ASYMPTOTIC INFERENCE FOR AR MODELS WITH HEAVY-TAILED G-GARCH NOISES

2014 ◽  
Vol 31 (4) ◽  
pp. 880-890 ◽  
Author(s):  
Rongmao Zhang ◽  
Shiqing Ling
Keyword(s):  
Least Squares ◽  
Tail Index ◽  
Least Squares Estimator ◽  
Consistent Estimator ◽  
Stable Processes ◽  
Slowly Varying Function ◽  
Asymptotic Inference ◽  
Heavy Tailed ◽  
Stable Vectors ◽  
Ar Models

It is well known that the least squares estimator (LSE) of an AR(p) model with i.i.d. (independent and identically distributed) noises is n1/αL(n)-consistent when the tail index α of the noise is within (0,2) and is n1/2-consistent when α ≥ 2, where L(n) is a slowly varying function. When the noises are not i.i.d., however, the case is far from clear. This paper studies the LSE of AR(p) models with heavy-tailed G-GARCH(1,1) noises. When the tail index α of G-GARCH is within (0,2), it is shown that the LSE is not a consistent estimator of the parameters, but converges to a ratio of stable vectors. When α ε [2,4], it is shown that the LSE is n1–2/α-consistent if α ε (2,4), logn-consistent if α = 2, and n1/2 / logn-consistent if α = 4, and its limiting distribution is a functional of stable processes. Our results are significantly different from those with i.i.d. noises and should warn practitioners in economics and finance of the implications, including inconsistency, of heavy-tailed errors in the presence of conditional heterogeneity.

scholarly journals Asymptotic inference for nearly unstable INAR(1) models

2003 ◽  
Vol 40 (3) ◽  
pp. 750-765 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Limiting Distribution ◽  
Least Squares Estimator ◽  
Asymptotic Inference ◽  
First Order ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

A New Regression-Based Tail Index Estimator

2019 ◽  
Vol 101 (4) ◽  
pp. 667-680 ◽  
Author(s):  
João Nicolau ◽  
Paulo M. M. Rodrigues
Keyword(s):  
Time Dependence ◽  
Pareto Distribution ◽  
Asymptotic Variance ◽  
Tail Length ◽  
Bias Reduction ◽  
Tail Index ◽  
Slowly Varying Function ◽  
Heavy Tailed Distributions ◽  
Unknown Form ◽  
Heavy Tailed

A new regression-based approach for the estimation of the tail index of heavy-tailed distributions with several important properties is introduced. First, it provides a bias reduction when compared to available regression-based methods; second, it is resilient to the choice of the tail length used for the estimation of the tail index; third, when the effect of the slowly varying function at infinity of the Pareto distribution vanishes slowly, it continues to perform satisfactorily; and fourth, it performs well under dependence of unknown form. An approach to compute the asymptotic variance under time dependence and conditional heteroskcedasticity is also provided.

scholarly journals Asymptotic inference for nearly unstable INAR(1) models

2003 ◽  
Vol 40 (03) ◽  
pp. 750-765 ◽  
Author(s):  
M. Ispány ◽  
G. Pap ◽  
M. C. A. van Zuijlen
Keyword(s):  
Least Squares ◽  
Rate Of Convergence ◽  
Limiting Distribution ◽  
Least Squares Estimator ◽  
Asymptotic Inference ◽  
First Order ◽  
Conditional Least Squares ◽  
Conditional Least Squares Estimator

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n 3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.

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

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 816
Author(s):  
Eunju Hwang
Keyword(s):  
Unit Root ◽  
Stable Distribution ◽  
Asymptotic Properties ◽  
Tail Index ◽  
Asymptotic Distributions ◽  
Uniform Limit ◽  
Limit Theory ◽  
Heavy Tailed ◽  
Heteroscedastic Variance ◽  
Ar Models

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 ρ=ρn is derived uniformly over stationary values in [0,1), focusing on ρn→1 as sample size n tends to infinity. For tail index α∈(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−ρn2, but no condition on the rate of ρn is required. It is shown that, for the tail index α∈(0,2), the LSE is inconsistent, for α=2, logn/(1−ρn2)-consistent, and for α∈(2,4), n1−2/α/(1−ρn2)-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 α∈(0,4); and no restriction on the rate of ρn is necessary.

Sign in / Sign up

Export Citation Format

Share Document