Generalized spectral tests for the martingale difference hypothesis

Abstract

This article proposes a test for the martingale difference hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n 1=2 ; with n the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach could be extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S&P 500 stock index and exchange rates highlights the merits of our approach. r 2005 Elsevier B.V. All rights reserved.JEL classification: C12

Testing the martingale difference hypothesis using integrated regression functions

“…Moreover, it can be shown that under mild assumptions our test based on D 2 n is asymptotically admissible, that is, there does not exist a test that is uniformly more powerful than our test for a general class of local alternatives to H 0 , see Corollary 4 in Escanciano and Velasco (2006). Thus, the test proposed in this paper should be viewed not as competing with but as a complement to that considered in Escanciano and Velasco (2006) since they have different power properties and both are asymptotically admissible.…”

“…Hong's (1999) approach involves the choice of a kernel, a bandwidth parameter and an integrating measure and, in general, statistical inferences are not robust to these choices. Escanciano and Velasco (2006) avoid bandwidth choices by means of a generalized spectral distribution function. In this paper we propose a MDH test which preserves the good properties of Escanciano and Velasco's (2006) test and that complements it because it leads to different power properties.…”

“…Escanciano and Velasco (2006) avoid bandwidth choices by means of a generalized spectral distribution function. In this paper we propose a MDH test which preserves the good properties of Escanciano and Velasco's (2006) test and that complements it because it leads to different power properties. The main methodology is assisted with the individual use of new nonlinear dependence measures defined in the paper, which can play an important role in explaining well-known stylized facts of economic and financial data, see Section 6 for examples.…”

“…For recent applications on convergence results on Hilbert spaces in the econometric literature see Politis and Romano (1994), White (1996, 1998), Chen and Fan (1999), Escanciano and Velasco (2006) and Escanciano (2006b)…”

“…The consistency and local power properties for D 2 n can be discussed following the lines of Section 3.2 in Escanciano and Velasco (2006). The reader is referred to the latter reference for the asymptotic power local properties of the CvM test based on D 2 n .…”