Testing the Martingale Hypothesis

Abstract

We propose new tests of the martingale hypothesis based on generalized versions of the Kolmogorov–Smirnov and Cramér–von Mises tests. The tests are distribution-free and allow for a weak drift in the null model. The methods do not require either smoothing parameters or bootstrap resampling for their implementation and so are well suited to practical work. The article develops limit theory for the tests under the null and shows that the tests are consistent against a wide class of nonlinear, nonmartingale processes. Simulations show that the tests have good finite sample properties in comparison with other tests particularly under conditional heteroscedasticity and mildly explosive alternatives. An empirical application to major exchange rate data finds strong evidence in favor of the martingale hypothesis, confirming much earlier research.

KEY WORDS:

• Brownian functional
• Cramér-von Mises test
• Exchange rates
• Explosive process
• Kolmogorov–Smirnov test

ACKNOWLEDGMENTS

The authors thank the Editor, two referees, Joon Park, and Yoon-Jae Whang for comments. P. C. B. Phillips thanks the NSF for support under Grant Nos. SES-0956687 and SES 12-58258. S. Jin acknowledges research support from the Sim Kee Boon Institute at Singapore Management University.

Notes

PW also consider a Markov switching model and Feigenbaum maps with system noise. We found that the results are similar for these models and so they are not reported.

We also tried EGARCH models as in Fong and Ouliaris (1995) and the results are again similar.