Estimating and Testing Nonlinear Local Dependence Between Two Time Series

The most common measure of dependence between two time series is the cross-correlation function. This measure gives a complete characterization of dependence for two linear and jointly Gaussian time series, but it often fails for nonlinear and non-Gaussian time series models, such as the ARCH-type models used in finance. The cross-correlation function is a global measure of dependence. In this article, we apply to bivariate time series the nonlinear local measure of dependence called local Gaussian correlation. It generally works well also for nonlinear models, and it can distinguish between positive and negative local dependence. We construct confidence intervals for the local Gaussian correlation and develop a test based on this measure of dependence. Asymptotic properties are derived for the parameter estimates, for the test functional and for a block bootstrap procedure. For both simulated and financial index data, we construct confidence intervals and we compare the proposed test with one based on the ordinary correlation and with one based on the Brownian distance correlation. Financial indexes are examined over a long time period and their local joint behavior, including tail behavior, is analyzed prior to, during and after the financial crisis. Supplementary material for this article is available online.