Nonlinearities in the CAPM: Evidence from Developed and Emerging Markets
| Author | Vasilios Sogiakas,John H. McColl,Duncan Lee,Serdar Neslihanoglu |
| DOI | http://doi.org/10.1002/for.2389 |
| Date | 01 December 2017 |
| Published date | 01 December 2017 |
Journal of Forecasting,J. Forecast. 36, 867–897 (2017)
Published online 25 January 2016 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/for.2389
Nonlinearities in the CAPM: Evidence from Developed
and Emerging Markets
SERDAR NESLIHANOGLU,1VASILIOS SOGIAKAS,2JOHN H. McCOLL AND
DUNCAN LEE3
1
Department of Statistics, Faculty of Science and Letters, Eskisehir Osmangazi University,
Eskisehir, Turkey
2
Adam Smith Business School (Economics), University of Glasgow, Glasgow, UK
3
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
ABSTRACT
This paper examines the forecasting ability of the nonlinear specifications of the market model. We propose a condi-
tional two-moment market model with a time-varying systematic covariance(beta) risk in the form of a mean reverting
process of the state space model via the Kalman filter algorithm. In addition, we account for the systematic compo-
nent of co-skewness and co-kurtosis by considering higher moments. The analysis is implemented using data from
the stock indices of several developed and emerging stock markets. The empirical findings favour the time-varying
market model approaches, which outperform linear model specifications both in terms of model fit and predictability.
Precisely, higher moments are necessary for datasets that involve structural changes and/or market inefficiencies which
are common in most of the emerging stock markets. Copyright © 2016 John Wiley & Sons, Ltd..
KEY WORDS CAPM; time-varying market model; co-skewness and co-kurtosis; quadratic and cubic
market model
INTRODUCTION
A lengthy criticism on the usefulness of the traditional capital asset pricing model (CAPM) model has been addressed
in the literature of arbitrage pricing models that propose risk factors on firm fundamentals (Fama and French, 2004)
or nonlinearities on the model specification. Specifically, researchers focus on the examination of the dynamics of
asset pricing models, either by addressing the importance and theoretical intuition of documented stylized factors or
by quantifying the time series properties of the data-generating process (DGP) and/or the estimated parameter set.
Consequently, some doubts on the mechanism with which informational efficiency of stock exchanges is examined
have been raised. The review papers of Schwert (2003) and Malkiel (2003) highlight this criticism and provide
evidence that several of the stylized facts tend to be weaker after the papers which highlighted them were published,
or viewed as short-term aberrations of a long-term efficient market.
Any decision about the usefulness of the two-moment CAPM on the examination of the efficient market hypoth-
esis should be made conditionally on the validity of the assumptions of this model and on the securities’ pricing
mechanism. Indeed, normally distributed returns, independence and homoscedasticity of security prices and linear-
ity of asset pricing models reflect a stock exchange of perfection. Additionally, the foundations behind the pricing
mechanism that CAPM suggests rely on the existence of rational investors with unconditional and unlimited leverage
opportunities who form a one-period investment and financing decision and on the existence of a market portfolio
which consists of all possible financial products of the underlying stock market.
Consequently, it is questionable whether investors care only about the one-period portfolio returns and whether
they do not care about the covariance of their portfolio returns with other factors relevantto labour income, investment
opportunities, business risk and political risk, thus diminishing the usefulness of CAPM. Moreover, the difficulty in
reflecting the minimum variance frontier on a market proxy of the market portfolio has ‘boomerang" effects. Several
extensions have been proposed in the literature, including Blume (1970) and Black et al. (1972), who implement
the CAPM on portfolios rather than on individual securities, Fama and MacBeth (1973), who test the significance
of the risk premia with a two-stage regression analysis, Black (1972), who proposes a zero-beta model, Merton
(1973) intertemporal CAPM, Breeden (1979) consumption CAPM and Roll (1977) with the arbitrage asset pricing
model. However, even if the authors of the above-mentioned papers by the empirical examination of the CAPM
model have verified a clear relationship between beta and asset returns, several asymmetrieswere detected against the
rationality of the CAPM. Specifically, aggressive firms underperform those expected by CAPM, while several other
firm fundamentals contribute substantially to the explanation of the cross-sectional asset returns, such as firm size
Correspondence to: Serdar Neslihanoglu, Department of Statistics, Faculty of Science and Letters, Eskisehir Osmangazi University, Eskisehir,
Turkey.E-mail: sneslihanoglu@ogu.edu.tr
Copyright © 2016 John Wiley & Sons, Ltd.
868 S. Neslihanoglu et al.
(Banz, 1981), liquidity (Amihud and Mendelson, 1986) and momentum (Jegadeesh and Titman, 1993). It is worth
mentioning that, as Subrahmanyam (2010) explains, more than 50 variables have been used in the literature to explain
asset returns. The intuition behind the utilization of these variables and/or factors is not crystal clear and this concerns
many economists about the usefulness of the extensions of the CAPM.
This paper addresses the limitations and the usefulness of the two-moment CAPM and proposes nonlinear exten-
sions with higher moments that account for the skewness and the kurtosis components of asset returns. This extension
is of crucial importance for trading, asset allocation, risk management, and the examination of the informational effi-
ciency of securities’ prices. Our objective is the comparative analysis of linear and nonlinear market models in terms
of predictive ability. We allow for exogenous structural changes, reflecting the October 2008 financial crisis period
and, most importantly, we utilize data from developed and emerging stock markets to identify potential patterns of
the predictability of asset pricing models in relation to the inefficiencies that are penetrated in less-developed or
emerging stock markets. Specifically, we compare the linear market model (consistent with the two-moment CAPM
including systematic covariance (beta)) with six nonlinear extensions. The first two are new reformulated forms of
higher-order DGPs as simple polynomial extensions of the linear market model that involve fractional moments, i.e.
the quadratic market model (consistent with the three-moment CAPM, including systematic covariance (beta)and
skewness (co-skewness)) and the cubic market model (consistent with the four-moment CAPM, including systematic
covariance, systematic skewness and systematic kurtosis (co-kurtosis)). The third model is the generalized addi-
tive model (GAM) (relaxing some of the assumptions underpinning polynomial models). The last three approaches
consist of the time-varying versions of the linear market model and polynomial extensions in the mean-reverting
specification of the state space model via the Kalman filter algorithm (Kalman, 1960), i.e. the time-varying lin-
ear market model (allowing for only time-varying systematic covariance), the time-varying quadratic market model
(allowing for time-varying systematic covariance and time-varying systematic skewness) and the time-varying cubic
market model (allowing for time-varying systematic covariance, time-varying systematic skewness and time-varying
systematic kurtosis).
The comparative analysis which is conducted in this paper sheds much light on the necessity of nonlinear models
in the explanation of asset prices. Time-varying model specifications outperform the unconditional models, while
structural changes of financial time series are better absorbed within higher moments of the CAPM. Finally, we
provide evidence in favour of higher-moment model specification when dealing with data of emergingstock markets,
underlying the importance of nonlinear models when analysing market inefficiencies.
Our paper contributes to the literature of CAPM usefulness in a number of ways. Firstly, it explains the usefulness
of CAPM using higher-order moments and nonlinearities to support the expected utility foundations of asset pricing
models. Secondly, it proposes an innovative GAM application in the CAPM framework. Thirdly,it proposes a nonlin-
ear model with fractional moments .ŒRmRf/,wheretakes any positive value) instead of integers that represent
the second and third moments. Finally, it applies the mean-revertingspecification of the state space model via Kalman
filter methodology in the proposed quadratic market model (QMM) and cubic market model (CMM) accounting for
the time-varying characteristics of the systematic covariance, the systematic skewness and the systematic kurtosis,
namely time-varying quadratic market model (TvQMM) and time-varying cubic market model (TvCMM).
The rest of the paper is organized as follows: the next section provides a revision of the literature review; the third
section considers that dataset for our analysis; the fourth section explains the research methodology; the fifth section
discusses our empirical findings; and the sixth section concludes the paper.
LITERATURE REVIEW
While Sharpe’s (1964) CAPM, under specific, and often heroic, assumptions, lay on the security market line that com-
prises exclusively the beta (systematic covariance) risk with respect to the market portfolio—a hypothetical portfolio
(Roll, 1977)—it fails to provide consistency through time and/or across firm fundamentals. A significant contribu-
tion on the former aspect of this literature is Merton’s (1973) intertemporal capital asset pricing model (ICAPM),
according to which investors optimize their portfolios considering the intertemporal relationship of expected returns
with future state variables. The latter inconsistency has motivated many researchers—among them, Fama and French
(1993)—to propose extensions that account for several stylized financial facts that associate investors’ expectations
with firm fundamentals. While the statistical significance of these characteristics, which do not, necessarily, repre-
sent state variables of concern to investors, on multifactor models, enhance the criticism against CAPM, Ang and
Chen (2007) argue that they could be fully accounted for by a one-factor model with time-varying factor loadings,
providing evidence in favour of the conditional CAPM.
Another significant contribution to the CAPM literature was developed by Kraus and Litzenberger (1976) in order
to relax the two restrictive assumptions of the CAPM; i.e. normally distributed asset returns and the quadratic utility
function (in terms of the mean and variance terms only; that is why it is called a two-moment CAPM in this paper). In
Copyright © 2016 John Wiley & Sons, Ltd. J. Forecast. 36, 867–897 (2017)
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