Stability of nonlinear AR(1) time series with delay (Q1613619)

Let \(\{e_t\}\) be a sequence of i.i.d. random variables. Consider the first-order nonlinear time series model with delay lag \(d\geq 1\) defined by \[ \xi_t =\varphi (\xi_{t-1},\dots , \xi_{t-d})\xi_{t-1} +\vartheta (\xi_{t-1},\dots ,\xi_{t-d}) +c(e_t;\;\xi_{t-1},\dots ,\xi_{t-d}),\quad t\geq 1, \] where the initial values \(\xi_{1-d},\dots ,\xi_0\) are independent of \(\{e_t\}\). The coefficient function \(\varphi \) and the intercept function \(\vartheta \) are bounded and measurable, and the function \(c\) is measurable. The model for \(\xi _t\) can be either nonparametric, partially parametrized (e.g. the EXPAR model), or fully parametric. Define \(X_t=(\xi_t,\dots , \xi_{t-d+1})\). Then \(\{X_t\}\) is the Markov chain associated with the process \(\{\xi_t\}\). The authors provide sufficient conditions for \(\{X_t\}\) to be geometrically ergodic or geometrically transient, and some weaker conditions for both ergodicity and transience. The conditions are sharp for threshold-like models.