Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations

Eunju Hwang
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Eunju Hwang: Department of Applied Statistics, Gachon University, Seongnam 13120, Korea

Mathematics, 2021, vol. 9, issue 8, 1-10

Abstract: This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient ? = ? n is derived uniformly over stationary values in [ 0 , 1 ) , focusing on ? n ? 1 as sample size n tends to infinity. For tail index ? ? ( 0 , 4 ) of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on 1 ? ? n 2 , but no condition on the rate of ? n is required. It is shown that, for the tail index ? ? ( 0 , 2 ) , the LSE is inconsistent, for ? = 2 , log n / ( 1 ? ? n 2 ) -consistent, and for ? ? ( 2 , 4 ) , n 1 ? 2 / ? / ( 1 ? ? n 2 ) -consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index ? ? ( 0 , 4 ) ; and no restriction on the rate of ? n is necessary.

Keywords: autoregression; augmented GARCH; heavy-tailed; limit theory (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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