A multiple-threshold AR(1) model

We consider the model Z, = +(0, k ) + +(I, k)Z,_, + a,(k) whenever r,_, < Z,_,S r,., 1S k k 1, with r, = -m and r, = m. Here {+(i, k); i =0 , l ; 1 5 k 5 1) is a sequence of real constants, not necessarily equal, and, for 1 5 k 5 I, {a,(k), t 2 1) is a sequence of i.i.d. random variables with mean 0 and with {a,(k), t 2 1) independent of {a,(j), t 2 1) for j # k. Necessary and sufficient conditions on the constants {+(i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal. NON-LINEAR TIME SERIES; SETAR MODELS; AUTOREGRESSIVE MODELS; MARKOV CHAINS