Non-standard inference for augmented double autoregressive models with null volatility coefficients

Abstract

This paper considers an augmented double autoregressive (DAR) model, which allows null volatility coefficients to circumvent the over-parameterization problem in the DAR model. Since the volatility coefficients might be on the boundary, the statistical inference methods based on the Gaussian quasi-maximum likelihood estimation (GQMLE) become non-standard, and their asymptotics require the data to have a finite sixth moment, which narrows applicable scope in studying heavy-tailed data. To overcome this deficiency, this paper develops a systematic statistical inference procedure based on the self-weighted GQMLE for the augmented DAR model. Except for the Lagrange multiplier test statistic, the Wald, quasi-likelihood ratio and portmanteau test statistics are all shown to have non-standard asymptotics. The entire procedure is valid as long as the data is stationary, and its usefulness is illustrated by simulation studies and one real example.where u, φ i ∈ R, ω > 0, α i > 0, {η t } is a sequence of independent and identically distributed (i.i.d.) random variables with zero mean and unit variance, and η t is independent of {y s ; s < t}. Model (1.1) was first termed by Ling (2004), and it is a subclass of ARMA-ARCH models in Weiss (1984) and of nonlinear AR models in Cline and Pu (2004), but it is different from Engle's ARCH model if some φ i = 0.

Asymmetric linear double autoregression

“…Note that the positive definiteness of D is satisfied for continuous η t with E(η 4 t ) < ∞; see Jiang et al (2020). If η t is normal, then κ 1 = 0, κ 2 = 2 and Ω = Σ, thus the QMLE reduces to the MLE and its asymptotics in Theorem 2 can be simplified to √ n( θ n − θ 0 ) → L N (0, Σ −1 ) as n → ∞.…”

“…where {η t } are standard normal, or follow standardized Student t 5 distribution with unit variance, or standardized skewed t distribution, denoted by st 5,−1.2 , with unit variance and skew parameter −1.2 (Jiang et al, 2020). The sample size is set to n = 500, 1000 or 2000, with 1000 replications for each sample size.…”

“…This important property makes model (1.1) suitable for handling heavy-tailed data in application. Many variants of DAR models are widely proposed and studied, such as the threshold DAR (Li et al, 2016), the mixture DAR (Li et al, 2017), the linear DAR (Zhu et al, 2018) and the augmented DAR (Jiang et al, 2020) models. Specifically, the linear DAR model has the form of…”

On the three‐step non‐Gaussian quasi‐maximum likelihood estimation of heavy‐tailed double autoregressive models

“…Recently, Guo et al (2019) studied an unconstrained global LADE in a unified framework of both stationary and explosive cases. Furthermore, for higher-order DAR models with extensions, see Nielsen and Rahbek (2014), Li et al (2015Li et al ( , 2016Li et al ( , 2017, Zhu et al (2018), and Jiang et al (2020), and so on.…”

Quasi-maximum Likelihood Inference for Linear Double Autoregressive Models

“…random variables. Model (1.2) is a special case of the ARMA-ARCH models in Weiss (1986), and it has received growing attention with many extensions such as the threshold DAR (Li et al, 2015(Li et al, , 2016, the mixture DAR (Li et al, 2017), the linear DAR (Zhu et al, 2018) and the augmented DAR (Jiang et al, 2020) models. Different with the linear DAR model (1.1), model…”