Invariance Principles for Tempered Fractionally Integrated Processes

TL;DR

This paper establishes invariance principles for tempered fractionally integrated processes with stable innovations, revealing different limit behaviors depending on the tempering regime, and applies these results to the distribution of AR(1) unit root estimators.

Contribution

It introduces new invariance principles for tempered fractional processes with stable innovations and characterizes their limit distributions under various tempering regimes.

Findings

Limit processes differ in weakly, strongly, and moderately tempered cases.

Derived the limit distribution of AR(1) unit root estimators with tempered errors.

Identified distinct asymptotic behaviors based on the tempering parameter.

Abstract

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 $\lambda \sim \lambda_*/N$. We show that the limit of the partial sums process takes a different form in the weakly tempered ($\lambda_* = 0$), strongly tempered ($\lambda_* = \infty$), 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.