Determination of the Number of Common Stochastic Trends under Conditional Heteroskedasticity

Permanent-transitory decompositions and the analysis of the time series prop- erties of economic variables at the business cycle frequencies strongly rely on the correct detection of the number of common stochastic trends (co-integration). Standard techniques for the determination of the number of common trends, such as the well-known sequential procedure proposed in Johansen (1996), are based on the assumption that shocks are homoskedastic. This contrasts with empirical evidence which documents that many of the key macro-economic and .nancial variables are driven by heteroskedastic shocks. In a recent paper, Cavaliere et al., (2010, Econometric Theory) demonstrate that Johansen.s (LR) trace statistic for co-integration rank and both its i.i.d. and wild boot- strap analogues are asymptotically valid in non-stationary systems driven by heteroskedastic (martingale di¤erence) innovations, but that the wild bootstrap performs substantially better than the other two tests in .nite samples. In this paper we analyse the behaviour of sequential procedures to determine the number of common stochastic trends present based on these tests. Numerical evidence suggests that the procedure based on the wild bootstrap tests performs best in small samples under a variety of heteroskedastic innovation processes.