On a class of threshold AR(k) processes

We consider the model $$Z_t = \sum\limits_{i = 1}^k {\phi (i,j)Z_{t - i} } + a_t (j)when\left {Z_{t - 1} ,Z_{t - 2,...,} Z_{t - k} } \right]^\prime \in R(j),$$ where {R(j);1?j? ?}is a partition of ? ^k , and for each 1?j??,{a _t (j);t? 0} are i.i.d. zero-mean random variables, having a strictly positive density. Sufficient conditions are obtained for this process to be transient. In addition, for a particular class of such models, necessary and sufficient conditions for ergodicity are obtained. Least-squares estimators of the parameters are obtained and are, under mild regularity conditions, shown to be strongly consistent and asymptotically normal.