On the testing multi-valued martingale difference hypothesis

Abstract

This paper presents a definition of Multi-Valued Martingale Difference (MVMD) based on Castaing representation of a multi-valued martingale that consists of martingale difference selections. Then testing the Multi-Valued Martingale Difference Hypothesis (MVMDH) is studied. Testing the Martingale Difference Hypothesis (MDH) earlier was based on linear measures, and later is developed in two directions to consider the existing nonlinearity in economic and financial data. First, the classical approaches have been modified by taking into account the possible nonlinear dependence. Second, the use of more sophisticated statistical tools such as those based on the generalized spectral analysis. In this paper, both these developments in MDH are modified for MVMDH and are applied to exchange rate data and returns of stock market data.