Estimation of a utility-based asset pricing model using normal mixture GARCH(1,1)

Abstract Brown and Gibbons [Brown, D.P., Gibbons, M.R., 1985. A simple econometric approach for utility-based asset pricing model. Journal of Finance 40, 359–381], Karson et al. [Karson, M., Cheng, D., Lee, C.F., 1995. Sampling distribution of the relative risk aversion estimator: theory and applications. Review of Quantitative Finance and Accounting 5, 43–54], and Lee et al. [Lee, C.F., Lee, J.C., Ni, H.F., Wu, C.C., 2004. On a simple econometric approach for utility-based asset pricing model. Review of Quantitative Finance and Accounting 22, 331–344] developed the theory and the distribution of unconditional relative risk aversion (RRA) estimates in utility-based asset pricing model by assuming normality for the log excess returns. While the normality assumption is not always appropriate for some security returns, Brown and Gibbons [Brown, D.P., Gibbons, M.R., 1985. A simple econometric approach for utility-based asset pricing model. Journal of Finance 40, 359–381] proposed generalized method of moments (GMM) to estimate unconditional RRA. However, RRA estimated by GMM is not statistically efficient with finite samples. The main purpose of this paper is to derive the process of estimating dynamic RRA with the maximum likelihood and a Bayesian method having a weakly informative prior density while assuming that the log excess returns on the market are distributed as normal mixture GARCH(1,1). This methodology will capture the variations of RRA across different periods. Empirical results are presented using market rates of returns and risk-free rates over the period 1941 to 2001.

[1]  Kenneth J. Singleton,et al.  Modeling the term structure of interest rates under non-separable utility and durability of goods , 1986 .

[2]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[3]  David P. Brown,et al.  A Simple Econometric Approach for Utility‐Based Asset Pricing Models , 1985 .

[4]  Anil K. Bera,et al.  Efficient tests for normality, homoscedasticity and serial independence of regression residuals , 1980 .

[5]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[6]  Sanford J. Grossman,et al.  The Determinants of the Variability of Stock Market Prices , 1980 .

[7]  J. Campbell.  Intertemporal Asset Pricing Without Consumption Data , 1992 .

[8]  M. Karson,et al.  Sampling distribution of the relative risk aversion estimator: Theory and applications , 1995, Handbook of Financial Econometrics, Mathematics, Statistics, and Machine Learning.

[9]  W. Gilks.  Markov Chain Monte Carlo , 2005 .

[10]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[11]  Alan Kraus,et al.  MARKET EQUILIBRIUM IN A MULTIPERIOD STATE PREFERENCE MODEL WITH LOGARITHMIC UTILITY , 1975 .

[12]  S. Viswanathan,et al.  No Arbitrage and Arbitrage Pricing: A New Approach , 1993 .

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

[14]  Ravi Jagannathan,et al.  Implications of Security Market Data for Models of Dynamic Economies , 1990, Journal of Political Economy.

[15]  L. Wasserman,et al.  Practical Bayesian Density Estimation Using Mixtures of Normals , 1997 .

[16]  Geoffrey J. McLachlan,et al.  Standard errors of fitted component means of normal mixtures , 1997 .

[17]  S. Newcomb.  A Generalized Theory of the Combination of Observations so as to Obtain the Best Result , 1886 .

[18]  M. Aitkin,et al.  Mixture Models, Outliers, and the EM Algorithm , 1980 .

[19]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[20]  G. Casella,et al.  Explaining the Gibbs Sampler , 1992 .

[21]  R. Engle,et al.  Empirical Pricing Kernels , 1999 .

[22]  Gonzalo Rubio,et al.  Asset pricing and risk aversion in the Spanish stock market , 1990 .

[23]  Mark Rubinstein,et al.  The Valuation of Uncertain Income Streams and the Pricing of Options , 1976 .

[24]  G. Box,et al.  On a measure of lack of fit in time series models , 1978 .

[25]  Leslie Godfrey,et al.  Testing the adequacy of a time series model , 1979 .

[26]  C. Robert,et al.  Estimation of Finite Mixture Distributions Through Bayesian Sampling , 1994 .

[27]  P. W. Zehna.  Invariance of Maximum Likelihood Estimators , 1966 .

[28]  David W. Wilcox.  The Construction of U.S. Consumption Data: Some Facts and Their Implications for Empirical Work , 1992 .

[29]  Lan Peter Hamen,et al.  Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models , 1982 .

[30]  Haim Levy,et al.  Financial decision making under uncertainty , 1979 .

[31]  M. Rothschild,et al.  Increasing risk II: Its economic consequences , 1971 .

[32]  Luigi Ermini,et al.  SOME NEW EVIDENCE ON THE TIMING OF CONSUMPTION DECISIONS AND ON THEIR GENERATING PROCESS , 1989 .

[33]  P. Boothe,et al.  The statistical distribution of exchange rates: Empirical evidence and economic implications , 1987 .

[34]  Carol Alexander,et al.  Normal Mixture Garch(1,1): Applications to Exchange Rate Modelling , 2004 .

[35]  Anil K. Bera,et al.  Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo Evidence , 1981 .

[36]  N. H. Hakansson..  OPTIMAL INVESTMENT AND CONSUMPTION STRATEGIES UNDER RISK FOR A CLASS OF UTILITY FUNCTIONS11This paper was presented at the winter meeting of the Econometric Society, San Francisco, California, December, 1966. , 1970 .

[37]  Geoffrey J. McLachlan,et al.  Finite Mixture Models , 2019, Annual Review of Statistics and Its Application.

[38]  L. Hansen,et al.  Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns , 1983, Journal of Political Economy.

[39]  Bart Cockx,et al.  Duration Dependence in the Exit Rate Out of Unemployment in Belgium: Is it True or Spurious? , 2002, SSRN Electronic Journal.

[40]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[41]  M. Rubinstein..  THE STRONG CASE FOR THE GENERALIZED LOGARITHMIC UTILITY MODEL AS THE PREMIER MODEL OF FINANCIAL MARKETS , 1976 .

[42]  S. Chib,et al.  Understanding the Metropolis-Hastings Algorithm , 1995 .

[43]  P. Green,et al.  On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion) , 1997 .

[44]  Stanley J. Kon.  Models of Stock Returns—A Comparison , 1984 .

[45]  Marc S. Paolella,et al.  Mixed Normal Conditional Heteroskedasticity , 2004 .

[46]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[47]  R. Lucas.  ASSET PRICES IN AN EXCHANGE ECONOMY , 1978 .

[48]  A. F. Smith,et al.  Statistical analysis of finite mixture distributions , 1986 .

[49]  D. Slesnick.  Are Our Data Relevant to the Theory? The Case of Aggregate Consumption , 1998 .

[50]  P. Green,et al.  Corrigendum: On Bayesian analysis of mixtures with an unknown number of components , 1997 .