[Paper Review] Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes with infinite variance

Pilipauskaitė, Vytautė, Skorniakov, Viktor+1 more|arXiv (Cornell University)|Jan 16, 2019

Complex Systems and Time Series Analysis7 citations

TL;DR

This paper studies joint temporal and contemporaneous aggregation of N independent random-coefficient AR(1) processes with infinite-variance innovations in the domain of normal attraction of an α-stable distribution (0 < α ≤ 2). As N and the time scale n tend to infinity at different rates, the limit behavior of the aggregate process depends on the tail index β of the random coefficient and the relative growth rates of N and n, yielding a rich variety of stable and non-stable limit distributions—including α-stable, (αβ)-stable, β-stable, and compound Poisson-type limits—extending prior finite-variance results to the infinite-variance setting with α < 2.

ABSTRACT

We discuss joint temporal and contemporaneous aggregation of $N$ independent copies of random-coefficient AR(1) process driven by i.i.d. innovations in the domain of normal attraction of an $\alpha$-stable distribution, $0< \alpha \le 2$, as both $N$ and the time scale $n$ tend to infinity, possibly at a different rate. Assuming that the tail distribution function of the random autoregressive coefficient regularly varies at the unit root with exponent $\beta > 0$, we show that, for $\beta < \max (\alpha, 1)$, the joint aggregate displays a variety of stable and non-stable limit behaviors with stability index depending on $\alpha$, $\beta$ and the mutual increase rate of $N$ and $n$. The paper extends the results of Pilipauskait\.e and Surgailis (2014) from $\alpha = 2$ to $0 < \alpha < 2$.

Motivation & Objective

• To study the joint asymptotic behavior of temporally and contemporaneously aggregated random-coefficient AR(1) processes with infinite-variance innovations.
• To extend prior results on finite-variance RCAR(1) panels (e.g., Pilipauskaitė & Surgailis, 2014) to the infinite-variance case with α < 2.
• To characterize the full range of limit distributions—stable and non-stable—arising from different mutual growth rates of N (number of processes) and n (time horizon), and the tail index β of the random coefficient.
• To establish limit theorems for the normalized aggregate SN,n(τ) in terms of α-stable, β-stable, (αβ)-stable, and Poisson-mixed stable processes.

Proposed method

• Models the RCAR(1) process as X(t) = aX(t−1) + ε(t), where a ∈ [0,1) has a density φ(x) ∼ ψ₁(1−x)β−1 near x=1, and {ε(t)} are i.i.d. α-stable innovations with 0 < α ≤ 2.
• Analyzes the joint aggregate SN,n(τ) = Σᵢ₌₁ᴺ Σₜ₌₁^[nτ] Xi(t) under joint limits as N, n → ∞ at possibly different rates.
• Uses weak convergence of finite-dimensional distributions and derives limit laws via characteristic function analysis and stochastic integral representations.
• Applies the Dambis–Dubins–Schwarz theorem and Poisson random measure techniques to represent limit processes as stochastic integrals with respect to α-stable random measures.
• Employs scaling and conditioning arguments to derive iterated limits and interpolate them into joint limits, identifying distinct regimes based on β and the N/n rate.
• Establishes convergence in finite-dimensional distributions (fdd) to α-stable, (αβ)-stable, β-stable, or compound Poisson-type limits depending on parameter regions defined by α, β, and the mutual growth rate of N and n.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the joint aggregate of N independent RCAR(1) processes with infinite-variance innovations as both N and the time scale n tend to infinity at different rates?
• RQ2 How does the tail index β of the random autoregressive coefficient influence the limit behavior, and what are the distinct regimes in the parameter space?
• RQ3 What types of stable and non-stable limit distributions emerge, and how do they depend on the relative growth rates of N and n?
• RQ4 How do the results extend previous findings for finite-variance innovations (α = 2) to the infinite-variance case (α < 2)?
• RQ5 What is the role of the parameter μ = lim N¹/β / n in determining the limit distribution, particularly in the intermediate regime?

Key findings

• For β < max(α, 1), the limit distribution of the normalized aggregate depends critically on the mutual growth rate of N and n, leading to distinct regimes: α-stable, (αβ)-stable, β-stable, or intermediate Poisson-type limits.
• When 1 ≤ β < α and N¹/β / n → ∞, the limit is α-stable; when 0 < β < min(α, 1) and N¹/β / n → ∞, the limit is (αβ)-stable.
• For 0 < β < α and N¹/β / n → 0, the limit is β-stable, indicating a heavy-tailed limit distinct from the α-stable case.
• In the intermediate regime where N¹/β / n → μ ∈ (0, ∞), the limit is a compound distribution involving a Poisson random measure, referred to as 'intermediate Poisson'.
• For α < β < 1, the limit is (αβ)-stable if N¹/γβ / n → ∞ (γ = 1−α / 1−β), α-stable if N¹/γβ / n → 0, and a convolution of (αβ)-stable and α-stable laws if N¹/γβ / n → μ ∈ (0, ∞).
• When β > max(α, 1), the limit is α-stable regardless of the growth rate of N and n, as the aggregate behaves like a sum of i.i.d. α-stable innovations scaled by E[(1−a)−α]¹/α.