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 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.
Highlights
Autoregressive (AR) time series models are extensively used in econometrics and financial markets and their theories are applicable to various time series models.Inference on autoregressive coefficient ρ in an AR model of order one, or more generally, on the sum of coefficients in an AR model of order p, is an important issue in times series analysis
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
One of the most common inferences in the AR models is given by the least squares estimator (LSE), and its limit theory has been developed gradually in time series studies from a simple AR model with i.i.d. errors to an advanced AR model with various types of errors, for example, Martingale difference errors or conditional heteroscedastic errors
Summary
Autoregressive (AR) time series models are extensively used in econometrics and financial markets and their theories are applicable to various time series models. For the error model of the stationary AR(1) model considered in the present work, the heavy-tailed augmented GARCH (or G-GARCH process) is adopted. For the case of heavy-tailed GARCH errors, it would be expected from the results of [21] that the LSE is inconsistent if the variance of GARCH process is infinite but is consistent with a slower convergence rate if it has finite variance and an infinite fourth moment. This has been theoretically verified by [13] for the AR models with the heavy-tailed G-GARCH errors.
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