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Momentum Effect Differs Across Stock Performances: Chinese Evidence

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  • The Chinese University of Hong Kong Shenzhen

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Prior empirical studies find positive and negative momentum effect across the global nations, but few focus on explaining the mixed results. In order to address this issue, we apply the quantile regression approach to analyze the momentum effect in the context of Chinese stock market in this paper. The evidence suggests that the momentum effect in Chinese stock is not stable across firms with different levels of performance. We find that negative momentum effect in the short and medium horizon (3 months and 9 months) increases with the quantile of stock returns. And the positive momentum effect is observed in the long horizon (12 months), which also intensifies for the high performing stocks. According to our study, momentum effect needs to be examined on the basis of stock returns. OLS estimation, which gives an exclusive and biased result, provides misguiding intuitions for momentum effect across the global nations. Based on the empirical results of quantile regression, effective risk control strategies can also be inspired by adjusting the proportion of assets with past performances.
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A Quantile Regression Analysis of Momentum
Effect in Chinese Stock Market
Sibo Liu
Business School
China University of
Political Science and Law
Beijing, 100088, China
Email: lsbxjtu@gmail.com
Zhaoyuan Li
Center for Applied Statistics
School of Statistics
Renmin University of China
Beijing, 100872, China
Email: lzyruc@gmail.com
Maozai Tian
Center for Applied Statistics
School of Statistics
Renmin University of China
Beijing, 100872, China
Email: mztian@263.net
Abstract—Empirical studies of momentum effect find mixed
results all across the global nations. In this paper, we apply
quantile regression approach to analyze the momentum effect in
context of Chinese stock market. Empirical results suggest that
the momentum effect in Chinese stock is not stable, while OLS
estimation results are misleading. Negative momentum effect in
short and medium horizon (3 months and 9 months) grows with
quantile of stock return increasing. Positive momentum effect can
be observed in long horizon (12 months) which also rises over
the high performing stocks. Based on our study, effective risk
control strategies can be inspired in Chinese stock market.
I. INTRODUCTION
In finance, the unpredictability of stock gains is supported
by traditional efficient-market hypothesis (EMH) which argues
investors cannot consistently achieve returns in excess of
average market returns on a risk-adjusted basis (Fama, E. F.
1970[1]). Extensive literature, however, shows that in a proper
term lagged cumulative stock return, to some extent, can ex-
plain the motion of stock returns. This phenomenon, as one of
the main interests of behavioral finance, is called ”momentum
effect”(L. Chan, N. Jegadeesh and J. Lakonishok,1995[2]).
Positive momentum effect means rising asset prices or returns
to rise further. If rising prices indicate a falling in the future,
negative momentum effect exists. Momentum effect is closely
bonded with investment strategy in financial practice. Momen-
tum and contrarian strategies are appropriate examples.
There are mainly two research branches for momentum
effect. First, portfolio-level analysis is applied extensively.
Consisted with high and low performing equities, equity
portfolios are considered by Jegadeesh, N. and S. Titman
(1993[3], 2001[4]), Antoniou, Lam and Paudyal (2007[5]) and
etc. Second, Brennan, Chordia and Subrahmanyam (1998[6])
and Chen (2003[7]) adopted firm-level momentum factors
for analyzing individual lagged stock returns as explanatory
variables to forecast future individual stock returns. In this
paper, we analyze the firm-level momentum in the framework
of quantile regression.
Empirical studies of momentum effect find mixed results
all across the global nations. For example, M. Hon and
I. Tonks(2001[8]) suggest that momentum is not a general
feature of the UK stock market, but is only apparent over
certain time periods; In the US financial market, Conrad and
Kaul (1998[9]) find that in the short-term (1-week) and the
long-term (24-36 months) negative momentum effects exist,
while in medium-term (3-12 months) momentum effects are
positive. In Chinese stock market, J. Kang, M. Liu and S.
Ni (2002[10]) observe the short-term (1-12 weeks) negative
momentum effects and positive medium-term (12-26 weeks)
momentum effects.
In this paper, we apply quantile regression approach to
analyze the momentum effect. To the best of our knowledge,
J. Lee and K. Chan et al (2010[11]) first introduce the quantile
regression idea to the analysis of Taiwan stock market. In
our work, we try to refine the model setting and the results
interpretation in context of Chinese stock market. There are
also several aspects which are different from the existing
literature. First, for the perspective of investment strategy,
the relation between different quantiles of equity returns and
degree of momentum effect is of great practical importance.
Moreover, by providing a wide spectrum of stock return
distribution, quantile regression may to some extent explain
the puzzling empirical results of momentum effect in global
financial markets. Second, through the ”share-split reform” in
2005, China’s stock market has experienced amazing growth
recently. As an emerging financial market, Chinese stock
market shows many unique characteristics to mature markets.
What we discover in this paper may represent a general feature
of financial conditions in developing countries.
The empirical results suggest that by traditional OLS es-
timation, momentum effect in one to three quarter-term is
negative and positive momentum effect is observed in the
four quarter-term. In the quantile regression framework, the
quantile of stock return in three and four quarter-term makes
significant difference. In three quarter-term, the coefficients of
momentum factor on a firm-level are decreasing across the
quantile of stock returns. That means high performing stocks
may indicate a strong negative momentum effect. However,
we find in four quarter-term, low performing stocks show
slight negative momentum, while high performing stocks are
of strong positive momentum effect. Hence, based on our
study, momentum effect in Chinese stock market is unstable
across different quantile of stock returns. And the reversal
phenomenon between three and four quarter-term is unlike
our prior knowledge. A risk control strategy can be inspired
through the results.
This paper proceeds as follow. Section II-A briefly intro-
duces the basic principles of quantile regression. Section II-
B documents the model and variables we use. Section III
describes Chinese data to be used in estimations. Section IV
reports the empirical results and related interpretation. Section
V concludes.
II. METHODOLOGY
A. Quantile regression
Quantile regression as introduced in Koenker and Bassett
(1978[12]) may be viewed as a natural extension of classical
least squares estimation of conditional mean models to the
estimation of an ensemble of models for conditional quantile
functions. Quantile regression methods offer a mechanism for
estimating models for the full range of conditional quantile
functions. As a result, quantile regression is capable of pro-
viding a more complete statistical analysis of the stochastic
relationships among random variables.
There are several equivalent mathematical definitions of
quantile regression and we document one of them below. A
check function (Koenker and Bassett, 1978[12]) is defined as:
min
βΘE{ρP(YX0β)|X=x}(1)
where IA(z)the usual indicator function of the set A,Θis a
parametric space for β.
ρp(z) = pzI[0,)(z)(1 p)I(−∞,0) (z)(2)
is called check function. In our analysis, the dependent vari-
able Yis the excess returns of individual stocks and the
independent variable Xdenotes the control variables as well
as momentum factors.
The restrictive assumption that the error terms are identi-
cally distributed at all points of the conditional distribution
of the dependent variable is avoided in quantile regression.
Hence, heterogeneity and the varying estimated slope param-
eters across different quantiles of the response variable dis-
tribution can be effectively solved. For modeling conditional
distribution of excess equity return conditional on momentum
factors, quantile regression is a proper strategy.
B. Model and variables
As explained above, we try to explore the behavior of the
excess return of individual stocks conditional on momentum
factors of different length terms. The basic regression equation
for the relation between momentum factor and stock returns
is as follows:
ERi,t =αi,T +
N
X
k=1
βi,tkMOMi,tk+εi,t ,
t= 1,2, . . . , T. (3)
Where ERi,t denotes the excess return of stock iat time t
which is calculated by
ERi,t =
n
Y
t0=1
(1 + ri,t0rfi,t0)1, t0= 1,2, . . . , n. (4)
Where ri,t0and rfi,t0are the raw return and risk-free rate
of day t0, respectively. nis the number of trading days in
time interval t. In our case, we adopt quarterly tfor time
interval. In particular, the raw return is defined by logarithmic
change as ri,t0= ln(pi,t0)ln(pi,t01), and we adopt three
month deposit rate as risk-free rate rfi,t0. Back in equation (3),
αi,T is the intercept item, MOMi,tkis klagged cumulative
stock return of stock i, denoting the firm-level momentum
factor at time tproposed by Brennan et al.(1998[13]) and
Chen (2003[14]) which is calculated by:
MOMi,tk=
k
Y
j=1
(1 + ERi,tj)1(5)
(note: indicator proposed by Brennan et al(1998[13]) and
Chen (2003[14]) does not consider the impact of risk-free rate.
In our analysis, we the remove the influence of risk-free rate
for .)
According to the time horizon of existing studies, we choose
N= 4 for momentum effect measuring. In other words, we
consider one quarter to four quarter cumulative stock return
to explain the stock return motion. εi,t are residuals.
In addition to controlling for other macroeconomic and firm-
level conditions, it is important to introduce some control
variables in our equation. Here, following the general factors
used by Carhart (1997[15]), market trend indicator, logarithm
of market value and book-to-market ratio are included to
extend the final estimation equation as:
ERi,t =αi,T +ϕi,T SCIi,t +γi,T ln MVi,t +ωi,T BMi,t
+
N
X
k=1
βi,tkMOMi,tk+εi,t , t = 1,2, . . . , T. (6)
Where SC Ii,t is the return of Shanghai Composite Index
which we adopt as the market trend indicator; ln MVi,t and
BMi,t denote the logarithm of market value and book-to-
market ratio of stock iat time t, respectively. The model can
thus be written as:
QERi,t (τ|x) = αi,t (τ) + γi,T (τ) ln MVi,tϕi,T SCIi,t
+ωi,T (τ)BMi,t +
N
X
k=1
βi,tk(τ)MOMi,tk
+εi,t(τ), t = 1,2, . . . , T. (7)
and can be estimated for any τ(0,1) by solving the equation
(1).
III. DATA
In order to capture the reliable information of momentum,
we remove the stocks which are suspended in a certain year for
more than cumulative 90 days in Chinese ”A” share. The final
TABLE I: Descriptive statistics
ER SCI lnMV BM MOM1 MOM2 MOM3 MOM4
Minimum -1.809 -0.139 12.427 0.000 -1.809 -2.126 -2.185 -2.649
1st Qu. -0.177 -0.030 14.401 0.497 -0.182 -0.255 -0.256 -0.360
Median 0.019 0.026 15.017 0.695 0.010 0.0400 0.058 0.029
Mean 0.049 0.015 15.143 0.683 0.044 0.084 0.134 0.181
3rd Qu. 0.252 0.064 15.733 0.891 0.246 0.361 0.438 0.558
Maximum 2.438 0.141 21.218 2.128 2.438 5.105 7.800 10.629
Std error 0.0025 0.0005 0.0077 0.0019 0.0025 0.0036 0.0046 0.0057
Std Dev. 0.340 0.074 1.072 0.264 0.341 0.497 0.634 0.788
Kurtosis 1.100 -0.747 1.197 -0.290 1.128 2.514 5.497 6.092
Skewness 0.515 -0.278 0.798 -0.257 0.543 0.872 1.385 1.616
Observations 19280 19280 19280 19280 19280 19280 19280 19280
0.0 0.2 0.4 0.6 0.8 1.0
0.6 0.2 0.2 0.6
(Intercept)
0.0 0.2 0.4 0.6 0.8 1.0
2.0 2.5 3.0 3.5 4.0 4.5
SCI
0.0 0.2 0.4 0.6 0.8 1.0
0.010 0.000 0.010 0.020
lnMV
0.0 0.2 0.4 0.6 0.8 1.0
0.20 0.10
BM
0.0 0.2 0.4 0.6 0.8 1.0
0.20 0.10 0.00
MOM1
0.0 0.2 0.4 0.6 0.8 1.0
0.10 0.06 0.02 0.02
MOM2
0.0 0.2 0.4 0.6 0.8 1.0
0.10 0.05 0.00
MOM3
0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.05 0.10 0.15
MOM4
Fig. 1: OLS and quantile regression estimates for momentum
effect model. The dots represents the 50 points estimates
according to quantile p ranging from 0.02 to 0.98. The grey
areas between the lower and upper confidence bound are 95%
confidence bands. The horizontal solid red line indicates the
ordinary least squares estimates of the mean effects with red
dots denoting 95% confidence bands. The solid grey line is
zero.
dataset consists of 964 individual stocks from January 2005
to December 2010. We calculate the indicators according to
the estimation model, for example quarterly excess return and
lagged cumulative factors. The descriptive statistics for our
simple are reported in Table 1. MOM1 to MOM4 in table 1
denote one to four lagged cumulative momentum factors. With
the increasing of lag number, the momentum factors exhibit
more right skewness.
IV. EMP IR IC AL R ES ULTS AND SLOPE EQUAL IT Y TEST
A. Empirical Results
Table 2 shows the comparison of quantile regression and
OLS results conditional on momentum factors as well as
0 0.2 0.4 0.6 0.8 1
0.1
0.05
0
0.05
0.1
0.15
quantile
slope
MOM1
MOM2
MOM3
MOM4
Fig. 2: Summary of quantile regression for momentum factors
other variables. The estimated standard errors are reported
in parentheses. The summary of OLS and quantile regression
estimates is plotted in Fig.1.
We first notice from the OLS results that all the factors
are significantly different from zero at 1% level except for
3rd lagged momentum which is significant at 10% level. The
trend of market return SCI and size measure of firm MV
are significantly positive to the returns of individual stocks,
while book-to-market ratios are observed to have a negative
contribution on stock return. These results all accord with
general economic intuitions. For momentum factors, we find
negative momentum effect in one, two and three quarter lagged
terms. The four lagged momentum factor, however, shows
positive influence. In other words, from the perspective of OLS
estimation, negative momentum effect does exist in the short
horizon, while in the long run positive momentum effect can
be observed. These results are in part different from J. Kang,
M. Liu and S. Ni (2002[10]).
On the other hand, from the quantile regression results, we
find more interesting feathers. The signs of control variables
SCI, MV and BM do not blur in quantile regression frame-
work. However, as to market trend indicator, high performing
stocks are more influenced by the trend of entire market. In
contrast, SCI indicator has less impact on motion of stock
returns at lower quantiles. For size indicator of firms lnMV,
positive magnitudes significantly drop in extreme quantiles of
stock returns. For the quantiles(0.1-0.3) of stock returns, book-
to-market are also connected to lower coefficients.
Next, we analyze the results of momentum factors in detail.
Estimation table shows that at quantile 0.1, two and three
lagged momentum factors are not significantly different from
zero. At quantile 0.3, in particular, it is also not significant for
three lagged momentum factor. Apart from these three unique
TABLE II: Empirical results
OLS Quantile()
Factors 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
SCI 2.998a2.165a2.358a2.680a2.854a3.030a3.167a3.280a3.524a3.902a
(0.029) (0.034) (0.035) (0.034) (0.034) (0.037) (0.038) (0.039) (0.043) (0.064)
MV 0.013a0.014a0.016a0.013a0.011a0.011a0.012a0.012a0.013a0.007a
(0.002) (0.003) (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) (0.003) (0.004)
BM 0.123a
0.182a
0.151a
0.115a
0.1036a
0.105a
0.104a
0.094a
0.086a
0.058a
(0.008) (0.010) (0.008) (0.007) (0.007) (0.007) (0.008) (0.010) (0.012) (0.016)
MOM1 0.066a
0.052a
0.035a
0.060a
0.067a
0.085a
0.086a
0.091a
0.090a
0.107a
(0.008) (0.012) (0.009) (0.008) (0.007) (0.008) (0.009) (0.010) (0.014) (0.023)
MOM2 0.0700a-0.017 0.038a
0.046a
0.049a
0.047a
0.049a
0.045a
0.052a
0.040a
(0.008) (0.014) (0.007) (0.006) (0.006) (0.007) (0.007) (0.008) (0.011) (0.019)
MOM3 0.013c0.007 0.016c0.003 0.011c
0.031a
0.049a
0.064a
0.073a
0.079a
(0.007) (0.016) (0.008) (0.006) (0.006) (0.007) (0.007) (0.008) (0.010) (0.017)
MOM4 0.030a
0.016c
0.008c0.013a0.027a0.045a0.058a0.069a0.086a0.102a
(0.005) (0.009) (0.005) (0.005) (0.005) (0.005) (0.005) (0.006) (0.007) (0.011)
intercept 0.107a
0.361a
0.311a
0.232a
0.148a
0.100a
0.061b-0.008 0.056 0.253a
(0.030) (0.040) (0.035) (0.028) (0.028) (0.029) (0.031) (0.034) (0.045) (0.053)
The estimated standard errors are reported in parentheses. a,b,cindicates
significance at the 1%, 5%and 10%level, respectively.
TABLE III: Slope equality test
Quantiles Variable Restr. Value Std. Error Prob.
0.2, 0.4 SCI 0.4964 0.0308 0.0000
lnMV 0.0049 0.0018 0.0061
BM 0.0476 0.0069 0.0000
MOM1 0.0324 0.0073 0.0000
MOM2 0.0111 0.0061 0.0686
MOM3 0.0267 0.0066 0.0001
MOM4 0.0349 0.0044 0.0000
0.4, 0.6 SCI 0.3129 0.0299 0.0000
lnMV 0.0012 0.0015 0.4442
BM 0.0000 0.0064 0.9942
MOM1 0.0189 0.0070 0.0067
MOM2 0.0004 0.0057 0.9374
MOM3 0.0378 0.0054 0.0000
MOM4 0.0307 0.0039 0.0000
0.6, 0.8 SCI 0.3575 0.0365 0.0000
lnMV 0.0007 0.0023 0.7630
BM 0.0172 0.0098 0.0800
MOM1 0.0038 0.0115 0.7393
MOM2 0.0027 0.0091 0.7641
MOM3 0.0238 0.0083 0.0041
MOM4 0.0286 0.0054 0.0000
ones, other factors are all significant across all quantiles we
consider. From Fig.1, we can discover that for one lagged
momentum factor, as a whole, in the condition of higher
quantiles of stock returns, higher coefficient can be observed.
This feather blurs for 2nd lagged momentum factor, which
indicates that different quantiles of stock returns make insignif-
icant difference for momentum effect. As to 3rd lagged factor,
apart from insignificant estimations of quantile 0.1 and 0.3, a
clearer trend similar to 1st lagged factor can be identified.
However, this rule is completely reversed in the circumstance
of 4th lagged momentum factor. As OLS shows, on the whole,
momentum effect on one year (12 months) has a significant
positive impact on individual stock returns. However, quantile
regression shows that in the condition of lower quantile (below
0.3), this effect is actually negative. As the quantile increases,
the positive influence grows. For comparison, we plot the
quantile regression results of lagged momentum factors in
Fig.2.
B. Slope equality test
In order to quantify whether coefficients are significantly
different across the neighboring quantiles, we use slope equal-
ity test. The restriction detail is b(τ=h)b(τ=k) = 0,
where b(τ=h)denotes the coefficient of a variable at quantile
h.hand kare neighboring quantiles. For simplicity, we
consider four quantiles, 0.2 to 0.8. Restriction value, standard
error and P-value are reported in table 3.
From table 3 we find that slopes of 3rd and 4th lagged mo-
mentum effect are significantly different across all the quan-
tiles we consider, while the slopes of 1st lagged momentum
are not significantly different between quantile 0.6 and 0.8. As
for slopes of 2nd lagged momentum factor, however, across
the quantiles they are not significantly different. The results
of slope equality test quantify the differences of momentum
effect and support the robustness of our findings in last section.
V. CONCLUSION
To sum up, there are mainly two findings. First, the mo-
mentum effect in Chinese stock is not stable. OLS estimation
results can be misleading. For example, at 0.1 and 0.3 quantile
of stock return, 3rd lagged momentum is insignificant. 0.2
quantile of stock return, 3rd lagged momentum effect is
positive, while OLS only gives a negative result. Moreover,
at quantile below 0.3 of stock return, a negative momentum
effect opposite to the OLS result exists. Hence, investment
strategy based on OLS estimation could be biased.
Second, we find distinct behaviors for different lagged
momentum factor across quantiles. Negative momentum effect
in short and medium horizon (3 months and 9 months) grows
with quantile increasing. On the other hand, long horizon
momentum effect (12 months) also rises over the high per-
forming stocks. According to the empirical results of quantile
regression, excess returns may be achieved by adjusting the
portions of assets with different past performance. Based on
our study, effective risk control strategies can be inspired.
ACKNOWLEDGMENT
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Past work suggests that momentum is among the most robust market anomalies, as well as momentum profitability concentrates in firms with high information uncertainty and high credit risk. This paper shows that such momentum concentrations naturally emerge in an equilibrium setting with mean-reverting dividend growth, persistent expected dividend growth, recursive preferences per Duffie and Epstein (1992), and leverage based on Abel (1999). In particular, when the representative agent is endowed with reasonable riskaversion and time preferences parameters, equilibrium momentum profitability is large in the interaction between high levered and risky cash flow firms. Momentum payoffs rapidly deteriorate and ultimately disappear as either leverage or cash flow risk diminishes.