Sequential unit root test for first-order autoregressive processes with initial values

This paper examines the effect of initial values and small-sample properties in sequential unit root tests of the first-order autoregressive (AR(1)) process with a coefficient expressed by a local parameter. Adopting a stopping rule based on observed Fisher information defined by Lai and Siegmund (1983), we use the sequential least squares estimator (LSE) of the local parameter as the test statistic. The sequential LSE is represented as a time-changed Brownian motion with drift. The stopping time is written as the integral of the reciprocal of twice of a Bessel process with drift generated by the time-changed Brownian motion. The time change is applied to the joint density and joint Laplace transform derived from the Bessel bridge of the squared Bessel process by Pitman and Yor (1982), by which we derive the limiting joint density and joint Laplace transform for the sequential LSE and stopping time. The joint Laplace transform is needed to calculate joint moments because the joint density oscillates wildly as the value of the stopping time approaches zero. Moreover, this paper also earns the exact distribution of stopping time by Imhof's formula for both normally distributed and fixed initial values. When the autoregressive coefficient is less than 1, the question arises as to whether the local-to-unity or the strong stationary model should be used. We make the decision by comparing joint moments for respective models with those calculated from the exact distribution or simulations.