Goodness-of-fit tests for Markovian time series models: Central limit theory and bootstrap approximations

TL;DR

This paper introduces new goodness-of-fit tests for Markovian time series models, utilizing nonparametric estimates and bootstrap methods, with theoretical validation and practical demonstrations.

Contribution

It develops novel goodness-of-fit tests for Markov models, including bootstrap schemes and a new CLT for dependent variables, with theoretical and empirical validation.

Findings

Test statistics have well-defined asymptotic properties.

Bootstrap methods effectively approximate critical values.

Numerical examples demonstrate good finite-sample performance.

Abstract

New goodness-of-fit tests for Markovian models in time series analysis are developed which are based on the difference between a fully nonparametric estimate of the one-step transition distribution function of the observed process and that of the model class postulated under the null hypothesis. The model specification under the null allows for Markovian models, the transition mechanisms of which depend on an unknown vector of parameters and an unspecified distribution of i.i.d. innovations. Asymptotic properties of the test statistic are derived and the critical values of the test are found using appropriate bootstrap schemes. General properties of the bootstrap for Markovian processes are derived. A new central limit theorem for triangular arrays of weakly dependent random variables is obtained. For the proof of stochastic equicontinuity of multidimensional empirical processes, we use a simple approach based on an anisotropic tiling of the space. The finite-sample behavior of the proposed test is illustrated by some numerical examples and a real-data application is given.