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. Testing the Multi-Valued Martingale Difference Hypothesis (MVMDH) then examined. Testing the Martingale Difference Hypothesis (MDH) earlier was based on linear measures then later developed two directions in order to account for the existing nonlinearity in economic and financial data. First, the classical approaches have been modified by take into account the possible nonlinear dependence. Second, the use of more sophisticated statistical tools such as those based generalized spectral analysis. According to this article, both these developments in MDH are modified for MVMDH and applies them to exchange rate data and returns of stock market data.

Weak Set-Valued Martingale Difference and Its Applications

“…This test is directly related to the efficient market, [16]. For computational convenience, instead of verifying whether {y t } is a martingale, we usually perform the testing of martingale difference hypothesis (called the MDH test) for its returns, meaning that E[d t+1 |F t ] = 0, where d t+1 = ln y t+1 − ln y t ≈ (y t+1 − y t )/y t , [12,29]. Various MDH tests depending on the relation of historical information F n−1 , F n−2 , • • • are found in [10,13].…”