Using Out-of-Sample Mean Squared Prediction Errors to Test the Martingale Difference Hypothesis

By Todd E. Clark and Kenneth D. West
May 2004 
RWP 04-03
Research Division 
Federal Reserve Bank of Kansas City 

Abstract

      We consider using out of sample mean squared prediction errors (MSPEs) to evaluate the null that a given series follows a zero mean martingale difference against the alternative that it is linearly predictable. Under the null of zero predictability, the population MSPE of the null “no change” model equals that of the linear alternative. We show analytically and via simulations that despite this equality, the alternative model’s sample MSPE is expected to be greater than the null’s. We propose and evaluate an asymptotically normal test that properly accounts for the upward shift of the sample MSPE of the alternative model. Our simulations indicate that our proposed procedure works well.

Keywords: Forecast evaluation, causality, exchange rates

JEL Codes: C52, C53, C12, F31

Todd Clark is a vice president and economist at the Federal Reserve Bank of Kansas City. Kenneth West is a Ragnar Frisch Professor of Economics at the University of Wisconsin. The authors thank Taisuke Nakata for research assistance and the following for helpful comments: Charles Engel, Bruce Hansen, Michael McCracken, seminar participants at the University of Wisconsin and Federal Reserve Bank of Kansas City, and attendees of Predictive Methodology and Application in Economics and Finance: A Conference in Honor of Clive W.J. Granger. West thanks the National Science Foundation for financial support. The views expressed herein are solely those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Kansas City or the Federal Reserve System.

Clark email:  todd.e.clark@kc.frb.org
West e-mail:  kdwest@wisc.edu

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