The Applications of Mixtures of Normal Distributions in Empirical Finance: A Selected Survey

This paper provides a selected review of the recent developments and applications of mixtures of normal (MN) distribution models in empirical finance. Once attractive property of the MN model is that it is flexible enough to accommodate various shapes of continuous distributions, and able to capture leptokurtic, skewed and multimodal characteristics of financial time series data. In addition, the MN-based analysis fits well with the related regime-switching literature. The survey is conducted under two broad themes: (1) minimum-distance estimation methods, and (2) financial modeling and its applications.

[1]  Bent E. Sørensen,et al.  GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study , 1996 .

[2]  Stochastic Conditional Duration Models with , 2004 .

[3]  Manabu Asai,et al.  Bayesian analysis of stochastic volatility models with mixture-of-normal distributions , 2009, Mathematics and Computers in Simulation.

[4]  A. Cohen,et al.  Estimation in Mixtures of Two Normal Distributions , 1967 .

[5]  D. M. Titterington,et al.  On the deter-mination of the number of components in a mixture , 1998 .

[6]  María Concepción Ausín,et al.  Bayesian estimation of the Gaussian mixture GARCH model , 2007, Comput. Stat. Data Anal..

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

[8]  Daniel B. Nelson.  CONDITIONAL HETEROSKEDASTICITY IN ASSET RETURNS: A NEW APPROACH , 1991 .

[9]  Lung-fei Lee,et al.  Estimation of some limited dependent variable models with application to housing demand , 1978 .

[10]  L. Bauwens,et al.  The stochastic conditional duration model: a latent variable model for the analysis of financial durations , 2004 .

[11]  N. Shephard,et al.  Stochastic Volatility: Likelihood Inference And Comparison With Arch Models , 1996 .

[12]  Carol Alexander,et al.  Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects , 2004 .

[13]  D. Chakrabarti,et al.  A fast fixed - point algorithm for independent component analysis , 1997 .

[14]  R. Quandt.  A New Approach to Estimating Switching Regressions , 1972 .

[15]  Charles S. Bos,et al.  Adaptive Polar Sampling With An Application To A Bayes Measure Of Value-At-Risk , 1999 .

[16]  Lung-fei Lee.  Unionism and Wage Rates: A Simultaneous Equations Model with Qualitative and Limited Dependent Variables , 1978 .

[17]  John Knight,et al.  Estimation of the stochastic conditional duration model via alternative methods , 2008 .

[18]  Dingan Feng,et al.  Stochastic Conditional Duration Models with ''Leverage Effect'' for Financial Transaction Data , 2004 .

[19]  Giampiero M. Gallo,et al.  Mixture Processes for Financial Intradaily Durations , 2004 .

[20]  William R. Melick,et al.  Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis , 1997, Journal of Financial and Quantitative Analysis.

[21]  D. Brigo,et al.  Displaced and Mixture Diffusions for Analytically-Tractable Smile Models , 2002 .

[22]  Giovanni De Luca,et al.  Time-Varying Mixing Weights in Mixture Autoregressive Conditional Duration Models , 2005 .

[23]  G. J. Jiang,et al.  Estimation of Continuous-Time Processes via the Empirical Characteristic Function , 2002 .

[24]  QianSheng Cheng,et al.  An approach to VaR for capital markets with Gaussian mixture , 2005, Appl. Math. Comput..

[25]  Andreas S. Weigend,et al.  Computing portfolio risk using Gaussian mixtures and independent component analysis , 1999, Proceedings of the IEEE/IAFE 1999 Conference on Computational Intelligence for Financial Engineering (CIFEr) (IEEE Cat. No.99TH8408).

[26]  Jun Yu,et al.  EMPIRICAL CHARACTERISTIC FUNCTION IN TIME SERIES ESTIMATION , 2001, Econometric Theory.

[27]  Portfolio optimization for alternative investments , 2003 .

[28]  B. Everitt,et al.  Finite Mixture Distributions , 1981 .

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

[30]  Stephen L Taylor,et al.  Modelling Financial Time Series , 1987 .

[31]  G. McLachlan.  On Bootstrapping the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture , 1987 .

[32]  Victor H. Lachos,et al.  Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions , 2010, Comput. Stat. Data Anal..

[33]  T. Wirjanto,et al.  An Empirical Characteristic Function Approach to VaR Under a Mixture-of-Normal Distribution with Time-Varying Volatility , 2010, The Journal of Derivatives.

[34]  Alan G. White,et al.  Value at Risk When Daily Changes in Market Variables are not Normally Distributed , 1998 .

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

[36]  H. Bozdogan.  Choosing the Number of Component Clusters in the Mixture-Model Using a New Informational Complexity Criterion of the Inverse-Fisher Information Matrix , 1993 .

[37]  N. Mendell,et al.  Simulated percentage points for the null distribution of the likelihood ratio test for a mixture of two normals. , 1988, Biometrics.

[38]  Damiano Brigo,et al.  Alternative asset-price dynamics and volatility smile , 2003 .

[39]  Jose A. Lopez,et al.  Methods for Evaluating Value-at-Risk Estimates , 1998 .

[40]  Byron J. T. Morgan,et al.  Integrated squared error estimation of normal mixtures , 2004, Comput. Stat. Data Anal..

[41]  N. Shephard,et al.  Stochastic volatility with leverage: Fast and efficient likelihood inference , 2007 .

[42]  Jeffrey R. Russell,et al.  Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data , 1998 .

[43]  Subu Venkataraman,et al.  Value at risk for a mixture of normal distributions: the use of quasi- Bayesian estimation techniques , 1997 .

[44]  Peter C. Schotman,et al.  An empirical application of stochastic volatility models , 1998 .

[45]  Z. D. Feng,et al.  Using Bootstrap Likelihood Ratio in Finite Mixture Models , 1994 .

[46]  A. Paulson,et al.  The estimation of the parameters of the stable laws , 1975 .

[47]  Franz C. Palm,et al.  The message in weekly exchange rates in the European Monetary System: mean reversion , 1993 .

[48]  Aapo Hyvärinen,et al.  A Fast Fixed-Point Algorithm for Independent Component Analysis , 1997, Neural Computation.

[49]  Empirical Characteristic Function in Time Series Estimation , 2001 .

[50]  Ray C. Fair,et al.  Methods of Estimation for Markets in Disequilibrium , 1972 .

[51]  C. Alexander,et al.  Symmetric Normal Mixture GARCH , 2004 .

[52]  J. Heckman.  Life Cycle Consumption and Labor Supply: An Explanation of the Relationship Between Income and Consumption Over the Life Cycle , 1974 .

[53]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[54]  A. Harvey,et al.  5 Stochastic volatility , 1996 .

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

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

[57]  Stephen L Taylor,et al.  Modelling Financial Time Series , 1987 .

[58]  W. Li,et al.  On a mixture autoregressive model , 2000 .

[59]  S. Csörgo.  Empirical characteristic functions , 1980 .

[60]  E. Ruiz,et al.  Estimation Methods for Stochastic Volatility Models: A Survey , 2004 .

[61]  Aapo Hyvärinen,et al.  Fast and robust fixed-point algorithms for independent component analysis , 1999, IEEE Trans. Neural Networks.

[62]  Jun Yu.  Empirical Characteristic Function Estimation and Its Applications , 2003 .

[63]  J. B. Ramsey,et al.  Estimating Mixtures of Normal Distributions and Switching Regressions , 1978 .

[64]  R. J. Ritchey,et al.  CALL OPTION VALUATION FOR DISCRETE NORMAL MIXTURES , 1990 .

[65]  Kien C. Tran.  Estimating mixtures of normal distributions via empirical characteristic function , 1998 .

[66]  C. R. Heathcote,et al.  The integrated squared error estimation of parameters , 1977 .

[67]  Tony S. Wirjanto,et al.  Asymmetric Stochastic Conditional Duration Model — A Mixture-of-Normal Approach , 2011 .

[68]  Richard E. Quandt,et al.  The Econometrics Of Disequilibrium , 1988 .

[69]  Jun Yu.  On Leverage in a Stochastic Volatility Model , 2004 .

[70]  Franz C. Palm,et al.  Simple diagnostic procedures for modeling financial time series , 1997 .

[71]  Wai Keung Li,et al.  On a Mixture Autoregressive Conditional Heteroscedastic Model , 2001 .

[72]  Reinhard Hujer,et al.  Econometric Analysis of Financial Trade Processes by Discrete Mixture Duration Models , 2005 .

[73]  A. Feuerverger,et al.  On the Efficiency of Empirical Characteristic Function Procedures , 1981 .

[74]  Peter F. Christoffersen.  Evaluating Interval Forecasts , 1998 .

[75]  George C. Tiao,et al.  Kurtosis of GARCH and stochastic volatility models with non-normal innovations , 2003 .

[76]  N. E. Day.  Estimating the components of a mixture of normal distributions , 1969 .

[77]  Wang Yan,et al.  Measuring the Information Content of Stock Trades , 2004 .

[78]  Peter Schmidt,et al.  An Improved Version of the Quandt-Ramsey MGE Estimator for Mixtures of Normal Distributions and Switching Regressions , 1982 .

[79]  An Efficient Estimation for Switching Regression Models , 2009 .

[80]  A. Feuerverger,et al.  The Empirical Characteristic Function and Its Applications , 1977 .

[81]  Dinghai Xu,et al.  Continuous Empirical Characteristic Function Estimation of Mixtures of Normal Parameters , 2010 .