Estimating and Testing Nonlinear Local Dependence Between Two Time Series

ABSTRACT

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.

KEY WORDS:

• Bivariate time series
• Block bootstrap
• Confidence intervals
• Independence testing
• Local Gaussian correlation
• Nonlinear dependence.

SUPPLEMENTARY MATERIALS

The supplementary material consists of the following parts: A: Details of singularity of the matrices in Section 2.1, B: Proofs of Theorems 2.1 and 2.2, C: Proofs of Theorems 3.1 and 3.2, D and E: Proof of validity of the block bootstrap, F: Choice of bandwidth and blocksize.

ACKNOWLEDGMENTS

The authors are indebted to the main editor and the referees for a number of useful suggestions that have improved our article.

Funding

The authors are grateful to the Norwegian Research Council for financial support through the grant 807442 “Nonlinear stochastic dependence models.” The authors are also grateful to Bård Støve and Håkon Otneim for supplying the financial dataset.