On the nearly nonstationary seasonal time series

A time series is said to be nearly nonstationary if some of its characteristic roots are close to the unit circle. For a seasonal time series, such a notion of near-nonstationarity is studied in a double-array setting. This approach not only furnishes a natural transition between stationarity and nonstationarity, but also unifies the corresponding asymptotic theories in a seasonal-time-series context. The general theory is expressed in terms of functionals of independent diffusion processes. The asymptotic results have applications to estimation and testing in a nearly nonstationary situation and serve as a useful alternative to the common practice of seasonal adjustment.