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

Abstract

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.

Acknowledgements

The authors would like to thank Clive W.J. Granger for his useful comments and encouragement. The usual disclaimer applies.

Notes

¹ Causality testing appears to be a popular methodology in applied research. A simple check in Econlit on the word ‘causality’ resulted in more than 2100 published journal articles. Among recent published articles that consider theoretical aspects of causality testing the following papers can be mentioned: Hecq (1996), Luintel (1999), Ostermark and Aaltonen (1999) and Lee et al. (2002).

² Dolado and Lütkepohl (1996) conducted a simulation study to check the power properties of the Toda–Yamamoto Wald test based on asymptotic distributions. However, Dolado and Lütkepohl dealt only with variables that are integrated of degree one.

³ A program procedure written in GAUSS by the authors for applications of this method is available from the authors on request.

⁴ For more information on the use of VAR models see Hatemi-J (1999, 2001).

⁵ The formal derivation of Equation 15 is based on Hatemi-J (2004b).

⁶ Yamada and Toda (1998) perform simulations close to ours for the homoscedastic case using the asymptotic distribution and similarly find over-rejection of the null hypothesis in finite samples with the over-rejection declining with greater sample size.

⁷ The frequency of the data is restricted by data availability since the effective exchange rate is available only on a monthly basis.

⁸ Doornik and Hansen (1994) test was used to test for multivariate normality. The null hypothesis of no ARCH effects was tested by applying a test suggested by Hacker and Hatemi-J (2005).