When is a copula constant? A test for changing relationships

A copula defines the probability that observations from two time series lie below given quantiles. It is proposed that stationarity tests constructed from indicator variables be used to test against the hypothesis that the copula is changing over time. Tests associated with different quantiles may point to changes in different parts of the copula, with the lower quantiles being of particular interest in financial applications concerned with risk. Tests located at the median provide an overall test of a changing relationship. The properties of various tests are compared and it is shown that they are still effective if pre-filtering is carried out to correct for changing volatility or, more generally, changing quantiles. Applying the tests to daily stock return indices in Korea and Thailand over the period 1995-9 indicates that the relationship between them is not constant over time.

[1]  W. Kruskal.  Ordinal Measures of Association , 1958 .

[2]  J. Nyblom,et al.  Comparisons of Tests for the Presence of Random Walk Coefficients in a Simple Linear Model , 1983 .

[3]  D. Picard.  Testing and estimating change-points in time series , 1985, Advances in Applied Probability.

[4]  J. Nyblom.  Testing for the Constancy of Parameters over Time , 1989 .

[5]  Harald Uhlig,et al.  Reasonable extreme bounds analysis , 1990 .

[6]  Jushan Bai,et al.  Weak Convergence of the Sequential Empirical Processes of Residuals in ARMA Models , 1994 .

[7]  P. Phillips,et al.  Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? , 1992 .

[8]  N. Shephard,et al.  Multivariate stochastic variance models , 1994 .

[9]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[10]  A. Harvey,et al.  Testing for a slowly changing level with special reference to stochastic volatility , 1996 .

[11]  Turalay Kenc,et al.  Ox: An Object-Oriented Matrix Language , 1997 .

[12]  M. C. Jones,et al.  Local Linear Quantile Regression , 1998 .

[13]  Jurgen A. Doornik,et al.  Statistical algorithms for models in state space using SsfPack 2.2 , 1999 .

[14]  R. Engle,et al.  CAViaR , 1999 .

[15]  Campbell R. Harvey,et al.  Autoregressive conditional skewness , 1999 .

[16]  Andrew Harvey,et al.  TESTS OF COMMON STOCHASTIC TRENDS , 2000, Econometric Theory.

[17]  A. Inoue.  TESTING FOR DISTRIBUTIONAL CHANGE IN TIME SERIES , 2001, Econometric Theory.

[18]  A. Harvey,et al.  Testing against smooth stochastic trends , 2001 .

[19]  Sanjiv Ranjan Das,et al.  Systemic Risk and International Portfolio Choice , 2002 .

[20]  R. Rigobón,et al.  No Contagion, Only Interdependence: Measuring Stock Market Comovements , 2002 .

[21]  C. Genest,et al.  Multivariate Option Pricing Using Dynamic Copula Models , 2005 .

[22]  P. Embrechts,et al.  Chapter 8 – Modelling Dependence with Copulas and Applications to Risk Management , 2003 .

[23]  J. C. Rodríguez,et al.  Measuring financial contagion:a copula approach , 2007 .

[24]  M. Wegkamp,et al.  Weak Convergence of Empirical Copula Processes , 2004 .

[25]  Sangyeol Lee,et al.  A nonparametric test for the change of the density function in strong mixing processes , 2004 .

[26]  E. Luciano,et al.  Copula methods in finance , 2004 .

[27]  Andrew J. Patton.  Modelling Asymmetric Exchange Rate Dependence , 2006 .

[28]  Xiaohong Chen,et al.  Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification , 2006 .

[29]  Giuliano De Rossi,et al.  Time-Varying Quantiles , 2006 .

[30]  Stephan Süss.  Alexander McNeil, Rüdiger Frey, Paul Embrechts (2005): “Quantitative Risk Management”, Princeton Series in Finance, $79.50.-. , 2006 .

[31]  A. Harvey,et al.  Tests of time-invariance , 2007 .

[32]  P. Schmidt,et al.  A robust version of the KPSS test based on indicators , 2007 .

[33]  Oliver Linton,et al.  The quantilogram: With an application to evaluating directional predictability , 2007 .

[34]  Andrew C. Harvey,et al.  Dynamic distributions and changing copulas , 2008 .

[35]  Andrew Harvey,et al.  Quantiles, expectiles and splines , 2009 .

[36]  Xiaohong Chen,et al.  STATISTICAL INFERENCE FOR MULTIVARIATE RESIDUAL COPULA OF GARCH MODELS , 2009 .

[37]  A. Harvey.  Tracking a changing copula , 2010 .

[38]  Paul Embrechts,et al.  Quantitative Risk Management , 2011, International Encyclopedia of Statistical Science.