Goodness-of-Fit tests with Dependent Observations

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.Comment: 26 page

Copulas and long memory

This paper focuses on the analysis of persistence propertiesof copula-based time series. We obtain theoretical results that demonstratethat Gaussian and Eyraud-Farlie-Gumbel-Morgenstern copulas always pro-duce short memory stationary Markov processes. We further show via sim-ulations that, in finite samples, stationary Markov processes, such as thosegenerated by Clayton copulas, may exhibit a spurious long memory-like be-havior on the level of copulas, as indicated by standard methods of inferenceand estimation for long memory time series. We also discuss applicationsof copula-based Markov processes to volatility modeling and the analysisof nonlinear dependence properties of returns in real financial markets thatprovide attractive generalizations of GARCH models. Among other conclu-sions, the results in the paper indicate non-robustness of the copula-levelanalogues of standard procedures for detecting long memory on the levelof copulas and emphasize the necessity of developing alternative inferencemethods