A New Approach to Goodness of Fit for Ergodic Markov Processes

2026-08-04

AI Summary

This study addresses the absence of tuning-free, density-based goodness-of-fit tests for ergodic Markov processes that do not require specifying an alternative hypothesis. The authors propose a nonparametric test based on comparing stationary densities, which evaluates whether any stationary distribution within the null model class matches the empirical density of the observed data. This approach obviates the need for explicit alternative hypotheses or smoothing parameter estimation. Theoretical analysis demonstrates that the test achieves nontrivial power against local alternatives converging at the $1/\sqrt{n}$ rate, while also possessing favorable asymptotic properties and computational tractability. Consequently, it offers a novel diagnostic tool for model validation in econometrics and financial modeling.

Abstract

We introduce a new density-based goodness of fit test for ergodic Markov processes. Our test compares the data against the class of models specified in the null hypothesis, and rejects if no model in the class yields a stationary density that matches with the data. No alternative needs to be specified in order to implement the test. Although our test compares densities, estimation of smoothing parameters is not required, and the test has nontrivial power against $1/\sqrt{n}$ local alternatives. The test provides new perspectives on some existing problems in econometric and financial modeling.

Problem

• goodness of fit
• ergodic Markov processes
• stationary density
• nonparametric test
• hypothesis testing

Innovation

• goodness of fit
• ergodic Markov processes
• density-based test
• stationary density
• nonparametric testing