Limiting distribution of the score statistic under moderate deviation from a unit root in MA(1)

This article derives an asymptotic distribution of Tanaka's score statistic under moderate deviation from a unit root in a moving average model of order one [MA(1)]. The limiting distribution is classified into three types depending on the order of deviation. In the fastest case, the convergence order of the asymptotic distribution continuously changes from the invertible process to the unit root one. In the slowest case, the limiting distribution coincides with one in the invertible process in the distribution sense. This implies that they share a common asymptotic property. The limiting distribution in the intermediate case has the boundary property between the fastest case and the slowest one.