Kolmogorov–Smirnov, Fluctuation, and Z_g Tests for Convergence of Markov Chain Monte Carlo Draws

Abstract

We examine the sizes and powers of three tests of convergence of Markov Chain Monte Carlo draws: the Kolmogorov–Smirnov test, fluctuation test, and Geweke's test. We show that the sizes and powers are sensitive to the existence of autocorrelation in the draws. We propose a filtered test that is corrected for autocorrelation. We present a numerical illustration using the Federal funds rate.

Keywords:

• Federal funds rates
• Fluctuation test
• Geweke's test
• Kolmogorov–Smirnov test
• Markov Chain Monte Carlo draws

Mathematics Subject Classification:

• 62F15
• 65C40
• 62-07
• 40A30

Acknowledgment

The authors thank an anonymous referee whose comments greatly help improve the article.

Notes

¹A proof for the asymptotic distribution of the FT is given in Ploberger et al. (1989).

Notes: (1) Sample size is 3,000 (N = 3,000).

(2) Number of replications is 3,000.

Notes: (1) Sample size is 3,000 (N = 3,000).

(2) Number of replications is 3,000.

AR(1) correction is made by Eq. (12)

.

²We may check stationarity of draws by a Bayesian procedure given in Goldman et al. (2001).