A random walk through seasonal adjustment : noninvertible moving averages and unit root tests

This paper examines the distributions of (zero frequency) unit root test statistics for I(1) processes in the presence of noninvertible moving average components. The analysis initially considers a noninvertible MA(1), for which the asymptotic distribution of the ADF test statistic under the unit root null hypothesis is shown to depend on the order of augmentation and can be shifted to either the right or the left, so that undersizing or oversizing problems may result. Although the distribution of the PP statistic depends on both the order of autocorrelation allowed and the weighting function used, it is always undersized with the Bartlett window. When extended to noninvertibility arising from X-11 seasonal adjustment of a random walk, the analytical features of the asymptotic distributions of these tests show corresponding characteristics as for the MA(1) case. These results are supported by a Monte Carlo analysis of the large sample distributions, and finite sample size properties of these unit root tests are also examined for a range of seasonal and nonseasonal I(1) processes.