Avramov & Chordia, 2006
Avramov & Chordia, 2006
Avramov & Chordia, 2006
Anomalies
Doron Avramov
R. H. Smith School of Business, University of Maryland
Tarun Chordia
Goizueta Business School, Emory University
The capital asset pricing model (CAPM) of Sharpe (1964) and Lintner
(1965) has long been a basic tenet of finance. However, subsequent work
by Basu (1977), Banz (1981), Jegadeesh (1990), and Fama and French
(FF) (1992) suggests that cross-sectional differences in average returns are
determined not only by the market risk, as prescribed by the CAPM, but
also by firm-level market capitalization, book-to-market, and prior
return. Some interpret the predictive ability of these variables as evidence
against market efficiency. Support for market efficiency has been pro-
vided by FamaFrench (1993, 1996) who show that, except for the
momentum effect, the impact of security characteristics on expected
returns can be explained within a risk-based multifactor model. However,
there is still an ongoing debate about whether expected returns are
explained by risk factors or by non-risk firm characteristics.
The failure of the CAPM has also been attributed to its static nature,
and, thus, to its incomplete description of asset prices. Indeed, both
theoretical and empirical work support the use of dynamic pricing mod-
els. For example, Hansen and Richard (1987) show that even if the static
CAPM fails, a dynamic version of the CAPM could be perfectly valid. In
addition, Gomes, Kogan, and Zhang (henceforth GKZ) (2003) develop a
We thank Yakov Amihud, Michael Brennan, Narasimhan Jegadeesh, Leonid Kogan, Jay Shanken, an
anonymous referee, and seminar participants at University of Florida, George Washington University, Tel
Aviv University, and Vanderbilt University for helpful comments. We are especially grateful to Cam Harvey
(editor) for numerous helpful suggestions. All errors are our own. Address correspondence to Doron
Avramov, University of Maryland, College Park, MD 20742, or email: davramov@rhsmith.umd.edu
The Author 2006. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights
reserved. For permissions, please email: journals.permissions@oxfordjournals.org.
doi:10.1093/rfs/hhj025 Advance Access publication February 20, 2006
The Review of Financial Studies / v 19 n 3 2006
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Asset Pricing Models and Financial Market Anomalies
1
We do not use the PastorStambaugh non-traded liquidity factor (non-traded innovations in liquidity)
but we use a traded portfolio that is long on stocks with high sensitivities and short on stocks with low
sensitivities to the non-traded liquidity factor.
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1004
Asset Pricing Models and Financial Market Anomalies
effect, though we note that the turnover effect is also robust when the
liquidity factor is formed as the return differential between high- and low-
turnover stocks.
Finally, we show that when model mispricing is allowed to vary
with business-cycle variables in the first-pass regression, then this
variation captures the impact of momentum on returns. Put differ-
ently, if in the cross-sectional regression the dependent variables is the
risk-adjusted return minus the time-varying component of alpha, then
past return variables become insignificant in this regression. This
suggests that there is indeed a business cycle pattern to the momen-
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The Review of Financial Studies / v 19 n 3 2006
2
Mankiw and Shapiro (1986) regress the average returns of the 464 NYSE stocks on their market betas, on
consumption growth betas, and on both betas. They find that the market betas are more strongly and
robustly associated with the cross-section of average returns. They also find that market beta drives out
consumption beta in multiple regressions. Breeden, Gibbons, and Litzenberger (1989) find comparable
performance of the CAPM and a model that uses a mimicking portfolio for consumption growth as the
single factor.
3
The international framework (Fama and French 1998) focuses on the market and value premia.
1006
Asset Pricing Models and Financial Market Anomalies
2. Methodology
Our asset pricing tests extend the approach of BCS (1998). BCS test
factor models by regressing risk-adjusted returns on firm-level attributes
such as size, book-to-market, and turnover. Under the null of exact
pricing, such attributes should be statistically insignificant in the cross-
section. The practice of using risk-adjusted returns, rather than gross or
excess returns, is also applied by Shanken (1992). It is intended to
address the finite sample bias attributable to errors in estimating factor
loadings in the first-pass time series regressions. As noted earlier, the
focus on single securities avoids the data-snooping biases that are
inherent in portfolio-based approaches, as noted by Lo and MacKinlay
1007
The Review of Financial Studies / v 19 n 3 2006
X
K
Rjt Et1 Rjt jkt1 f kt ejt , 1
k1
X
K
Et1 Rjt RFt kt1 jkt1 , 2
k1
where RFt is the risk-free rate and kt is the risk premium for factor k at
time t. The estimated risk-adjusted return on each security for month t is
then calculated as
X
K
R*jt Rjt RFt ^jkt1 F kt , 3
k1
where Fkt fkt kt1 is the sum of the factor innovation and its corre-
sponding risk premium and ^jkt is the conditional beta estimated by a
first-pass time-series regression over the entire sample period as per the
specification given below. Our risk adjustment procedure imposes the
assumptions that the conditional zero-beta return equals the conditional
1008
Asset Pricing Models and Financial Market Anomalies
risk-free rate, and that the factor premium is equal to the excess return on
the factor, as is the case when factors are portfolio based.
Next, we run the cross-sectional regression
X
M
R*jt c0t cmt Zmjt1 ejt , 4
m1
0
^ct Zt1 Zt1 1 Zt1
0
R*t , 5
where Xjt1 and Zjt1 are vectors of firm characteristics, zt1 denotes a
vector of macroeconomic variables, and represents the parameters that
capture the dependence of on the macroeconomic variables and the
firm characteristics. Ultimately, the null to test is ct 0. While we have
checked the robustness of our results for the general case where
Xjt1 Zjt1 , we will focus on the case where the factor loadings depend
upon firm-level size, book-to-market, and business-cycle-related vari-
ables. That is, the vector Xjt1 stands for size and book-to-market and
the vector Zjt1 for size, book-to-market, turnover, and various lagged
return variables.
The dependence on size and book-to-market is motivated by the gen-
eral equilibrium model of GKZ (2003), which justifies separate roles for
size and book-to-market as determinants of beta. In particular, firm size
captures the component of a firms systematic risk attributable to its
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The Review of Financial Studies / v 19 n 3 2006
growth option, and the book-to-market ratio serves as a proxy for risk of
existing projects. Incorporating business-cycle variables follows the exten-
sive evidence on time series predictability (Keim and Stambaugh 1986;
Fama and French 1989; and Chen 1991).
To describe the beta modeling in the time series regressions, let us focus
on the one factor CAPM, and let us assume that there is a single macro-
economic predictor zt1 . The conditional beta of security j is modeled as
rjt j j1 rmt j2 zt1 rmt j3 Sizejt1 rmt j4 zt1 Sizejt1 rmt
j5 BM jt1 rmt j6 zt1 BM jt1 rmt ujt , 8
where rjt Rjt RFt and rmt is excess return on the value-weighted
market index. Note that the firm characteristics size and book-to-market
ratio and the macroeconomic predictor are all lagged one period as
compared to the excess market and individual stock returns. Moreover,
not only do we condition betas on size and the book-to-market ratio, but
we also allow this conditioning to vary over time with zt1 .
Then, R*jt in Equation (4), the dependent variable in the cross-sec-
tion regression, is given by j ujt . The time series regression (8) is run
over the entire sample. While this entails the use of future data in
calculating the factor loadings, Fama and French (1992) have shown
that this forward looking does not impact any of the results. We can
confirm that this is indeed true for a handful of random cases we have
examined.
For perspective, it is useful to compare our approach to earlier studies
. Fama and French (1992) estimate beta by assigning the firm to one of
100 size-beta sorted portfolios. Firms beta (proxied by the portfo-
lios beta) is allowed to evolve over time when the firm changes its
portfolio classification.
. Fama and French (1993) focus on 25 size and book-to-market sorted
portfolios, which allow firms beta to change over time as they move
between portfolios.
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Asset Pricing Models and Financial Market Anomalies
ft a b0 Bt t 9
ft a b0 Bt c0 zt1 t , 10
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3. Data
The basic data consist of monthly returns, size, book-to-market, turn-
over, and lagged returns for a sample of common stocks of NYSE,
AMEX, and NASDAQ-listed companies. The sample spans the period
July 1964 through December 2001. NASDAQ stocks enter the sample
only from January 1983 since the NASDAQ trading volume, one of the
firm characteristics, is unavailable in Center for Research in Security
Prices (CRSP) before November 1982.
We run the analysis using monthly observations for the most part. We
test the CCAPM using quarterly data because data on consumption
growth is available only at the quarterly frequency. To be included in
the monthly analysis, a stock has to satisfy the following criteria. First, its
return in the current month, t, and over the past 36 months has to be
available from CRSP. Second, sufficient data has to be available to
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Asset Pricing Models and Financial Market Anomalies
4
We have confirmed the robustness of our results to different minimum data requirements ranging from 20
through 72 months. The requirement of the full 72 data points is mostly important for the most general
specification in Equation (6).
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Table 1
Summary statistics
This table presents the time-series averages of the cross-sectional means, medians, and standard devia-
tions for an average of 2871 NYSE-AMEX and NASDAQ stocks in Panel A and 1818 NYSE and
AMEX stocks in Panel B over 450 months from July 1964 through December 2001. The last column
represents the time-series averages of slope coefficients in cross-sectional OLS regressions of excess return
on various equity characteristics, t ratios (in parenthesis), and the adjusted R2 , denoted R 2 . Size
represents the market capitalization in billions of dollars. Turnover is the monthly share trading volume
divided by shares outstanding in percent. RET2-3, RET4-6, and RET7-12 are the cumulative returns
over the second through third, fourth through sixth, and seventh through twelfth months before the
current quarter, respectively. To be included in the sample, each stock has to satisfy the following
criteria: (i) its return in the current month t and in the previous 36 months be available from Center for
Research in Security Prices (CRSP), (ii) sufficient data be available to calculate the size and turnover as
of the month t 2, and (iii) sufficient data be available on the Compustat tapes to calculate the book-to-
market ratio as of December of the previous year. The book-to-market ratio provides summary statistics
for this variable after book-to-market values greater than the 0.995 fractile or less than the 0.005 fractile
are set to equal the 0.995 and 0.005 fractile values, respectively.
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Asset Pricing Models and Financial Market Anomalies
4. Results
Here, we empirically assess the pricing abilities of various time-varying
beta models. We also report evidence based on fixed beta models as a
benchmark for comparison. Here are the models examined: (i) CAPM, (ii)
FamaFrench three-factor model, (iii) FamaFrench augmented by the
liquidity factor of Pastor and Stambaugh (2003), (iv) FamaFrench
5
Note that the sign of the coefficient on firm size is different depending on whether dollar trading volume or
turnover is used as a measure of liquidity. See Chordia, Subrahmanyam, and Anshuman (2001) for details.
6
Avramov (2002) and Brennan and Xia (2005) show that the cay variable displays an impressive predictive
power only when the shares of asset wealth and labor income (in total wealth) are based on data realized
subsequent to the prediction period. However, it has poor predictive power when constructed using
quantities available at the time of prediction.
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4.1 CAPM
Table 2 summarizes the FamaMacBeth coefficient estimates for the
cross-sectional regressions with risk-adjusted returns as the dependent
variable and excess market return as the risk factor. In panel A, the
cross-sectional regressions are conducted at a monthly frequency. The
second column presents the fixed-beta results, in column three, we scale
betas by size and the book-to-market ratio, in column four, betas are
scaled by the default spread, and in the last column, the scaling is as
described in Equation (7).
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Asset Pricing Models and Financial Market Anomalies
Table 2
FamaMacBeth regression estimates with excess market return as the risk factor
1017
The Review of Financial Studies / v 19 n 3 2006
Table 2
(continued)
Unscaled Size + BM def (Size + BM) def
This table presents the time-series averages of individual stock cross-sectional OLS regression coefficient
estimates. Panel A (B) presents the monthly results for NYSE-AMEX and NASDAQ (NYSE-AMEX)
stocks over 450 months from July 1964 through December 2001. In the second column, the dependent
variable is the excess return risk-adjusted using the excess market return as the risk factor. In the third, the
dependent variable is the excess return risk-adjusted using the excess market return when the market beta
jt1 j1 j2 zt1 j3 j4 zt1 Sizejt1 j5 j6 zt1 BM jt1 ,
where z is the default spread. Size represents the logarithm of market capitalization in billions of dollars.
BM is the logarithm of the book-to-market ratio with the exception that book-to-market ratios greater
than the 0.995 fractile or less than the 0.005 fractile are set equal to the 0.995 and the 0.005 fractile
values, respectively. NYTURN (NASDTURN) is turnover of NYSE-AMEX (NASDAQ) stocks.
RET2-3, RET4-6, and RET7-12 are the cumulative returns over the second through third, fourth
through sixth, and seventh through twelfth months before the current month or quarter respectively.
2 is the time-series average of the monthly adjusted R2 . The t-statistics in parenthesis use standard
R
errors as per Shanken (1990), and the t-statistics in the square brackets use standard errors as in
Jagannathan and Wang (1998). All coefficients are multiplied by 100.
1018
Asset Pricing Models and Financial Market Anomalies
7
We present the OLS, Shanken (1992) corrected and Jagannathan and Wang (1998) corrected t-statistics.
In general, the Jagannathan and Wang (1998) corrected t-statistics are higher than those using the
Shanken correction and are often higher than the OLS t-statistics.
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The Review of Financial Studies / v 19 n 3 2006
Table 3
FamaMacBeth regression estimates with excess market return, SMB, and HML as risk factors
1020
Asset Pricing Models and Financial Market Anomalies
Table 3
(continued)
Unscaled Size + BM def (Size + BM) def
This table presents the time-series averages of individual stock cross-sectional OLS regression coefficient
estimates. Panel A (B) presents the monthly results for NYSE-AMEX and NASDAQ (NYSE-AMEX)
stocks over 450 months from July 1964 through December 2001. In the second column, the dependent
variable is the excess return risk-adjusted using the FamaFrench (1993) factors. In the third, the
dependent variable is the excess return risk-adjusted using the FamaFrench factors with loadings scaled
by size and book-to-market ratio. In the fourth, default spread (yield differential between BBB- and
AAA-rated bonds) is used as a scaling variable. In the last column, each of the factor loadings of stock j is
modeled as
jt1 j1 j2 zt1 j3 j4 zt1 Sizejt1 j5 j6 zt1 BM jt1 ;
where z is the default spread. Size represents the logarithm of market capitalization in billions of dollars.
BM is the logarithm of the book-to-market ratio with the exception that book-to-market ratios greater
than the 0.995 fractile or less than the 0.005 fractile are set equal to the 0.995 and the 0.005 fractile
values, respectively. NYTURN (NASDTURN) is turnover of NYSE-AMEX (NASDAQ) stocks.
RET2-3, RET4-6, and RET7-12 are the cumulative returns over the second through third, fourth
through sixth, and seventh through twelfth months before the current month or quarter, respectively.
2 is the time-series average of the monthly adjusted R2 . The OLS t-statistics are presented under the
R
coefficient estimates, t-statistics in parenthesis use standard errors as per Shanken (1990), and the
t-statistics in the square brackets use standard errors as in Jagannathan and Wang (1998). T-statistics
in curly brackets are unadjusted by a potential error-in-variables bias. All coefficients are multiplied
by 100.
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The Review of Financial Studies / v 19 n 3 2006
Also, observe from Table 2 that scaling the market beta by size and book-
to-market does not capture any of the size and book-to-market effects. It is
only when the excess market return is augmented by SMB and HML and
when the factor loadings on SMB and HML are scaled by macroeconomic
and firm-level variables, does the impact of size and book-to-market dis-
appear.8 Note also from the unconditional FamaFrench model that SMB
and HML do not capture the size and book-to-market effects in individual
stock returns, even when Fama and French (1996) show that SMB and
HML capture the size and book-to-market effects on portfolio returns.9
Why does the conditional FamaFrench model capture the impact of
MKT beta
1
SMB beta
beta
0.5
HML beta
-0.5
64:07 68:07 72:07 76:07 80:07 84:07 88:07 92:07 96:07 00:07
Figure 1
The figure plots the cross-sectional averages of the FamaFrench factor loadings over the sample period
when beta is scaled by firm-level size and book-to-market and by the default spread.
8
We have also conditioned the excess market return and the FamaFrench factors by turnover and the
past twelve-month returns to find that the strong impact of turnover and momentum on the cross-section
of individual stock returns remains undiminished.
9
Even Fama and French (1993) use the Gibbons, Ross, and Shanken test statistic to show that their three-
factor model rejects the null of (joint) zero intercepts for their test portfolios (see page 41 in their paper).
However, they do argue that the rejection is marginal and suggest that the model can be useful in practical
applications.
1022
Asset Pricing Models and Financial Market Anomalies
10
The business cycle expansions and contractions are dated by the NBER and can be obtained from http://
www.nber.com/cycles/cyclesmain.html.
11
We thank Lubos Pastor for providing this factor.
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The Review of Financial Studies / v 19 n 3 2006
Table 4
FamaMacBeth regression estimates with excess market return, SMB, HML, and liquidity as risk factors
1024
Asset Pricing Models and Financial Market Anomalies
Table 4
(continued)
Unscaled Size + BM def (Size + BM) def
This table presents the time-series averages of individual stock cross-sectional OLS regression coefficient
estimates. Panel A (B) presents the monthly results for NYSE-AMEX and NASDAQ (NYSE-AMEX)
stocks over 450 months from July 1964 through December 2001. The dependent variable in the second
column (unscaled) is the excess return risk-adjusted using the FamaFrench (1993) factors augmented by
the PastorStambaugh (2003) liquidity factor. In the third, the dependent variable is the excess return
risk-adjusted using the above factors with loadings scaled by size and book-to-market ratio. In the fourth,
default spread (yield differential between BBB- and AAA-rated bonds) is used as a scaling variable. In the
last column, each of the factor loadings of stock j is modeled as
jt1 j1 j2 zt1 j3 j4 zt1 Sizejt1 j5 j6 zt1 BM jt1 ;
where z is the default spread. Size represents the logarithm of market capitalization in billions of dollars.
BM is the logarithm of the book-to-market ratio with the exception that book-to-market ratios greater
than the 0.995 fractile or less than the 0.005 fractile are set equal to the 0.995 and the 0.005 fractile
values, respectively. NYTURN (NASDTURN) is turnover of NYSE-AMEX (NASDAQ) stocks.
RET2-3, RET4-6, and RET7-12 are the cumulative returns over the second through third, fourth
through sixth, and seventh through twelfth months before the current month or quarter, respectively.
2 is the time-series average of the monthly adjusted R2 . The t-statistics in parenthesis use standard
R
errors as per Shanken (1990), and the t-statistics in the square brackets use standard errors as in
Jagannathan and Wang (1998). All coefficients are multiplied by 100.
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The Review of Financial Studies / v 19 n 3 2006
While we have shown that the liquidity factor does not capture the
impact of turnover on expected returns, we would like to issue some
caveats. With turnover as an explanatory variable in the second
stage regression, we have focused on the impact of the level of
liquidity on returns, whereas the PastorStambaugh measure is
designed to capture the impact of liquidity risk.12 We have used the
value-weighted average return on stocks with high sensitivities to
liquidity less the value-weighted average return on stocks with low
sensitivities to liquidity as a proxy for the PastorStambaughs non-
traded liquidity factor. We do not rule out a case where a different
12
We have also experimented with a value-weighted portfolio that is long on high turnover decile stocks
and is short on low turnover decile stocks. This liquidity factor performed worse than the Pastor
Stambaugh factor in terms of capturing the impact of characteristics on the cross-section of returns.
13
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.
1026
Asset Pricing Models and Financial Market Anomalies
Table 5
FamaMacBeth regression estimates with excess market return, SMB, HML, and momentum as risk
factors
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Table 5
(continued)
Unscaled Size + BM def (Size + BM) def
This table presents the timeseries averages of individual stock crosssectional OLS regression coefficient
jt1 j1 j2 zt1 j3 j4 zt1 Sizejt1 j5 j6 zt1 BM jt1 ;
where z is the default spread. Size represents the logarithm of market capitalization in billions of dollars.
BM is the logarithm of the book-to-market ratio with the exception that book-to-market ratios greater
than the 0.995 fractile or less than the 0.005 fractile are set equal to the 0.995 and the 0.005 fractile
values, respectively. NYTURN (NASDTURN) is turnover of NYSE-AMEX (NASDAQ) stocks.
RET2-3, RET4-6, and RET7-12 are the cumulative returns over the second through third, fourth
through sixth, and seventh through twelfth months before the current month or quarter, respectively.
2 is the time-series average of the monthly adjusted R2 . The t-statistics in parenthesis use standard
R
errors as per Shanken (1990), and the t-statistics in the square brackets use standard errors as in
Jagannathan and Wang (1998). All coefficients are multiplied by 100.
1028
Asset Pricing Models and Financial Market Anomalies
Table 6
FamaMacBeth regression estimates with the market portfolio and the maximally correlated portfolio with
labor income as risk factors
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Table 6
(continued)
Unscaled Size + BM def (Size + BM) def
This table presents the time-series averages of individual stock cross-sectional OLS regression coefficient
estimates. Panel A (B) presents the monthly results for NYSE-AMEX and NASDAQ (NYSE-AMEX)
stocks over 450 months from July 1964 through December 2001. In the second column, the dependent
variable is the excess return risk-adjusted using the excess market return as well as a factor that is
maximally correlated with labor income growth. In the third, the dependent variable is the excess return
jt1 j1 j2 zt1 j3 j4 zt1 Sizejt1 j5 j6 zt1 BM jt1 ;
where z is the default spread. Size represents the logarithm of market capitalization in billions of dollars,
BM is the logarithm of the book-to-market ratio with the exception that book-to-market ratios greater
than the 0.995 fractile or less than the 0.005 fractile are set equal to the 0.995 and the 0.005 fractile
values, respectively. NYTURN (NASDTURN) is turnover of NYSE-AMEX (NASDAQ) stocks.
RET2-3, RET4-6, and RET7-12 are the cumulative returns over the second through third, fourth
through sixth, and seventh through twelfth months before the current month or quarter, respectively.
2 is the time-series average of the monthly adjusted R2 . The t-statistics in parenthesis use standard
R
errors as per Shanken (1990), and the t-statistics in the square brackets use standard errors as in
Jagannathan and Wang (1998). All coefficients are multiplied by 100.
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Asset Pricing Models and Financial Market Anomalies
Table 7
FamaMacBeth regression estimates with the maximally correlated portfolio with consumption growth as
the risk factor
This table presents the time-series averages of individual stock cross-sectional OLS regression coefficient
estimates. The sample contains 150 quarterly observations from the third quarter of 1964 through the
fourth quarter of 2001. In the second column, the dependent variable is the excess return risk-adjusted
using a factor that is maximally correlated with consumption growth. In the third, the dependent variable
is the excess return risk-adjusted using the above factor with loadings scaled by size and book-to-market
ratio. In the fourth, default spread (yield differential between BBB- and AAA-rated bonds) is used as a
scaling variable. In the last column, each of the factor loadings of stock j is modeled as
jt1 j1 j2 zt1 j3 j4 zt1 Sizejt1 j5 j6 zt1 BM jt1 ;
where z is the default spread, def. Size represents the logarithm of market capitalization in billions of
dollars, BM is the logarithm of the book-to-market ratio with the exception that book-to-market ratios
greater than the 0.995 fractile or less than the 0.005 fractile are set equal to the 0.995 and the 0.005
fractile values, respectively. NYTURN (NASDTURN) is turnover of NYSE-AMEX (NASDAQ)
stocks. RET2-3, RET4-6, and RET7-12 are the cumulative returns over the second through third, fourth
through sixth, and seventh through twelfth months before the current month or quarter, respectively. R 2
is the time-series average of the monthly adjusted R2 . The t-statistics in parenthesis use standard errors as
per Shanken (1990), and the t-statistics in the square brackets use standard errors as in Jagannathan and
Wang (1998). All coefficients are multiplied by 100.
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The Review of Financial Studies / v 19 n 3 2006
14
We present results for NYSE-AMEX and NASDAQ stocks. We have checked the results for NYSE-
AMEX stocks alone, and the results are qualitatively identical.
1032
Asset Pricing Models and Financial Market Anomalies
Table 8
FamaMacBeth regression estimates with excess market return, SMB, HML, liquidity, and momentum as
risk factors
This table presents the time-series averages of individual stock cross-sectional OLS regression coefficient
estimates. The sample includes NYSE-AMEX and NASDAQ stocks over 450 months from July 1964
through December 2001. The dependent variable in the second column (unscaled) is the excess return risk-
adjusted using the FamaFrench (1993) factors augmented by the PastorStambaugh liquidity factor and
a momentum factor obtained from Ken Frenchs web page. In the third, the dependent variable is the
excess return risk-adjusted using the above factors with loadings scaled by firm-level size, book-to-market
ratio, turnover, and the past twelve month returns. In the fourth, default spread (yield differential
between BBB- and AAA-rated bonds) is used as a scaling variable. In the last column, each factor
loading of stock j is modeled as
jt1 j1 j2 zt1 j3 j4 zt1 Sizejt1 j5 j6 zt1 BM jt1
j7 j8 zt1 Turnjt1 j9 j10 zt1 Momjt1 ;
where z is the default spread. Size represents the logarithm of market capitalization in billions of dollars.
BM is the logarithm of the book-to-market ratio with the exception that book-to-market ratios greater
than the 0.995 fractile or less than the 0.005 fractile are set equal to the 0.995 and the 0.005 fractile values,
respectively. Turn is turnover, and Mom is past twelve-month return. NYTURN (NASDTURN) is
turnover of NYSE-AMEX (NASDAQ) stocks. RET2-3, RET4-6, and RET7-12 are the cumulative
returns over the second through third, fourth through sixth, and seventh through twelfth months before
the current month or quarter, respectively. R 2 is the time-series average of the monthly adjusted R2 . The
t-statistics in parenthesis use standard errors as per Shanken (1990) and, the t-statistics in the square
brackets use standard errors as in Jagannathan and Wang (1998). All coefficients are multiplied by 100.
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The Review of Financial Studies / v 19 n 3 2006
rjt j0 j1 BCt1 j1 FFt j2 zt1 FFt j3 Sizejt1 FFt
j4 zt1 Sizejt1 FFt j5 BM jt1 FFt j6 zt1 BM jt1 FFt ujt ; 11
15
Similarly, Avramov and Chordia (2006) show that optimal portfolios based on time-varying alpha hold
stock with high previous year returns. That is, firm-level momentum is captured by alpha that varies with
aggregate macroeconomic variables.
1034
Asset Pricing Models and Financial Market Anomalies
Table 9
FamaMacBeth regression estimates with the FamaFrench three factor model and time-varying alpha
This table presents the time-series averages of individual stock cross-sectional OLS regression coefficient
estimates for NYSE-AMEX and NASDAQ stocks over 450 months from July 1964 through December
2001. The dependent variable in the cross-sectional regression is j0 ujt1 based on the following time
series regression
rjt j0 j1 BCt1 j1 FFt j2 zt1 FFt j3 Sizejt1 FFt j4 zt1 Sizejt1 FFt
j5 BMjt1 FFt j6 zt1 BMjt1 FFt ujt ;
where z is the default spread, and BC is a vector of business cycle variables including, default spread,
term spread, dividend yield, and the three-month Treasury bill yield. In the second column, the
dependent variable has no conditioning variables. In the third, size and book-to-market ratio are the
conditioning variables for the factor loadings. In the fourth, default spread is used as a scaling variable
for the betas. In the last column, scaling by size and book-to-market is allowed to vary over the business
cycle. Size represents the logarithm of market capitalization in billions of dollars. BM is the logarithm of
the book-to-market ratio with the exception that book-to-market ratios greater than the 0.995 fractile or
less than the 0.005 fractile are set equal to the 0.995 and the 0.005 fractile values, respectively. NYTURN
(NASDTURN) is turnover of NYSE-AMEX (NASDAQ) stocks. RET2-3, RET4-6, and RET7-12 are
the cumulative returns over the second through third, fourth through sixth, and seventh through twelfth
months before the current month or quarter, respectively. R 2 is the time-series average of the monthly
adjusted R2 . The t-statistics in parenthesis use standard errors as per Shanken (1990), and the t-statistics
in the square brackets use standard errors as in Jagannathan and Wang (1998). All coefficients are
multiplied by 100.
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5. Conclusions
This article develops and applies a framework in which to examine
whether the predictive ability of size, book-to-market, turnover, and
past returns is explained by various asset pricing models. Following
recent developments in economic theory, our framework allows factor
loadings in first-pass time series regressions to change with firm-level size
and book-to-market as well as business cycle-related variables. Risk-
adjusted returns based on the first-pass regressions are then regressed
on size, book-to-market, turnover, and prior returns. If the predictive
power of such firm-level variables is explained by asset pricing models,
then they should be statistically insignificant in the second-pass cross-
sectional regressions.
16
Note that the difference in the signs and significance of some of the variables in Table 9 as compared to
(say) Table 3 should not be surprising. The dependent variable in the cross-sectional regressions is
different. In Table 3, the dependent variable is the risk-adjusted return while in Table 9 the dependent
variable is the asset mispricing purged of the business cycle variation.
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Asset Pricing Models and Financial Market Anomalies
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