GEOMETRIC ERGODICITY AND CONDITIONAL SELF-WEIGHTED M-ESTIMATOR OF A GRCAR(P) MODEL WITH HEAVY-TAILED ERRORS

We establish the geometric ergodicity for general stochastic functional autoregressive (linear and nonlinear) models with heavy-tailed errors. The stationarity conditions for a generalized random coefficient autoregressive model (GRCAR( p )) are presented as a corollary. And then, a conditional self-weighted M-estimator for parameters in the GRCAR( p ) is proposed. The asymptotic normality of this estimator is discussed by allowing infinite variance innovations. Simulation experiments are car-ried out to assess the finite-sample performance of the proposed methodology and theory, and a real heavy-tailed data example is given as illustration.

Model (1) with p = 1 (GRCAR(1)) was first introduced by Hwang and Basawa (1998), and it includes Markovian bilinear model and random coefficient exponential autoregressive model as special cases. When Var( t ) = 0, model (1) becomes the ordinary autoregressive (AR(p)) model. There have been a lot of perfect theoretical achievements about it. For example, Ling (2005) proposed a self-weighted least absolute deviation estimator and showed its asymptotic normality. The method has been used in many references, such as Pan et al. (2007), Pan and Chen (2013), Pan et al. (2015). Wang and Hu (2017) proposed a self-weighted M-estimator for the AR(p) model and established the asymptotic normality of this estimator. When Var( t ) ≠ 0 with p = 1, and t is independent 419 of t , model (1) becomes the first-order random coefficient autoregressive (RCAR(1)) model (see Nicholls and Quinn (1982)), which has been frequently used to describe the random perturbations of dynamical systems in economics and biology (see Tong (1990), Yu et al. (2011), Zhang et al. (2015), Araveeporn (2017)). As a generalization of RCAR model and AR models, GRCAR model has become one type of important models in nonlinear time series, since it allows dependence between random errors and random coefficients. Estimation of parameters and asymptotic properties of GRCAR models have been studied in the literature. For instance, Hwang and Basawa (1997) established the local asymptotic normality of a class of GRCAR models. Zhao and Wang (2012) constructed confidence regions for the parameters by using empirical likelihood method. Zhao et al. (2013) considered the problem of testing the constancy of coefficients in a GRCAR model by empirical likelihood method. Zhao et al. (2018) studied the variable selection problem in GRCAR models. Zhao et al. (2019) proposed a weighted least squares estimate and empirical likelihood (EL) based weights through using some auxiliary information for GRCAR models. Moreover, time series models with heavy-tailed errors, even when E( 2 t ) is infinite, are often found and studied in economic and financial modeling. Wu (2013) studied M-estimation for general ARMA processes with infinite variance. Yang and Ling (2017) investigated the self-weighted least absolute deviation estimation for heavy-tailed threshold autoregressive models. Fu et al. (2021) studied the asymptotic properties for the conditional self-weighted M-estimator of GRCAR(1) model with possibly heavy-tailed errors. However, general easy-to-check conditions for stationarity and limiting distributions of robust parameter estimators for statistical inference of GRCAR(p) with heavy-tailed errors are still open problems.
This article aims to reach two targets. First, we establish the geometric ergodicity of general stochastic functional autoregressive (linear and nonlinear) models with possibly heavy-tailed error terms under a mild moment condition. Moreover, the stationarity conditions for GRCAR(p) are implied as a corollary of our general result. Second, motivated by Yang and Ling (2017), Ling (2005), Wang and Hu (2017) and Fu et al. (2021), we prove the asymptotic property of a self-weighted M-estimator (SM-estimator) for GRCAR(p) with possible infinite variance, and show that the limiting distribution of SM-estimator is asymptotically normal. Simulation results and a real data example are given to support our methodology.
The contents of this article are organized as follows. Section 2 presents the main results. Section 3 reports the simulation results. Section 4 shows a real data example. All proofs of our main results are given in Section 5.

Geometric Ergodicity
We first establish geometric ergodicity of general stochastic functional autoregressive models (including linear and nonlinear) under a mild moment condition. Then the geometric ergodicity which can imply the stationarity conditions of model (1) is given as a corollary of the main theorem.
Consider a general stochastic functional autoregressive model defined as follows It can be seen that both linear and nonlinear autoregressive models are included. This model can be rewritten in vector form as follows: (i) There exists some norm ‖⋅‖ v on the p-dimensional vector space,constants 0 < < 1, 0 < < 1 and c ≥ 0, such that (ii) The density function of t is continuous and positive everywhere,and E | | t | | < ∞ for in (i). Then, model (3) is geometrically ergodic, which implies {y t } in model (2) is stationary and geometrically ergodic. Under the conditions of Theorem 2.1, more concretely, we have the following corollary.
Corollary 2.2. Suppose model (2) satisfies (i) There exists a constant vector = ( 1 , … , p ) ′ and a 0 < < 1 satisfying and for any K > 0, (ii) The density function of t is continuous and positive everywhere,and E | | t | | < ∞ for in (i). Then, {y t } in model (2) is stationary and geometrically ergodic.
In Corollary 2.2, when t (x) = t0 + ( t1 , … , tp ) ′ x, x ∈  p , we can get the stationarity conditions for GRCAR(p) as another corollary of our general result.
Corollary 2.3. Suppose model (1) satisfies The density function of t is continuous and positive everywhere, and E | | t | | < ∞ for in (ii). Then, {y t } in model (1) is stationary and geometrically ergodic. Remark 1. Theorem 2.1 establishes the geometric ergodicity of general stochastic functional autoregressive (linear and nonlinear) models with possibly heavy-tailed error terms under a mild moment condition. The stationarity conditions for GRCAR(p) in Corollary 2.3 is a consequence of Theorem 2.1. In Corollary 2.3, the moment condition is very weak. We only require a finite moment of order (0 < < 1) about the error t , which includes the Cauchy distribution. The condition on random coefficients makes the model not too far away from the linear AR. This is a reasonable requirement for any (non-parametric or parametric) AR-type model.

Remark 2.
We note that the GRCAR is a quite broad kind of models for time series data. A special case of this type of models can be used to describe conditionally heteroscedastic structure. For example, consider the model It is shown that this very special case of the GRCAR has similar function to AR(1)-ARCH(1), but we don't need to restrict the parameters 0 and 1 to be non-negative. Furthermore, if the assumption on the distribution of error is changed to t ∼ t 2 , the model becomes a model with infinite variance.

Conditional Self-weighted M-estimation
where Θ ⊂  p+1 is the parameter space containing the true value 0 .

Asymptotic Normality of SM-estimation
To derive the asymptotic property of̂S M , we need the following assumptions: (C.2) Let (⋅) be a convex function on R with left derivative − and right derivative + . Choose a function such that − ≤ ≤ + .
and Zhu (2018). Examples of (x) satisfying assumptions include (x) = x 2 , (x) = |x| and (x) = 1 2 x 2 I(|x| ≤ m) + (m |x| − 1 2 m 2 )I(|x| > m), which then correspond to the conditional self-weighted least-squares estimator, conditional self-weighted least absolute deviation estimator and conditional self-weighted Huber estimator respectively. Assumption (C.5) is standard on the weight t for the self-weighted method in IVAR models which allows Ey 2 t to be infinite by properly choosing weight function t . Firstly, the purpose of the weight t is to downweight the leverage points in X t such that the covariance matrices and in Theorem 2.4 above are finite. Secondly, the t allow us to approximate L n (̂n) by a quadratic form. In addition, Theorem 2.4 generalizes the results of Ling (2005), Wang and Hu (2017) and Fu et al. (2021).
which shows the asymptotic property of the conditional self-weighted least-squares estimator of parameters in a GRCAR(p) model with finite variance.
For the case (x) = |x| and E 2 1 = ∞, taking (x) = sign(x). Suppose that the errors t have zero median with a density f (x) satisfying sup which shows the asymptotic property of the conditional self-weighted least absolute deviation estimator for a GRCAR(p) model with infinite variance.
Using Theorem 2.4, we have which includes the asymptotic property of the conditional self-weighted Huber estimator for a GRCAR(p) model with finite variance or infinite variance.

SIMULATION STUDIES
We conduct some simulation studies in finite samples through Monte Carlo experiments. What we are interested in are the accuracy and sampling distribution of the proposed estimator. The results show that our method performs well.
Data are generated from the following GRCAR (p) models: of the error process in model B. Here three distributions are given for reference: N(0, 1) has finite expectation and finite variance; t 2 has finite expectation but infinite variance; Cauchy(0,1) only have a finite moment of order (0 < < 1). We set the sample sizes n = 200 and n = 400. The number of replications is 2000. Tables I and II list the biases, standard deviations (SD) and asymptotic standard deviations (ADs) of the conditional self-weighted least absolute deviation estimator (SM 1 ) and conditional self-weighted Huber estimator (SM 2 ) with the following choice of weight functions respectively: where K is the 0.9 quantile of data { |y 1 |, … , |y n | } . The weight w 1t is similar to Ling (2005). The weights w 2t , w 3t and w 4t were considered by Yang and Ling (2017). The tuning parameter of Huber estimator is taken as m = 1.5. We definê= is the derivative ofĜ(r) at r = 0 and {̂t} is the sequence of residuals in GRCAR(p). The ADs are calculated by (10)-(14). We estimate f (0) byf (1 + e −x ) 2 and b n = 1.06 × n −1∕5 . For the choice of the optimal bandwidth and its motivation, we refer to Silverman (1986,p.40)and Pan et al. (2007). Tables I and II show that all the biases are very small and all the SDs and ADs are very close no matter E 2 t is finite or infinite and no matter the random coefficients are correlated with the error process or independent of the error process. All the biases, SDs and ADs become smaller, when n increases from 200 to 400. And the SM 2 estimators perform better than the SM 1 estimators. All of the estimators based on w 1t are more efficient than the others. To get an overall view on the sampling distributions of the SM 1 estimators and the SM 2 estimators, we simulate 2000 replications for the case 1 = 0.5 and n = 400 when the error distributions are N(0, 1) and t 2 for model A, and for the case 1 = −0.5 and n = 400 when the error distributions are N(0, 1), t 2 and Cauchy for model B. respectively. Figure 1 shows the density curves of model A. The density curves of model B are presented in Figure 2. We can see that the density of N(0, 1) is approximated reasonably well by those of N n1 , N n2 , N t1 , N t2 and N c1 , N c2 in both model A and model B.
In conclusion, the numeric results show that the conditional self-weighted M-estimators perform well in finite sample no matter with finite variance or infinite variance.

REAL DATA ANALYSIS
The proposed methodology is applied to modeling a real dataset. We consider the Hang Seng Index (HSI) for the stock market which is the most influential index in the Hong Kong stock market and one of the most important indices in the Asian financial markets as well. This index has been extensively investigated in the literature. Our dataset consists of the daily Hang Seng closing index from 7 May 2020 to 31 December 2021, which was downloaded at https://cn.investing.com/. There are 412 available observations in total, which are denoted by x 1 , x 2 , … , x 412 . The first 392 data are selected as the training sample to build model, and next 20 data are used as test sample to evaluate the model. We take the following steps to analyze this dataset by the GRCAR model and method proposed in this article.
Step 1. Data transformation: The sample time plot for the data {x t } is shown in Figure 3(a). It can be seen that the time series is not stationary because of various levels. To get a stationary time series, let y t = 100 × (log(x t ∕x t−1 )). The sample path plot for the data {y t } is shown in Figure 3(b). Figure 3(b) indicates that {y t } is close to stationarity.
Step 2. Model identification: The plot of sample autocorrelation function (ACF) and sample partial autocorrelation function (PACF) of {y t } are presented in Figure 3(c) and (d) respectively. Figure 3 can provide some important information for tentative identification of the orders of a stable AR model. Based on the sample ACF and PACF plots, it is reasonable to consider fitting an AR(3) autocorrelation structure to {y t }. Since stock data are affected by various factors, the coefficients of the autoregressive model may also change randomly over Figure 2. The sampling distribution for model B time, even the coefficients may be correlated with the error, so we can try to fit a GRCAR(3) model instead of AR(3). However, we realize that the data may be heavy-tailed and how to determine the order of autoregression for a time series with infinite variance is a problem which needs further study. Here we just use the sample PACF to give a rough indication of the order in which a GRCAR model might be fitted.
Step 3. Heavy-tail test: To test whether {y t } has a heavy-tailed distribution, we use Hill's estimator (see Drees et al. (2000) and Resnick (2000)) to estimate the tail index of y t . Let y (1) > y (2) > · · · > y (n) be the order statistics of y t , t = 1, … , n. The estimators of the right-tail and left-tail indices are defined as ) } −1 , Figure 3. (a) Sample path of data x t , (b) sample path of data y t , (c) sample ACF of y t , (d) sample PACF of y t respectively. Figure 4 displays the Hill estimates of the right-tail and left-tail indices when 1 ≤ k ≤ 200. From Figure 4, we can see that both the right and left tail indices are most likely less than 2. Hence, y t should have a heavy tail. Therefore, it may be more appropriate to suppose these data are generated from a process with infinite variance rather than to assume this data have finite variance.
Based on the above discussion, we can fit a GRCAR(3) model to the data: Step 4. Parameter estimation: The unknown parameters are estimated by different methods on the training data. We calculated the mean absolute errors (MAEs) of predicted values for transformed data based on 1000 repetitions. The results are shown in Table III. From them, we can see that the self-weighted estimators perform better, especially SM 2 . So we choose the SM 2 for further analysis. The estimates are ( 0 , 1 , 2 , 3 ) = (0.0493, 0.0059, 0.0214, −0.1608), whose corresponding asymptotic standard deviations are 0.0028, 0.0019, 0.0018, 0.0018 respectively.
Step 5. Model diagnostics: The absolute value of the eigenvalues for the corresponding matrix in model (15) are 0.5549, 0.5383 and 0.5383, which are all less than one. Therefore, this model satisfies the stationary conditions of Corollary 2.3. Figure 5 presents the residuals of the fitted model (15), the normal Q-Q plot of the residuals, the sample autocorrelation function (ACF) of the residuals and the sample ACF of the squared residuals. From that, we can see the model (15) fits the data reasonably well.
Step 6. Prediction: We use the above model (15) (3)-ARCH(2) model. The predictive performance of different models is presented in Figure 6. We can see that the predicted value of model (15) captures the change trend of the real value and most of the predicted values are very close to the true values. Figure 6 and   In summary, our model and method perform well in analysis and forecasting of time series data with heavy-tailed distributions.

PROOFS OF THEORETICAL RESULTS
This section presents the proofs of our theoretical results.
Proof of Theorem 2.1. It is easy to see that {X t } defined by (3) is a homogenous Markov chain. By the conditions of Theorem 2.1, this Markov chain is u p -irreducible and aperiodic, and all bounded sets with positive u p -measure in R p are small sets. Take the test function g(x) = ‖x‖ v . It holds that , then C is a small set, and The condition (5) of Corollary 2.2 is equivalent to p − 1 p−1 − · · · − p ≠ 0, | | ≥ 1. It implies the roots Proof. By applying the conditions (C.1) with y t being stationary and ergodic, it is easy to get (a) and (b). So we omit the proofs of (a) and (b), and only give the proof of (c). Put nt = 1 √ n t C ′ X t ( t ). Then and nt , 1 ≤ t ≤ n is a sequence of martingale differences with respect to  t−1 . By (a), it follows that where = C ′ ΩC. Put t = t C ′ X t , then for any > 0, we have Hence (25) can be rewritten to be Because of Lemma 5.2, we can get where A ∼ N(0, ). Note that V( ) has a unique minimum at = 1 −1 A almost surely and V n ( ) has convex sample paths due to the condition (C.2). Applying Lemma 5.1, we havê So the proof of Theorem 2.4 is completed. ◾

CONCLUDING REMARKS
This article establishes the geometric ergodicity of general stochastic functional autoregressive models (including linear and nonlinear) under a broad condition. Furthermore, the stationary conditions and a self-weighted M-estimator for GRCAR(p) models with possibly heavy-tailed errors are proposed. The proposed estimator is shown to be asymptotically normal. The simulation study and a real data example showed that our theory and methodology perform well in practice. A general approach to stationarity and estimation for GRCAR models with heavy-tailed errors is presented. The methodology and results could be extended further to other time series models such as heavy-tailed GRCARMA models.