20

Testing the Martingale Hypothesis

J. Carlos Escanciano and Ignacio N. Lobato

Abstract

This chapter examines testing the Martingale difference hypothesis (MDH) and related statistical
inference issues. The earlier literature on testing the MDH was based on linear measures of depen-
dence, such as sample autocorrelations; for example, the classic Box–Pierce portmanteau test and
the variance ratio test. In order to account for the existing nonlinearity in economic and financial
data, two directions have been entertained. First, to modify these classical approaches by taking
into account possible nonlinear dependence. Second, to use more sophisticated statistical tools
such as those based on empirical process theory or the use of generalized spectral analysis. This
chapter discusses these developments and applies them to exchange rate data.

20.1 Introduction 972
20.2 Preliminaries 973
20.3 Tests based on linear measures of dependence 975
20.3.1 Tests based on a finite-dimensional conditioning set 977
20.3.2 Tests based on an infinite-dimensional conditioning set 980
20.4 Tests based on nonlinear measures of dependence 984
20.4.1 Tests based on a finite-dimensional conditioning set 985
20.4.2 Tests based on an infinite-dimensional information set 988
20.5 Related hypotheses 996
20.6 Conclusions 997

20.1 Introduction

Martingale testing has received enormous attention in econometrics. One of the
main reasons is the efficient market hypothesis and the many ideas related to
it. In addition, many economic theories examining dynamic contexts in which
expectations are assumed to be rational lead to dependence restrictions of this kind
in the underlying economic variables (see, e.g., Hall, 1978; Fama, 1991; LeRoy,
1989; Lo, 1997; Cochrane, 2005). These theories have prompted a great deal of
research in macro- and financial economics which has stimulated a huge interest in
developing suitable econometric techniques. This econometric research has grown
around the theme of the lack of predictability of macro- or financial series, but this

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