Testing the martingale difference hypothesis using integrated regression functions

Abstract

An omnibus test for testing a generalized version of the martingale difference hypothesis (MDH) is proposed. This generalized hypothesis includes the usual MDH, testing for conditional moments constancy such as conditional homoscedasticity (ARCH effects) or testing for directional predictability. A unified approach for dealing with all of these testing problems is proposed. These hypotheses are long standing problems in econometric time series analysis, and typically have been tested using the sample autocorrelations or in the spectral domain using the periodogram. Since these hypotheses cover also nonlinear predictability, tests based on those second order statistics are inconsistent against uncorrelated processes in the alternative hypothesis. In order to circumvent this problem pairwise integrated regression functions are introduced as measures of linear and nonlinear dependence. The proposed test does not require to chose a lag order depending on sample size, to smooth the data or to formulate a parametric alternative model. Moreover, the test is robust to higher order dependence, in particular to conditional heteroskedasticity. Under general dependence the asymptotic null distribution depends on the data generating process, so a bootstrap procedure is considered and a Monte Carlo study examines its finite sample performance. Then, the martingale and conditional heteroskedasticity properties of the Pound/Dollar exchange rate are investigated.

The Evolution of Stock Market Efficiency Over Time: A Survey of the Empirical Literature

“…We highlight in this section two groups of time series analytic tools yet to be adopted in the extant stock market studies, except by the developers of the tests themselves. First, several test statistics have been proposed for testing whether stock returns are martingale difference sequence, or equivalently, whether stock prices follow a martingale process (see Hinich and Patterson, 1992; Domínguez and Lobato, 2003; Kuan and Lee, 2004; Hong and Lee, 2005; Escanciano and Velasco, 2006a, b). Statistical tests of martingale difference sequence are designed to capture linear and nonlinear serial dependence in mean, but they do not impose any restrictions on the dynamics in conditional variance and other higher‐order conditional moments 10,11 .…”

Persistence in Nonlinear Time Series: A Nonparametric Approach

“…where E (Y t ) = t for all t, are called the Integrated Pairwise Regression Functions (IPRF) in general, and the Integrated Pairwise Autoregression Functions (IPAF) when X t = Y t ; see Escanciano and Velasco (2006a). As previously noted in the literature (cf.…”

Empirical likelihood based testing for regression

“…For the calibration of the critical points associated to the test statistics, two approximations were mainly used: one is based on the bootstrap (see [17]) and the other is based on martingale transformations (see [19]). For alternative testing procedures based on smoothing techniques, see [23], and for recent generalizations of the empirical regression process ideas to dependent data, see the papers by [18], [6], [7] and [8].…”