Multiscale test of CAPM for three Central and Eastern European stock markets.
Dajcman, Silvo ; Festic, Mejra ; Kavkler, Alenka 等
1. Introduction
The basic capital asset pricing model (CAPM), developed
independently by Sharpe (1964), Lintner (1965) and Mossin (1966),
building on the earlier work of Markowitz (1952) on mean-variance
portfolio theory, has been the corner-stone of modern finance for the
last four decades. Although the CAPM has been extensively empirically
studied, the debate about its validity is continuing due to its
simplicity, and other alternative asset pricing models not being without
theoretical and/or empirical weaknesses, it has been widely applied in
financial practice to evaluate not just securities, but any investment.
Finding evidence in favor or against the validity of CAPM is therefore
of great interest for financial public.
CAPM predicts that the risk premium of an individual asset (i.e.
excess return of an asset over the risk-free return) should be
proportional to the market premium (i.e. excess return of the market
portfolio over the risk-free return). The factor of proportionality is
known as systematic risk or beta (P) of an asset. Since the specific
risk of an asset can be diversified away, investors in an asset are
compensated only for bearing the systematic risk of an asset. Knowing
the beta of an asset and the market premium, one can calculate the
expected rate of return for any asset.
The CAPM theory generates four main testable implications: i) The
risk premium for any asset with positive beta is positive; ii) There
should be a linear relationship between the beta and the excess return
of an asset; iii) An asset that is uncorrelated with the market
portfolio has an expected return equal to the risk-free rate; and iv)
There should be no systematic effect of non-beta risk on the excess
returns of an asset. Early empirical studies on CAPM (Douglas 1968;
Black 1972; Black et al. 1972; Miller, Scholes 1972; Blume, Friend 1973;
Fama, MacBeth 1973) were partially supportive of the implications of the
model. They found that the relationship between beta and expected
returns is positive; however, the studies consistently found that
empirical models underestimated the market premium expected from the
theoretical CAPM (Campbell 2000). Many studies in the eighties and
nineties questioned the validity of the Sharpe-Lintner-Mossin's
CAPM. The empirical studies of Banz (1981), Reinganum (1981), Gibbons
(1982), Shanken (1985), and Fama and French (1992) found that the return
generation process depended not only on the beta of an asset but also on
other variables like size, the book to market ratio and the
earnings/price ratio. The further development of the basic CAPM model
took two directions. Some authors worked on theoretically extending the
basic CAPM model (for instance the zero-beta CAPM of Black (1972), the
intertemporal CAPM of Merton (1973), the consumer CAPM of Lucas (1978)
and Breeden (1979)) and developing new models of asset pricing (the
arbitrage pricing model of Ross (1976) which became the basis for the
multifactor asset pricing models). Other authors tried to improve the
methods of empirically testing the CAPM.
One strand of empirical studies that try to improve the empirical
testing of CAPM, which this paper aims to contribute to, has pointed out
the importance of time interval of returns on estimation of beta
(Levhari, Levy 1977; Handa et al. 1989, 1993; Brailsford, Faff 1997;
Lynch, Zumbach 2003; Gencay et al. 2005; Fernandez 2006; Rhaeim et al.
2007; Aktan et al. 2009). These studies are built on the assumption that
economic and financial phenomena may exhibit different characteristics
over different time scales as economic agents make decisions about
consumption, saving and investing with heterogeneous time horizons.
Therefore, not only may systematic risk across time scales differ
(Levhari, Levy 1977; Lynch, Zumbach 2003), but also the validity of CAPM
may be more relevant for some time scales than for others (Handa et al.
1993; Gencay et al. 2005; Fernandez 2006; Rhaeim et al. 2007; Aktan et
al. 2009). To obtain beta estimates for interval returns these studies
either apply the data of asset returns for different frequencies, e.g.
weekly, monthly or annual returns (Handa et al. 1993; Lynch, Zumbach
2003), or apply a wavelet technique to obtain the multiscale estimates
of the beta (Gencay et al. 2005; Fernandez 2006; Rhaeim et al. 2007;
Aktan et al. 2009). The CAPM is then tested in a two-stage procedure: in
the first step the betas of the assets are estimated and in the second
the predictions of the CAPM model are tested by either the Fama and
MacBeth's (1973) methodology within the ordinary least squares
(OLS) regression framework (Gencay et al. 2005; Fernandez 2006; Rhaeim
et al. 2007; Aktan et al. 2009; Handa et al. 1993) or by conditional
CAPM testing methodology in the GARCH framework (Brailsford, Faff 1997;
Lynch, Zumbach 2003).
The aim of the paper is to test the validity of CAPM for three
Central and Eastern European stock markets (i.e. for Slovenia, Hungary
and the Czech Republic) on a scale-by-scale basis. For this purpose, we
propose to test CAPM implications in a modified two-step procedure of
Fama and MacBeth (1973).
In the first stage, we follow the studies of Gencay et al. (2005),
Fernandez (2006), Rhaeim et al. (2007) and Aktan et al. (2009) to obtain
beta estimates for particular shares via a wavelet methodology. Wavelet
analysis is a novel technique that enables us to investigate the
multiscale features of the systematic risk of the assets. As wavelets
are localized in both time and scale, unlike the Fourier analysis and
spectral analysis, they thus provide a convenient and efficient way of
representing complex variables or signals (Ramsey 1999). The wavelet
analysis become extensively used in finance either as a signal
decomposition tool (e.g. Mallat, Zhang 1993; Gencay et al. 2001a, 2001b,
2003) or a tool to detect interdependence between variables (Gencay et
al. 2005; In, Kim 2006; In et al. 2008; Fernandez 2006; Rhaeim et al.
2007).
In the second stage, we test the CAPM implications in the
generalized method of moments (GMM) framework. Fama and MacBeth (1973)
proposed testing CAPM in the OLS framework. In order to correct for
cross-sectional correlation of standard error they ran the second stage
equation for portfolio of stocks rather than for individual stocks. The
other studies that obtain betas in the first stage by wavelet analysis
(Gencay et al. 2005; Fernandez 2006; Rhaeim et al. 2007; Aktan et al.
2009) use the OLS framework in the second stage testing of the CAPM
implications by following the Fama and MacBeth suggestion of forming
portfolio of stocks. However, this procedure assumes there is no serial
correlation in residual returns and suffers from the errors-in-variables
problem (MacKinlay, Richardson 1991; Cochrane 2000; Shanken, Zhou 2006),
since the betas used in the second stage regression are estimates of the
true, unknown betas. The associated tests of CAPM, based on
t-statistics, may no longer be valid (Shanken, Zhou 2006). One way to
deal with this problem is to use the Shanken's (1992) asymptotic
standard errors with a correction factor. The other way, as applied in
our study, is to test CAPM using GMM, which yields estimates robust to
both the errors-in-variables problem and serial correlation of standard
errors.
The paper provides two primary scientific contributions. It
examines the validity of CAPM implications for three Central and Eastern
European stock markets on a multiscale basis. To our knowledge, there
are no other empirical studies providing multiscale evidence on CAPM
validity for these stock markets. Next, a contribution to the literature
of empirical testing of CAPM is made--a two-stage methodology of testing
the CAPM implications is proposed, building on the Fama and MacBeth
(1973) methodology, and applying two modern econometric techniques
(wavelet analysis and the generalized method of moments) which are
robust to statistical problems of previous multiscale CAPM tests. Based
on the results of the empirical tests on multiscale validity of CAPM, we
argue that financial investments, based on CAPM calculation of asset
prices, should resort to multiscale estimation of systematic risk,
corresponding to investment horizon of the financial investment.
2. Methodology
2.1. The Capital Asset Pricing Model (CAPM)
The CAPM model of Sharpe (1964), Lintner (1965) and Mossin (1966)
emerges from the maximization problem for an economic agent in an
environment of uncertainty. Following Blanchard and Fischer (1989) an
economic agent with a horizon of T periods wants to maximize his present
(discounted) value of expected utility of consumption:
max [E.sub.0] [T-1.summation over (t=0)] [1/[(1 + [theta]).sup.t]]
U([c.sub.t])], (1)
where [E.sub.0] denotes the expectation conditional on information
available at time 0, where [theta] is a subjective discount rate, which
expresses time preference of the agent. U([c.sub.t]) is the utility
function, dependent on consumption level cf. Let us assume that for a
given time t the agent can allocate his wealth among n - 1 risky assets
with a stochastic rate of return [r.sub.it] and a riskless asset with a
rate of return [r.sub.0t]. The maximization results in n first order
conditions:
U'([c.sub.t]) = E[U'([c.sub.t+1])(1 + [r.sub.it])]/1 +
[theta], i = 0,..., n - 1. (2)
The economic agent that wants to maximize his total present value
consumption utility should choose a consumption path where the marginal
utility of the consumption in the present period t equals the discounted
expected marginal utility of the consumption for the next period. The
first order conditions should hold, regardless of riskiness of the asset
(i.e. also for the riskless asset).
By rearranging the first order conditions of (2):
E[U'([c.sub.t+1])]E[([r.sub.it] - [r.sub.0t])] = 0, i = 1,...,
n - 1, (3)
equation (2) can be rewritten as:
E[U'([c.sub.t+1])]E[([r.sub.it] - [r.sub.0t])] + cov
[U'([c.sub.t+1]), [r.sub.it] = 0, i = 1,..., n - 1. (4)
At equilibrium, the rate of return from an asset must satisfy the
following equation:
E([r.sub.it]) = [r.sub.0t] - cov[U'([c.sub.t+1]),
[r.sub.it]]/E(U'([c.sub.t+1])], i = 1,..., n - 1. (5)
According to equation (5), the investor will invest in an asset
with an expected rate of return less than the riskless rate of return if
the asset return has a positive relationship with the marginal utility
of consumption.
Suppose there exists an asset m with a rate of return that is
negatively related with the marginal utility of consumption in the next
period, so that U'([c.sub.t+1]) = -[gamma][r.sub.mt] for some
positive [gamma]. It follows that cov [U' ([c.sub.t+1]),
[r.sub.it]] = -[gamma]cov([r.sub.it], [r.sub.mt]). Equation (5) is also
valid for the asset m:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
where [[sigma].sup.2.sub.m] is the variance of the asset m return.
It follows that:
E([r.sub.it]) = [r.sub.0t] -
[cov([r.sub.it],[r.sub.mt])/[[sigma].sup.2.sub.m]][E([r.sub.mt]) -
[r.sub.0t]]. (7)
If we assume that m is a market portfolio (consisting of all traded
assets in the market), equation (7) represents the security market line
in the CAPM. Equation (7) implies that excess returns from asset i (i.e.
in excess of the risk-free asset return) should be proportional to the
market premium (i.e. market return in excess of the risk-free asset
return). The proportionality factor is called the beta (Pi) or
systematic risk of the asset i:
[[beta].sub.i] = [cov([r.sub.it],
[r.sub.mt])/[[sigma].sup.2.sub.m]]. (8)
With a definition of beta, equation (7) can be rewritten as:
E([r.sub.it]) = [r.sub.0t] + [[beta].sub.i]E([r.sub.mt] -
[r.sub.0t]). (9)
The term E([r.sub.mt] - [r.sub.0t]) is referred to as the market
risk premium, given that it represents the return over the risk-free
rate required by investors to hold the market portfolio.
Rearranging equation (9), we obtain:
E ([r.sub.it]) - [r.sub.0t] = [[beta].sub.i]E([r.sub.mt] -
[r.sub.0t]), (10)
from which it follows that the risk premium on an individual asset
equals its beta time the market risk premium.
In empirical studies [[beta].sub.i] is usually estimated by
ordinary least squares (OLS) from the following regression (Fernandez
2006):
[r.sub.it] - [r.sub.0t] = [[alpha].sub.i] +
[[beta].sub.i]([r.sub.mt] - [r.sub.0t]) + [[epsilon].sub.it] or
alternatively [er.sub.it] = [[alpha].sub.i] + [[beta].sub.i][er.sub.mt]
+ [[epsilon].sub.it], (11)
where [er.sub.it] is the excess return of asset i over the riskless
asset return in time period t, [[alpha].sub.i] is a regression constant,
which according to CAPM should be zero for all assets, [er.sub.mt] is
the excess return of market portfolio over riskless asset return in time
period t and [[epsilon].sub.it] is a random error term.
2.2. Empirical testing of the CAPM implications
Following the procedure of Fama and MacBeth (1973), the validity of
CAPM can be tested in a two-stage procedure. In the first stage, the
time series regressions of equation (11) are run to obtain beta
estimates for each stock i for time periods t (time period is usually a
year; t = 1,..., T). In the second stage, a cross-sectional regression
is run:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)
over all assets i (i = 1, N), where [[beta].sub.it] are estimates
of the betas of the first stage regressions, [RV.sub.i,t] are residual
variances of the first stage equation and [[epsilon].sub.i,t] is a
random error term.
The CAPM theory generates four main testable implications (Campbell
et al. 1997):
1) [H.sub.0] : [[gamma].sub.0,t] = 0 ([[gamma].sub.0,t], called
also Jensen's alpha, in Sharpe-Lintner-Mossin's CAPM should be
zero);
2) [H.sub.0] : [[gamma].sub.1,t] > 0 (CAPM implicates that the
risk-return trade-off should be positive; stocks with higher beta should
generate higher excess returns);
3) [H.sub.0] : [[gamma].sub.2,t] = 0 (CAPM implicates linear
relationship between the beta and the excess return of an asset i);
4) [H.sub.0] : [[gamma].sub.3,t] = 0 (CAPM implicates no systematic
effect of non-beta risk on excess return of an asset i).
To test the CAPM implications, Fama and MacBeth (1973) suggested
using time-series averages as estimates of expected values, and then to
test whether these are significantly different from zero with standard
t-test. However, this procedure assumes there is no serial correlation
in residual returns and suffers from the errors-in-variables problem
(MacKinlay, Richardson 1991; Cochrane 2000; Shanken, Zhou 2006), since
the betas used in the second stage regression are estimates of the true,
unknown betas. The associated tests of CAPM, based on t-statistics, may
no longer be valid (Shanken, Zhou 2006). In order to correct for
cross-sectional correlation of the standard errors, Fama and MacBeth
(1973) suggested to run second stage equations for portfolio of stocks
rather than for individual stocks. However, by this procedure the serial
correlation in the residual returns and the errors-in-variables problems
are still unresolved (MacKinlay, Richardson 1991; Cochrane 2000;
Shanken, Zhou 2006). One way to deal with the remaining problems is to
use the Shanken's (1992) asymptotic standard errors with a
correction factor. The other way is to estimate the second stage
equation as a pooled time-series cross-section by generalized method of
moments (Cochrane 2000).
2.3. Empirical testing of CAPM in the Generalized Method of Moments
framework
The Generalized Method of Moments (GMM), originally developed by
Hansen (1982), refers to a class of estimators that are constructed by
exploiting the sample moment counterparts of population moment
conditions (also known as orthogonality conditions) of the data
generating model. They became widely used methods in economics and
finance. Stock market features like volatility clustering1, non-normal
distribution of returns, or serial correlation make ordinary least
squares an inappropriate estimator for the capital asset pricing model.
If the returns exhibit heteroscedasticity conditional on the factors or
serial correlation, the standard errors of the parameter estimates may
not be correct, even asymptotically, and the associated tests may no
longer be valid (Shanken, Zhou 2006). As argued by MacKinlay and
Richardson (1991), Cochrane (2000), Shanken and Zhou (2006), and Lozano
and Rubio (2009), a robust test of the CAPM can be constructed using the
GMM because its estimates are robust to both conditional
heteroscedasticity and serial correlation in the return residuals.
Cochrane (2000) showed that the problem of serially correlated errors
and the errors-in-variables can be comprehensively tackled by estimating
the CAPM by the GMM. We will use a GMM estimator to test the robustness
of the OLS estimates of the validity of the CAPM hypotheses.
The second stage equation of Fama and MacBeth (equation (12)) will
be estimated by a two-step GMM estimator.
Let us consider a linear regression model:
[y.sub.t] = [z'.sub.t][[delta].sub.0] + [[epsilon].sub.t], t =
1, 2,..., n, (13)
where [z.sub.t] is a L x 1 vector of explanatory variables,
[[delta].sub.0] is a vector of unknown coefficients and
[[epsilon].sub.t] is a random error term. This model allows for the
possibility that some or all of the elements zt may be correlated with
the error term et (i.e. E[[z.sub.tk] [[epsilon].sub.t]] [not equal to] 0
for some k). Further, let us assume that there exists a K x 1 vector of
instrumental variables xt which may contain some or all of the elements
of [z.sub.t]. Instrumental variables [x.sub.t] satisfy a set of K
orthogonality conditions:
E[[g.sub.t](w.sub.t], [[delta].sub.0])] = E[[x.sub.t]
[[epsilon].sub.t]] = E[[x.sub.t]([y.sub.t] -
[z'.sub.t][[delta].suib.0])] = 0, (14)
where [g.sub.t]([w.sub.t], [[delta].sub.0]) = [x.sub.t]
[[epsilon].sub.t] = [x.sub.t]([y.sub.t] - [z'.sub.t]
[[delta].sub.0]). The GMM estimator of the parameter vector is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (15)
where [??] is a weight matrix, whereas [S.sub.xy] = [n.summation
over (t=1)] [x.sub.t][y.sub.t] and [S.sub.xz] = [n.summation over (t=1)]
[x.sub.t][z'.sub.t] are the sample moments. Notation is the same as
in Hayashi (2000) and Zivot and Wang (2006).
An efficient two-step GMM estimator utilizes the result that a
consistent estimate of [delta] may be computed by GMM with an
arbitrarily positive definite and symmetric weight matrix [??], such
that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as n [right
arrow] [infinity] (W is symmetric and positive definite). The most
common choices for [??] are [??] = [I.sub.k] ([I.sub.k] is an identity
matrix) or [??] = [S.sup.-1.sub.xx] = [([n.sup.-1X'X).sup.-1],
where X is dimension n x k with t-th row equal [x'.sub.t]. Let us
denote [g.sub.n] ([delta]) = 1/n - [n.summation over (t=1)] [x.sub.t]
([y.sub.t] - [z'.sub.t][delta]). A first step consistent estimate
of S (i.e. asymptotic variance-covariance matrix of the sample moment
gn(8)) is obtained by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)
The Newey-West estimator with Bartlett kernel weights is used to
estimate the GMM asymptotic variance-covariance matrix.
2.4. Multiscale analysis of systematic risk
In order to test the CAPM on a multiscale basis, the irst stage
scale-by-scale estimates of Pi t will be obtained by the maximal overlap
discrete wavelet transform (MODWT) of irst stage regression variables.
This procedure will be explained next.
2.4.1. A basic concept of wavelets
Similar to Fourier analysis, wavelet analysis involves the
projection of the original time series onto a sequence of basis
functions, which are known as wavelets. There are two basic wavelet
functions: the father wavelet (also known as a scaling function), [phi],
and the mother wavelet (also known as a wavelet function), [psi], which
can be scaled and translated to form a basis for the Hilbert space
[L.sup.2](R) of square integrable functions. The father and mother
wavelets are defined by the functions:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)
where j = 1,..., J is the scaling parameter in a j-level
decomposition and k is a translation parameter (j, k [member of] Z ).
The long term trend of the time series is captured by the father
wavelet, which integrates to 1, while the mother wavelet, which
integrates to 0, describes fluctuations from the trend. The continuous
wavelet transform of a square integrable time series X(t) consists of
the scaling, [[alpha].sub.J,k], and wavelet coefficients,
[[beta].sub.j,k], (Craigmile, Percival 2002):
[[alpha].sub.J,k] = [integral][[phi].sub.J,k](t) X(t) and
[[beta].sub.j,k] = [integral][[psi].sub.j,k](t)X(t). (18)
It is possible to reconstruct X(t) from these transform coeficients
using:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)
2.4.2. Multiscale beta estimation by the maximal overlap discrete
wavelet transform
In practice, we observe a time series for a inite number of
regularly spaced times, so we can make use of a maximal overlap discrete
wavelet transform (MODWT). The MODWT is a linear iltering operation that
transforms a series into coeficients related to variations over a set of
scales. It is similar to the discrete wavelet transform (DWT), but it
gives up the orthogonality property of the DWT to gain other features
that render MODWT more suitable for the aims of our study. As noted by
Percival and Mojfeld (1997), this includes: i) the ability to handle any
sample size regardless of whether the series is dyadic (that is of size
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), or not; ii)
increased resolution at coarser scales as the MODWT oversamples the
data; iii) translation-invariance, which ensures that MODWT wavelet
coefficients do not change if the time series is shifted in a
"circular" fashion; and iv) the MODWT produces a more
asymptotically efficient wavelet variance estimator than the DWT.
Let (2) X be an N dimensional vector whose elements represent the
real-valued time series {Xt : t = 0,..., N -1}. For any positive
integer, J0, the level J0 MODWT of X is a transform consisting of the
[J.sub.0] + 1 vectors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] all of
which have dimension N. The vector [[??].sub.j] contains the MODWT
wavelet coefficients associated with changes at scale [[tau].sub.j] =
[2.sup.j-1] (for j = 1, [J.sub.0]), while [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII] contains MODWT scaling coefficients associated
with averages on scale (3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII]. Based upon definition of MODWT coefficients we can write
(Percival, Walden 2000):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)
where [[??].sub.j] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] are N x N matrices. Vectors are denoted by bold.
By definition, the elements of [[??].sub.j] and [MATHEMATICAL
EXPRESSION NOT REPRODUCIBLE IN ASCII] are outputs obtained by filtering
X, namely:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)
for t = 0,..., N - 1, where [[??].sub.j,l] and [[??].sub.j,l] are
the jth MODWT wavelet and scaling filters. The MODWT treats the series
as if it were periodic, whereby the unobserved samples of the
real-valued time series [X.sub.-1], [X.sub.-2],... [X.sub.-N] are
assigned the observed values at [X.sub.N-1], [X.sub.N-2],... [X.sub.0].
The MODWT coefficients are thus given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)
for t = 0,..., N - 1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are
periodization of [[??].sub.j,l] and [[??].sub.j,l] to circular filters
of length N.
This periodic extension of the time series is known as analyzing
{[X.sub.t]} using "circular boundary conditions" (Percival,
Walden 2000; Cornish et al. 2006). There are [L.sub.j] - 1 wavelet and
scaling coefficients that are influenced by the extension ("the
boundary coefficients"). Since [L.sub.j] increases with j, the
number of boundary coefficients increases with scale. Exclusion of
boundary coefficients in the wavelet variance, wavelet correlation and
covariance provides unbiased estimates (Cornish et al. 2006).
Given that market portfolio return time series, [r.sub.mt], and
stock i return time series, [r.sub.it] are stationary processes, a MODWT
transformation of the two series can be performed to obtain vectors of
wavelet coefficients [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The estimate of
the beta for a stock i at scale [[tau].sub.j],
[[beta].sub.i]([[tau].sub.j]), can be obtained by an ordinary least
squares regression:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the
scale [[tau].sub.j] vector of wavelet coefficients, not affected by
boundary condition, obtained by transforming the time series [er.sub.it]
by MODWT; [er.sub.it] is the excess return of stock i over the return of
risk-free asset at time t; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII] is the scale [[tau].sub.j] vector of wavelet coefficients, not
affected by the boundary condition, obtained by transforming [er.sub.mt]
by MODWT; [er.sub.mt] is the excess market return over return of a
risk-free asset (i.e. the market premium) at time t;
[[epsilon].sub.it]([[tau].sub.j]) is a white noise error term of the OLS
regression at scale ([[tau].sub.j]), and [[alpha].sub.i]([[tau].sub.j])
is a regression constant term at scale ([[tau].sub.j]).
3. Empirical results
3.1. Data
Three Central and Eastern European stock markets were considered:
the Slovenian, Hungarian and Czech stock markets. The stock markets of
Slovenia, the Czech Republic and Hungary are chosen as they share some
common characteristics: they are markets with a short post-communist
era, relatively small market capitalization (4); they have a relatively
small number of listed companies (5), and the stock exchanges are owned
by a common holding company (together with the Vienna stock exchange,
these three CEE stock exchanges form the CEE Stock Exchange Group).
These are also the markets with most developed economies (for instance
by GDP per capita) in the Central and Eastern Europe. There are also
some important differences between them: Czech and Hungarian stocks have
attracted many foreign investors (Caporale, Spagnolo 2010), the
Slovenian market less so: the stock market turnover and liquidity of
shares listed on the Ljubljana stock exchange is smaller than on the
Budapest and Prague stock exchanges (6). An important reason for
choosing only these three stock markets was also the availability of
historical data for long enough time period. The data set consisted of
stocks quoted in the main stock index of the investigated stock markets
(LJSEX for Slovenia, BUX for Hungary and PX for the Czech Republic).
These stock indices were taken as proxies for the market portfolio
returns. We endeavored to take the longest possible time series of stock
(stock index) returns, but at the same time we had to consider the
availability of the risk-free asset return time series. A major drawback
of testing CAPM in these markets is the low number of quoted stocks and
the relatively short historical time series (7). The daily returns of
the 3-month money market rates of the considered countries are taken as
proxies for the countries' risk-free returns (8). Given that we
worked with nominal returns, we used a nominal proxy for the risk-free
rate.
The first date of observation for the Slovenian stock market was
January 1, 2002, for Hungarian stock market it was April 1, 1997 and for
the Czech stock market it was January 10, 1995. Stock (and stock
indices) returns were calculated as the differences of logarithmic daily
closing prices of the stocks or stock indices (ln ([P.sub.t]) - ln
([P.sub.t-1]), where P is a closing price). In cases when there was no
trading with a particular stock on a specific day, we took the closing
price of the last trading day. We considered stock splits and reverse
stock splits and accordingly adjusted prices of the stocks. The data for
stock (stock indices) prices were taken from the web pages of Ljubljana,
Budapest and Prague stock exchanges.
Tables 1 to 3 present some descriptive statistics of the data. The
data appear extremely non-normal. The majority of the return
distributions are negatively skewed (especially in the Hungarian and the
Czech stock market), possibly due to the large negative returns
associated with the financial crises in the observed period (9). The
data also display a high degree of excess kurtosis. Such skewness and
kurtosis are common features in asset return distributions, which are
repeatedly found to be leptokurtic (Henry 2002). The Jarque-Bera test
rejects the hypothesis of normally distributed returns for all stocks as
well as stock indices.
The stationarity of stock returns, interest rates, excess returns
and market premiums was checked using the Augmented Dickey-Fuller (ADF)
test, the Phillips-Perron (PP) and the Kwiatkowski-Phillips-Schmidt-Shin
(KPSS) tests. The returns of all the stocks listed on the Budapest and
Prague stock exchanges were found to be stationary, whereas the most of
the stocks listed on the Ljubljana stock exchange were fractionally
integrated (i.e. with long range dependence) (10). Within the GMM
framework it is assumed that excess returns and market premiums are
stationary and ergodic with finite fourth moments (Hansen 1982;
MacKinlay, Richardson 1991). The results (not presented here, but
abtainable from the authors) of testing these conditions show that
stationarity hypothesis cannot be rejected for the excess returns of
stocks and market premiums for the Hungarian and Czech and stock
markets, while the excess returns of stocks and a market premium for
Slovenian stock market exert long range dependence. The fourth moment
(kurtosis) is finite for all the investigated stock returns and market
premiums.
3.2. Results of testing CAPM implications
In order to test the CAPM implications in a proposed two-step
procedure, we in the first-stage estimated systematic risk (i.e. beta)
of stocks in the stock markets on a scale-by-scale basis. The beta for
each individual stock was re-calculated for each subsample of 250
trading days (approximately one trading year) over the full observation
period. For Slovenia's stock market, the effective observation
period, for which the CAPM was tested, was therefore January 3,
2002--January 8, 2010 and was the same for all stocks. There were in
total 72 observations in the second stage regression to test CAPM
implications for the Slovenian stock market. For the Hungarian stock
market, the effective observation period for the calculation of the
betas stretched from April 5, 2000--April 2, 2010, and was the same for
all observed stocks. The total number of observations in the second
stage equation amounted to 100 observations. For the Czech stock market,
the effective period of observation was from January 9, 1996--December
21, 2009; however due to a data availability issue it was different for
individual stocks. The total number of observations in the second stage
equation of testing implications of CAPM was 78 (11).
The MODWT transformation of excess returns of the stocks and market
premiums was performed using a Daubechies least asymmetric filter with a
wavelet filter length of 8 (LA8). This is a common wavelet filter in
other empirical studies on financial markets (Gencay et al. 2005;
Fernandez 2006; Rhaeim et al. 2007; Ranta 2010). Since the daily data of
excess returns and market premiums were used in our analysis, the
maximum level of decomposition we used was [J.sub.0] = 6, (j = 1,...,
6). The wavelet scale [[tau].sub.1] measures the dynamics of returns
over 2-4 days, scale [[tau].sub.2] over 4-8 days, scale [[tau].sub.3]
over 8-16 days, scale [[tau].sub.4] over 16-32 days, scale [[tau].sub.5]
over 32-64 days and scale [[tau].sub.6] over 64-128 days. To obtain the
unbiased estimates of the betas, the MODWT boundary condition was
handled by using a "reflection boundary condition" (Percival,
Walden 2000). We found that betas of stocks vary from scale to scale,
confirming the findings in literature that systematic risk is a
multiscale phenomenon. The results are not presented here as they are
only an input in the second stage regression of testing CAPM
implications.
In the second stage of testing the CAPM implications, we applied
the GMM framework. For the purpose of comparison, we also performed an
OLS regression and reported these results as well. To assert that the
CAPM is valid for a stock market, the null hypotheses must not be
rejected.
Hypotheses are tested based on two-sided t-test, except for the
hypothesis [H.sub.0] : [[gamma].sub.1,t] > 0 where one-sided t-test
is used. We tested the CAPM for raw returns (in this CAPM model, the
betas in the first step were obtained on raw, MODWT non-transformed,
returns), and for return dynamics of wavelet scales [[tau].sub.1] to
[[tau].sub.6]. We also report the average daily market premium in the
period for which the CAPM was tested.
The results of the OLS regression show that the explanatory power
(as measured by [R.sup.2]) of the CAPM for the raw (i.e. daily) returns
is weak, but increases higher wavelet scales (see Tables 4 through 6).
We notice that for Slovenian stock market the greatest explanatory power
of CAPM is achieved for wavelet scale [[tau].sub.6] and for Hungarian
stock market for wavelet scale [[tau].sub.1]. For the Czech stock
market, the average of [R.sup.2] across all CAPM models is higher than
for the Slovenian and Hungarian stock market, meaning the CAPM has
greater explanatory power for the excess returns of stocks in the Czech
stock market. CAPM in the Czech stock market best explains scale
[[tau].sub.6] (corresponding to investment horizon over 32-64 days)
excess return dynamics.
One can see that the t-statistics of the regression coefficients
estimated by OLS and GMM differ. The Durbin-Watson statistics point out
the problem of serial correlation. As we previously noted, the problems
of estimating pooled time-series cross-section version of the CAPM by
OLS are due to heteroscedasticity and the serial correlation in returns,
which in turn makes the standard errors of the parameter estimates
incorrect, even asymptotically. The associated tests based on
t-statistics may no longer be valid; therefore inferences regarding the
CAPM hypotheses should be made on the basis of the robust results of the
GMM estimator.
Regarding the CAPM hypotheses, the following conclusions may be
drawn from the GMM estimator results:
--The hypothesis of zero Jensen's alpha, [H.sub.0] :
[[gamma].sub.0,t] = 0, that must not be rejected if CAPM is valid, is
generally rejected for the Slovenian stock market. For the Hungarian
stock market; however, the hypothesis is not rejected for wavelet scales
[[tau].sub.3] to [[tau].sub.6] and for the Czech stock market for the
raw returns and wavelet scales [[tau].sub.3], [[tau].sub.4] and
[[tau].sub.6].
--According to CAPM one should expect a positive relationship
between risk (as measured by beta) and return, meaning that the stocks
with higher beta should generate higher excess returns. This in turn
means that the security market line has a positive slope (12). For the
CAPM to be valid in investigated stock markets, the hypothesis [H.sub.0]
: [[gamma].sub.1,t] > 0 must not be rejected (13). The hypothesis can
be rejected for all stock markets.
--The hypothesis of a linear relationship between the betas of the
stocks and their excess returns ([H.sub.0] : [[gamma].sub.2,t] = 0)
cannot be rejected for any of the CAPM models for the Slovenian and
Hungarian stock market. For the Czech stock market, the non-linear
relationship between the betas and excess returns could be identified
for wavelet scales [[tau].sub.5] and [[tau].sub.6].
--The results of the second stage regression of testing the CAPM
implications show that there are significant other factors, beside the
beta of the stocks that can significantly explain excess returns of
stocks in the three CEE stock markets. These are especially important
for the Hungarian and Czech stock market as [[gamma].sub.3,t] is
significantly different from zero for all investment horizons (scales),
which is inconsistent with the CAPM theory. In the Slovenian stock
market, the systematic effect of non-beta factors on the excess returns
on stocks is found for wavelet scales [[tau].sub.1] to [[tau].sub.4].
Based on these results, we may conclude that support for the CAPM
implications in the investigated CEE stock markets is weak. For the
Slovenian stock market, the raw returns model seems to best support the
CAPM hypotheses, however the intercept is significantly different from
zero, and the market premium is not significantly positive which is
inconsistent with the CAPM. There are at least two violations of the
CAPM implications at each wavelet scale. For Hungarian stock market, the
evidence against the CAPM is the strongest for the raw returns and
wavelet scales [[tau].sub.1] and [[tau].sub.2]. The CAPM implications
seem to be more relevant for investors with longer investment horizons
(of at least 8-16 days). However, the rejection of the hypothesis of
positive market premium and significant relevance of unobservable
factors, not captured by the beta, weakens the validity of the CAPM in
the Hungarian stock market. For the Czech stock market, the results
indicate that the beta does not fully explain all variability in the
excess returns of the stocks in the market. The weakest evidence against
CAPM is found in the shortest and medium term investment horizons (i.e.
for the raw returns and scales [[tau].sub.3] and [[tau].sub.4]), which
suggest that CAPM may be more relevant for these investment horizons
which is an important implication for financial public in the
investigated CEE countries. Financial investments based on CAPM
calculation of asset prices should resort to multiscale estimation of
systematic risk, corresponding to investment horizon of the financial
investment.
The CAPM hypotheses that seem to be most often violated are the
zero Jensen's alpha condition, a positive market premium and
non-systematic influence of non-observable variables on excess returns
of stocks in the market. The Jensen's alpha (intercept) in the
estimated CAPM models is found to be positive. As shown by Jarrow and
Protter (2011), a non-zero alpha implies an arbitrage opportunity, which
is a stronger violation of market efficiency. For Slovenian stock
market, the estimated intercepts are significantly different from zero
(by t-test) for all CAPM models, whereas for the Hungarian and the Czech
stock market the intercepts are not significantly different from zero
for higher wavelet scales (scales [[tau].sub.3] to [[tau].sub.6];
exception being CAPM model of wavelet scale [[tau].sub.3] for the Czech
stock market). Further, according to CAPM, the systematic risk is the
only factor that determines the excess returns of stocks in an efficient
stock market. However, the significant parameter estimates
[[tau].sub.3,t] show that the beta (i.e. systematic or market risk)
cannot fully explain the variability in excess returns in the markets.
There are other, by model non-observable variables, that influence
excess returns of stocks in all the investigated markets, regardless of
time scale. Violations of these two CAPM hypotheses indicate that the
efficiency of the investigated stock markets is questionable (14).
Our results support the findings of other studies that systematic
risk and CAPM are multiscale phenomena (Gencay et al. 2005; Fernandez
2006; Rhaeim et al. 2007; Bortoluzzo et al. 2010). However these studies
use different methodology and test only predictions regarding
Jensen's alpha and positive market premium. Gencay et al. (2005)
found that for the United States stock market, the implications of CAPM
are more relevant for investors with medium to long-term (wavelet scales
[[tau].sub.2] and [[tau].sub.3]) horizons as compared to those with
short-term horizons as estimates of [[tau].sub.1,t] for these scales
best approximated the historical market premium. For the German stock
market, the CAPM predictions are the most relevant for wavelet scale
[[tau].sub.3], corresponding to 8-16 days investment horizon dynamics.
Mid and higher wavelet scales were also found to better capture
risk-return relationship in the United Kingdom's stock market.
Fernandez (2006), who analyzed the stock market in Chile, observed that
CAPM is the most relevant for wavelet scale [[tau].sub.2], because the
estimated market risk premium for this scale is the closest to the
actual risk premium ([[tau].sub.1.t] in her study, however, was not
significantly larger than zero). Rhaeim et al. (2007) studied the French
stock market and concluded that the predictions of the CAPM are more
relevant in the short term than in the long term, which makes the French
market different from those of the United States, Germany, and the
United Kingdom. Aktan et al. (2009) found that for the Turkish stock
market, CAPM is most relevant for wavelet scale [[tau].sub.3].
Bortoluzzo et al. (2010) showed that based on the t-test and [R.sup.2],
the CAPM performed the best for wavelet scales [[tau].sub.1] to
[[tau].sub.3].
4. Conclusion
In this paper the systematic risk and validity of four testable
implications of CAPM for three Central and Eastern European stock
markets (namely in Slovenia, Hungary and the Czech Republic) was
investigated on a multiscale basis. A modification of the Fama and
MacBeth (1973) methodology was proposed, where in the first stage the
systematic risk of individual stocks in the stock markets was calculated
applying a wavelet methodology. In the second stage regression, four
CAPM implications were tested in the generalized method of moments
framework. We found that the systematic risk and validity of the CAPM
implications is a multiscale phenomenon. Empirical evidence in support
of the CAPM implications in the investigated CEE stock markets was found
to be weak. The CAPM hypotheses that seem to be the most commonly
violated in these stock markets are the zero Jensen's alpha
condition, a positive market premium and non-systematic influence of
non-observable variables on the excess returns of stocks in these stock
markets.
The results of our study indicate that CAPM implications in the
Slovenian stock market may be more relevant for investors with daily
rather than longer-term investment horizons. For the Hungarian stock
market, the CAPM may be more relevant for investors with investment
horizons of at least 8-16 days while for the Czech stock market, CAPM is
of the most relevance for investors with investment horizons of 8-32
days.
doi:10.3846/16111699.2011.633097
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Silvo Dajcman [1], Mejra Festic [2], Alenka Kavkler [3]
[1] School of Economics and Business, Finance, University of
Maribor, Razlagova 14, Maribor, Slovenia
[2] Banka Slovenije, Slovenska 35, 1505 Ljubljana, Slovenia
[3] School of Economics and Business, Quantitative Economic
Analysis, University of Maribor, Razlagova 14, 2000 Maribor, Slovenia
E-mails: [1] silvo.dajcman@uni-mb.si (corresponding author); [2]
mejra.festic@bsi.si; [3] alenka.kavkler@uni-mb.si
Received 06 June 2011; accepted 14 October 2011
(1) Volatility clustering can be described as departure from the
volatility's long-term mean over an extended period of time
(Teresiene 2009).
(2) Concepts and notations as in Percival and Walden (2000) are
used. Another thorough description of MODWT using matrix algebra is
found in Gencay et al. (2002).
(3) Percival and Walden (2000) denote scales of MODWT obtained
wavelet coefficients with a letter t and scales of scaling coefficients
with X. We use the same notations.
(4) The stock market capitalization of all the shares listed on the
Ljubljana stock exchange at the end of the year 2010 was, according to
FESE (2010), 6.99 billion EUR. The stock market capitalization of all
the shares listed on the Prague stock exchange, at the same time,
reached 31.92 billion EUR and on the Budapest stock exchange 20.62
billion EUR. To compare this with some developed European stock markets:
the stock market capitalization of all the shares listed on the Deutsche
Borse at the end of 2010 was 1,066 billion EUR.
(5) At the end of 2010, the Ljubljana stock exchange had 72
different companies listed, the Budapest stock market had 52, and the
Prague stock market had 27. According to FESE (2010) NYSE Euronext had
1,135 stock companies listed, the Deutsche Borse 765, and the Vienna
stock market 110.
(6) The equity turnover of Ljubljana's stock exchange in 2010
was 0.7 billion [euro], of the Prague's stock exchange 30.5 billion
[euro] and of the Budapest's stock exchange 39.9 billion [euro]
(CEEG 2011).
(7) Pooling time-series and cross-section data enlarges the
dataset, thereby increasing the variability of the data and increasing
the efficiency of the GMM estimator.
(8) In empirical literature different proxies are used for
risk-free rates. Most often these include: 10-year treasury bills rates
(Gencay et al. 2005), 3-month treasury bills (Michailidis et al. 2006),
3-month money market rates (Gencay et al. 2005; Rhaeim et al. 2007) and
interest rates paid on bank deposits of diverse maturity (Aktan et al.
2009; Fernandez 2006). We use 3-month money market rates due to the
availability of historical data.
(9) The Russian financial crisis (in 1998), the dot-com crisis (in
2000), the internet companies stocks bubble burst (in 2002), the Middle
East financial crisis (in 2006) and the Global financial crisis (in
2007-2008).
(10) These results are not presented here, but can be obtained from
the authors.
(11) For the Czech stock market we have an unbalanced panel. This
is because for the Czech stock market a relatively long time series of
return series is available only for 5 different stocks (see Table 3). To
have large enough number of observations to test the CAPM implications,
we estimated an unbalanced panel of pooled time-series cross-section
data.
(12) In fact, if all of the observed stocks well represent the
market portfolio, then on the basis of the CAPM theory we should expect
an average daily excess return to equal the average daily historical
market premium (i.e. excess return of the market portfolio over the
risk-free rate). The later was positive for all stock markets: an
average daily market premium in the effective observation period was
0.00333% for the Slovenian stock market, 0.003313% for the Hungarian
stock market and 0.002519% for the Czech stock market. As the average
(weighted) beta of all the stocks representing market portfolio is per
definition 1, then [[gamma].sub.1,t] should equal the historical market
premium. Therefore, we should expect [[gamma].sub.1,t] to be positive
and close to the historical market premium.
(13) The empirical null hypothesis is [H'.sub.0] :
[[gamma].sub.1] = 0 and the alternative hypothesis [H'.sub.0] :
[[gamma].sub.1]> 0, what is the opposite from the null hypothesis
[H.sub.0] : [[gamma].sub.1,t] > 0 written above.
(14) There are some studies of stock market efficiency of CEE
countries (Gilmore, McManus 2002; Worthington, Higgs 2004; Kasman et al.
2009), mainly rejecting their efficiency.
Silvo DAJCMAN, PhD, a Teaching Assistant of Finance and Banking,
School of Economics and Business, University of Maribor. His areas of
interest are financial markets, banking and monetary policy.
Mejra FESTIC, PhD, is Associate Professor of Economic Theory and
Policy, Banking and Finance employed at the University of Maribor. She
has been a Vice-rector for the field of academic affairs at the
University of Maribor and the principal of the institute EIPF Economic
Institute, established in 1963, specialized for economic dynamics
forecasts in world economies, Euro area and Europe. Her research field
regards to financial stability, monetary systems, banking, finance,
capital structure, bank risks management and coordination of economic
policies. From the first March 2011 she is employed as a vice-governor
at the Bank of Slovenia.
Alenka KAVKLER, PhD, is Assistant Professor for Quantitative
Economic Analysis at the Faculty for Economics and Business, University
of Maribor, Slovenia. She finished her PhD in the field of Mathematical
Methods in Economics at the Vienna University of Technology. Her
research interests involve smooth transition regression models, duration
models, matching methods and tree models.
Table 1. Descriptive statistics for returns series of stocks listed
at Slovenian stock exchange and its representative national stock index
Stock/ Period of Number of
stock index observation observations Min Max
Aerodrom 3.1.2002- 2,132 -0.1557 0.1656
Ljubljana 20.7.2010
Gorenje 3.1.2002- 2.132 -0.0830 0.0831
20.7.2010
Intereuropa 3.1.2002- 2,132 -0.1016 0.1542
20.7.2010
Krka 3.1.2002- 2,132 -0.1025 0.1984
20.7.2010
Lasko 3.1.2002- 2,132 -0.1504 0.1263
20.7.2010
Luka Koper 3.1.2002- 2,132 -0.0965 0.1281
20.7.2010
Mercator 3.1.2002- 2,132 -0.0949 0.1129
20.7.2010
Petrol 3.1.2002- 2,132 -0.1020 0.1328
20.7.2010
Sava 3.1.2002- 2,132 -0.1274 0.1535
20.7.2010
LJSEX 3.1.2002- 2,132 -0.0830 0.0831
(index) 20.7.2010
3-month
money 4.1.2002- 2,132 0.0000 0.0004
market 20.7.2010
interest rate
Stock/ Std.
stock index Mean deviation Skewness Kurtosis
Aerodrom 0.00022 0.02059 -0.01 10.20
Ljubljana
Gorenje 0.00022 0.01056 0.03 6.51
Intereuropa -0.00073 0.01769 0.41 12.14
Krka 0.00079 0.01591 0.75 19.41
Lasko -0.00017 0.02110 -0.16 9.01
Luka Koper 0.00004 0.01813 -0.08 7.31
Mercator 0.00032 0.01682 0.02 8.94
Petrol 0.000402 0.01691 0.32 12.06
Sava 0.00040 0.01949 -0.00 8.91
LJSEX 0.00021 0.01056 -0.47 15.38
(index)
3-month
money 0.00018 0.00009 0.067 2.60
market
interest rate
Stock/ Jarque-Bera
stock index statistics
Aerodrom 4,605.59 ***
Ljubljana
Gorenje 1.092.22 ***
Intereuropa 7,476.29 ***
Krka 24,131.11 ***
Lasko 3,215.55 ***
Luka Koper 1,651.34 ***
Mercator 3,133.19 ***
Petrol 7,329.22 ***
Sava 3,102,66 ***
LJSEX 13,701.78 ***
(index)
3-month
money 15.85 ***
market
interest rate
Notes: Jarque-Bera statistics: *** indicate that the null hypothesis
(of normal distribution) is rejected at the 1% significance level;
** indicate that the null hypothesis is rejected at the 5%
significance level and * indicate that the null hypothesis is
rejected at the 10% significance level.
Table 2. Descriptive statistics for returns series of stocks
listed at Hungarian stock exchange
and its representative national stock index
Stock/ Period of Number of Min Max
stock index observation observations
Egis 1.4.1997- 3,276 -0.3567 0.1944
12.5.2010
Fotex 1.4.1997- 3,276 -0.3365 0.2346
12.5.2010
MOL 1.4.1997- 3,276 -0.2231 0.1403
12.5.2010
MTelekom 14.11.1997- 3,119 -0.1257 0.1199
12.5.2010
OTP 1.4.1997- 3,276 -0.2513 0.2092
12.5.2010
Pannergy 1.4.1997- 3,276 -0.2076 0.2343
12.5.2010
Raba 17.12.1997- 3,096 -0.2501 0.1999
12.5.2010
Richte 1.4.1997- 3,276 -0.231 0.2178
12.5.2010
Synergon 5.5.1999- 2,759 -0.1625 0.1526
12.5.2010
TVK 1.4.1997- 3,276 -0.2231 0.2068
12.5.2010
BUX (index) 1.4.1997- 3,276 -0.1803 0.1362
12.5.2010
3-month
money 2.4.1997- 3,276 0.0002 0.0009
market 12.5.2010
interest
rate
Stock/ Mean Std. deviation Skewness Kurtosis
stock index
Egis 0.00022 0.02676 -0.97 20.41
Fotex 0.00029 0.03281 0.40 13.60
MOL 0.00058 0.02449 -0.21 9.70
MTelekom -0.000031 0.02136 -0.20 6.67
OTP 0.00087 0.02782 -0.22 10.80
Pannergy -0.00019 0.02674 0.20 11.68
Raba -0.00037 0.02600 -0.14 12.56
Richte 0.00040 0.02620 -0.63 16.39
Synergon -0.00056 0.02986 0.41 8.63
TVK 0.00010 0.02755 -0.15 11.84
BUX (index) 0.00045 0.01924 -0.64 13.18
3-month
money 0.00043 0.00016 1.03 3.15
market
interest
rate
Stock/ Jarque-Bera
stock index statistics
Egis 41,904.65 ***
Fotex 15,419.85 ***
MOL 6,153.19 ***
MTelekom 1,769.98 ***
OTP 8,321.11 ***
Pannergy 10,304.38 ***
Raba 11,794.59 ***
Richte 24,698.68 ***
Synergon 3,724.70 ***
TVK 10,683.4 ***
BUX (index) 14,367.97 ***
3-month
money 581.93 ***
market
interest
rate
Note: See notes for Table 1.
Table 3. Descriptive statistics for returns series of stocks
listed at the Czech stock exchange
and its representative national stock index
Stock/ Period of Number of Min Max
stock index observation observations
Auto Group 26.9.2007- 654 -0.1777 0.3830
12.5.2010
CETV 26.9.2007- 655 -0.237 0.3075
12.5.2010
CEZ 10.1.1995- 3,596 -0.1539 0.2040
12.5.2010
ECM Real 1.11.1997- 629 -0.2707 0.3381
Estate 12.5.2010
Erste 26.9.2007- 654 -0.1836 0.1632
Group 12.5.2010
Komercni 10.1.1995- 3,596 -0.2076 0.1594
Banka 12.5.2010
ORCO 26.9.2007- 654 -0.3185 0.2646
12.5.2010
Philip 10.1.1995- 3,596 -0.1634 0.1263
Moris 12.5.2010
Telefonica 28.3.1995- 3,596 -0.1281 0.1299
12.5.2010
Unipetrol 26.8.1997- 3,187 -0.1704 0.1799
12.5.2010
PX (index) 10.1.1995- 3,596 -0.1619 0.1236
12.5.2010
3-month
money 10.1.1995- 3,597 0.0001 0.0014
market 12.5.2010
interest
rate
Stock/ Mean Std. deviation Skewness Kurtosis
stock index
Auto Group -0.00182 0.03774 1.69 25.23
CETV -0.00180 0.04803 0.28 9.01
CEZ 0.00062 0.02407 -0.32 8.52
ECM Real -0.00282 0.04049 0.64 17.94
Estate
Erste -0.00088 0.03751 -0.11 7.37
Group
Komercni 0.00024 0.02684 -0.31 7.62
Banka
ORCO -0.00427 0.05067 -0.06 9.55
Philip 0.00020 0.02435 -0.31 6.94
Moris
Telefonica 0.00012 0.02184 -0.02 6.93
Unipetrol 0.00015 0.0263 -0.13 7.61
PX (index) 0.00028 0.01492 -0.41 14.88
3-month
money 0.00023 0.00019 1.75 6.64
market
interest
rate
Stock/ Jarque-Bera
stock index statistics
Auto Group 13,760.99 ***
CETV 991.58 ***
CEZ 4,626.25 ***
ECM Real 5,895.57 ***
Estate
Erste 521,34 ***
Group
Komercni 3,263.82 ***
Banka
ORCO 1,169.40 ***
Philip 2,386.91 ***
Moris
Telefonica 2,316.04 ***
Unipetrol 2,829.18 ***
PX (index) 21,256.18 ***
3-month
money 3,823.02 ***
market
interest
rate
Note: See notes for Table 1.
Table 4. OLS and GMM results of testing the CAPM implications
for the Slovenian stock market
[[gamma].sub.0] [[gamma].sub.1]
The raw 0.0017226 -0.0028493
returns (1.5842) (-0.9188)
model (4.0906) *** (-1.1368)
0.0030379 -0.0046549
Scale [[tau].sub.1] (3.2739) *** (-1.7047)
(3.9668) *** (-1.6664)
0.0026197 -0.0037492
Scale [[tau].sub.2] (2.7954) *** (-1.6487)
(6.6680) *** (-3.1191)
0.0026623 -0.0026533
Scale [[tau].sub.3] (2.4106) ** (-1.0619)
(5.3394) *** (-1.3943)
0.0016307 -0.000967
Scale [[tau].sub.4] (2.3055) ** (-0.645)
(5.1414) *** (-1.0791)
0.0010343 -0.0008610
Scale [[tau].sub.5] (1.8088) * (-1.5395)
(2.4308) ** (-4.5264)
0.0008870 -0.0007791
Scale [[tau].sub.6] (1.7634) * (-1.1321)
(3.1269) *** (-2.4630)
[[gamma].sub.2] [[gamma].sub.3]
The raw 0.0010036 -0.0003844
returns (0.4667) (-0.1748)
model (0.4574) (-1.9077)
0.0023998 -7.710092
Scale [[tau].sub.1] (1.2449) (-3.964) ***
(1.0606) (-3.8933) ***
0.001870 -17.2757427
Scale [[tau].sub.2] (1.3045) (-2.8209) ***
(1.7976) (-3.5271) ***
0.0010027 -51.4099179
Scale [[tau].sub.3] (0.6635) (-3.7277) ***
(0.7065) (-4.4703) ***
0.00000635 -71.6956619
Scale [[tau].sub.4] (0.0069) (-2.0858) **
(0.007900) (-2.5624) **
0.0000468 -45.1373426
Scale [[tau].sub.5] (0.1216) (-0.7427)
(0.1996) (-0.7714)
-0.00008766 23.9927004
Scale [[tau].sub.6] (-0.2117) (0.2183)
(-0.2266) (0.1955)
Statistical parameters
of OLS regression
The raw [R.sup.2] = 0.0474
returns DW =2.3611
model
[R.sup.2] = 0.2532
Scale [[tau].sub.1] DW = 2.4535
[R.sup.2] = 0.1487
Scale [[tau].sub.2] DW = 2.5909
[R.sup.2] = 0.1886
Scale [[tau].sub.3] DW = 2.5633
[R.sup.2] = 0.102
Scale [[tau].sub.4] DW = 2.5641
[R.sup.2] = 0.0494
Scale [[tau].sub.5] DW = 2.4818
[R.sup.2] = 0.6791
Scale [[tau].sub.6] DW = 1.0401
Average daily market premium in the effective period
of testing the CAPM = 0.00003333
Notes: The raw returns model presents the results of testing
the CAPM implications where betas in the first stage of the
procedure are calculated on raw returns data (i.e. wavelet
untransformed data or daily returns). As the GMM is just
identified, the OLS and GMM estimates of gammas are equal.
In the first parenthesis the t-statistics based on the OLS
estimates of the gammas are presented and in the
second parentheses under the gamma estimates t-statistics
based on GMM estimates of the gammas are presented. Exceeded
critical values for the rejection of the null hypotheses
are indicated by *** for a 1% significance level, by
** for a 5% significance level and by * for a 10%
significance level.
Table 5. OLS and GMM results of testing the CAPM
implications for Hungarian stock market
[[gamma].sub.0] [[gamma].sub.1]
The raw 0.0016614 -0.0022957
returns (1.9750) * (-1.0615)
model (2.9183) *** (-1.3952)
0.0016817 -0.0020719
Scale [[tau].sub.1] (2.2575) ** (-1.0641)
(3.1465) *** (-1.3956)
0.0015469 -0.0023548
Scale [[tau].sub.2] (1.9495) * (-1.2522)
(2.4867) ** (-1.6735)
0.0013178 -0.0020981
Scale [[tau].sub.3] (1.9414) * (-1.1923)
(1.7363) (-0.9028)
0.0008520 -0.0001620
Scale [[tau].sub.4] (1.1968) (-0.0994)
(1.6319) (-0.1389)
0.0004296 -0.0006927
Scale [[tau].sub.5] (1.0204) (-1.0849)
(1.1809) (-1.0809)
0.0004375 -0.0007058
Scale [[tau].sub.6] (1.0455) (-0.9819)
(1.3744) (-1.3072)
[[gamma].sub.2] [[gamma].sub.3]
The raw 0.0010059 -2.327127
returns (0.7140) (-3.4344) ***
model (0.8684) (-2.3450) **
0.0009235 -5.2062058
Scale [[tau].sub.1] (0.6953) (-3.8781) ***
(0.8023) (-2.67822) ***
0.0012020 -8.9222896
Scale [[tau].sub.2] (1.0117) (-3.3656) ***
(1.3053) (-1.9197) *
0.0010014 -16.6414480
Scale [[tau].sub.3] (0.8834) (-3.2448) ***
(0.6931) (-2.9870) ***
-0.0005302 -29.5496919
Scale [[tau].sub.4] (-0.5522) (-2.8795) ***
(-0.7645) (-2.5956) ***
0.0002313 -35.5379933
Scale [[tau].sub.5] (0.5365) (-2.2386) **
(0.4650) (-2.4041) **
0.0002957 -106.8265938
Scale [[tau].sub.6] (0.8646) (-3.5265) ***
(1.1958) (-3.3503) ***
Statistical parameters
of OLS regression
The raw [R.sup.2] = 0.1199
returns DW = 2.348
model
[R.sup.2] = 0.147
Scale [[tau].sub.1] DW = 2.3111
[R.sup.2] = 0.1127
Scale [[tau].sub.2] DW = 2.296
[R.sup.2] = 0.1129
Scale [[tau].sub.3] DW = 2.4454
[R.sup.2] = 0.1051
Scale [[tau].sub.4] DW = 2.4703
[R.sup.2] = 0.0603
Scale [[tau].sub.5] DW = 2.5344
[R.sup.2] = 0.1185
Scale [[tau].sub.6] DW = 2.3455
Average daily market premium in the effective period
of testing rhe CAPM = 0.00003313
Notes: See notes for Table 4.
Table 6. OLS and GMM results of testing the CAPM
implications for the Czech stock market
[[gamma].sub.0] [[gamma].sub.1]
The raw 0.0013403 -0.0016320
returns (1.3969) (-0.8152)
model (1.7263) (-0.7263)
0.0012513 -0.0011470
Scale [[tau].sub.1] (1.6069) (-0.8043)
(2.0825) ** (-0.8583)
0.0014114 -0.0022774
Scale [[tau].sub.2] (1.5786) (-1.1663)
(2.8825) *** (-1.3740)
0.0011302 -0.0016343
Scale [[tau].sub.3] (1.3047) (-0.9592)
(1.7626) * (-1.07580)
0.0005087 -0.00000362
Scale [[tau].sub.4] (0.6005) (-0.0023)
(0.9999) (-0.0026)
0.0007546 -0.0017561
Scale [[tau].sub.5] (1.2384) (-1.6084)
(2.213851) ** (-2.6671)
0.0002361 -0.0011234
Scale [[tau].sub.6] (0.5086) (-1.6993)
(0.8393) (-2.6049)
[[gamma].sub.2] [[gamma].sub.3]
The raw 0.0008760 -1.9239822
returns (0.8098) (-3.8927) ***
model (0.5976) (-3.3691) ***
0.00060268 -3.8715827
Scale [[tau].sub.1] (0.6978) (-3.4946) ***
(0.6054) (-3.2663) ***
0.0011587 -6.3959734
Scale [[tau].sub.2] (1.0789) (-3.4339) ***
(1.0217) (-2.9857) ***
0.0006552 -10.8993042
Scale [[tau].sub.3] (0.8323) (-3.4776) ***
(0.7906) (-2.3686) ***
0.0000216 -32.9996030
Scale [[tau].sub.4] (0.031795) (-4.3256) ***
(0.0274) (-4.138774) ***
0.0011311 -68.219816
Scale [[tau].sub.5] (2.3295) ** (-4.7254) ***
(3.4117) *** (-4.3507) ***
0.0007235 -100.5335394
Scale [[tau].sub.6] (2.2280) ** (-3.2369) ***
(3.3360) *** (-2.171988) **
Statistical parameters
of OLS regression
The raw [R.sup.2] = 0.1701
returns DW = 2.6387
model
[R.sup.2] = 0.1464
Scale [[tau].sub.1] DW = 2.6498
[R.sup.2] = 0.1481
Scale [[tau].sub.2] DW = 2.5717
[R.sup.2] = 0.1535
Scale [[tau].sub.3] DW = 2.5225
[R.sup.2] = 0.2022
Scale [[tau].sub.4] DW = 2.5291
[R.sup.2] = 0.2576
Scale [[tau].sub.5] DW = 2.5072
[R.sup.2] = 0.1331
Scale [[tau].sub.6] DW = 2.4782
Average daily market premium in the effective period of
testing the CAPM = 0.00002519
Notes: See notes for Table 4.
