Invariance Principles for Tempered Fractionally Integrated Processes

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in $\alpha$-stable $(1<\alpha \le 2)$ i.i.d. innovations and related tempered linear processes with vanishing tempering parameter $\lambda \sim {\lambda }_{\ast }\mathrm{/}N$. We show that the limit of the partial sums process takes a different form in the weakly tempered (${\lambda }_{\ast }=0$), strongly tempered (${\lambda }_{\ast }=\mathrm{\infty }$), and moderately tempered ($0<\lambda_* < \infty$) cases. These results are used to derive the limit distribution of the OLS estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.