A threshold AR(1) model

We consider the model where φ 1 , φ 2 are real coefficients, not necessarily equal, and the a t ,'s are a sequence of i.i.d. random variables with mean 0. Necessary and sufficient conditions on the φ 's are given for stationarity of the process. Least squares estimators of the φ 's are derived and, under mild regularity conditions, are shown to be consistent and asymptotically normal. An hypothesis test is given to differentiate between an AR(1) (the case φ 1 = φ 2 ) and this threshold model. The asymptotic behavior of the test statistic is derived. Small-sample behavior of the estimators and the hypothesis test are studied via simulated data.