Smooth Transition Quantile Capital Asset Pricing Models with Heteroscedasticity

Capital asset pricing model (CAPM) has become a fundamental tool in finance for assessing the cost of capital, risk management, portfolio diversification and other financial assets. It is generally believed that the market risks of the assets, often denoted by a beta coefficient, should change over time. In this paper, we model timevarying market betas in CAPM by a smooth transition regime switching CAPM with heteroscedasticity, which provides flexible nonlinear representation of market betas as well as flexible asymmetry and clustering in volatility. We also employ the quantile regression to investigate the nonlinear behavior in the market betas and volatility under various market conditions represented by different quantile levels. Parameter estimation is done by a Bayesian approach. Finally, we analyze some Dow Jones Industrial stocks to demonstrate our proposed models. The model selection method shows that the proposed smooth transition quantile CAPM–GARCH model is strongly preferred over a sharp threshold transition and a symmetric CAPM–GARCH model.

[1]  Roger Koenker,et al.  An Empirical Quantile Function for Linear Models with | operatornameiid Errors , 1982 .

[2]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[3]  Roger Koenker,et al.  An empirical quantile function for linear models with iid errors , 1981 .

[4]  Cathy W. S. Chen,et al.  Multi-regime nonlinear capital asset pricing models , 2009 .

[5]  Cathy W. S. Chen,et al.  Comparison of nonnested asymmetric heteroskedastic models , 2006, Comput. Stat. Data Anal..

[6]  Clive W. J. Granger,et al.  Large returns, conditional correlation and portfolio diversification: a value-at-risk approach , 2001 .

[7]  Eric Ghysels.  On Stable Factor Structurs in the Pricing of Risk. , 1995 .

[8]  Cathy W. S. Chen,et al.  A Bayesian threshold nonlinearity test for financial time series , 2005 .

[9]  Roger Koenker,et al.  Conditional Quantile Estimation and Inference for Arch Models , 1996, Econometric Theory.

[10]  Robert A. Levy,et al.  Beta Coefficients as Predictors of Return , 1974 .

[11]  C. Granger,et al.  Modelling Nonlinear Economic Relationships , 1995 .

[12]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[13]  Cathy W. S. Chen,et al.  Bayesian inference and model comparison for asymmetric smooth transition heteroskedastic models , 2008, Stat. Comput..

[14]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[15]  R. Koenker,et al.  Goodness of Fit and Related Inference Processes for Quantile Regression , 1999 .

[16]  T. Teräsvirta.  Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models , 1994 .

[17]  Cathy W. S. Chen,et al.  Semi-parametric quantile estimation for double threshold autoregressive models with heteroskedasticity , 2013, Comput. Stat..

[18]  David W. Bacon,et al.  Estimating the transition between two intersecting straight lines , 1971 .

[19]  R. Banz,et al.  The relationship between return and market value of common stocks , 1981 .

[20]  E. Fama,et al.  Common risk factors in the returns on stocks and bonds , 1993 .

[21]  H. Tong.  On a threshold model , 1978 .

[22]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[23]  Campbell R. Harvey,et al.  The Risk and Predictability of International Equity Returns , 1993 .

[24]  H. Tong,et al.  Threshold Autoregression, Limit Cycles and Cyclical Data , 1980 .

[25]  R. Chou,et al.  ARCH modeling in finance: A review of the theory and empirical evidence , 1992 .

[26]  R. Koenker.  Quantile Regression: Name Index , 2005 .

[27]  Keming Yu,et al.  Bayesian quantile regression , 2001 .

[28]  Eric Ghysels,et al.  On Stable Factor Structures in the Pricing of Risk: Do Time-Varying Betas Help or Hurt? , 1998 .

[29]  Timo Teräsvirta,et al.  Smooth transition autoregressive models - A survey of recent developments , 2000 .

[30]  R. Jagannathan,et al.  The Conditional CAPM and the Cross-Section of Expected Returns , 1996 .

[31]  H. Tong,et al.  ON ESTIMATING THRESHOLDS IN AUTOREGRESSIVE MODELS , 1986 .

[32]  Mehmet Caner,et al.  Time-Varying Betas Help in Asset Pricing: The Threshold CAPM , 2003 .

[33]  J. Lintner.  THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS , 1965 .

[34]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[35]  Cathy W. S. Chen,et al.  Bayesian Causal Effects in Quantiles: Accounting for Heteroscedasticity , 2009, Comput. Stat. Data Anal..

[36]  W. Sharpe.  CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* , 1964 .

[37]  E. Fama,et al.  The Cross‐Section of Expected Stock Returns , 1992 .