Report

OLS Limit Theory for Drifting Sequences of Parameters on the Explosive Side of Unity

Abstract: A limit theory is developed for the least squares estimator for mildly and purely explosive autoregressions under drifting sequences of parameters with autoregressive roots ρn satisfying ρn → ρ ∈ (—∞, —1] ∪ [1, ∞) and n (|ρn| —1) → ∞. Drifting sequences of innovations and initial conditions are also considered. A standard specification of a short memory linear process for the autoregressive innovations is extended to a triangular array formulation both for the deterministic weights and for the primitive innovations of the linear process, which are allowed to be heteroskedastic L1-mixingales. The paper provides conditions that guarantee the validity of Cauchy limit distribution for the OLS estimator and standard Gaussian limit distribution for the t-statistic under this extended explosive and mildly explosive framework.

Keywords: triangular array; explosive autoregression; linear process; conditional heteroskedasticity; mixingale; Cauchy distribution;

JEL Classification: C12; C18; C22;

https://doi.org/10.59576/sr.1113

Bibliographic Information

Provider: Federal Reserve Bank of New York

Part of Series: Staff Reports

Publication Date: 2024-08-01

Number: 1113