Vector Autoregressions and Causality

TLDR

The study aims to develop a limit theory for Wald tests of Granger causality in levels VARs and ECMs that accommodates stochastic trends and cointegration. The authors extend previous work to the general case, deriving conditions under which these Wald tests are asymptotically valid as chi‑square criteria. The resulting limit theory reveals that inference from unrestricted levels VARs and ECMs is problematic, with nuisance parameters and nonstandard distributions undermining the statistical basis for Granger causality tests, a surprising departure from earlier expectations. © 1993 by The Econometric Society.

Abstract

A limit theory for Wald tests of Granger causality in levels vector autoregressions (VAR's) and error correction models (ECM's) is developed, which allows for stochastic trends and cointegration. Earlier work is extended to the general case, thereby characterizing when these Wald tests are asymptotically valid as 'x'(superscript 2) criteria. Our results for inference from unrestricted levels VAR are not encouraging: the limit theory often involves nuisance parameters and nonstandard distributions, a situation offering no satisfactory statistical basis for these tests. Granger causality tests in ECM's also suffer from nuisance parameter dependencies asymptotically and in some cases nonstandard limit theory. Both these results are somewhat surprising in light of earlier research. Copyright 1993 by The Econometric Society.