A New Wald Test for Hypothesis Testing Based on MCMC outputs

TL;DR

This paper introduces a new Wald test based on MCMC outputs that is easy to implement, well-defined under improper priors, and applicable to latent variable models in economics and finance.

Contribution

It proposes a novel MCMC-based Wald test that overcomes limitations of traditional methods, including handling improper priors and providing easy calibration.

Findings

Test follows chi-squared distribution asymptotically

Applicable to models with improper priors

Demonstrated usefulness in economic and financial models

Abstract

In this paper, a new and convenient $\chi^2$ wald test based on MCMC outputs is proposed for hypothesis testing. The new statistic can be explained as MCMC version of Wald test and has several important advantages that make it very convenient in practical applications. First, it is well-defined under improper prior distributions and avoids Jeffrey-Lindley's paradox. Second, it's asymptotic distribution can be proved to follow the $\chi^2$ distribution so that the threshold values can be easily calibrated from this distribution. Third, it's statistical error can be derived using the Markov chain Monte Carlo (MCMC) approach. Fourth, most importantly, it is only based on the posterior MCMC random samples drawn from the posterior distribution. Hence, it is only the by-product of the posterior outputs and very easy to compute. In addition, when the prior information is available, the finite sample theory is derived for the proposed test statistic. At last, the usefulness of the test is illustrated with several applications to latent variable models widely used in economics and finance.