This paper analyzes whether inclusion of a statistically independent random walk in a vector autoregression can result in spurious inference. The problem was raised originally by Ohanian (1988). In a Monte Carlo simulation based on the VAR’s estimated by Sims (1980b, 1982), Ohanian found that block exogeneity of the genuine variables with respect to an artiﬁcially generated random walk variable was rejected too often. In the present paper we attempt a full analytical study of this problem. It can be shown that if the genuine variables are nonstationary, the Wald statistic for testing the block exogeneity hypothesis does not have the usual asymptotic chi-square distribution. This result is consistent with Ohanian’s ﬁnding. Furthermore, the derived asymptotic distribution is free of nuisance parameters so that we can unambiguously determine the eﬀect of including the random walk. Interestingly, it can also be shown that if the genuine variables of the model are stationary, the asymptotic distribution is still chi-square in spite of the inclusion of the random walk.
Toda, Hiro Y. and Phillips, Peter C.B., "The Spurious Effect of Unit Roots on Exogeneity Tests in Vector Autoregressions: An Analytical Study" (1991). Cowles Foundation Discussion Papers. 1221.