This paper investigates the limiting properties of the Canova
and Hansen test, testing for the null hypothesis of no unit
root against seasonal unit roots, under a sequence of local
alternatives with the model extended to have seasonal dummies
and trends or no deterministic term and also only seasonal dummies.
We derive the limiting distribution of the test statistic and
its characteristic function under local alternatives. We find
that the local limiting power is an inverse function of the
spectral density at frequency π (π/2) when we test
against a negative unit root (annual unit roots). We also
theoretically show that the local limiting power of the Canova
and Hansen test against a negative unit root (annual unit roots)
does not increase when the true process has annual unit roots
(a negative unit root) but not a negative unit root (annual unit
roots), which has been observed in Monte Carlo simulations in such
research as Caner (1998, Journal of Business and Economic
Statistics 16, 349–356), Canova and Hansen (1995,
Journal of Business and Economic Statistics 13, 237–252),
and Hylleberg (1995, Journal of Econometrics 69, 5–25).