Contemporaneous aggregation of triangular array of random-coefficient AR(1) processes

TL;DR

This paper studies the aggregation behavior of random-coefficient AR(1) processes, characterizing their limiting processes, limit behaviors of partial sums, and proposing an estimator for the mixing distribution.

Contribution

It provides a comprehensive analysis of the limiting aggregated process, including its existence, representation, and limit behaviors, along with a disaggregation estimation method.

Findings

Limiting aggregated process exists under general conditions.

Partial sums exhibit four different limit behaviors based on parameters.

Estimator for the mixing distribution is weakly consistent.

Abstract

We discuss contemporaneous aggregation of independent copies of a triangular array of random-coefficient AR(1) processes with i.i.d. innovations belonging to the domain of attraction of an infinitely divisible law W. The limiting aggregated process is shown to exist, under general assumptions on W and the mixing distribution, and is represented as a mixed infinitely divisible moving-average. Partial sums process of $ is discussed under the assumption E(W^2) is finite and a mixing density regularly varying at the "unit root" x=1 with exponent \beta >0. We show that the above partial sums process may exhibit four different limit behaviors depending on \beta and the L\'evy triplet of W. Finally, we study the disaggregation problem in spirit of Leipus et al. (2006) and obtain the weak consistency of the corresponding estimator of the mixing distribution in a suitable L_2-space.