%Paper: ewp-fin/9307001
%From: choup@wuecon.wustl.edu (Pin-Huang Chou)
%Date: Thu, 22 Jul 1993 19:47:03 -0500 (CDT)
%Date (revised): Sun, 25 Jul 1993 00:53:10 -0500 (CDT)
%Date (revised): Mon, 26 Jul 1993 20:56:44 -0500 (CDT)
%Date (revised): Tue, 27 Jul 1993 22:52:04 -0500 (CDT)
%Date (revised): Fri, 15 Apr 1994 16:29:06 -0500 (CDT)
%Date (revised): Mon, 18 Apr 1994 17:01:55 -0500 (CDT)

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\title{A Further Re-Examination of the Contrarian Investment Strategy:
Evidence from Multivariate Tests\thanks{First version: November 16, 1992. We
are grateful to Professor Guofu Zhou for several helpful discussions and
comments. The comments from participants at the annual meetings of the
1994 Midwest Finance Association and the 1993 Southern Finance Association are
also acknowledged.}}
\author{Pin-Huang Chou\\ Department
of Economics \\Washington University\\One Brookings Drive \\St. Louis, MO 63130
\and Robert P. Parks \\
Department of Economics \\Washington
University\\One Brookings Drive\\St. Louis, MO 63130}
\maketitle
\begin{abstract}
%\baselineskip 8mm
This paper investigates the performance of the contrarian investment strategy
using the CAPM and APT. The results from multivariate tests
of structural changes based on the CAPM show that the systematic risks of
contrarian portfolios are not stable over time. In addition, the mean-variance
efficiency of two market indexes, equal-weighted and value-weighted, cannot be
rejected. This confirms Chan's (1988) findings that the abnormal returns
documented in previous studies can be attributed to changes in systematic
risks
of the contrarian stocks. Using the asymptotic principal components technique
to extract the APT factors and subsequently running regressions of contrarian
stock returns on those factors, we find that the APT models explain the
returns
of contrarian stocks very well. The empirical results suggest that the
contrarian investment strategy does not  outperform the market, and the CAPM
and the APT work equally well in evaluating the contrarian performance.\\
\end{abstract}
%\baselineskip 8mm

\newpage
%{\Large
%A Further Re-Examination of the Contrarian Investment Strategy:
%Evidence from Multivariate Tests}

\section{Introduction and Overview of the Literature}
Most previous studies have attributed the abnormal returns of contrarian
investment strategies to market overreaction or the inability of the
underlying model to correctly measure the systematic risk of the
contrarian portfolio [see De Bondt and Thaler (1985, 1987), Chan (1988),
Ball, Kothari and Shanken (1992), and Jones (1993)]. Lo and MacKinlay (1990),
on the other hand, assert that contrarian profits may not necessarily be
induced by
market overreaction, and they show that in theory the contrarian profits
can result from the cross-autocovariance among stocks. Using weekly return
data, they find that a major proportion of the contrarian abnormal returns
are attributed to the cross effect, rather than market overreaction. Based
on daily return data of stocks listed on the NYSE/AMEX, Jegadeesh and
Titman (1993) even find that a ``counter-contrarian strategy'' that buys
past winners and sells past losers generates positive returns over three-
and twelve-month holding periods. Using the equally weighted index as the
proxy for market portfolio in the Capital Asset Pricing Model (CAPM), Chan
(1988) finds that the risks of losers and winners are not constant over
time, and he asserts that ``only small abnormal returns'' are found after
the changes in risks are controlled. These empirical findings appear to
suggest that the performance of contrarian strategies does not persist
over time, and the performance of the contrarian strategy seems to depend
on the particular portfolios that are formed (i.e., based on different
time-interval, different holding periods, etc).\footnote {See also Fama
(1991) for a brief review. }

As is well known, in a ``perfect world'' of the CAPM, it is not possible
for an investor to construct a portfolio with a significant Jensen
performance measure. Significant (positive or negative) Jensen measures
may be due to the inefficiency of the market index used in empirical
studies or reflect
the fact that there are some factors that are not absorbed by the market
index. Basically, the contrarian strategy is viewed as a passive strategy
in the sense that no additional information other than historical return
data is required to determine the composition of the invested portfolio.
Thus, it is clear that such a strategy does not possess selectivity as
well as timing abilities, and should not be able to outperform the
market.\footnote {See Grinblatt and Titman (1990) for the definitions and
discussion of selectivity and timing abilities.} If the contrarian
strategy earns significant abnormal returns, given that the underlying
market index is mean-variance efficient, it is evident that the market is
not informationally efficient in the weak form.\footnote{See, for
example, Fama (1976) for a definition of the weak form market efficiency.
Note that there are two different concepts of efficiency here. The
information efficiency concerns the speed of stock prices adjusting to
market information, whereas the mean-variance efficiency, implied by the
CAPM, emphasizes the risk-return tradeoff under the assumption of perfect
market (perfect information). However, it is well known that the market
efficiency and the correctness of the underlying (equilibrium) model are
empirically indistinguishable. See Fama (1991) for a discussion. In this
study, we will assume that information efficiency holds so that the
mean-variance efficiency of market indexes can be tested.}

Capital market researchers and practitioners are often concerned with the
risk-return tradeoff and the equilibrium risk-return relationship in
capital markets. Using the standard event study method, De Bondt and Thaler
(1985) calculate the average cumulative abnormal residuals (CARs) of the
contrarian portfolio over a ``post-formation period,'' and they find that the
contrarian strategy earns a substantial positive abnormal return. However, it
should be noted that if there is significant changes in intercept and
systematic risk, the CARs will be significant in general. More specifically,
the average CAR over the observation period will be the sum of two
terms:
\begin{eqnarray}
CAR = \hat{\alpha}^e + \hat{\beta}^e \bar{R}_{m2}, \label{eq:car}
\end{eqnarray}
where $\hat{\alpha}^e$ and $\hat{\beta}^e$ are respectively the OLS estimates
of changes in intercept and changes in systematic risk; $\bar{R}_{m2}$ is the
average excess return of the underlying market index over the observation
period [see, for example, Chou (1993) for a detailed explanation]. Thus, it is
clear that if $\hat{\alpha}^e$ and/or $\hat{\beta}^e$ are significantly
different from zero, the CAR will always be significant. The sign of the CAR,
however, depends on the magnitude of each term in (\ref{eq:car}). In the case,
the CAR is no longer an appropriate measure of abnormal return because the
changes in intercept and systematic risk are ignored. Chan (1988) has
investigated the systematic risks of the ``extreme performers.'' Assuming that
the equal-weighted index is mean-variance efficient, he reports that both
winners and losers experience significant changes in systematic
risks, and the results from regressions show that the intercepts (i.e.,
the Jensen's performance measure) are not significantly different from
zero for most of the ``test periods.''\footnote{Chan (1988) studies
the period from 1926 to 1989. He divides the sample into eighteen 6-year
sub-periods; each sub-period is then divided into two periods: the first
three years being the ranking period and the remaining the test period. }
However, in all ranking periods these extreme performers are found to have
highly significant intercepts (abnormal returns). This suggests that
the market index he used may not be mean-variance efficient with respect to
the
contrarian stocks in the ranking periods, if the selection bias over
the ranking period can be ignored.\footnote{The reason is that
the CAPM asserts that in equilibrium the expected excess returns of {\it
all} stocks and that of the market index should have a linear
relationship with no intercept term, given that the underlying market
index is mean-variance efficient. Here an implicit assumption behind the
assertion is that the ``information efficiency'' also holds. See also a
brief discussion in footnote 3.} Consequently, concluding that contrarian
portfolios' systematic risks are not stable over time is somewhat
ambiguous because the systematic risk measures may not be appropriate when the
underlying market index is not mean-variance efficient. Recently, Ball,
Kothari and Shanken (1992) (hereafter BKS) have provided a thorough review
on the performance of this strategy, and they also address the potential bias
caused by the inefficiency of the underlying market index. In addition, they
argue that a major proportion of the abnormal returns are attributed to some
microstructure factors that affect the measurement of contrarian portfolio
returns and to the inaccurate measurement of zero-beta returns. Still, in
several sub-periods of their study, the regression analysis based on the CAPM
specification yields significant regression intercepts. Therefore, we are
motivated to further re-examine the efficiency of different market indexes
over time and the stability of contrarian stocks' systematic risks.

Another issue addressed in the literature is the potential bias of the CAPM
systematic risk measure if the underlying true model is multifactor [see
Jones (1993)]. Somewhat surprisingly, the contrarian strategy performance has
not been examined using the APT.\footnote {Jegadeesh and Titman (1993) use a
one-factor model in their study. Their emphasis, however, is on the
decomposition and identification of sources of excess returns on the trading
strategies employed in their study, rather than  on the ``correctness'' of
asset pricing theories.} We thus examine whether the abnormal performance
disappears when the APT is used as the underlying pricing model. Equivalently,
we intend to examine whether those stocks that comprise the contrarian
investment strategy are properly priced by the APT.

The purpose of this paper is twofold: first, we re-examine the stability of
contrarian stocks' risks and the efficiency of the market indexes based on
the CAPM; and second, using the APT as an alternative model, we examine if
the contrarian stocks are properly priced by the APT. The first model is
essentially a model of structural changes, and is the same as Chan's (1988).
However, our tests differ from Chan (1988) in that our tests are based on a
multivariate framework so that we can test whether the Jensen measures
(intercepts) for all contrarian stocks equal zero simultaneously (i.e.,
$H_0:\;\alpha_i = 0\; \forall\; i$, a joint hypothesis), rather than test that
the summation of individual Jensen measures equals zero (i.e., $H_0: \; \sum
\alpha_i=0$), as in Chan (1988). It is possible that the former is rejected,
while the latter cannot be rejected. Roughly, one may say that our test has a
higher ``power,'' given the underlying null hypothesis being $H_0:\;\alpha_i =
0\; \forall\; i$. Testing the zero-intercept hypothesis is equivalent to
examining a necessary condition for a market index to be {\it ex ante}
mean-variance efficient [see, e.g., Gibbons, Ross and Shanken (1989)(hereafter
GRS)]. If we cannot reject the efficiency of the market index, it then is more
meaningful to look at the stability of the systematic risk measures. Otherwise,
looking at the stability of betas is ambiguous.

The second model based on the APT follows Connor and Korajczyk
(hereafter CK) (1986, 1988, 1990), in which they use asymptotic principal
components technique to extract the APT factors. We apply this method to see
if the APT can explain the performance of the contrarian investment
strategy.

It should be noted that our purpose in this paper is not to investigate the
factors that can be attributed to the behavior of systematic risk or factor
loadings. Instead, we intend to see which model, the CAPM or the APT, provides
a better account of the contrarian returns based on a multivariate statistical
framework.

Similar to most previous studies [e.g., BKS (1992) and Chan (1988)], our
study uses monthly return data. We investigate the performance of the
contrarian strategy for the period from 1940 to 1989, which is divided
equally into 10 5-year sub-periods. Following BKS (1992) closely, we
identify 50 winner and 50 loser stocks based on the average returns during
the first 30 months of each sub-period; then, their performance during the
following 30 months is evaluated.

Similar to Chan (1988), our empirical results also show
that the contrarian portfolio stocks have experienced significant changes in
systematic risks. Besides, the zero-intercept hypotheses cannot be
rejected for almost all sub-periods studied, using both the equal-weighted and
the value-weighted indexes. This result suggests that these two indexes are
indeed mean-variance efficient with respect to the contrarian stocks, and
the hypothesis that all contrarian stocks earn no abnormal return over the
testing period therefore cannot be rejected.

Moreover, as the result based on the CAPM, in most sub-periods under study, we
find that using the APT models the joint hypothesis that intercepts equal zero
for all contrarian stocks cannot be rejected. Thus, we conclude that the
contrarian investment strategy overall earns no abnormal returns.
%This
%also implies that the abnormal returns reported in previous studies may be
%due to the inefficiency of the market index used and/or instability of the
%index to absorb all variations in market factors.

The remainder of the paper is organized as follows: Section 2 describes
the data, and section 3 contains the methodology and empirical procedures.
The results are presented in section 4, while the last section contains
the conclusions.\\

%\newpage
\section{The Data}

We use stock returns from the monthly CRSP
(Center for Research in Security Prices) data base. The data for riskfree
rates are from the Fama tape. We
study the period from 1940 to 1989. This 40-year period is divided into
ten 5-year sub-periods. All stocks listed on NYSE/AMEX which do not have
missing data during each sub-period are included in the sample. Thus, each
sub-period contains 60 monthly observations for all stocks.\footnote {
Choosing 5-year time interval as study period is somewhat arbitrary.
Basically, the rationale for choosing 5-year data is that several studies
have suggested that systematic risks remain stable in a 5 year time
interval. See, for example, the discussions in MacKinlay (1987) and GRS
(1989).} We then divide each sub-period equally into two intervals: the
first 30 months are used as the ranking period and the remaining 30
months as the testing period. The ranking period is used to
identify winners and losers. Specifically, all stocks are ranked in ascending
order according to their (average) returns during the ranking period. The best
(worst) 50 performing stocks are identified as winners (losers). Table 1
presents the average returns of losers, winners, and the contrarian portfolio
that holds a long position on losers and a short position on winners.
The results indicate that the contrarian strategy earns positive average
returns for seven of the ten subperiods (see the last column in Table 1). It
appears that on average the performance of this particular strategy earns a
positive return. In all subperiods, loser (winner) portfolio has better (worse)
performance in the testing period than in ranking period (i.e., the second
column in Table 1 is less than the fourth and the third is greater than the
first). This conforms to the ``price reversal'' phenomenon documented in the
literature. The result also appears to support Lo and Mackinlay's finding
(1990) that individual stocks' monthly returns are (weakly) negatively
autocorrelated.
\begin{table}
\caption{The Average Monthly Returns of Loser and Winner
Portfolios in Each Subperiod}
\vspace{3mm}
%\baselineskip 5mm
\footnotesize
\begin{center}
\begin{tabular}{p{11cm}}
The average monthly returns of loser and winner portfolios are the
sample averages on raw return data. Each number in the cell below is the
average of 50 stocks over a 30-month period (ranking or testing period).
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c||r|r|r|r|r|}
\hline
{} & \multicolumn{2}{c|}{Ranking Period} & \multicolumn{2}{c|}{Testing
Period} & Contrarian \\ \hline
Subperiod & Loser & Winner & Loser & Winner & (Loser-Winner) \\ \hline
40-44 & -1.2278\% & 8.3067\% &  1.3241\% & -0.0583\% &  1.3824\% \\ \hline
45-49 & -1.1552\% & 0.8276\% &  3.9045\% &  0.6508\% &  3.2537\% \\ \hline
50-54 & -0.5920\% & 1.4705\% &  4.6697\% &  1.9714\% &  2.6983\% \\ \hline
55-59 & -1.5812\% & 2.4165\% &  3.8790\% &  1.2202\% &  2.6588\% \\ \hline
60-64 & -2.7116\% & 2.1566\% &  2.6401\% &  1.3054\% &  1.3347\% \\ \hline
65-69 & -2.6424\% & 5.5634\% &  2.6078\% &  3.9617\% & -1.3539\% \\ \hline
70-74 & -3.4524\% & 5.2680\% & -2.7479\% & -3.9317\% &  1.1838\%  \\ \hline
75-79 & -0.6226\% & 8.3997\% &  1.9615\% &  1.8318\% &  0.1297\%  \\ \hline
80-84 & -3.1638\% & 5.3486\% &  1.9425\% &  2.3344\% & -0.3919\% \\ \hline
85-89 & -2.1162\% & 6.4755\% & -0.3921\% & -0.0182\% & -0.3739\% \\ \hline
\end{tabular}
\end{center}
\normalsize
\clearpage
\end{table}

To form the portfolios used in our study, we again rank these extreme stocks in
ascending order based on their average return during the ranking period.
Then, the 50 losers (winners) are divided evenly into 5 loser (winner)
portfolios. That is, the first loser portfolio contains the first ten worst
stocks, the second loser portfolio contains the subsequent ten stocks,
and so on. In the same manner, the first ten of the bottom 50 stocks are
grouped as the sixth portfolio, then the following ten stocks comprise the
seventh portfolio, etc. The tenth portfolio contains the ten stocks that have
the best performance among all stocks during the ranking period. Thus, in
total we have 10 portfolios: 5 winners and 5 losers. In the following, we will
call these portfolios extreme portfolios or contrarian portfolios
interchangeably. Next, their
performance over the testing period is tested by the two models which we
present in the next section.\\


%\newpage
\section{The Models and Empirical Procedures}

This section summarizes the two models used in this study. The first model
is from Chou (1993). Some testable hypotheses with exact testing
statistics are derived there. The second model is proposed by CK (1988).

\vspace{10mm}
\subsection{Model 1: A Model of Structural Changes Based on The CAPM}

This model extends GRS's model (1989) to the case in which systematic
risks of portfolios or individual assets are allowed to change over two
different periods. Therefore, the model can be viewed as a model for
testing structural changes.

Assume that the returns of each portfolio follow the linear relationship
below:
\begin{equation}
R_{it} = \alpha_i + \beta_i R_{mt} + \alpha_i^e \delta_t + \beta_i^e R_{mt}
\delta _t + \varepsilon_{it} \hspace{8mm} i=1,...,N; t=1,...,T_1+T_2
\label{eq:capm}
\end{equation}
\noindent
where
\begin{center}
\begin{tabular}{rcp{4.5in}}
$R_{it}$ & = & excess return on portfolio i in period t,\\
$R_{mt}$ & = & excess return on the market index m in period t,\\
$\delta_t$ & = & a dummy variable, which equals 1 when $t > T_1$, and
       zero otherwise.\\
$\varepsilon_{it}$ & = & a normally distributed random disturbance term with
zero mean, conditional on $R_{mt}$;\\
$\alpha_i, \beta_i$ & : & parameters for portfolio i,\\
$\alpha_i^e, \beta_i^e$ & : & parameters measuring the effect captured by
the dummy variable.
\end{tabular}
\end{center}

\noindent
Let $T \equiv T_1+T_2$, where $T_1$ is the length of the
ranking period and $T_2$ the length of the testing period. Here,
$T_1=T_2=30$, and N=10 (10 contrarian portfolios). Also,
assume that the covariance matrix is stationary over time,
and the returns of all securities are only contemporaneously
correlated. More specifically, we assume that
\begin{equation}
cov(\varepsilon_{is}, \varepsilon_{jt}) = \left \{ \begin{array} {ll}
     \sigma_{ij}&\quad\mbox{if $s=t$} \\
     0&\quad\mbox{otherwise.} \end{array}
\right .
\end{equation}
%\newpage
%\baselineskip 10mm
We can rewrite (1) and (2) in a more compact form as follows:
\begin{equation}
\BPbf{ R} = \BPbf{ XB} + \BPbf{ E}, \hspace{12mm} \mbox{where} \; \BPbf{ E}
\sim N(0,
I_T \otimes \Sigma )
\end{equation}
\noindent
where
\begin{eqnarray}
\BPbf{ R} & = & \left ( \begin{array}{c} \BPbf{ R_1} \\\BPbf{ R_2} \end{array}
\right
) \;
\BPbf{ E} = \left ( \begin{array}{c} \BPbf{ E_1} \\\BPbf{ E_2} \end{array}
\right )
\nonumber\\
\BPbf{ X} & = & \left ( \begin{array}{cc} \BPbf{ X_1} & \BPbf{ 0} \\\BPbf{ X_2}
& \BPbf{
X_2} \end{array} \right ) \nonumber\\
\BPbf{ B} & = & \left ( \begin{array}{ccc} \alpha_1 & \ldots & \alpha_N \\
\beta_1 & \ldots & \beta_N \\ \alpha_1^e & \ldots & \alpha_N^e  \\
\beta_1^e &  \ldots & \beta_N^e
\end{array} \right )\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\BPbf{ R_1} & = & \left ( \begin{array}{cccc} R_{11} & R_{12} & \ldots &
R_{1N}\\
                                        \vdots & \vdots & \ddots & \vdots\\
                                  R_{T_1 1}& R_{T_1 2} & \ldots & R_{T_1 N}
 \end{array} \right ) \hspace{10mm}
\BPbf{ X_1} = \left ( \begin{array}{cc} 1 & R_{1p} \\  \vdots & \vdots \\
1 & R_{T_1 p} \end{array} \right ) \nonumber\\
\BPbf{ E_1}  & = & \left ( \begin{array}{cccc} \varepsilon_{11} &
\varepsilon_{12}
& \ldots & \varepsilon_{1N}\\ \vdots & \vdots & \ddots & \vdots\\
\varepsilon_{T_1 1}& \varepsilon_{T_1 2} & \ldots & \varepsilon_{T_1 N}
\end{array} \right ) \nonumber
\end{eqnarray}
\noindent
$R_2$, $X_2$, and $E_2$ are defined in the same manner.

The model can be estimated by ordinary least square, i.e., $\hat{B} =
(X'X)^{-1}X'R$, and the covariance matrix can be estimated as:
\begin{eqnarray}
\hat{\Sigma} = \frac {1}{T-4} (R-X \hat{B})'(R-X \hat{B})
\end{eqnarray}
%\baselineskip 8mm
Some testable hypotheses of interest are summarized in Table 2. As is well
known, the CAPM asserts that if a portfolio, say {\it p}, is
mean-variance efficient, then the expected excess returns of individual
assets, $E(R_i)$, and that of the portfolio, $E(R_p)$, will have the following
relationship:
\begin{eqnarray}
E(R_i) = \beta_{ip} E(R_p) \hspace{10mm} \forall~i,\nonumber
\end{eqnarray}
\noindent
which is referred to as the {\it security market line} (SML) in the
literature. This implies that if we regress the excess returns of
individual securities on those of an efficient portfolio, the resulting
intercepts should be indistinguishable from zero. Thus, it is clear that
hypotheses 1 and 2 can be used to test the {\it necessary condition} for
a given portfolio, in this study the equal-weighted or value-weighted
index, to be mean variance efficient over the ranking period and testing
period, respectively. The hypotheses can be tested based on a statistic
derived by GRS (1989). Specifically, by exploiting Hotelling $T^2$
statistic, they derive that the following statistic has a central F
distribution under the null hypothesis:
\begin{equation}
GRS = \frac {T_j (T_j - N-1)}{N(T_j-2)} \frac
{\hat{\alpha}'\hat{\Sigma}^{-1}\hat{\alpha}} {1 + \bar{r}_{mj}^2/S_{mj}^2}
\label{eq:grs}
\end{equation}
\noindent
where $\bar{r}_{mj}$ and $S_{mj}^2$ are the average return and sample variance
of the market index over period {\it j}, respectively. Note that all
statistics
in Table 2 are very similar to the statistic in (\ref{eq:grs}). In addition,
hypothesis 3 is designed to examine the changes in intercepts, while
hypothesis
4 examines the changes in systematic risks. It should be noted that
only under the condition that the underlying market index is efficient over
time does it make sense to test the stability of the systematic risk
measure, provided that all underlying assumptions are fulfilled.

\begin{table}[t]
\caption{Summary of Hypotheses Underlying Model 1}

\footnotesize
\begin{center}
\begin{tabular}{|l|l|l|}
\hline \hline
 Hypothesis       &  Statistics${}^*$  & Distribution${}^{**}$ \\ \hline
$H_1: \alpha=0$ & $\frac {T-N-3}{N(T-4)}m_{11}^{-1} \hat{\alpha}^{\prime}
\hat{\Sigma}^{-1}\hat{\alpha}$  & $F_{N, T-N-3}$\\ \hline
$H_2: \alpha + \alpha^e=0$ & $\frac {T-N-3}{N(T-4)} (m_{11} + 2m_{31}
+m_{33})^{-1} (\hat{\alpha}+\hat{\alpha}^e)'
\hat{\Sigma}^{-1}(\hat{\alpha}+\hat{\alpha}^e)$  & $F_{N, T-N-3}$ \\ \hline
$H_3: \alpha^e=0$ & $\frac {T-N-3}{N(T-4)}m_{33}^{-1} \hat{\alpha}^{e \prime}
\hat{\Sigma}^{-1}\hat{\alpha}^e$  & $F_{N, T-N-3}$\\ \hline
$H_4: \beta^e=0$ & $\frac {T-N-3}{N(T-4)}m_{44}^{-1} \hat{\beta}^{e \prime}
\hat{\Sigma}^{-1}\hat{\beta}^e$  & $F_{N, T-N-3}$\\ \hline
\hline
\end{tabular}
\end{center}
%\vspace{1mm}
\begin{center}
\begin{tabular}{p{13cm}}
${}^*$ $m_{ij}$ in each statistic is the $(i,j)$th element of $(X'X)^{-1}$.
Solving it explicitly, we have the following:\\
$m_{11} = T_1^{-1}(1+\bar{R}_{m1}^2/S_{m1}^2) $ \\
$m_{22} = T_1^{-1} S_{m1}^{-2} $ \\
$m_{33} = m_{11} + T_2^{-1}(1+\bar{R}_{m2}^2/S_{m2}^2) $ \\
$m_{44} = m_{22}+T_2^{-1} S_{m2}^{-2}$ \\
$m_{11}+2m_{31}+m_{33} = T_2^{-1}(1+\bar{R}_{m2}^2/S_{m2}^2) $ \\
Here $T_1 = T_2 = 30$. Subscript 1 denotes the ranking period, and subscript 2
denotes the test period. $\bar{R}_{mt}$ and $S_{mt}^2$ ($t = 1, 2$) are the
average and sample variance of the excess return on the market index $m$ over
the period $i$, respectively.
\end{tabular}
\end{center}
\normalsize
\vspace{5mm}
\end{table}

\normalsize
%\baselineskip 8mm
It is also worth mentioning that in effect we are testing if the
market index significantly deviates from the {\it ex post} efficient
frontier, spanned by different linear combinations of all sampled
portfolios and the underlying market index [see GRS(1989) for detail].
Thus, it is possible that the compositions of the ``true'' market
portfolio has changed, while the constant-weight market index still
maintains its efficiency (i.e., $H_4$ is accepted and $H_3$ is rejected).

\subsubsection*{A Comparison to Chan's (1988) Hypotheses}

The model specification (\ref{eq:capm}) is essentially the same as Chan (1988).
However, we have a different design on the tests of the contrarian profits.
Using the CAPM as the underlying model, Chan (1988) chooses the
equal-weighted index as the proxy for the market portfolio. Assuming that
the equal-weighted index is efficient and using the same model as equation
(\ref{eq:capm}), Chan (1988) tests the following three kinds of hypotheses for
abnormal returns:
\begin{eqnarray}
H_{w}: \sum_{winners} \alpha_i &=& 0 \nonumber\\
H_{l}: \sum_{losers} \alpha_i& =& 0 \nonumber\\
H_{c}: \sum_{losers} \alpha_i - \sum_{winners} \alpha_i& = &0\nonumber
\end{eqnarray}
\begin{figure}
\caption{An Example}
\vspace{7.2in}
\end{figure}
What Chan tests is a hypothesis that the sum of the individual stocks'
intercepts equals zero. However, as we have explained previously, the CAPM
asserts that {\it all} stocks should have zero intercepts, if the market index
is efficient. It is possible that in fact some stocks have significant
intercepts, but the sum of the intercepts are not significantly different from
zero. To see this, we draw a graph demonstrating the test in Chan (1988).
Figure 1 draws the joint parameter space for the intercept of loser portfolio
and the negative intercept of winner portfolio. The 45 degree dashed line
represents Chan's null hypothesis on the zero abnormal return of the contrarian
portfolio: $H: \alpha_c = \alpha_l + (-\alpha_w) = 0$, where $\alpha_c$ refers
to the Jensen's measure for the contrarian portfolio, and $\alpha_l$ and
$\alpha_w$ are the Jensen's measures for the loser and winner portfolios
respectively. For both possible estimates in Figure 1, one will not be able to
reject Chan's hypothesis because both the confidence intervals contain part of
the 45 degree line. However, here one will reject the null hypothesis that both
winner and loser portfolios have zero intercepts because none of the
confidence intervals contain the origin, which represents the joint hypothesis.
{}From this demonstration, one may say, roughly, that our test has a
higher ``power'' than Chan's, given the underlying null being the joint
hypothesis specified in our model.

In Chan's (1988) study, he finds that abnormal returns obtained by the
contrarian investment strategy
are due to the changes in systematic risk of winner and loser portfolios. His
empirical results also show that the regression intercepts are significant
in the ranking periods. This suggests that the equal-weighted index
may not be mean-variance efficient with respect to the contrarian
stocks.\footnote{Alternatively, if one adopts Chan's assumption on
the ex ante efficiency of the underlying index, the resulting intercepts
(Jensen's performance measures) can be viewed as abnormal returns
adjusted for risks. In this sense, then, contrarian portfolios are considered
to have abnormal performance. The sign of the abnormal return for the
contrarian strategy depends on the magnitude of the resulting intercepts
for the loser and winner portfolios.}
If it is so, the systematic risk measure obtained by using the excess returns
of the inefficient index is somehow inaccurate.\footnote{See, for example, the
arguments in Roll (1977). Of course, if we know the correlation between the
market index and the true market portfolio, then the resulting risk measure
can still be an indication of riskiness. However, this requires imposing a
strong assumption on the characteristics of missing assets and therefore is
considered implausible [see, e.g., Dybvig and Ross (1985).]}

However, since in each sub-period the contrarian stocks are chosen based on
their realized average returns over the ranking period which is also included
in our model here, the estimation of $\alpha$'s and $\beta$'s over the ranking
period may be biased due to a selection bias. In particular, in this case it
is possible that the covariance matrices are different between ranking and
testing periods. Therefore, we also use the standard GRS test (\ref{eq:grs}) to
test the zero-intercept hypothesis using the data over the testing periods
only. The results can be used to see if the empirical results of our model is
``robust.'' Note that in this case the GRS statistic will have F a
distribution with degrees of freedom 10 and 19, rather than 10 and 47. We use
equal-weighted and value-weighted indexes as proxies for the market
portfolio.\\

\subsection{Model 2: A Model Based on the APT}

This section summarizes the equilibrium APT and the asymptotic principal
components method proposed by CK (1988). The empirical procedure
implementing the test is also presented. Readers familiar with their studies
may skip this section.\\

\subsubsection{The Equilibrium APT}

CK (1988) assume that there are an {\it infinite} number of risky assets whose
returns are generated by the linear relationship below:
\begin{equation}
\tilde{r}_t = E(\tilde{r}_t) + B \tilde{f}_t +\tilde{\varepsilon}_t
\hspace{15mm} t=T_1+1,...,T_1+T_2;   \label{eq:apt}
\end{equation}
where
\begin{equation}
E(\tilde{\varepsilon} | \tilde{f}_t ) = 0,
E(\tilde{f}_t)=0, E(\tilde{\varepsilon}_t \tilde{\varepsilon}_t^{\prime}
)=V.
\end{equation}
\noindent
Here, $\tilde{r}_t$ is the $( \infty \times 1)$ vector of asset returns, $B$
is the $(\infty \times k)$ factor loading matrix, $\tilde{f}_t$ is a $(k
\times 1)$ vector of unobserved factors, and the number of factors, $k$, is
assumed known. The equilibrium APT implies:
\begin{equation}
E(\tilde{r}_t) = r_{Ft}\; e + B \gamma_t, \hspace{10mm}t=T_1+1,...,T_1+T_2,
\label{eq:apt1}
\end{equation}
where $e$ is a column vector of ones, and $r_{Ft}$ is the riskfree rate at
time $t$. Combining (\ref{eq:apt}) and (\ref{eq:apt1}), we have:
\begin{equation}
\tilde{r}_t - r_{Ft} = B(\gamma_t + \tilde{f}_t) +\tilde{\varepsilon}_t
\hspace{15mm} t=T_1+1,...,T_1+T_2; \label{eq:apt2}
\end{equation}

\noindent
With a finite sample of $n$ assets, (\ref{eq:apt2}) can be written as:
\begin{equation}
R^n = B^n F + \varepsilon^n,  \label{eq:apt3}
\end{equation}
where $R^n$ is an $(n \times T_2)$ vector of excess returns, $B^n$ is
the corresponding $(n \times k)$ submatrix of $B$, and $F \equiv
(F_1,...,F_t,...,F_{T_2})$, with $F_t \equiv \gamma_t + \tilde{f}_t$. Equation
(\ref{eq:apt3}) implies that performing regressions of $R^n$ on $F$
(and on a constant term) should yield insignificant intercepts, if $F$ is
known. Since the factors are not observable, CK (1988) propose an
asymptotic principal components method, suggested by Chamberlain
and Rothschild (1983), to find a linear transformation of the factors.
The method extracts the factors from the time series of all stock returns,
which a priori cannot be identified or related to any known factors such as
GNP or inflation. We summarize the method in the following.\\

\subsubsection{The Asymptotic Principal Components for Extracting the APT
Factors}

Define $\Omega \equiv (1/n)R^{n \prime} R^n$. Let $G_k$ denote the orthonormal
$(k \times T_2)$ matrix consisting of
the first $k$ eigenvectors of $\Omega$ . CK (1988) prove that $G_k$ is a
linear
transformation of the factors. Specifically, they prove that
\begin{eqnarray}
G_k = L^n F + \phi ^n \nonumber
\end{eqnarray}
\noindent
where $L^n$ is a nonsingular matrix for all $n$ and $plim_{n \to
{\infty}} \phi^n = 0$. Therefore, as $n$, the number of assets, approaches
infinity, $G_k$ will be a linear transformation of the true factors $F$. We
may
estimate the factor loadings of individual stocks by running time-series
regressions of their excess returns on $G_k$ and a constant term. The
(equilibrium) APT implies that the resulting intercepts should not be
significantly different from zero. Also, since a linear transformation of the
factors, the design matrix in a regression, will not affect $R^2$ and the
significance of regression coefficients, using $G_k$ yields the same result as
using the original factors. Assuming that the regression errors are jointly
normally distributed, the joint hypothesis that the intercepts are zero for
all
sampled stocks can be tested based upon the Hotelling $T^2$ statistic, which
is
distributed as an F variate if the hypothesis is true.\footnote
{Theoretically,
the APT does not require the assumption of normality. If the normality
assumption is not imposed, the resulting statistic will have a $\chi^2$
distribution asymptotically. However, CK (1988) still use F instead of
$\chi^2$
to test the joint hypotheses because the F test is more conservative, while
using $\chi^2$ tends to reject the null hypotheses too often. See footnote 6
of CK (1988) for a discussion on this issue.}\\


\subsubsection{The Empirical Procedure}

The empirical procedure is:\footnote{ All empirical tests are
carried out using SAS. The asymptotic principal components and most
testing statistics are calculated using SAS/IML procedure.}

\begin{enumerate}
\item Estimate $\hat{\Omega}$ using all stock returns over the testing-period
data ($T_2= 30$).

\item Calculate the first $k$ ($k$ = 1, 5, 10) eigenvectors. These $k$ vectors
are used to construct the orthonormal matrix, $G_k$ .\footnote {
CK (1988) propose an iterative procedure to extract the factors. However,
they also point out that there is no need to iterate the estimation
because a large sample will result in a good estimate of $G_k$ in the first
estimation. Here, we follow their procedure.}

\item Run a regression of each winner (and loser) portfolio on a vector
of ones, $e$, and $G_k'$. Then, check the significance of the
resulting intercepts.
\end{enumerate}

We explain step three and derive the statistics for testing
the zero-intercepts hypothesis in the appendix.\\

%\newpage
\section{Empirical Results}
\subsection{Empirical Results of the CAPM}

Table 3 presents the empirical results of the first model. It shows that
almost all hypotheses for all subperiods are rejected at the 1\% significance
level. Recall that rejection of hypotheses 1 and 2 implies that the
underlying market index, equal-weighted (EWI) or value-weighted (VWI), is
not mean-variance efficient with respect to the contrarian stocks. Thus,
the rejection of hypothesis 1 indicates that both indexes are not
mean-variance efficient with respect to the contrarian stocks over all
ranking periods. This conforms to Chan's empirical results that both
winner and loser portfolios have significant intercepts (i.e., ``abnormal
returns'') for all ranking periods. However, as we have mentioned in section
2, there is a potential problem of selection bias.

\begin{table}
\caption{Empirical Results of Model 1}
%\baselineskip 5mm
\footnotesize
\begin{center}
\begin{tabular}{p{14cm}}
This table presents the results of the four hypothesis tests based on
model 1, presented in Table 1. Both equal-weighted and value-weighted indexes
are used. The first statistic in each cell is calculated based on the formula
in Table 1, and the second statistic in parenthesis is the corresponding p
value. All p values are calculated based on an F distribution
with degrees of freedom 10 and 47.\\
\end{tabular}
\end{center}
\begin{center}
Value Weighted Index\\
\vspace{2mm}
\begin{tabular}{|c|r|r|r|r|}
\hline
Period & $H_1: \alpha = 0$ & $H_2: \alpha + \alpha^e =0$ & $H_3:
\alpha^e = 0$ & $H_4: \beta^e = 0$ \\ \hline
40-44 &  6.32566 &  0.61225  &   1.84292  &   18.0487   \\
 {}   &(4.6208E-6)&(0.7955443)&(0.0786876)& (6.316E-13)\\ \hline
45-49 &  7.0790  &  0.88029  &    3.8180  &   7.31245   \\
 {}   &(1.1614E-6)&(0.5576855)&(0.000806)& (7.6798E-7)\\ \hline
50-54 &  6.46711 &   0.52015 &    3.73872&    2.87372  \\
 {}   &(3.5455E-6)&(0.8672496)&(0.0009629)& (0.0070372)\\ \hline
55-59 &  3.39708 &  1.49541  &   2.61861  &   6.76788   \\
 {}   &(0.0020914)&(0.1711112)&(0.0128125)& (2.0363E-6)\\ \hline
60-64 &  9.37220 &  1.06316  &   4.95490  &   5.27711   \\
 {}   &(2.606E-8 )&(0.4086332)&(0.0000694)& (0.0000358)\\ \hline
65-69 &  8.28417 &   1.34421 &    5.71062 &    2.08789  \\
 {}   &(1.4707E-7)&(0.2357903)&(0.0000151)& (0.0446233)\\ \hline
70-74 &  11.6232 &   1.34376 &    8.53072 &    8.28837  \\
 {}   &(1.0247E-9)&(0.2360081)&(9.8309E-8)& (1.4605E-7)\\ \hline
75-79 &  4.55101 &   0.87117 &    3.35329 &     8.76498 \\
 {}   &(0.0001623)&(0.5656874)& (0.0023124)& (6.7446E-8)\\ \hline
80-84 &  8.54719 &    1.11660 &   4.35369 &  7.14574 \\
 {}   &(9.5721E-8)&(0.3702058)& (0.000248)& (1.0311E-6)\\ \hline
85-89 &  11.7041 &   1.30890 &    6.39829 &     3.58735 \\
 {}   &(9.192E-10)&(0.2535866)& (4.0318E-6)& (0.0013553)\\ \hline
\end{tabular}
\end{center}
%\normalsize
%\end{table}
%
%%\newpage
%\begin{table}
%(Table 3 continue)\\
\begin{center}
Equally Weighted Index\\
\vspace{2mm}
\footnotesize
\begin{tabular}{|c|r|r|r|r|}
\hline
Period & $H_1: \alpha = 0$ & $H_2: \alpha + \alpha^e =0$ & $H_3:
\alpha^e = 0$ & $H_4: \beta^e = 0$ \\ \hline
40-44 & 6.04124  &  0.80315   &   1.48152  &   13.84195      \\
 {}   &(7.9349E-6) & (0.6263996)  & (0.1763221) & (6.127E-11)   \\ \hline
45-49 & 7.18540  &  0.72344   &  3.58216  &   8.06797      \\
 {}   &(9.6103E-7) & (0.6984365)  & (0.0013714) & (2.1049E-7)   \\ \hline
50-54 & 7.42860  &  0.66284   &  3.83489  &   4.61534      \\
 {}   &(6.2671E-7) & (0.7522942)  & (0.0007762) & (0.0001416)   \\ \hline
55-59 & 3.27871  &  1.40800   &  2.50397  &  9.53297       \\
 {}   &(0.0027452) & (0.206304)  & (0.0167847) & (2.0381E-8)   \\ \hline
60-64 & 9.28927  &  1.45177   &  5.17698  &   7.37692      \\
 {}   &(2.9611E-8) & (0.187958)  & (0.0000439) & (6.859E-7)   \\ \hline
65-69 & 6.86939  &   1.41183 &   5.48995 &    5.21942     \\
 {}   &(1.6931E-6) & (0.2046374) & (0.0000234)& (0.0000403)   \\ \hline
70-74 & 11.3164  &  1.31677   &  8.14813  &   12.0858      \\
 {}   &(1.5544E-9) & (0.2495249)  & (1.8418E-7) & (5.54E-10)   \\ \hline
75-79 & 4.75099  &   0.86238 &   3.17310&     20.90000     \\
 {}   &(0.0001063) & (0.5734431) & (0.0035037)& (4.385E-14)   \\ \hline
80-84 & 9.03391  &   2.0848 &   4.82438&    9.49186     \\
 {}   &(4.4066E-8) & (0.044946)  & (0.0000911)& (2.1698E-8)   \\ \hline
85-89 & 11.6357  &  1.25257 &   5.97680&     5.08336     \\
 {}   &(1.0077E-9) & (0.2842456) & (8.9825E-6)& (0.0000532)   \\ \hline
\end{tabular}
\end{center}
\vspace{10mm}
%\clearpage
\normalsize
\end{table}

The results of testing hypothesis 2 in Table 3 overall quite support Chan's
findings. Chan (1988) reports that both winner and loser portfolios have
insignificant intercepts for most of the testing periods. His result seems
to suggest that the equal-weighted index may be efficient with respect to
these extreme portfolios over the testing periods. Our empirical
results also show that the efficiency of both indexes cannot be rejected for
almost all of the sub-periods.

In addition, the empirical results of hypothesis 3: $\alpha^e=0$ show that
there are also significant changes in intercepts. In fact, OLS regressions
show that in the ranking periods losers have negative intercepts, whereas the
winners have positive intercepts. This suggests that winners (losers) earn
positive (negative) ``abnormal returns'' even after adjusting for the
systematic risks, provided that the assumption of efficiency of the market
indexes is imposed. In the testing periods, we find that there is a
``reversal'' effect; the sign of changes in intercept, $\alpha^e$, is
positive for losers and negative for winners.\footnote{ We do not
report all the regression results here. They are, of course, available
from the author upon request.}

In addition, the results of testing hypothesis 4 indicate that there
are very significant changes in systematic risks for the contrarian stocks
over the ranking and testing periods, which confirm Chan's (1988) findings.

Thus, the results suggest that abnormal returns of the contrarian stocks are
eliminated through changes in intercepts and systematic risks.
There is ``market reaction,'' but no ``overreaction.'' The results seem to
indicate that there is a tendency for the
indexes to reduce their ``inefficiency'' with respect to the contrarian
stocks. Or alternatively, one may consider that the market reacts in a way such
that these extreme performers' abnormal returns are eliminated.

\begin{table}
\caption{Empirical Results of GRS test}
%\baselineskip 5mm
\footnotesize
\begin{center}
\begin{tabular}{p{11cm}}
   This table presents the results based on the GRS test [Gibbons, Ross,
and Shanken (1989)]. The p-values are calculated based on an F
distribution with degrees of freedom 10 and 19.\\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|rr|rr|}
\hline
{} & \multicolumn{2}{c|}{VWI}   &  \multicolumn{2}{c|}{EWI}  \\\hline
Period &  Statistic & (p-value) & Statistic & (p-value) \\ \hline
40-44  &  1.43389 & (0.23922) &  1.64292 & (0.16879) \\ \hline
45-49  &  0.79859 & (0.63200) &  0.79860 & (0.63200) \\ \hline
50-54  &  1.46508 & (0.22716) &  1.24246 & (0.32721) \\ \hline
55-59  &  1.61232 & (0.17766) &  1.56834 & (0.19123) \\ \hline
60-64  &  1.85914 & (0.11749) &  1.56544 & (0.19216) \\ \hline
65-69  &  1.37796 & (0.26238) &  1.52854 & (0.20440)\\ \hline
70-74  &  1.99083 & (0.09433) &  2.05127 & (0.08532) \\ \hline
75-79  &  0.99515 & (0.48043) &  1.08226 & (0.42120) \\ \hline
80-84  &  1.29556 & (0.30026) &  1.32323 & (0.28702) \\ \hline
85-89  &  2.13933 & (0.07378) &  2.38383 & (0.04953) \\ \hline
\end{tabular}
\end{center}
%\clearpage
\normalsize
\vspace{3mm}
\end{table}

In Table 4 we reports the empirical results of the standard GRS tests using
the testing period data only. Again,
we cannot reject the zero-intercept hypothesis for all subperiods for both
indexes at 5\% significance level, except that the hypothesis for the
equal-weighted index at the last sub-period is marginally rejected (p-value
0.04953). However, the resulting statistics are different from Table 3, which
may be due to changes in the covariance matrix. However, since the degrees of
freedom here are different, there is not much we can say about the
nonstationarity of the covariance matrix. The GRS test results, though,
suggest that our model is ``robust'' here.\\

%\vspace{10mm}
%\newpage
\subsection{Empirical Results of APT Model}

After deleting stocks that have missing data during each subperiod, the
remaining stock return data are used to extract the APT factors. The number of
stocks used for each of the ten sub-periods is, respectively, 716, 803,
945, 938, 931, 1507,  1872, 1723, 1724, and 1509. Note that in each subperiod
under study only the data over the testing period are used to extract the APT
factors. The empirical results are summarized in Table 5.
\begin{table}
\caption{Empirical Results of the APT}
%\baselineskip 5mm
\footnotesize
\begin{center}
\begin{tabular}{p{12cm}}
       This table presents the testing results of the APT models. The first
statistic in each cell is calculated based on the formula derived on section
3.2, and the second statistic in parenthesis is the corresponding p value. In
this case, the p values for 1-factor model are calculated based on an F
distribution with degrees of freedom 10 and 19. The p values for 2
factor model is based on F distribution with degrees of freedom 10 and
18, and for 5-factor model degrees of freedom are 10 and 15.\\
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|c|r|r|r|}
\hline
Period  &      1-factor &  2-factor &   5-factor   \\ \hline
42/7-44 &    0.9759    &  0.8405   &  0.6304   \\
{}      &   (0.4943)   & (0.5985)  & (0.7671)  \\  \hline
47/7-49 &    0.6350    &  0.8608   &  1.0315   \\
{}      &   (0.7668)   & (0.5825)  & (0.4635)  \\  \hline
52/7-54 &    0.75992    &  0.4723   &  0.5376   \\
{}      &   (0.76680)   & (0.8868)  & (0.8378)  \\  \hline
57/7-59 &    1.7364    &  1.5095   &  1.8347   \\
{}      &   (0.1443)   & (0.2147)  & (0.1396)  \\  \hline
62/7-64 &    2.3015    &  1.9652   &  1.9028   \\
{}      &   (0.0566)   & (0.1019)  & (0.1261)  \\  \hline
67/7-69 &      2.2387    &  1.6101&   2.0143  \\
{}      &     (0.0627)   &  (0.1821)&  (0.1068)  \\ \hline
72/7-74 &     1.9238     & 1.8423 &  1.5613  \\
{}      &    (0.1055)    & (0.1245) & (0.2109) \\ \hline
77/7-79 &     2.7144     & 2.2697 & 3.6082   \\
{}      &    (0.0293)  &  (0.0625) &(0.0126)  \\ \hline
82/7-84 &     2.3015     & 1.9652 &  1.9028  \\
{}      &    (0.0566)   &  (0.1019) &(0.1261) \\ \hline
87/7-89 &     1.8053     & 1.2849 &  1.5451    \\
{}      &    (0.1286)    & (0.3087) & (0.2162) \\ \hline
\end{tabular}
\vspace{10mm}

\begin{tabular}{|r|r|r|r|}
\hline
Significance level  &      1-factor &  2-factor &   5-factor   \\ \hline
1\%   &  0 & 0 & 0 \\\hline
5\%   &  1 & 0 & 1 \\\hline
10\%  &  4 & 1 & 1 \\\hline
\end{tabular}
\end{center}
\normalsize
\end{table}
%
%\subsubsection{The APT Models Without the January Effect}

Table 5 shows that none of the zero-intercept hypothesis can be rejected at
1\% significance level for all subperiods in the 1-, 2- and 5-factor APT
models. At 5\% significance The 1-factor APT model rejects the
zero-intercept hypothesis only for one sub-period, while four sub-periods have
significant intercepts at 10\% significance level.

The 2-factor APT seems to explain the contrarian performance very well. The
hypothesis is rejected only in one sub-period at 10\% significance level,
while none of the statistics have p-value smaller than 5\%.

The empirical results of the 5-factor APT model are very similar to those of
the 2-factor APT model. As in the 2-factor model, the result shows that the
zero-hypothesis for the eighth sub-period (7/77-79) is rejected (p-value
0.01262). The empirical results somehow confirm some findings in the APT
literature that the number of the APT factors is between 2 and 6.\footnote{We
have also examined if the January effect can account for the performance of the
contrarian stocks. The results are similar to the APT without including the
January effect. Also, overall we found no significant January effect for the
contrarian stocks. However, since each subperiod only contains 30 months, this
means that the dummy variable that represents the January observations only
takes value on for two observations. This may cause a problem in estimating
the regression coefficient and its standard deviation. Thus, we do not report
the results here. They are available from the authors.}\\

%\subsubsection{The APT Models With the January Effect}
%
%By including a dummy variable which take value one for January observations,
%we
%are able to test if the January effect can account for the performance of the
%contrarian portfolios, a problem having been of interest in the contrarian
%literature. As in the zero-intercept joint hypothesis in the
%standard APT model, we can test the joint hypothesis that the coefficients of
%the January dummy variable equal zero for all contrarian stock. The results
%are summarized in Table 6.
%
%\vspace{5mm}
%\begin{center}
%[Table 6 here]
%\end{center}
%\vspace{5mm}
%
%Table 6 shows that in 1-factor APT model three subperiods have significant
%January effect at 5\% significance level, and six sub-periods at 10\%
%significance level. Moreover, the zero-intercept hypothesis is rejected for
%two sub-periods at 5\% significance level.
%
%However, no significant intercept or significant January effect can be found
% in the 2-factor
%APT model, the result being similar to the APT model without including
%the
%January dummy variable. In the 5-factor APT model, only one sub-period has
%January effect, but only significant at 10\% level.
%
%Note that in the eighth sub-period (7/77-79) the 1- and 5-factor APT models
%fail to explain the contrarian performance, with or without including the
%January dummy variable. Thus the ``abnormal'' performance can not be explained
%by
%January observations. However, the 2-factor APT model seems to have captured
%the ``abnormal'' returns very well; the zero-intercept hypothesis and the
%hypothesis of no January effect cannot be rejected for all sub-periods.
%
%\newpage
\section{Conclusions}

In this study, we have reexamined the performance of the contrarian
investment strategy by introducing a model of structural changes based on
the CAPM and a model based on the APT. Our study differs from previous
studies mainly in two aspects. First, our test is based on a multivariate
framework. This allows us to test some joint hypotheses by further considering
the correlation among the contrarian stocks. First, we have found that for
all sub-periods in our study, the winner and loser stocks have experienced
significant changes in intercepts and systematic risks. Since the
zero-intercept hypothesis cannot be rejected for almost all sub-periods, the
abnormal returns documented in previous studies can be attributed to changes
in systematic risks and intercepts of contrarian stocks.
Second, we use the APT as the alternative pricing model, which so far has not
been done by other studies. We found that overall the APT models have
explained the contrarian performance very well. The empirical results based on
the 2-factor APT is especially satisfactory.

To conclude, we found that overall the contrarian investment strategy does not
earn no abnormal returns, and the CAPM and the APT models have explained its
performance equally well.

\newpage
\appendix
\section*{Appendix A}
This appendix derives the statistic for testing the zero-intercept
hypothesis described in section 3.2. The statistic we derive below is
actually very similar to the statistics described in section 7 of GRS
(1989), but differs in that here we are using ``principal components''
rather than a set of ``efficient portfolios.'' Let $R_c$ be the $(10 \times
T_2)$ matrix of excess returns of the 10 contrarian portfolios (5 winners
and 5 losers) over the testing period. Thus, in step three we run the
following regression:
\begin{equation}
R_c = \alpha_c e _{T_2}^{\prime} + B_c G_k + \varepsilon_c  \label{eq:a1}
\end{equation}
where
\begin{eqnarray}
\alpha_c & = & (\alpha_1,..., \alpha_{10})'\nonumber\\
e_{T_2} & = & (1,...,1)' \hspace{10mm} \nonumber
\end{eqnarray}
and $B_c$ is the corresponding $(10 \times k)$ factor-loading
matrix.\footnote[16]{ Here $B_c$ actually is the true factor-loading
matrix multiplied by a nonsingular matrix (i.e., $B_c \times L^{n-1}$).}
Therefore, equation (\ref{eq:apt2}) implies a testable
hypothesis: $\alpha_c = 0$, analogous to the zero-intercept
hypothesis in the CAPM test. This hypothesis can be viewed, roughly, as a
joint hypothesis of the validity of the APT and zero abnormal returns of
the contrarian portfolios. If the hypothesis is rejected, it may suggest
that the APT is not an appropriate pricing model, in spite of the fact
that the APT {\it does not} assure that {\it all} assets will be
correctly priced. However, if we fail to reject this hypothesis, it then
suggests that the contrarian strategies do not earn abnormal returns.
This hypothesis can also be tested based on a statistic similar to the GRS
test. We now derive the statistic for testing this hypothesis. Equation
(\ref{eq:a1}) can be written as the following:

\begin{equation}
\BPbf{ R}_c^{\prime} = \BPbf{ X}\BPbf{ \Gamma} + \BPbf{
\varepsilon}_c^{\prime},
\end{equation}
\noindent
where
\begin{eqnarray}
X & = & [e_{T_2} : \; G_k^{\prime}]_{T_2 \times (k+1)} \nonumber\\
\Gamma & = & [\alpha_c : \; B_c ]_{(k+1) \times 10} \nonumber
\end{eqnarray}
Thus, based on the assumptions above, we have the unbiased
estimators: $\hat{\Gamma} = (X'X)^{-1} X' R_c^{\prime}$.
Note here that using the OLS yields the estimators with desirable
small-sample properties, since in the case where explanatory variables are
identical across equations GLS method (or SUR method) yields the same
results as OLS. Thus, we have
\begin{equation}
\hat{\Gamma} \sim N( \Gamma, (X'X)^{-1} \otimes \Sigma ) \label{eq:a6}
\end{equation}
where $\Sigma$ is the variance-covariance matrix of the residuals. From
(\ref{eq:a6}), we have $\hat{\alpha}_c \sim N(\alpha_c, a_{11}\Sigma)$, where
$a_{11}$ is the (1,1) element of $(X'X)^{-1}$. In addition, the covariance
matrix can be estimated as:
\begin{equation}
\hat{\Omega} = \frac {1}{T_2 - k - 1} (R_c^{\prime} - X \hat{\Gamma}
)'(R_c^{\prime} - X \hat{\Gamma})
\end{equation}
$(T_2-k-1)\hat{\Omega}$ has a Wishart distribution, denoted as
$W_{10}(T_2-k-1,
\Sigma)$.\footnote {For an introduction
of the Wishart distribution and its properties, see, for example, Anderson
(1984) and Muirhead (1982).} It can be shown that $\hat{\alpha}_c$ and
$\hat{\Omega}$ are independent.
Then, by a theorem from Muirhead (1982, p.211), we have the following
statistic a standard Hotelling $T^2$, which has a central $F$
distribution under the null hypothesis:
\begin{equation}
J \equiv \frac {T_2-k-10}{10 a_{11} T_2} \hat{\alpha}_c^{\prime}
\hat{\Omega}^{-1} \hat{\alpha}_c \sim F(10, T_2-k-10)
\end{equation}
Note that the statistic $J$ has a very similar form as the GRS.



\newpage
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\end{document}
%\newpage
%\begin{center}
%Table 6. Empirical Results of the APT including January Effect
%\end{center}
%
%       This table presents the testing results of the APT models by including
%a dummy variable for capturing the January effect.
%The first statistic in each cell is calculated based on the formula derived on
%section 3.2, and
%the second statistic in parenthesis is the corresponding p value. In this
%case, the p values for 1-factor model are calculated based on an F
%distribution with degrees of freedom 10 and 18. The p values for 2
%factor model is based on F distribution with degrees of freedom 10 and
%17, and for 5-factor model degrees of freedom are 10 and 14.\\
%
%\begin{center}
%\begin{tabular}{|c|r|ll|ll|ll|}
%\hline
%{} & {}  &  \multicolumn{2}{c|}{1-factor}  &
%\multicolumn{2}{c|}{2-factor} & \multicolumn{2}{c|}{5-factor}\\ \hline
%Period  & Hypo. &  statistic &  p-value & statistic &  p-value & statistic &
%p-value \\ \hline
%42/7-44 & $\alpha=0$ & 1.0684 & (0.4346)& 0.4190& (0.9180)&   0.8063& (0.6260)
%\\
%{}      & $J=0$     & 3.0377 & (0.0194)& 1.0129& (0.4715)& 1.7475 &
%(0.1648)  \\ \hline
%47/7-49 & $\alpha=0$& 0.5308 & (0.8454)& 0.2548& (0.9837)&  0.5067
%& (0.8625) \\
%{}      & $J=0$     & 3.13595 & (0.0169)& 0.7762& (0.6505)& 1.0563 &
%(0.4504) \\  \hline
%52/7-54 & $\alpha=0$& 1.0208 & (0.4661)& 0.1565& (0.9975)&  0.5535
%& (0.8286) \\
%{}      & $J=0$     & 0.8046 & (0.6273)& 0.2751& (0.9786)& 0.5959 &
%(0.7929) \\  \hline
%57/7-59 & $\alpha=0$& 1.6673 & (0.1700)& 0.4428& (0.9042)&  1.6468
%& (0.1756) \\
%{}      & $J=0$     & 2.5811 & (0.0387)& 0.1590& (0.9973)& 1.994  &
%(0.1155) \\  \hline
%62/7-64 & $\alpha=0$& 1.4966 & (0.2233)& 0.4885& (0.8750) &    1.002
%& (0.4787) \\
%{}      & $J=0$     & 2.1101 & (0.0807)& 0.4275& (0.9132) & 1.7568
%& (0.1626) \\ \hline
%67/7-69 & $\alpha=0$& 1.8779 & (0.1215)& 0.5522& (0.8300) &   1.817
%& (0.1340)  \\
%{}      & $J=0$     & 1.6549 & (0.1692)& 0.8041& (0.6280) &
%2.1484 & (0.0928) \\ \hline
%72/7-74 & $\alpha=0$& 1.6543 & (0.1735)& 0.4585& (0.8945) &  1.3842 &
%(0.2670) \\
%{}      & $J=0$     & 1.495  & (0.2199)& 0.2091& (0.9921) & 0.1522
%& (0.9975) \\ \hline
%77/7-79 & $\alpha=0$& 2.733  & (0.0328)& 0.7304& (0.6877) &  3.4209
%& (0.0126) \\
%{}      & $J=0$     & 0.9550 & (0.5109)& 0.2306& (0.9886) & 0.7937
%& (0.6371) \\ \hline
%82/7-84 & $\alpha=0$& 2.7608 & (0.0315)& 0.6898& (0.7209) &  2.2554 &
%(0.0673) \\
%{}      & $J=0$     & 2.0400 & (0.0903)& 0.3513& (0.9519) & 1.649
%& (0.1904) \\ \hline
%87/7-89 & $\alpha=0$& 1.8337 & (0.1303)& 0.2964& (0.9724) &
%1.2038 & (0.3540) \\
%{}      & $J=0$     & 2.3445 & (0.0556)& 0.2547& (0.9837) & 0.4998
%&  (0.8632) \\ \hline
%\end{tabular}
%
%\vspace{10mm}
%
%\begin{tabular}{|r|r|r|r|r|}
%\hline
%Significance level & Hypo.  &      1-factor &  2-factor &   5-factor   \\
%\hline
%1\% & $\alpha=0$ &  0 & 0 & 0 \\
%{}  & $J=0$      &  0 & 0 & 0 \\ \hline
%5\% & $\alpha=0$ &  2 & 0 & 1 \\
%{}  & $J=0$      &  3 & 0 & 0 \\ \hline
%10\%& $\alpha=0$ &  2 & 0 & 1 \\
%{}  & $J=0$      &  6 & 0 & 1 \\ \hline
%\end{tabular}
%\end{center}

