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.

摘要

两个时间序列之间相关性的最常见度量是互相关函数。该度量完全描述了两个线性和联合高斯时间序列的相关性，但是对于非线性和非高斯时间序列模型（例如金融中使用的ARCH型模型），它通常会失败。的互相关函数是一个全球性的依赖量度。在本文中，我们将非线性局部变量应用于双变量时间序列相关性的度量称为局部高斯相关。它通常也适用于非线性模型，并且可以区分正局部依赖和负局部依赖。我们构造了局部高斯相关性的置信区间，并基于这种依赖性度量开发了一个检验。为参数估计，测试功能和块自举程序派生出渐近性质。对于模拟和金融指标数据，我们构建置信区间，然后将建议的测试与基于普通相关性的测试和基于布朗距离相关性的测试进行比较。长期检查财务指标，并在金融危机之前，之中和之后分析其局部联合行为，包括尾部行为。