This article obtains the asymptotic distributions of the seasonal variance ratio tests proposed by A.M.R. Taylor (2005,Journal of Econometrics 124, 33) when these tests are applied to a periodically integrated process [PI(1)]. In contrast to the situation where the process is seasonally integrated [SI(1)], all test statistics in the PI(1) case are driven by a single stochastic trend and hence follow the distribution obtained by Breitung (2002, Journal of Econometrics 108, 343) for the original (non-seasonal) variance ratio test. The multivariate non-parametric cointegration test of Breitung (2002 Journal of Econometrics 108, 343) is also investigated to distinguish between PI and SI processes. A Monte Carlo analysis shows how these results apply in finite samples for both SI and PI processes and an empirical application investigates seasonally unadjusted quarterly US industrial production series.