When is a Copula Constant? A Test for Changing Relationships

Abstract

A copula de…nes the probability that the observations from two time series are both below a given quantile. 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 di¤erent quantiles may point to changes in di¤erent parts of the copula, with the lower quantiles being of particular interest in …nancial 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 e¤ective if pre-…ltering is carried out to correct for changing volatility or, more generally, changing quantiles. Applying the tests to data on IBM and General Motors stock returns indicates that the relationship is not constant over time.

Large portfolio risk management and optimal portfolio allocation with dynamic elliptical copulas

“…We further test the dynamic dependence with the BQ test ( [11]). These tests do not require a model for the copula but can be regarded as stationarity tests for time-varying bivariate quantiles.…”

A simple nonparametric test for structural change in joint tail probabilities

“…Following Busetti & Harvey (2008), we let ξ(τ ) denote the τ -quantile of some univariate probability distribution. To avoid unnecessary notational complications, we consider continuous distributions only, so ξ(τ ) is uniquely defined.…”

“…Of course, other functionals of the BIQ τ (t) such as the sum of absolute values might also be used as test statistics, but we focus here on the performance of the maximum and the range statistic (compared to the sum of squares statistic proposed by Busetti & Harvey (2008)). …”

“…Following Busetti & Harvey (2008), we base our test on joint exceedences of certain quantiles of the marginal distributions. Instead of using sum of squares of a normalized indicator function, we propose two alternative test statistics.…”

Monitoring Test for Stability of Dependence Structure in Multivariate Data Based on Copula

“…However, there is empirical evidence suggesting that the dependence structure is likely to change due to some financial adjustments and critical social events (see, for example, Longin and Solnik [5], Patton [6] and Rodriguez [7]). To cope with this, Dias and Embrechts [8] and Guegan and Zhang [9] suggested a likelihood ratio test for copula parameter changes in specific copula families, Harvey [10] and Busetti and Harvey [11] developed a nonparametric stationarity test for a constant copula based on time-varying quantiles, and Na et al [12] studied a cusum test for detecting the copula parameter change.…”