A test for independence based on the correlation dimension

TLDR

The paper introduces an independence test applicable to the residuals of any time‑series model that can be transformed into one driven by independent, identically distributed errors. The test is implemented via Dechert and LeBaron's software, which is fast enough to estimate the null distribution using bootstrap methods. Its first‑order asymptotic distribution is independent of estimation error under consistent parameter estimation, allowing it to serve as a model‑selection and specification test and acting as a nonlinear analogue of the Box‑Pierce Q statistic for ARIMA models.

Abstract

This paper presents a test of independence that can be applied to the estimated residuals of any time series model that can be transformed into a model driven by independent and identically distributed errors. The first order asymptotic distribution of the test statistic is independent of estimation error provided that the parameters of the model under test can be estimated -consistently. Because of this, our method can be used as a model selection tool and as a specification test. Widely used software1 written by Dechert and LeBaron can be used to implement the test. Also, this software is fast enough that the null distribution of our test statistic can be estimated with bootstrap methods. Our method can be viewed as a nonlinear analog of the Box-Pierce Q statistic used in ARIMA analysis.