)
0.0186
(4.27)***
0.0117
(2.51)**
0.0043
(0.93)
0.0007
(0.12)
-0.0062
(-1.04)
-0.0062
(-0.83)
-0.0053
(-0.66)
-0.0028
(-0.32)
-0.0047
(-0.41)
49
Mean adj. R2
[F-stat]
0.494
[11.25]***
0.499
[4.08]***
0.506
[1.78]
0.507
[1.52]
0.512
[1.90]
0.515
[1.55]
0.515
[0.90]
0.515
[0.18]
0.474
[0.91]
Panel B: book-to-market portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
Mean adj. R2
[F-stat]
0.0071
(4.12)***
0.0079
(3.56)***
0.0033
(1.44)
0.0035
(1.29)
0.0038
(1.30)
0.0050
(1.38)
0.0034
(0.85)
0.0060
(1.11)
0.0068
(0.89)
-0.0018
(-1.07)
-0.0030
(-1.31)
-0.0027
(-1.03)
-0.0011
(-0.33)
0.0025
(0.53)
0.0034
(0.69)
0.0083
(1.28)
0.0061
(0.86)
0.0060
(0.68)
0.0034
(1.48)
0.0044
(1.55)
0.0069
(2.33)**
0.0083
(2.37)
0.0058
(1.49)
0.0034
(0.84)
0.0009
(0.20)
-0.0057
(-0.95)
0.0008
(0.10)
0.0218
(5.38)***
0.0243
(5.04)***
0.0143
(3.01)***
0.0097
(1.75)
0.0100
(1.63)
0.0074
(1.05)
0.0044
(0.56)
0.0068
(0.77)
0.0051
(0.46)
0.464
[9.22]***
0.470
[7.12]***
0.477
[3.38]***
0.480
[2.13]*
0.486
[2.09]*
0.489
[1.63]
0.491
[1.60]
0.492
[1.52]
0.446
[0.84]
s
h
m
l
Mean adj. R2
[F-stat]
0.0035
(1.70)*
0.0010
(0.43)
0.0013
(0.52)
-0.0044
(-1.34)
-0.0020
(-0.57)
-0.0015
(-0.32)
-0.0024
(-0.48)
-0.0044
(-0.63)
-0.0017
(-0.18)
-0.0041
(-2.02)**
-0.0025
(-1.06)
-0.0065
(-2.32)**
0.0003
(0.09)
-0.0052
(-0.91)
-0.0033
(-0.52)
-0.0069
(-0.78)
-0.0022
(-0.23)
-0.0036
(-0.33)
0.0109
(4.38)***
0.0077
(2.51)**
0.0054
(1.57)
0.0047
(1.03)
0.0019
(0.36)
0.0011
(0.19)
0.0012
(0.20)
0.0017
(0.22)
-0.0004
(-0.04)
0.0076
(1.44)
0.0011
(0.21)
0.0024
(0.43)
-0.0068
(-1.07)
-0.0019
(-0.26)
-0.0011
(-0.13)
0.0002
(0.02)
-0.0038
(-0.33)
-0.0005
(-0.04)
0.446
[6.84]***
0.451
[1.90]*
0.455
[1.94]*
0.460
[0.96]
0.465
[0.55]
0.470
[0.90]
0.471
[0.62]
0.473
[0.88]
0.424
[0.98]
Panel C: momentum portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
50
Panel D: liquidity portfolios
Systematic
Co-moments
2nd to 3rd
2nd to 4th
2nd to 5th
2nd to 6th
2nd to 7th
2nd to 8th
2nd to 9th
2nd to 10th
2nd to 15th
s
h
m
l
Mean adj. R2
[F-stat]
0.0018
(1.07)
0.0014
(0.70)
0.0002
(0.10)
-0.0001
(-0.04)
0.0014
(0.48)
-0.0038
(-1.06)
-0.0053
(-1.24)
-0.0023
(-0.47)
-0.0081
(-1.21)
-0.0023
(-1.31)
-0.0027
(-1.24)
-0.0030
(-1.25)
-0.0043
(-1.26)
-0.0072
(-1.61)
-0.0044
(-0.88)
-0.0023
(-0.39)
-0.0082
(-1.28)
-0.0040
(-0.53)
0.0047
(2.04)
0.0025
(0.96)
0.0026
(0.94)
0.0045
(1.26)
0.0041
(1.01)
0.0049
(1.10)
0.0070
(1.35)
0.0052
(0.92)
0.0052
(0.64)
0.0044
(1.03)
0.0046
(0.95)
0.0017
(0.35)
0.0011
(0.20)
0.0066
(1.09)
-0.0034
(-0.49)
-0.0051
(-0.67)
-0.0016
(-0.20)
-0.0071
(-0.71)
0.375
[1.50]
0.383
[0.68]
0.388
[0.67]
0.393
[0.97]
0.399
[1.07]
0.406
[1.35]
0.407
[1.15]
0.405
[1.20]
0.348
[1.12]
51