Goodness-of-Fit tests with Dependent Observations

TL;DR

This paper extends classical goodness-of-fit tests to dependent data by introducing self-copulas, showing dependence reduces effective sample size, with applications demonstrated on financial time series revealing long-range dependence.

Contribution

It generalizes GoF tests to dependent observations using self-copulas, providing analytical results and applying them to financial data to reveal long-range dependence.

Findings

Dependence reduces the effective number of independent observations.

Self-copulas encode all non-linear temporal dependencies.

Financial time series exhibit long-range dependence confirmed by self-copulas.

Abstract

We revisit the Kolmogorov-Smirnov and Cram\'er-von Mises goodness-of-fit (GoF) tests and propose a generalisation to identically distributed, but dependent univariate random variables. We show that the dependence leads to a reduction of the "effective" number of independent observations. The generalised GoF tests are not distribution-free but rather depend on all the lagged bivariate copulas. These objects, that we call "self-copulas", encode all the non-linear temporal dependences. We introduce a specific, log-normal model for these self-copulas, for which a number of analytical results are derived. An application to financial time series is provided. As is well known, the dependence is to be long-ranged in this case, a finding that we confirm using self-copulas. As a consequence, the acceptance rates for GoF tests are substantially higher than if the returns were iid random variables.