Article1992

Testing Causality Between Two Vectors in Multivariate Autoregressive Moving Average Models

Hafida Boudjellaba · Jean‐Marie Dufour · Roch Roy

86 citations0/0 files storedJournal of the American Statistical Association

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Monetary Policy and Economic Impact Financial Risk and Volatility Modeling Diverse Scientific and Economic Studies

Abstract

In the analysis of economic time series, a question often raised is whether a vector of variables causes another one in the sense of Granger. Most of the literature on this topic is concerned with bivariate relationships or uses finite-order autoregressive specifications. The purpose of this article is to develop a causality analysis in the sense of Granger for general vector autoregressive moving average (ARMA) models. We give a definition of Granger noncausality between vectors, which is a natural and simple extension of the notion of Granger noncausality between two variables. In our context, this definition is shown to be equivalent to a more complex definition proposed by Tjostheim. For the class of linear invertible processes, we derive a necessary and sufficient condition for noncausality between two vectors of variables when the latter do not necessarily include all the variables considered in the analysis. This result is then specialized to the class of stationary invertible ARMA processes. Further, relatively simple necessary and sufficient conditions are obtained for two important cases: (1) the case where the two vectors reduce to two variables inside a larger vector including other variables; and (2) the case where the two vectors embody all the variables considered. Test procedures for these necessary and sufficient conditions are discussed. Among other things, it is noted that the necessary and sufficient conditions for noncausality may involve singularities at which standard asymptotic regularity conditions do not hold. To deal with such situations, we propose a sequential approach that leads to bounds tests. Finally, the tests suggested are applied to Canadian money and income data. The tests are based on bivariate and trivariate models of changes in nominal income and two money stocks (M1 and M2). In contrast with the evidence based on bivariate models, we find from the trivariate model that money causes income unidirectionally.

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References

1979Distribution of the Estimators for Autoregressive Time Series with a Unit Root 1969Investigating Causal Relations by Econometric Models and Cross-spectral Methods 1988Testing for a Unit Root in Time Series Regression 1981Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root 1980Macroeconomics and Reality 1972MONEY, INCOME, AND CAUSALITY 1982Measurement of Linear Dependence and Feedback Between Multiple Time Series 1983Forecasting and Conditional Projection Using Realistic Prior Distributions

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