Estimation of Copula Models for Time Series of Possibly Different Lengths

The theory of conditional copulas provides a means of constructing flexible multivariate density models, allowing for time varying conditional densities of each individual variable, and for time-varying conditional dependence between the variables. Further, the use of copulas in constructing these models often allows for the partitioning of the parameter vector into elements relating only to a marginal distribution, and elements relating to the copula. This paper presents a two-stage (or multi-stage) maximum likelihood estimator for the case that such a partition is possible. We extend the existing statistics literature on the estimation of copula models to consider data that exhibit temporal dependence and heterogeneity. The estimator is flexible enough that the case that unequal amounts of data are available on each variable is easily handled. We investigate the small sample properties of the estimator in a Monte Carlo study, and find that it performs well in comparisons with the standard (one-stage) maximum likelihood estimator. Finally, we present an application of the estimator to a model of the joint distribution of daily Japanese yen - U.S. dollar and euro - U.S. dollar exchange rates. We find some evidence that a copula that captures asymmetric dependence performs better than those that assume symmetric dependence.

[1]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[2]  T. Louis,et al.  Inferences on the association parameter in copula models for bivariate survival data. , 1995, Biometrics.

[3]  Re-Emerging Markets , 1997 .

[4]  J. Rosenberg.  Non-Parametric Pricing of Multivariate Contingent Claims , 2003 .

[5]  Asymptotic Properties of One-step Estimator Obtained from an Optimal Step-size , 1987 .

[6]  M. E. Johnson,et al.  A Family of Distributions for Modelling Non‐Elliptically Symmetric Multivariate Data , 1981 .

[7]  Campbell R. Harvey,et al.  Forecasting International Equity Correlations , 1994 .

[8]  T. Bollerslev,et al.  A CONDITIONALLY HETEROSKEDASTIC TIME SERIES MODEL FOR SPECULATIVE PRICES AND RATES OF RETURN , 1987 .

[9]  D. Andrews.  Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables , 1988, Econometric Theory.

[10]  H. Joe,et al.  The Estimation Method of Inference Functions for Margins for Multivariate Models , 1996 .

[11]  David V. Glidden,et al.  A Two-Stage Estimator of the Dependence Parameter for the Clayton-Oakes Model , 2000, Lifetime data analysis.

[12]  Allan Timmermann,et al.  Dangers of Data-Driven Inference: The Case of Calendar Effects in Stock Returns , 1998 .

[13]  Increased Correlation in Bear Markets: A Downside Risk Perspective , 2002 .

[14]  C. Gouriéroux,et al.  PSEUDO MAXIMUM LIKELIHOOD METHODS: THEORY , 1984 .

[15]  V. P. Godambe.  An Optimum Property of Regular Maximum Likelihood Estimation , 1960 .

[16]  R. Stambaugh,et al.  Analyzing Investments Whose Histories Differ in Length , 1997 .

[17]  L. L. Cam,et al.  On the Asymptotic Theory of Estimation and Testing Hypotheses , 1956 .

[18]  J. Rosenberg.  Nonparametric Pricing of Multivariate Contingent Claims , 2000 .

[19]  David X. Li.  On Default Correlation: A Copula Function Approach , 1999 .

[20]  Joseph P. Romano,et al.  The stationary bootstrap , 1994 .

[21]  C. Genest,et al.  A semiparametric estimation procedure of dependence parameters in multivariate families of distributions , 1995 .

[22]  M. Kendall.  Statistical Methods for Research Workers , 1937, Nature.

[23]  F. Longin,et al.  Extreme Correlation of International Equity Markets , 2000 .

[24]  D. Clayton.  A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence , 1978 .

[25]  P. Hansen,et al.  Technical Appendix A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH(1,1)? , 2005 .

[26]  Nicole A. Lazar,et al.  Statistical Analysis With Missing Data , 2003, Technometrics.

[27]  O. Scaillet,et al.  Nonparametric estimation of copulas for time series , 2003 .

[28]  C. Small,et al.  The theory and applications of statistical inference functions , 1988 .

[29]  R. Farebrother.  A Class of Bivariate Density Functions with Common Marginals , 1993, Econometric Theory.

[30]  H. White,et al.  A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models , 1988 .

[31]  C. Genest,et al.  An Extension of Osuna's Model for Stress Caused by Waiting. , 2001, Journal of mathematical psychology.

[32]  James Jianmeng Xu.  Statistical modelling and inference for multivariate and longitudinal discrete response data , 1996 .

[33]  R. Plackett.  A Class of Bivariate Distributions , 1965 .

[34]  Jason P. Fine,et al.  On association in a copula with time transformations , 2000 .

[35]  B. Mandlebrot.  The Variation of Certain Speculative Prices , 1963 .

[36]  D. Oakes.  A Model for Association in Bivariate Survival Data , 1982 .

[37]  Paul Embrechts,et al.  Using copulae to bound the Value-at-Risk for functions of dependent risks , 2003, Finance Stochastics.

[38]  Anthony S. Tay,et al.  Evaluating Density Forecasts with Applications to Financial Risk Management , 1998 .

[39]  Thierry Roncalli,et al.  Copulas for finance , 2000 .

[40]  Kee-Hong Bae,et al.  A New Approach to Measuring Financial Contagion , 2000 .

[41]  Tom Smith,et al.  A Test for Multivariate Normality in Stock Returns , 1993 .

[42]  J. Wooldridge,et al.  Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances , 1992 .

[43]  Christian Genest,et al.  A nonparametric estimation procedure for bivariate extreme value copulas , 1997 .

[44]  Halbert White,et al.  Estimation, inference, and specification analysis , 1996 .

[45]  Financial Analysts Federation.  The financial analysts journal , 1960 .

[46]  Andrew Ang,et al.  Asymmetric Correlations of Equity Portfolios , 2001 .

[47]  P. Kofman,et al.  Imputation Methods for Incomplete Dependent Variables in Finance , 2000 .

[48]  D. Oakes,et al.  Bivariate survival models induced by frailties , 1989 .

[49]  Eric Jondeau,et al.  Conditional Dependency of Financial Series: An Application of Copulas , 2001 .

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

[51]  Eric Bouyé,et al.  Copulas for Finance - A Reading Guide and Some Applications , 2000 .

[52]  J. Rosenberg.  Semiparametric Pricing of Multivariate Contingent Claims , 1999 .

[53]  C. Genest,et al.  Statistical Inference Procedures for Bivariate Archimedean Copulas , 1993 .

[54]  T. W. Anderson.  Maximum Likelihood Estimates for a Multivariate Normal Distribution when Some Observations are Missing , 1957 .

[55]  Andrew J. Patton.  Modelling Time-Varying Exchange Rate Dependence Using the Conditional Copula , 2001 .

[56]  W. Newey,et al.  Large sample estimation and hypothesis testing , 1986 .