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

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 $\rho ={\rho }_n$ is derived uniformly over stationary values in $[0,1)$, focusing on ${\rho }_n\to 1$ as sample size n tends to infinity. For tail index $\alpha \in (0,4)$ 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 $1-{\rho }_n^2$, but no condition on the rate of ${\rho }_n$ is required. It is shown that, for the tail index $\alpha \in (0,2)$, the LSE is inconsistent, for $\alpha =2$, $logn/(1-{\rho }_n^2)$-consistent, and for $\alpha \in (2,4)$, $n^{1-2/\alpha }/(1-{\rho }_n^2)$-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 $\alpha \in (0,4)$; and no restriction on the rate of ${\rho }_n$ is necessary.

1. Introduction

Autoregressive (AR) time series models are not only extensively used in econometrics and financial markets but also their theories are applicable to various time series models. Inference on autoregressive coefficient $\rho$ 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. 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.

We consider a stationary AR(1) model with conditional heteroscedastic errors. In particular, the error model adopted in this work is a more general form than the autoregressive conditional heteroscedastic (ARCH) model of Engle [1] and the generalized ARCH (GARCH) model of Bollerslev [2]. Since the introduction of the (G)ARCH model of Engle [1] and Bollerslev [2], many time series models in the GARCH family have been proposed in modeling financial time series data with applications to financial volatility: for instance, we refer to multiplicative GARCH in [3], exponential GARCH in [4], nonlinear asymmetric GARCH in [5], GJR-GARCH in [6], threshold GARCH in [7], etc.

To unify and generalize such parametric GARCH models, Ref. [8] defined the so-called augmented GARCH process and established its diffusion limit. Independently, Ref. [9] suggested a general GARCH (G-GARCH) process with nonparametric functionals and studied its moment properties, which is a first-order simple form of the augmented GARCH model of [8]. Ref. [10] presented strong approximation for the sum of squares of augmented GARCH sequences with applications to change-point analysis, and Ref. [11] developed a functional central limit theorem for a polynomial form of the augmented GARCH model. Ref. [12] studied the probabilistic structure of the augmented GARCH(1,1) sequences and asymptotic distribution of their various functionals. Meanwhile, Ref. [13] dealt with asymptotic inference for AR models with heavy-tailed G-GARCH noises, and Ref. [14] studied weak convergence of renorming volatilities in the G-GARCH models.

In the mean time, Ref. [15] developed the Gaussian limit theory of the least squares estimator (LSE) for a stationary autoregression with Martingale difference errors having homoscedastic variance, which holds uniformly over stationary values of the autoregression coefficient ${\rho }_n$, including local vicinities of unity with condition $(1-{\rho }_n)n\to \infty$ as sample size n tends to infinity. In this paper, we provide a limit theory for the stationary autoregression, like [15], but by allowing for the errors to have conditional heteroscedastic variance. 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.

Several authors studied analysis and inference for time series models with GARCH errors. For example, Ref. [16] provided a review of theoretical results for time series models with GARCH errors. The LSEs for ARMA-ARCH models have been done by [17,18] under the finite fourth moment conditions, and their results can be easily extended to the ARMA-GARCH models. Ref. [19] dealt with the OLS estimation for AR-GARCH models under non-finite fourth moments, and Ref. [20] considered the likelihood-based inference for AR-ARCH models. 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. The notable work of [13] make us extend to limit theory combined with an interesting result of [15], who dealt with the theory on condition $(1-{\rho }_n)n\to \infty$ as $n\to \infty$.

In this work, as an extension of [15], we establish uniform limit theory for the LSEs of autoregression coefficient ${\rho }_n$ in the stationary AR(1) model with heavy-tailed G-GARCH(1,1) errors, for all ${\rho }_n\in [0,1)$, but focusing on ${\rho }_n\to 1$ as $n\to \infty$. The asymptotic distributions of the LSEs, involved with the stable distributions depending on the tail index, are established for stationary ${\rho }_n$ that includes neighborhood of the unity, ${\rho }_n\to 1$, but any condition on the rate of ${\rho }_n$ is not required. The tail index $\alpha$ of the G-GARCH(1,1) process is considered in range $(0,4)$ in this work. Our limit theory shows that, for the tail index $\alpha \in (0,2)$, the LSE is inconsistent, for $\alpha =2$, $logn/(1-{\rho }_n^2)$-consistent and for $\alpha \in (2,4)$, $n^{1-2/\alpha }/(1-{\rho }_n^2)$-consistent. This work might have the novelty that the condition ${\rho }_n\to 1$ plays a role of bridge between pure stationary and unit-root processes. In addition, this paper is a notable extension by adding three issues to the work of [15]: first, the errors have conditional heteroscedastic variance; second, the errors have heavy-tailed feature with specific tail indices; and, finally, any condition on the rate of ${\rho }_n$ is not required.

The remainder of the paper is organized as follows: in Section 2, the model, assumptions, and existing theory are described. Our main results are presented in Section 3 while their proofs are given in Section 4. Concluding remarks are stated in Section 5.

2. Assumption and Existing Theory

We consider an AR(1) process $\{y_t:t=1,2,\dots ,n\}$ defined by

$$y_t={\rho }_n{y}_{t-1}+{ϵ}_t$$

(1)

with initialization $y_0$ and coefficient $\rho ={\rho }_n\in [0,1)$, where $\{{ϵ}_t:t=1,2,\dots \}$ is generated by the general GARCH(1,1) process, G-GARCH(1,1):

$${ϵ}_t=h_t{\eta }_t\mathrm{and}{h}_t^{\delta }=g\left({\eta }_{t-1}\right)+c\left({\eta }_{t-1}\right)h_{t-1}^{\delta }$$

(2)

where $\delta >0$, $P\{h_t^{\delta }>0\}=1$, $c\left(0\right)<1$, $c(·)$ and $g(·)$ are nonnegative functions, $\left\{{\eta }_t\right\}$ is a sequence of i.i.d. symmetric random variables with $E{\eta }_1=0$, $E{\eta }_1^2=1$. Formally, $\{y_t=y_{t,n}:t=1,2,\dots ,n\}$ is a triangular array but n is omitted for notational simplicity. Throughout the paper, we make the following assumptions:

(A1)

$Elog\left(c\left({\eta }_t\right)\right)<0$.

(A2)

There exists a $k_0>0$ such that $E{\left(c\left({\eta }_t\right)\right)}^{k_0}\ge 1$, $E\left[{\left(c\left({\eta }_t\right)\right)}^{k_0}{log}^{+}\left(c\left({\eta }_t\right)\right)\right]<\infty$, and $E(g\left({\eta }_t\right)+|{\eta }_t{{|}^{\delta })}^{k_0}<\infty$, where ${log}^{+}\left(x\right)=max\{0,log\left(x\right)\}$.

(A3)

The density $f\left(x\right)$ of ${\eta }_1$ is positive in the neighborhood of zero.

Conditions (A1)–(A3) are adopted from [13], who derived asymptotic theory for AR models with the heavy-tailed G-GARCH noises. The remarkable work of [13] led us to extend an interesting limit theory along with one of [15], who studied uniform limit theory with condition $(1-{\rho }_n)n\to \infty$ as $n\to \infty$.

For $\rho ={\rho }_n\in [0,1)$, model (1) is stationary and has the following expansion:

$$y_t=\sum _{l=0}^{\infty }{\rho }_n^l{ϵ}_{t-l}.$$

(3)

Given observations $\{y_0,y_1,\dots ,y_n\}$, the LSE ${\widehat{\rho }}_n$ of ${\rho }_n$ in model (1) and their difference are respectively given by

$${\widehat{\rho }}_n=\frac{{\sum }_{t=1}^n{y}_t{y}_{t-1}}{{\sum }_{t=1}^n{y}_{t-1}^2}\mathrm{and}{\widehat{\rho }}_n-{\rho }_n=\frac{{\sum }_{t=1}^n{y}_{t-1}{ϵ}_t}{{\sum }_{t=1}^n{y}_{t-1}^2}.$$

(4)

The limiting distribution of ${\widehat{\rho }}_n-{\rho }_n$ is obtained by the asymptotic behaviors of ${\sum }_{t=1}^n{y}_{t-1}{ϵ}_t$ and ${\sum }_{t=1}^n{y}_{t-1}^2$. By (3), ${\sum }_{t=1}^n{y}_{t-1}{ϵ}_t$ and ${\sum }_{t=1}^n{y}_{t-1}^2$ can be approximated by

$$\sum _{l=0}^H{\rho }_n^l\sum _{t=1}^n{ϵ}_{t-1-l}{ϵ}_t\mathrm{and}\sum _{l=0}^H\sum _{m=0}^H{\rho }_n^{l+m}\sum _{t=1}^n{ϵ}_{t-1-l}{ϵ}_{t-1-m},$$

respectively, as $H\to \infty$. We need to see the behavior of the heavy-tailed G-GARCH process $\left\{{ϵ}_t\right\}$ as well as the convergence of ${\sum }_{t=1}^n{ϵ}_{t-l}{ϵ}_{t-m}$, for $l,m\in \{1,2,\dots ,H\}$. We first give the following lemma for the tail index of $\left\{{ϵ}_t\right\}$, presented by [13].

Lemma 1

(Zhang and Ling (2015) [13]). Under Conditions (A1)–(A3), there exists a unique $\alpha \in (0,\delta k_0]$ such that $E{\left(c\left({\eta }_t\right)\right)}^{\alpha /\delta }=1$ and

$$P\left(\right|{ϵ}_1|>x)\sim c_{0}E{\left|{\eta }_1\right|}^{\alpha }{x}^{-\alpha },$$

(5)

where

$$c_0=c_0^{\left(\alpha \right)}=\frac{E[{\{g\left({\eta }_1\right)+c\left({\eta }_1\right)h_1^{\delta }\}}^{\alpha /\delta }-{\left\{c\left({\eta }_1\right)h_1^{\delta }\right\}}^{\alpha /\delta }]}{\alpha E\left[c{\left({\eta }_1\right)}^{\alpha /\delta }{log}^{+}\left(c\left({\eta }_1\right)\right)\right]}.$$

In (5) or in the following, $a_n\sim c{b}_n$ means that ${lim}_{n\to \infty }{a}_n/b_n=c$.

For tail index $\alpha$ in (5), this work considers three cases with different asymptotic results: $0<\alpha <2$, $\alpha =2$ and $2<\alpha <4$. As a case of $\alpha =4$ in the simple heavy-tailed GARCH noise, but not G-GARCH, Ref. [22] recently established a Gaussian limit theory as ${\rho }_n$ converges to the unity with condition $(1-{\rho }_n)n\to \infty$ as $n\to \infty$. This present work excludes case $\alpha =4$ in the G-GRACH model $\left\{{ϵ}_t\right\}$ because the asymptotic result is not a stable limit, which is a main focus of this work, but it is a Gaussian limit with such a lengthy analysis as in [22]. This work is an extension of [15], and, moreover, our limit theory plays a role of a bridge between the inferences in [13,23], who have dealt with unit-root and stationary AR models with heavy-tailed GARCH errors, respectively.

We here summarize the discussion in [13] along with results of [21]. The same notations as those in [13] are used, but some complicated notations are referred to in their paper.

For any positive integer $H>0$, let the $(H+1)$-dimensional stable random vector with stable index $\alpha /2\in (0,2]$ be denoted by

$${\mathbf{S}}_H\left(\alpha \right):=(S^0\left(\alpha \right),S^1\left(\alpha \right),\dots ,S^H\left(\alpha \right)).$$

For any given integers l and H, we define two $(H+1)$-dimensional random vectors:

$${\mathcal{E}}_{t,l,H}^{\left(1\right)}=({ϵ}_{t-l},{ϵ}_{t-l-1},\dots ,{ϵ}_{t-l-H}),{\mathcal{E}}_{t,l,H}^{\left(2\right)}=\left({ϵ}_{t-l}^2-c_n,{ϵ}_{t-l}{ϵ}_{t-l-1},\dots ,{ϵ}_{t-l}{ϵ}_{t-l-H}\right),$$

where

$$c_n=c_n^{\left(\alpha \right)}=\left\{\begin{array}{cc}0\hfill & \mathrm{for}0<\alpha <2\hfill \\ E{ϵ}_1^2{I}_{\left(\right|{ϵ}_1|\le \sqrt{c_{0}n})}=c_{0}logn\hfill & \mathrm{for}\alpha =2\hfill \\ E{ϵ}_1^2\hfill & \mathrm{for}2<\alpha <4.\hfill \end{array}\right.$$

(6)

Under the conditions in (A1)–(A3), $\left\{{\mathcal{E}}_{t,l,H}^{\left(1\right)},t\ge 1\right\}$ is a varying regular random vector sequence with index $\alpha$. Let

$$U\left(\alpha \right)=\underset{H\to \infty }{lim}\underset{n\to \infty }{lim}\frac{1}{a_n^2}\left[\sum _{t=1}^n({ϵ}_{t-l}^2-c_n)+\sum _{h=1}^H\sum _{t=1}^n{ϵ}_{t-l}{ϵ}_{t-l-h}\right]$$

(7)

where

$$a_n=a_n^{\left(\alpha \right)}={\left(c_0^{\left(\alpha \right)}E{\left|{\eta }_1\right|}^{\alpha }n\right)}^{1/\alpha }$$

(8)

$U\left(\alpha \right)$ is the limit of sum of all components in ${\sum }_{t=1}^n{\mathcal{E}}_{t,l,H}^{\left(2\right)}$ divided by $a_n^2$, and it is independent of l by the stationarity. Note that $U\left(\alpha \right)$ exists by Equation (3.5) of [13]. According to Theorem 2.8 and Proposition 3.3 of [21], we have

$$\frac{1}{a_n^2}\sum _{t=1}^n{\mathcal{E}}_{t,l,H}^{\left(2\right)}{\to }^d(S^0\left(\alpha \right),S^1\left(\alpha \right),\dots ,S^H\left(\alpha \right))={\mathbf{S}}_H\left(\alpha \right),$$

(9)

where ${\to }^d$ denotes convergence in distribution. The limiting stable distribution in (9) has somewhat different forms for $0<\alpha <2$ and for $2\le \alpha <4$, whose explicit expressions are given in Equations (3.2) and (3.3) of [13], respectively. See [13,21] for its details. Note that the limiting distribution ${\mathbf{S}}_H\left(\alpha \right)$ in (9) does not depend on l. Ref. [13] has discussed the following result.

Lemma 2

(Zhang and Ling (2015) [13]). Under Conditions (A1)–(A3), for any positive integers K and H, we have

$$\frac{1}{a_n^2}\left(\sum _{t=1}^n{\mathcal{E}}_{t,0,H}^{\left(2\right)},\sum _{t=1}^n{\mathcal{E}}_{t,1,H}^{\left(2\right)},\dots ,\sum _{t=1}^n{\mathcal{E}}_{t,K,H}^{\left(2\right)}\right){\to }^d{({\mathbf{S}}_H\left(\alpha \right),{\mathbf{S}}_H\left(\alpha \right),\dots ,{\mathbf{S}}_H\left(\alpha \right))}_{1\times (K+1)}$$

as $n\to \infty$, where $0<\alpha <4$ and ${\mathbf{S}}_H$ is a $(H+1)$-dimensional stable random vector with stable index $\alpha /2\in (0,2)$.

The next section presents our main limit theories with their proofs. The stable limit in Lemma 2 is used to derive our results.

3. Main Results

In this section, we provide the limit theory, involved with the stable distribution, of the LSE ${\widehat{\rho }}_n$ for $0<\alpha <4$. The limit theory is uniform over all stationary values of $\rho ={\rho }_n\in [0,1)$, including ${\rho }_n\to 1$, on which this present work focuses. The convergence rate of ${\widehat{\rho }}_n$ depends on $1-{\rho }_n^2$, but any condition of the rate of ${\rho }_n\to 1$ is not required. As seen in the asymptotic results below, if $0<\alpha <2$, the LSE is inconsistent as the limiting represents a ratio of $\alpha /2$-stable random variables, like the estimators of autocorrelation function of the heavy-tailed process with $0<\alpha <2$ in [21]. If $\alpha =2$, the LSE is $(logn)/(1-{\rho }_n^2)$-consistent, while, if $2<\alpha <4$, it is $n^{1-2/\alpha }/(1-{\rho }_n^2)$-consistent. We now present our main results as follows:

Theorem 1.

Assume (A1)–(A3) and let $0<\alpha <2$. If ${\rho }_n\to 1$ as $n\to \infty$, then we have

$$\frac{1}{1-{\rho }_n^2}({\widehat{\rho }}_n-{\rho }_n){\to }^d\frac{U\left(\alpha \right)}{S_0\left(\alpha \right)+2U\left(\alpha \right)}.$$

Theorem 2.

Assume (A1)–(A3) and let $\alpha =2$. If ${\rho }_n\to 1$ as $n\to \infty$, then we have

$$\frac{logn}{1-{\rho }_n^2}({\widehat{\rho }}_n-{\rho }_n){\to }^{d}U\left(\alpha \right).$$

Theorem 3.

Assume (A1)–(A3) and let $2<\alpha <4$. If ${\rho }_n\to 1$ as $n\to \infty$, then we have

$$\frac{n^{1-2/\alpha }}{1-{\rho }_n^2}({\widehat{\rho }}_n-{\rho }_n){\to }^d\frac{(c_{0}E|{\eta }_1{{|}^{\alpha })}^{2/\alpha }}{E{ϵ}_1^2}U\left(\alpha \right).$$

To derive the asymptotic results in Theorems above, we need to observe the joint limiting distribution of the numerator and the denominator of ${\widehat{\rho }}_n-{\rho }_n$ in (4). In the following lemma, the joint asymptotic distribution is established, which unify all the three cases considered: $0<\alpha <2,$ $\alpha =2$ and $2<\alpha <4$.

Lemma 3.

We consider the AR(1) model with coefficient ${\rho }_n$ in (1) and assume (A1)–(A3). For $0<\alpha <4$ and for $c_n$ given as in (6), if ${\rho }_n\to 1$ as $n\to \infty$, then we have

$$\left\{\frac{1-{\rho }_n^2}{n^{2/\alpha }}\sum _{t=1}^n\left(y_{t-1}^2-\frac{c_n}{1-{\rho }_n^2}\right),\frac{1}{n^{2/\alpha }}\sum _{t=1}^n{y}_{t-1}{ϵ}_t\right\}{\to }^d\left(Z_{\alpha /2}^{\left(1\right)},Z_{\alpha /2}^{\left(2\right)}\right)$$

where

$$Z_{\alpha /2}^{\left(1\right)}=(c_0^{\left(\alpha \right)}E|{\eta }_1{|}^{\alpha }{)}^{2/\alpha }\left[S^0\left(\alpha \right)+2U\left(\alpha \right)\right]\mathrm{and}{Z}_{\alpha /2}^{\left(2\right)}=(c_0^{\left(\alpha \right)}E|{\eta }_1{{|}^{\alpha })}^{2/\alpha }U\left(\alpha \right).$$

4. Proofs

This section gives proofs of the main results. Thanks to Lemma 3, which includes all the three cases of tail index $\alpha$ considered, the results in Theorems 1–3 are derived neatly and concisely, and thus we state the proof of Lemma 3 prior to those of Theorems 1–3. Indeed, the proof of Lemma 3 is a main contribution to the limit theory in this study.

Proof of Lemma 3.

To prove the desired limiting distribution, by Cramer’s Wold Device, we may show that, for any real numbers $(u,v)$,

$$u\frac{1-{\rho }_n^2}{n^{2/\alpha }}\sum _{t=1}^n\left(y_{t-1}^2-\frac{c_n}{1-{\rho }_n^2}\right)+v\frac{1}{n^{2/\alpha }}\sum _{t=1}^n{y}_{t-1}{ϵ}_t{\to }^{d}u{Z}_{\alpha /2}^{\left(1\right)}+v{Z}_{\alpha /2}^{\left(2\right)}.$$

Recalling (8), we will verify that $A_n^{\left(1\right)}\left(u\right)+A_n^{\left(2\right)}\left(v\right):=$

$$u\frac{1-{\rho }_n^2}{a_n^2}\sum _{t=1}^n\left(y_{t-1}^2-\frac{c_n}{1-{\rho }_n^2}\right)+v\frac{1}{a_n^2}\sum _{t=1}^n{y}_{t-1}{ϵ}_t{\to }^{d}u\left(S^0\left(\alpha \right)+2U\left(\alpha \right)\right)+vU\left(\alpha \right).$$

(10)

First, for $A_n^{\left(1\right)}\left(u\right)$, we write

$$\sum _{t=1}^n\left(y_{t-1}^2-\frac{c_n}{1-{\rho }_n^2}\right)=\sum _{t=1}^n\left(\sum _{l=0}^{\infty }\sum _{m=0}^{\infty }{\rho }_n^{l+m}{ϵ}_{t-1-l}{ϵ}_{t-1-m}-\frac{c_n}{1-{\rho }_n^2}\right)$$

$$=\sum _{t=1}^n\left[\sum _{l=0}^H\sum _{m=0}^H{\rho }_n^{l+m}{ϵ}_{t-1-l}{ϵ}_{t-1-m}-\frac{c_n}{1-{\rho }_n^2}+I_t^{\left(1\right)}\left(H\right)\right]$$

(11)

where $I_t^{\left(1\right)}\left(H\right)$ is given as three double-summations as follows:

$$I_t^{\left(1\right)}\left(H\right)=\sum _{l=H+1}^{\infty }\sum _{m=0}^H{\rho }_n^{l+m}{ϵ}_{t-1-l}{ϵ}_{t-1-m}+$$

$$\sum _{l=0}^H\sum _{m=H+1}^{\infty }{\rho }_n^{l+m}{ϵ}_{t-1-l}{ϵ}_{t-1-m}+\sum _{l=H+1}^{\infty }\sum _{m=H+1}^{\infty }{\rho }_n^{l+m}{ϵ}_{t-1-l}{ϵ}_{t-1-m}.$$

Note that, for any $\delta >0$, $P\left(\frac{1}{a_n^2}\left|{\sum }_{t=1}^n{I}_t^{\left(1\right)}\left(H\right)\right|>\delta \right)\le$

$$P\left(\frac{1}{a_n^2}\left|\sum _{l=H+1}^{\infty }{\rho }_n^l\sum _{m=0}^H{\rho }_n^m\sum _{t=1}^n{ϵ}_{t-1-l}{ϵ}_{t-1-m}\right|>\delta /3\right)$$

$$+P\left(\frac{1}{a_n^2}\left|\sum _{l=0}^H{\rho }_n^l\sum _{m=H+1}^{\infty }{\rho }_n^m\sum _{t=1}^n{ϵ}_{t-1-l}{ϵ}_{t-1-m}\right|>\delta /3\right)$$

$$+P\left(\frac{1}{a_n^2}\left|\sum _{l=H+1}^{\infty }{\rho }_n^l\sum _{m=H+1}^{\infty }{\rho }_n^m\sum _{t=1}^n{ϵ}_{t-1-l}{ϵ}_{t-1-m}\right|>\delta /3\right)\to 0\mathrm{as}n\to \infty ,(H\to \infty )$$

and thus

$$\frac{1}{a_n^2}\sum _{t=1}^n{I}_t^{\left(1\right)}\left(H\right){\to }^{p}0\mathrm{as}n\to \infty .$$

(12)

In addition, (11) is written as

$$\sum _{t=1}^n\left[\sum _{l=0}^H{\rho }_n^{2l}{ϵ}_{t-1-l}^2-\frac{c_n}{1-{\rho }_n^2}+B_t\left(H\right)+I_t^{\left(1\right)}\left(H\right)\right]$$

(13)

where $B_t\left(H\right)=\sum {\sum }_{l\ne m}{\rho }_n^{l+m}{ϵ}_{t-1-l}{ϵ}_{t-1-m}$

$$=\sum \sum _{l>m}{\rho }_n^{l+m}{ϵ}_{t-1-l}{ϵ}_{t-1-m}+\sum \sum _{l<m}{\rho }_n^{l+m}{ϵ}_{t-1-l}{ϵ}_{t-1-m}=:B_t^{\left(1\right)}\left(H\right)+B_t^{\left(2\right)}\left(H\right).$$

In the summation of $B_t^{\left(1\right)}\left(H\right)$, set $j=l-m$, and in the summation of $B_t^{\left(2\right)}\left(H\right)$, set $i=m-l$, then we get $B_t\left(H\right)=B_t^{\left(1\right)}\left(H\right)+B_t^{\left(2\right)}\left(H\right)=$

$$=\sum _{j=1}^H\sum _{m=0}^H{\rho }_n^{2m+j}{ϵ}_{t-1-m}{ϵ}_{t-1-m-j}+\sum _{i=1}^H\sum _{l=0}^H{\rho }_n^{2l+i}{ϵ}_{t-1-l}{ϵ}_{t-1-l-i}$$

$$=2\sum _{j=1}^H\sum _{m=0}^H{\rho }_n^{2m+j}{ϵ}_{t-1-m}{ϵ}_{t-1-m-j}.$$

(14)

The summation from $t=1$ to $t=n$ of the first two expressions in (13) is given by

$$\sum _{t=1}^n\left[\sum _{l=0}^H{\rho }_n^{2l}({ϵ}_{t-1-l}^2-c_n)+\sum _{l=0}^H{\rho }_n^{2l}{c}_n-\frac{c_n}{1-{\rho }_n^2}\right]$$

$$=\sum _{l=0}^H{\rho }_n^{2l}\sum _{t=1}^n\left({ϵ}_{t-1-l}^2-c_n\right)+\sum _{t=1}^n{c}_n\left(\sum _{l=0}^H{\rho }_n^{2l}-\frac{1}{1-{\rho }_n^2}\right)$$

$$=\sum _{l=0}^H{\rho }_n^{2l}\sum _{t=1}^n\left({ϵ}_{t-1-l}^2-c_n\right)+n{c}_n\frac{{\rho }_n^{2H+2}}{1-{\rho }_n^2}.$$

(15)

Therefore, we see that the first term $A_n^{\left(1\right)}\left(u\right)$ of the left-hand side in (10) becomes, by (11)–(15),

$$u\frac{1-{\rho }_n^2}{a_n^2}\left[\sum _{l=0}^H{\rho }_n^{2l}\sum _{t=1}^n\left({ϵ}_{t-1-l}^2-c_n\right)+n{c}_n\frac{{\rho }_n^{2H+2}}{1-{\rho }_n^2}+2\sum _{t=1}^n\sum _{j=1}^H\sum _{m=0}^H{\rho }_n^{2m+j}{ϵ}_{t-1-m}{ϵ}_{t-1-m-j}\right]+o_p\left(1\right)$$

(16)

Second, for $A_n^{\left(2\right)}\left(v\right)$, we write

$$\sum _{t=1}^n{y}_{t-1}{ϵ}_t=\sum _{t=1}^n{ϵ}_t\left(\sum _{l=0}^{\infty }{\rho }_n^l{ϵ}_{t-1-l}\right)=\sum _{l=0}^H{\rho }_n^l\sum _{t=1}^n{ϵ}_t{ϵ}_{t-1-l}+\sum _{t=1}^n{I}_t^{\left(2\right)}\left(H\right)$$

where $I_t^{\left(2\right)}\left(H\right)={\sum }_{l=H+1}^{\infty }{\rho }_n^l{ϵ}_t{ϵ}_{t-1-l}$. It is clear that $\frac{1}{a_n^2}{\sum }_{t=1}^n{I}_t^{\left(2\right)}\left(H\right){\to }^{p}0$ as $n\to \infty$, ($H\to \infty$).

Thus, the second term $A_n^{\left(2\right)}\left(v\right)$ of the left-hand side in (10) is

$$v\frac{1}{a_n^2}\left[\sum _{l=0}^H{\rho }_n^l\sum _{t=1}^n{ϵ}_t{ϵ}_{t-1-l}\right]+o_p\left(1\right).$$

(17)

Hence, by (16) and (17), in (10), $A_n^{\left(1\right)}\left(u\right)+A_n^{\left(2\right)}\left(v\right)=$

$$u(1-{\rho }_n^2)\left[\sum _{l=0}^H{\rho }_n^{2l}\left(\frac{1}{a_n^2}\sum _{t=1}^n\left({ϵ}_{t-1-l}^2-c_n\right)\right)+2\sum _{j=1}^H\sum _{m=0}^H{\rho }_n^{2m+j}\left(\frac{1}{a_n^2}\sum _{t=1}^n{ϵ}_{t-1-m}{ϵ}_{t-1-m-j}\right)\right]$$

$$+v\sum _{l=0}^H{\rho }_n^l\left(\frac{1}{a_n^2}\sum _{t=1}^n{ϵ}_t{ϵ}_{t-1-l}\right)+O\left(\frac{n{c}_n{\rho }_n^{2H}}{a_n^2}\right)+o_p\left(1\right).$$

(18)

Now, we choose $H=H_{n,\alpha }$, depending on n and $\alpha$, such that $H\to \infty$ and $O\left(n{c}_n{\rho }_n^{2H}/a_n^2\right)$$\to 0$ as $n\to \infty$ and ${\rho }_n\to 1$. When $0<\alpha <2$, $c_n=0$ and thus any sequence $H\to \infty$ may be taken. If $\alpha =2$, then $O\left(n{c}_n{\rho }_n^{2H}/a_n^2\right)=O({\rho }_n^{2H}logn)$ by (6). Hence, we choose $H=\lfloor -loglogn/log{\rho }_n\rfloor$, the integer part of $-loglogn/log{\rho }_n$. Since $log{\rho }_n<0$ and $log{\rho }_n\to 0$, $H\to \infty$ as $n\to \infty$, and it follows that $2Hlog{\rho }_n=-2loglogn=log{(logn)}^{-2}$ and thus ${\rho }_n^{2H}logn=1/logn\to 0$. Therefore, on behalf of the following discussion, if $0<\alpha \le 2$, then we choose

$$H=⌊-\frac{loglogn}{log{\rho }_n}⌋.$$

(19)

If $2<\alpha <4$, then we choose

$$H=⌊-\left(1-\frac{2}{\alpha }\right)\frac{logn}{log{\rho }_n}⌋,$$

(20)

which tends to ∞ and satisfies $O\left(n{c}_n{\rho }_n^{2H}/a_n^2\right)=O\left(n^{1-2/\alpha }{\rho }_n^{2H}\right)=O(1/n^{1-2/\alpha })\to 0$.

By (9) and Lemma 2, Ref. (18) has the same limiting distribution as

$$u(1-{\rho }_n^2)\left[\sum _{l=0}^H{\rho }_n^{2l}{S}^0\left(\alpha \right)+2\sum _{m=0}^H{\rho }_n^{2m}\sum _{j=1}^H{\rho }_n^j{S}^j\left(\alpha \right)\right]+v\sum _{l=0}^H{\rho }_n^l{S}^{l+1}\left(\alpha \right).$$

(21)

We see that $(1-{\rho }_n^2){\sum }_{l=0}^H{\rho }_n^{2l}=1-{\rho }_n^{H+2}\to 1$ as ${\rho }_n\to 1$ and $H\to \infty$, since $log{\rho }_n^H=Hlog{\rho }_n\to -\infty$ for chosen H in both (19) and (20), and thus ${\rho }_n^H\to 0$. Letting $j=l+1$ in the third term of (21), we see that, as $H\to \infty$, (21) tends to $u\left(S^0\left(\alpha \right)+2{\sum }_{j=1}^{\infty }{S}^j\left(\alpha \right)\right)+v{\sum }_{j=1}^{\infty }{S}^j\left(\alpha \right)$. By (7) and (9), we obtain the desired result in (10). □

As mentioned above, the main contribution of the limit theory in the present paper is the proof of Lemma 3, which is somewhat lengthy and challenging. Due to the combination of Lemma 3 and Slutsky’s theorem, we can see simple validations of Theorems 1–3. When we prove Theorems 1–3 in the following, Slutsky’s theorem is used.

Proof of Theorem 1.

For $0<\alpha <2$, $c_n=0$ and

$$\frac{1}{1-{\rho }_n^2}({\widehat{\rho }}_n-{\rho }_n)=\frac{{\sum }_{t=1}^n{y}_{t-1}{ϵ}_t/n^{\frac{2}{\alpha }}}{(1-{\rho }_n^2){\sum }_{t=1}^n{y}_{t-1}^2/n^{\frac{2}{\alpha }}}{\to }^d\frac{Z_{\alpha /2}^{\left(2\right)}}{Z_{\alpha /2}^{\left(1\right)}}=\frac{U\left(\alpha \right)}{S_0\left(\alpha \right)+2U\left(\alpha \right)}$$

by Lemma 3, the desired result holds. □

Proof of Theorem 2.

For $\alpha =2$, $c_n=c_{0}logn$, and we write

$$\frac{logn}{1-{\rho }_n^2}({\widehat{\rho }}_n-{\rho }_n)=\frac{\frac{1}{n}{\sum }_{t=1}^n{y}_{t-1}{ϵ}_t}{\frac{1-{\rho }_n^2}{nlogn}{\sum }_{t=1}^n{y}_{t-1}^2}.$$

The first component of the left side in Lemma 3 implies

$$\frac{1-{\rho }_n^2}{n}\sum _{t=1}^n\left(y_{t-1}^2-\frac{c_n}{1-{\rho }_n^2}\right){\to }^d{Z}_{\alpha /2}^{\left(1\right)}$$

and thus

$$\frac{1-{\rho }_n^2}{nlogn}\sum _{t=1}^n{y}_{t-1}^2=\frac{Z_{\alpha /2}^{\left(1\right)}}{logn}+\frac{c_n}{logn}+o_p\left(1\right).$$

By Lemma 3, we have

$$\frac{\frac{1}{n}{\sum }_{t=1}^n{y}_{t-1}{ϵ}_t}{\frac{1-{\rho }_n^2}{nlogn}{\sum }_{t=1}^n{y}_{t-1}^2}=\frac{Z_{\alpha /2}^{\left(2\right)}}{Z_{\alpha /2}^{\left(1\right)}/logn+c_0+o_p\left(1\right)}+o_p\left(1\right){\to }^d\frac{1}{c_0}{Z}_{\alpha /2}^{\left(2\right)}=U\left(\alpha \right),$$

noting $E|{\eta }_1{|}^2=1$. The desired result is obtained. □

Proof of Theorem 3.

For $2<\alpha <4$, $c_n=E{ϵ}_1^2$. We see

$$\frac{n^{1-2/\alpha }}{1-{\rho }_n^2}({\widehat{\rho }}_n-{\rho }_n)=\frac{n^{1-2/\alpha }{\sum }_{t=1}^n{y}_{t-1}{ϵ}_t/n^{\frac{2}{\alpha }}}{(1-{\rho }_n^2){\sum }_{t=1}^n{y}_{t-1}^2/n^{\frac{2}{\alpha }}}.$$

(22)

By the first component of the left side in Lemma 3, we have

$$\frac{1-{\rho }_n^2}{n^{2/\alpha }}\sum _{t=1}^n{y}_{t-1}^2=Z_{\alpha /2}^{\left(1\right)}+c_n{n}^{1-2/\alpha }+o_p\left(1\right)$$

and thus (22) is

$$\frac{n^{1-2/\alpha }{Z}_{\alpha /2}^{\left(2\right)}}{Z_{\alpha /2}^{\left(1\right)}+c_n{n}^{1-2/\alpha }+o_p\left(1\right)}+o_p\left(1\right)=\frac{Z_{\alpha /2}^{\left(2\right)}}{Z_{\alpha /2}^{\left(1\right)}/n^{1-2/\alpha }+E{ϵ}_1^2+o_p\left(1\right)}+o_p\left(1\right){\to }^d\frac{1}{E{ϵ}_1^2}{Z}_{\alpha /2}^{\left(2\right)}.$$

The desired result is obtained. □

5. Concluding Remarks

This work considered an AR-G-GARCH model with tail index $\alpha \in (0,4)$ and established limit theory for LSE of autoregression coefficient $\rho ={\rho }_n\in [0,1)$ with condition ${\rho }_n\to 1$ as $n\to \infty$. For the tail index $\alpha \in (0,4)$ of G-GARCH errors, asymptotic distributions of the LSEs are involved with the stable distribution. The convergence rate of the LSE depends on $1-{\rho }_n^2$, but no condition on the rate of ${\rho }_n$ is required. It is shown that, for the tail index $\alpha \in (0,2)$, the LSE is inconsistent, for $\alpha =2$, $logn/(1-{\rho }_n^2)$-consistent, and for $\alpha \in (2,4)$, $n^{1-2/\alpha }/(1-{\rho }_n^2)$-consistent. This paper extends the uniform limit theory of [15] with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed; and no restriction on the rate of ${\rho }_n$ is necessary. The results can be applicable to financial volatility analysis. Volatility reveals heavy-tail and long-memory features as seen in [24], who discussed volatility of realized volatility. Ref. [24] claimed that adopting a GARCH process in the errors improves the goodness-of-fit, especially for realized volatility with non-Gaussianity and heavy-tails. Empirical analysis with financial market data, based on the results in this work, would be interesting and challenging.

This work focuses on convergence condition of ${\rho }_n\to 1$ but similar derivations for fixed coefficients $\rho$ can be done with results of uniform limit theory, which is a special case of [13]. If tail index $\alpha =4$, Ref. [22] investigated the limit theory of the LSE of AR(1)-GARCH(1,1) models, which is a simple linear case of (2). For the case of G-GARCH process in (2) with tail index $\alpha =4$, the asymptotic result for the LSE in model (1) provides a Gaussian limit theory with convergence rate ${(n/logn)}^{1/2}/(1-{\rho }_n^2)$. Its proof for case $\alpha =4$ is completely different from arguments in this work dealing with the cases $0<\alpha <4$, and might be so lengthy as in [22]. Thus, the Gaussian limit theory for $\alpha =4$ remains for future study at this moment.

As the convergence ${\rho }_n\to 1$ with a slower rate is accounted for, the asymptotic results provide a bridge between pure stationary or explosive process and unit root (or local to unity) process as mentioned by [25], who has developed limit theory for a mildly integrated process with intercept; $y_t=d+{\rho }_n{y}_{t-1}+u_t$ with assumption ${\rho }_n=1+c/k_n,k_n=o\left(n\right)$ as $n\to \infty$, where c and d are nonzero constants, and $\left\{u_t\right\}$ is a sequence of i.i.d. errors. The limit theory has been established for mildly stationary case $c<0$ and for mildly explosive case $c>0$. Earlier, Ref. [26] developed limit theory for a mildly integrated process, including both mildly stationary and mildly explosive cases. The extension of the mildly integrated process with nonzero intercept by adopting the heavy-tailed G-GARCH errors with tail index $\alpha \in (0,4)$ will be an interesting study. It will make a bridge between the inferences in [13,23], who have dealt with, respectively, unit-root and stationary AR models with heavy-tailed G-GARCH errors. This extension will be considered as future research.