Testing the Martingale Difference Hypothesis

Abstract

In this paper we consider testing that an economic time series follows a martingale difference process. The martingale difference hypothesis has typically been tested using information contained in the second moments of a process, that is, using test statistics based on the sample autocovariances or periodograms. Tests based on these statistics are inconsistent since they cannot detect nonlinear alternatives. In this paper we consider tests that detect linear and nonlinear alternatives. Given that the asymptotic distributions of the considered tests statistics depend on the data generating process, we propose to implement the tests using a modified wild bootstrap procedure. The paper theoretically justifies the proposed tests and examines their finite sample behavior by means of Monte Carlo experiments.

Keywords:

• Nonlinear dependence
• Nonparametric
• Correlation
• Bootstrap

JEL Classification:

• C12 and C52

Acknowledgment

We thank the editor, associate editor, and two referees for useful suggestions. We also thank Herman Bierens, Tim Bollerslev, John Nankervis, Gene Savin, and Winfried Stute for useful conversations. Finally, we thank seminar participants at the 2000 Econometric Society World Meeting for useful comments. Domínguez acknowledges financial support from Consejo Nacional de Ciencia y Tecnología (CONACYT) under project grant J38276D and Lobato acknowledges financial support from Asociación Mexicana de Cultura.

Notes

^aAn uncorrelated process cannot be forecasted using linear functions of lagged values, while an MDS cannot be forecasted using either linear or nonlinear functions of past values.

^bWe call [A] the compactification of Aand, in case the function R ^W is not defined for every ãin the frontier of A, we extend the process by considering that R ^W (ã) = lim _{ã n →ã} R ^W (ã _n ). For the cases of interest, this limit always exists.

^cNotice that y¯ = y¯ ^C  + o _p (n ^−1∣2) where y¯ ^C is the usual definition for the sample mean that takes into account all the available observations (y¯ ^C  = (n + p)⁻¹∑ _t=−p+1 ⁿ y _t ).

^dIn the previous covariance formula and in the rest of the paper, τ˜ ∧ υ˜ denotes the vector whose ith component is the minimum of the ith components of the vectors τ˜ and υ˜.

^eFor instance, Box and Pierce's Q, or Q* or the statistics proposed by [Anderson (1993)], [Durlauf (1991)], [Hong (1996)], or [Robinson (1991)].