Granger-Causal Analysis of Conditional Mean and Volatility Models

Abstract

Recent economic developments have shown the importance of spillover and contagion effects in financial markets as well as in macroeconomic reality. Such effects are not limited to relations between the levels of variables but also impact on the volatility and the distributions. Granger causality in conditional means and conditional variances of time series is investigated in the framework of several popular multivariate econometric models. Bayesian inference is proposed as a method of assessment of the hypotheses of Granger noncausality. First, the family of ECCC-GARCH models is used in order to perform inference about Granger-causal relations in second conditional moments. The restrictions for second-order Granger noncausality between two vectors of variables are derived. Further, in order to investigate Granger causality in conditional mean and conditional variances of time series VARMA-GARCH models are employed. Parametric restrictions for the hypothesis of noncausality in conditional variances between two groups of variables, when there are other variables in the system as well are derived. These novel conditions are convenient for the analysis of potentially large systems of economic variables. Bayesian testing procedures applied to these two problems, Bayes factors and a Lindley-type test, make the testing possible regardless of the form of the restrictions on the parameters of the model. This approach also enables the assumptions about the existence of higher-order moments of the processes required by classical tests to be relaxed. Finally, a method of testing restrictions for Granger noncausality in mean, variance and distribution in the framework of Markov-switching VAR models is proposed. Due to the nonlinearity of the restrictions derived by Warne (2000), classical tests have limited use. Bayesian inference consists of a novel Block Metropolis-Hastings sampling algorithm for the estimation of the restricted models, and of standard methods of computing posterior odds ratios. The analysis may be applied to financial and macroeconomic time series with changes of parameter values over time and heteroskedasticity.