Goodness-of-Fit Testing for Time Series Models via Distance Covariance

TL;DR

This paper introduces a new goodness-of-fit test for time series models using the auto-distance covariance function (ADCV) to better detect dependence in residuals, especially for financial data, accounting for parameter estimation effects.

Contribution

It develops a novel ADCV-based test for independence of residuals in time series, with a general theoretical framework and adjustments for parameter estimation.

Findings

The ADCV test effectively detects dependence in residuals.

Theoretical limits of the test are derived for various models.

Simulation studies demonstrate the test's practical performance.

Abstract

In many statistical modeling frameworks, goodness-of-fit tests are typically administered to the estimated residuals. In the time series setting, whiteness of the residuals is assessed using the sample autocorrelation function. For many time series models, especially those used for financial time series, the key assumption on the residuals is that they are in fact independent and not just uncorrelated. In this paper, we apply the auto-distance covariance function (ADCV) to evaluate the serial dependence of the estimated residuals. Distance covariance can discriminate between dependence and independence of two random vectors. The limit behavior of the test statistic based on the ADCV is derived for a general class of time series models. One of the key aspects in this theory is adjusting for the dependence that arises due to parameter estimation. This adjustment has essentially the same form regardless of the model specification. We illustrate the results in simulated examples.