Advances in Management & Applied Economics, vol.3, no.2, 2013, 179-192
ISSN: 1792-7544 (print version), 1792-7552 (online)
Scienpress Ltd
Threshold effects in the capital asset pricing model
using panel smooth transition regression (PSTR)
Evidence from net oil export and import groups
Chien-Chung Nieh 1 and Hsueh-Chu Yao 2
Abstract
In this study, we used the PSTR (panel smooth transition regression) model to
investigate the nonlinear relationship between beta (systematic risk) and returns
(world market excess returns) for net oil export and net oil import groups. We set
the volatility of world market excess return as the threshold variable and the
percentage changes of crude oil price and exchange rate as the control variables.
Our results support the use of a nonlinear model to elucidate the behavior both
groups. We found that all beta values are positive and higher in the low regime
(i.e., volatility of world market excess return is low) and lower in the high regime
(i.e., volatility of world market excess return is high). For the net oil export group,
the crude oil price change percentage is positive in the high regime, but the
exchange rate percentage change is positive in the low regime. For the net oil
import group, in both the low and high regimes, changes in crude oil price and
exchange rate have equally positive effects on the individual market excess return.
JEL classification numbers: G12, Q49, F3, C33
Keywords: CAPM, crude oil price, exchange rate, panel smooth transition
regression model (PSTR), Schwarz's inequality, Triangle inequality
1
2
Department of Banking and Finance of Tamkang University, Taiwane,
e-mail: niehcc@mail.tku.edu.tw
Department of Banking and Finance of Tamkang University, Taiwane,
e-mail: courtis_yao@hotmail.com
Article Info: Received : December 5, 2012. Revised : January 3, 2013
Published online : March 15, 2013
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Threshold effects in the capital asset pricing model using PSTR...
1 Introduction
The well-known and widely used capital asset pricing model (CAPM) has
been one of the most important models in finance over the last 40 years. Studies
using the CAPM have demonstrated that beta or systematic risk is the only
suitable risk measure, and additionally, that a positive linear relationship exists
between beta and expected returns.
Early empirical studies such as Fama and MacBeth (1973) tested the validity
of the CAPM using a two-step approach. 3 They support the applicability of the
CAPM for analyses of the U.S. stock market. However, a number of later studies
have indicated that beta is not significantly related to return. In particular, Fama
and Fench (1992) found that by analyzing a sample of U.S. stock returns data from
1963–1990, the results revealed not only a flat relationship between return and
beta, but also announced the death of beta. Subsequently, many studies have
shown that beta and return may not be related empirically because of bias created
by the combination of positive and negative returns. Thus, Pettengill et al. (1995)
used an alternative model to examine the validity of beta in measuring risk. In that
study, the authors assumed that when the realized market returns exceeded the
risk-free rate of interest (i.e., the realized market excess returns are positive or
up-market), a positive relationship should exist between beta and return. Similarly,
they also assumed that when the realized market excess returns are negative or
down-market, a negative relationship should exist between beta and return. Hence,
Pettengill et al. (1995) adjusted the approach used by Fama and MacBeth (1973)
and developed a conditional relationship between beta and realized returns by
separating periods of positive and negative market excess returns. The results of
this adjustment show that for the period 1936–1990, there is a significant positive
(negative) relationship between beta and realized returns when market excess
returns are positive (negative) in U.S. stock returns.
Following the statistical methodology by Pettengill et al. (1995), Fletcher
(2000) examined the conditional relationship between beta and returns in (a)
international stock returns from January 1970 to July 1998, and (b) the monthly
returns of MSCI equity indices of 18 developed markets. Fletcher (2000) found a
significant positive relationship in up-market months and a significant negative
relationship in down-market months. Moreover, the relationships appear to be
symmetric, and there is also a positive mean excess return on the world index.
These results are consistent with those obtained by Pettengill et al. (1995). In
addition, Tang and Shum (2003) employed the conditional framework to
international stock markets and examined the conditional relationship between
beta and returns for the period from January 1991 to December 2000. They also
found a significant positive relationship between beta and returns in up-markets
3
In the first step, a CAPM is used to estimate beta. In the second step, portfolio returns
are regressed on portfolio betas to test the relationship between beta and returns.
Chien-Chung Nieh and Hsueh-Chu Yao
181
(positive market excess returns); however, they found a significant negative
relationship in down-markets (negative market excess returns).
As mentioned, these results imply the existence of a nonlinear risk-return
relationship. Woodward and Marisetty (2005) applied a logistic smooth transition
autoregressive (STAR) model to examine the beta non-stationarity issue in the
Australian stock market. In doing so, they not only found evidence of a smooth
transition between regimes using the 4-month-lagged yield spread as the transition
variable, but also demonstrated the applicability of the STAR model. Similarly,
Chen and Huang (2007) investigated the relation between stock returns and the
world index for four Pacific Rim economies and found that the International
Capital Asset Pricing Model (ICAPM) betas displayed regime switching based on
underlying market volatility. Thus, the authors suggested that estimates of the
ICAPM should account for the changes in betas over time and over different
variance regimes. Finally, Woodward and Brooks (2009) examined the nonlinear
relationship between beta and returns in 39 U.S. industry portfolios. Their
empirical results also support the presence of a nonlinear relationship.
With the growing interest in CAPM issues, a number of studies have further
introduced alternate pricing factors, such as crude oil price and exchange rate, to
investigate the relationship between beta and return. Basher and Sadorsky (2006)
constructed an international multifactor model, which accommodated both
unconditional and conditional risk factors, to examine the relationship between oil
price risks and emerging stock market returns. Their results provide significant
evidence that oil price risks affect stock price returns in emerging markets and
suggest that the exchange rate risk factor is statistically non-significant.
Additionally, the authors found that changes in oil price volatility have an
asymmetric effect on stock returns. Nandha and Hammoudeh (2007) examined the
relationship between beta and return with oil price and exchange rate factors for
15 Asia-Pacific countries by using the international factor model. Similarly, they
indicated that oil price and exchange rate factors are both important for pricing
individual stock market returns. Ramos and Veiga (2011) analyzed the exposure
of the oil and gas industry to oil prices in 34 countries. Using a multifactor panel
model to estimate the oil and gas excess stock returns, they strongly supported the
view that oil price is a globally priced factor for the oil industry. Furthermore, beta,
exchange rate, and oil price factors also have a significant impact on the oil
industry’s excess returns.
Thus, following both the nonlinear frame and the concept of alternate pricing
factors shown in the literature, we apply the PSTR model, which was developed
by González et al. (2004, 2005), to investigate the influence and nonlinear
relationship between beta (systematic risk) and returns (world market excess
returns) for net oil export and import groups. In our study, the volatility of the
world market excess return is the threshold variable, and the percentage change of
crude oil price and the percentage change of exchange rate are the control
variables. Among panel data studies, Hansen (1999) initially introduced threshold
effects, which included using a panel threshold regression (PTR) model. This PTR
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Threshold effects in the capital asset pricing model using PSTR...
model assumes that a transition between different regimes depends on the value of
a selected threshold variable. Because the transition from one regime to another
displays a sudden jumping effect when using the PTR model, and because the
PTR model assumes that the threshold for regime switching is clearly defined and
distinguished, results drawn by the PTR model may not accurately represent the
market situation being investigated. To revise these drawbacks, we apply a PSTR
model. This model allows for a gradual adjustment of the intermediate regime as
the system moves from one regime to another among extreme cases.
The remainder of this paper is organized as follows: Section II introduces the
specifications of the PSTR model, and Section III presents a description of the
data used in this study. Section IV shows the empirical results, and Section Ⅴ
offers a conclusion.
2 Methodology
González et al. (2004, 2005) developed the general panel smooth transition
regression (PSTR) model, which is defined as follows:
yit =
µi + β 0' xit + β1' xit h(qit ; γ , c) + ε it
(1)
where i=1,…,N, and t=1,…,T. yit is the dependent variable, xit is the
k-dimensional vector of explicative variables, µi proxies the fixed individual
effect, and the error term is denoted by ε it . The transition function h(qit ; γ , c) is
Following
a continuous and bounded function of the transition variable qit .
the work of Granger and Teräsvirta (1993), González et al. (2004, 2005)
considered the following logistic transition function:
−1
m



with γ > 0 and c1 ≤  ≤ cm
(2)
h(qit ; γ , c) =+
1 exp  −γ ∏(qit − c j  
j =1

 

where c = (c1 , , cm )' is an m-dimensional vector of location parameters, and the
slope of transition function is denoted by γ , which proxies the smoothness of
transitions.
Generally, considering m = 1 or m = 2 is sufficient because these values
allow for the commonly encountered types of variation in the parameters. 4 For
m = 1 , the model implies that the two extreme regimes are associated with low and
high values of qit . When γ → ∞ , the logistic transition function h(qit ; γ , c)
4
m = 1 corresponds to a logistic function with an S-shape, and m = 2 corresponds to a
exponential function with a U-shape.
Chien-Chung Nieh and Hsueh-Chu Yao
183
becomes an indicator function I [ A] , which possesses a value of 1 when event A
occurs, and is otherwise 0. Thus, the PSTR model in (1) is reduced to Hansen
(1999)’s two-regime panel threshold model. For m = 2 , the transition function
has a minimum of (c1 + c2 ) 2 and possesses a value of 1 for both low and high
values of qit . Therefore, if γ → ∞ , the model becomes a three-regime threshold
model. Finally, for any value of m , the transition function (2) becomes constant
when γ → 0 , in which case the model is reduced to a homogenous or linear fixed
effects panel regression.
González et al. (2004, 2005) generalized the PSTR model to allow more than
two different regimes, and constructed the additive model as follows:
r
yit =
µi + β 0' xit + ∑ β 'j xit h j (qit( j ) ; γ j , c j ) + ε it
(3)
j =1
where the transition functions h j (qit( j ) ; γ j , c j ) , j = 1, , r , are of the logistic type
(2). 5
The PSTR model building procedure consists of specification, estimation,
and evaluation stages. Specification includes testing homogeneity and selecting
the transition variable qit . If the tests fail to show homogeneity, Eq. (2) is
employed to determine the appropriate form of transition function, that is, the
proper value of m . A nonlinear least square method is used to estimate the
parameters. During the evaluation stage, the estimated model is subjected to
misspecification tests to examine whether it provides an adequate data description.
The null hypotheses tested at this stage include parameter constancy, remaining
heterogeneity, and autocorrelation of the errors. Finally, the number of regimes in
the panel must be determined, meaning that a value must be selected for r in Eq.
(3). 6
3 Data
In this study, we set the volatility of world market excess return as the
threshold variable and the percentage changes of crude oil price and exchange rate
as the control variables to investigate the nonlinear relationship between beta
(systematic risk) and return (world market excess returns) among 14 emerging
markets. Next, according to data provided by the Energy Information
5
As Hansen (1999) highlighted, if m = 1 , qit( j ) = qit , and γ j → ∞ for all j = 1, , r ,
the model in (3) becomes a panel threshold regression (PTR) model with r + 1
regimes.
6
For a more detail, see González et al. (2004, 2005) or Colletaz and Hurlin (2006).
184
Threshold effects in the capital asset pricing model using PSTR...
Administration (EIA), 7 we divided all markets into two groups 8: oil exporters
(e.g., Colombia, Indonesia, Malaysia, Mexico, Russia, and Venezuela) and net oil
importers (e.g., India, Israel, the Philippines, South Africa, South Korea, Sri
Lanka, Taiwan, and Thailand). This study features a monthly sample period that
spans September to May of 2011, with 117 observations for each variable. Sample
markets information is shown in Table 1.
Table 1. Stock price indices and risk-free rate employed in the study
Market
Colombia
Indonesia
Malaysia
Mexico
Russia
Venezuela
India
Israel
Philippines
South
Africa
South
Korea
Sri Lanka
Taiwan
Thailand
Market index
IGBC INDEX
Market risk-free rate
1-month
interbank
offered rate
IDX COMPOSITE
3-month
interbank
offered rate
FTSE BURSA MALAYSIA 3-month
interbank
KLCI
offered rate
MEXICO IPC
90-day money market
RUSSIAN MICEX INDEX
90-day money market
VENEZUELA SE GENERAL 30-day money market
INDIA
BSE
(100) 3-month India T-bills
NATIONAL
TEL AVIV SE GENERAL
3-month Israel T-bills
PHILIPPINE SE I(PSEI)
3-month
Philippines
T-bills
FTSE/JSE
3-month South Africa
T-bills
KOREA SE COMPOSITE 3-month South Korea
(KOSPI)
T-bills
COLOMBO SE
3-month
interbank
offered rate
TAIWAN SE WEIGHTED
90-day money market
BANGKOK S.E.T.
3-month
interbank
offered rate
7
Source data are available from their homepage: <http://wia.doe.gov>.
8
If a country’s oil export volume is greater than its oil import volume, it is categorized as
a net oil export group; conversely, if a country’s oil import volume is greater than its
oil export volume, it is categorized as a net oil import group.
Chien-Chung Nieh and Hsueh-Chu Yao
185
To account for the volatility of world market excess returns, we used the
MSCI world index 9 and constructed a proxy of the moving-average standard
deviation for the percentage change of the world market excess return. 10 The
percentage change of crude oil price data in the model is represented by the
logarithmic change in West Texas Intermediate (WTI), priced in U.S. dollars per
barrel. The percentage change of exchange rate data for all currencies is defined as
the logarithmic change in currency rates against the U.S. dollar. Except for the
crude oil price data, all data are extracted from the DATASTREAM database. The
crude oil price data were obtained from the EIA database. Table 2 shows the
descriptive statistics of all variables.
Table 2: Summary statistics
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Sum
Sum Sq. Dev.
Observations
Cross
sections
IERit
0.881255
1.173347
38.79845
-31.61199
7.101660
-0.315026
5.047692
313.2681
1443.496
WERit
-0.333787
0.418272
13.48965
-22.44858
5.339821
-0.953404
5.318842
615.1334
-546.7429
VWERit
5.359381
3.940894
20.70533
0.451495
3.941785
1.951098
6.588259
1918.012
8778.665
∆OPit
∆ERit
1.229173
1.944766
27.34475
-42.93591
9.893556
-0.883681
5.488764
635.9202
2013.386
-0.001574
-0.005686
69.31588
-34.89809
4.054526
4.650950
76.27072
372312.2
-2.577990
82559.77
1638
46676.92
1638
25435.17
1638
160233.6
1638
26910.94
1638
14
14
14
14
14
Notes: In this table we report the descriptive statistics of monthly individual market
excess return ( ), monthly world market excess return ( ), monthly volatility of world
market excess return ( ), monthly percentage change of crude oil price ( ) and monthly
percentage change of exchange rate ( ). *** denote significance at 1% levels, respectively.
9
10
Following Tang and Shum (2003).
Following Nieh (2002), a time-varying proxy for σ t to measure the volatility of
world market excess return can be specified as:
0.5
m


σ it = (1 m)∑ (log WERt +i -1 - log WERt +i -2 ) 2  , where WER is the world market
i =1


excess return ( i =1, ,14 denotes the market and t denotes the time period) and
m = 3 considering seasonal considerations.
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Threshold effects in the capital asset pricing model using PSTR...
4 Empirical results
Table 3 shows the stationary results of each variable using LLC (Levin et al.,
2002) and IPS (Im et al., 2003) tests. 11 The results indicate that the null variable
among the non-stationary variables can be rejected for any series level, which
ensures an I(0)-type series.
Table 3: Panel unit root tests
IERit
WERit
VWERit
OPit
 ERit
LLC
IPS
Intercept
Intercept&Trend
Intercept
Intercept&Trend
-31.9166***
-34.9949***
-32.8845***
-33.2761***
-24.0072***
-23.9369***
-30.9666***
-31.1013***
-3.52020***
-2.83301***
-8.98482***
-7.96964***
-37.2324***
-41.6300***
-29.7843***
-29.7480***
-39.5655***
-44.5602***
-35.7457***
-36.6405***
Notes: In this table we report the descriptive statistics of monthly individual market
excess return ( IERit ), monthly world market excess return ( WERit ), monthly volatility of
world market excess return ( VWERit ), monthly percentage change of crude oil price
( ∆OPit ) and monthly percentage change of exchange rate ( ∆ERit ). LLC and IPS represent
the Levin et al. (2002) and Im et al. (2003) panel unit root techniques. The 10, 5, and 1%
critical values for LLC and IPS are: -1.65, -1.96, and -2.58. *, **, and *** denote
significance at 10%, 5% and 1% levels, respectively.
Next, we applied the PSTR model developed by González et al. (2004, 2005)
to not only (a) set the volatility of world market excess return ( VWERit ) as the
threshold variable and (b) the percentage change of crude oil price ( ∆OPit ) and the
percentage change of exchange rate ( ∆ERit ) as control variables, but also to define
the nonlinear relationship between beta ( β ) and individual market excess return
( IERit ), as follows:
IERit = µi + α 0 ∆OPit + α1∆ERit + α 2WERit
+ ( β 0 ∆OPit + β1∆ERit + β 2WERit )
g (VWERit ; γ , c) + ε it
11
LLC is a modified version of the LL (Levin & Lin 1992, 1993) panel unit root
technique.
(4)
Chien-Chung Nieh and Hsueh-Chu Yao
187
Before beginning the PSTR test, we examined a linear model and tested it
against a nonlinear model with one threshold. By rejecting the null hypothesis, a
single threshold model was estimated and tested against a double-threshold model.
This process continued until the hypothesis of no additional threshold was not
rejected. Table 4 shows the results of the linearity and specification tests for no
remaining nonlinearity.
Table 4: LM Test for Remaining Nonlinearity
Country
Number of Location
Parameters
Net Oil Export
Net Oil Import
m =1
m=2
m =1
m=2
=
H 0 : r 0=
vs H1 : r 1
17.091***
(0.000)
22.362***
(0.000)
40.909***
(0.000)
41.630***
(0.000)
=
H 0 : r 1=
vs H1 : r 2
0.750
(0.862)
21.148***
(0.002)
3.509
(0.321)
12.517*
(0.053)
=
H 0 : r 2=
vs H1 : r 3
-
24.004***
(0.001)
-
-
Table 5: Determination of the Number of Location Parameters
Number
Parameters
Country
of
Net Oil Export
Location
Net Oil Import
m =1
m=2
m =1
m=2
1
2
1
1
Residual Sum of Squares
25060
23806
28071
28084
AIC Criterion
3.6108
3.5896
3.4276
3.4313
Schwarz Criterion
3.6627
3.6869
3.4690
3.4779
Optimal
Number
Thresholds
*
r ( m)
of
The linearity tests clearly rejected the null hypothesis for linearity in both the
net oil exporting and net oil importing groups, but the null hypothesis of r = 1
was not rejected at 1% significance for m = 2 concerning the net oil import
group. In the next step, we selected the optimal number of location parameters for
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Threshold effects in the capital asset pricing model using PSTR...
the transition functions based on the Akaike criterion (AIC) and Schwartz’s
Bayesian (SBC) criterion. Consequently, we consider all groups with m = 1 .
These results are shown in Table 5. 12
Table 6: Parameter Estimates for the Final PSTR Models
Country
Net Oil Export
Net Oil Import
(m, r * )
(1,1)
(1,1)
Variables
Coefficient
estimate
t-statistic
Coefficient
estimate
t-statistic
α0
-0.0728
-0.5846
0.0575***
2.4088
α1
0.6760***
2.2021
-0.0127
-0.1545
α2
1.7030***
5.4551
0.9532***
15.5308
β0
0.2890
1.6066
0.2168***
3.1700
β1
-1.0620***
-2.2853
-0.3017
-1.4508
β2
-1.4147***
-3.6415
-0.7485***
-5.4687
c
2.2356
9.8750
γ
0.2512
1.3148
RSS
25060
28071
AIC
3.6108
3.4276
BIC
3.6627
3.4690
The parameter estimation results appear in Tables 6 and 7. For the net oil
export group, the relationship between the individual market excess return and
beta is shown to be positively related in two different regimes. The relationship is
higher in the low regime (threshold value ≤ 2.24%) and lower in the high regime
(threshold value>2.24%) . This suggests that a 1% increase in the world market
excess return leads to an incremental 1.7030% increase in the individual market
excess return in the low regime and a 0.2883% increase in the high regime. For
12
For more detail, see Enders (2004), pp. 69-70.
Chien-Chung Nieh and Hsueh-Chu Yao
189
the control variables, the percentage change of crude oil price is negatively related
(-0.0728%) to individual market excess returns in the low regime and positively
related (0.2462%) in the high regime. Conversely, the percentage change of
exchange rate is positively related (0.6760%) to individual market excess returns
in the low regime and negatively related (-0.3860%) in the high regime.
For the net oil export group, both the individual market excess returns and
beta are positive; moreover, they are higher in the low regime (threshold value <
9.88%) and lower in the high regime (threshold value > 9.88%) for the net oil
import group. The crude oil price percentage change is positively related in two
different regimes, where the percentage change of exchange rate is negatively
related.
Table 7: Estimation of Coefficients of Control Variables in PSTR Models
Country
Net Oil Export
Low regime
Variables
(c ≤ 2.24%)
Beta
1.7030
-0.0728
0.6760
∆ Crude Oil Price
∆ Exchange Rate
Country
High regime
(c >2.24%)
0.2883
0.2162
-0.3860
Net Oil Import
Low regime
High regime
(c >9.88%)
Variables
(c ≤ 9.88%)
Beta
0.9532
0.0575
0.2047
0.2743
-0.0127
-0.3144
∆ Crude Oil Price
∆ Exchange Rate
Our main empirical findings are as follows. First, we found that all beta
values are positive, and they are higher in the low regime and lower in the high
regime. This suggests that beta has asymmetric effects on excess returns in
individual markets. 13 Specifically, investors in stable circumstances (i.e., world
market excess return is less volatile) can earn more risk premium than in unstable
circumstances (i.e., world market excess returns are more volatile).
Second, the result of changes in crude oil price has a positive coefficient in
the net oil export group, which suggests that rising crude oil prices result in an oil
risk premium increase. This outcome is in line with actual economic behavior.
Generally, when oil-producing countries announce a reduction in oil production,
13
This finding is consistent with previous research. See Woodward and Marisetty (2005),
Chen and Huang (2007), and Woodward and Brooks (2009).
190
Threshold effects in the capital asset pricing model using PSTR...
the result is marked by market volatility. An oil price increase caused by a
decrease in oil supply is ultimately beneficial for a particular stock market. In
contrast, for the net oil-importing group, regardless of stable or unstable
circumstances, changes in crude oil price have a positive effect on the excess
return of individual markets. Furthermore, oil-dependent countries, oil producers,
and oil investors are more concerned with information concerning changes in oil
price because this information affects key production and investment decisions.
Therefore, when oil prices increase, investors demand a higher oil risk premium.
This is consistent with the evidence presented by Basher and Sadorsky (2006).
Finally, as Narayan and Narayan (2010) argued, an appreciation in the
exchange rate of an export dominant country reduces the competiveness of exports
and is, furthermore, disadvantageous to the domestic stock price. For an import
dominant country, an appreciation in the exchange rate caused by reducing input
costs increases domestic stock prices. Therefore, for the net oil export group, our
results demonstrate that changes in the exchange rate have a positive effect on
individual market excess returns under stable circumstances. The implication here
is that only when the exchange rate depreciates during stable conditions can an
exchange risk premium exist. Moreover, for the net oil import group, individual
markets experience exchange rate risk premiums, regardless of conditions.
5 Conclusion
In this study, we used the PSTR model, developed by González et al. (2004,
2005), to investigate the relationship between beta (systematic risk) and returns
(world market excess returns) for net oil export and import groups over the period
from September to May of 2011. We used the volatility of world market excess
returns as the threshold variable and the percentage change of crude oil price and
the percentage change of exchange rate as control variables in our analysis.
The results support a nonlinear relationship between systematic risk and
world market excess returns, based on the volatility of excess world market
returns. This suggests that beta has asymmetric effects on individual market
excess returns regarding two groups: net oil exporters and net oil importers.
Investors in the low regime (i.e., world market excess return is less volatile) can
take more risk premium than investors in the high regime (i.e., world market
excess returns are more volatile).
For the net oil export group, we find that changes in the crude oil price that
benefit individual market excess returns exist only in the high regime. In addition,
exchange rate fluctuations have a positive effect on individual market excess
returns in the low regime. Finally, for the net oil export group, the results indicate
that for both the low and high regime, changes in crude oil price and exchange rate
have a positive effect on individual market excess return.
Chien-Chung Nieh and Hsueh-Chu Yao
191
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