On Testing for Randomized Unit Root and Seasonal Unit Root

A lot of time series analysis in economics and nance is to determine whether a unit root and/or seasonal unit root is present in the data. These tests are usually based on unit root tests orginally developed by Dickey & Fuller(1981). Testing for the presence of a seasonal root has been considered by Dickey, Hasza & Fuller (1984). Li(1991) considered tests for the existence of a seasonal and a regular unit root or just one of the kinds in the data. An alternative to the unit root autoregression is the Random Coe cient Autoregressive model (RCAR). It can be viewed as a representation of ARCH model which has the same conditional variance pattern. Leybourne, McCabe and Tremayne(1996), Leybourne, McCabe and Mills(1996), and Granger and Swanson(1994) suggested that it could be sensible to focus attention on modelling the levels of stock prices, rather than, say, their logarithmic rst di erences. Dickey, Hasza and Fuller(1984) considered testing for a seasonal root for a SARIMA(p; 0; 0) (0; 1; 0)s model (using the standard notation of Box and Jenkins, 1976). In this paper, a slightly more elaborate model a SARIMA(p; 1; 0) (0; 1; 0)s model, will be considered which may be regarded as a generalized airline model in the sense of Box and Jenkins. The existence of randomized unit roots is addressed via the Lagrange Multiplier test. The asymptotic representation of the test statistics will be in terms of Brownian processes. Sample critical values and empirical sizes and powers of the test will be provided in this paper.