Tests for causality between integrated variables using asymptotic and bootstrap distributions: theory and application

Causality tests in the Granger's sense are increasingly applied in empirical research. Since the unit root revolution in time-series analysis, several modifications of tests for causality have been introduced in the literature. One of the recent developments is the Toda-Yamamoto modified Wald (MWALD) test, which is attractive due to its simple application, its absence of pre-testing distortions, and its basis on a standard asymptotical distribution irrespective of the number of unit roots and the cointegrating properties of the data. This study investigates the size properties of the MWALD test and finds that in small sample sizes this test performs poorly on those properties when using its asymptotical distribution, the chi-square. It is suggested that use be made of a leveraged bootstrap distribution to lower the size distortions. Monte Carlo simulation results show that an MWALD test based on a bootstrap distribution has much smaller size distortions than corresponding cases when the asymptotic distribution is used. These results hold for different sample sizes, integration orders, and error term processes (homoscedastic or ARCH). This new method is applied to the testing of the efficient market hypothesis.