Abstract
This paper investigates the problem of testing for linear Granger causality in mean when the number of parameters is high with the possible presence of nonlinear dynamics. Dependent innovations are taken into account by considering tests which asymptotic distributions is a weighted sum of chi-squares and tests with modified weight matrices. Wald, Lagrange Multiplier (LM) and Likelihood Ratio (LR) tests for linear causality in mean are studied. It is found that the LM tests based on restricted estimators significantly improve the analysis of linear Granger causality in mean relations when the dimension is high or when the autoregressive order is large. We also see that the tests based on a modified asymptotic distribution have a better control of the error of first kind when compared to the tests with modified statistic in finite samples. An application to international finance data is proposed to illustrate the robustness to the presence of nonlinearities of the studied tests.
Highlights
Since the paper by Granger [18], the use of linear causality in mean to study relations between subsets of variables is a common practice
In this paper we considered the problem of testing the linear causality in mean in situations where the dimension is high or the autoregressive order is large
Concerning the Wald tests with modified statistics, it emerges that this kind of tests poorly perform when the dimension is high or the autoregressive order is large in small samples
Summary
Since the paper by Granger [18], the use of linear causality in mean to study relations between subsets of variables is a common practice (see [33, 35], or [19] among other references). This can be explained by the fact that linear causality in mean is based on linear predictors, and can be tested by considering tests of zero restrictions on the parameters of VAR models with iid innovations. The symbol ⇒ denotes the convergence in distribution and the almost surely convergence is denoted by −a.→s
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