Quasi-maximum Likelihood Inference for Linear Double Autoregressive Models

Abstract

This paper investigates the quasi-maximum likelihood inference including estimation, model selection and diagnostic checking for linear double autoregressive (DAR) models, where all asymptotic properties are established under only fractional moment of the observed process. We propose a Gaussian quasi-maximum likelihood estimator (G-QMLE) and an exponential quasi-maximum likelihood estimator (E-QMLE) for the linear DAR model, and establish the consistency and asymptotic normality for both estimators. Based on the G-QMLE and E-QMLE, two Bayesian information criteria are proposed for model selection, and two mixed portmanteau tests are constructed to check the adequacy of fitted models. Moreover, we compare the proposed G-QMLE and E-QMLE with the existing doubly weighted quantile regression estimator in terms of the asymptotic efficiency and numerical performance. Simulation studies illustrate the finite-sample performance of the proposed inference tools, and a real example on the Bitcoin return series shows the usefulness of the proposed inference tools.

Figures (8)

• Figure 1: The $\text{ARE}(\widehat{\bm\theta}_n,\widetilde{\bm\theta}_n)$ (left panel) and $\text{ARE}(\widehat{\bm\theta}_n^{\star},\check{\bm\theta}^{\star opt}_n)$ (right panel) for $\delta=k/20~(k=0,1,\ldots,20)$, where $m(x)$ is the pdf of the $N(0,6)$ ($\circ$), standard Laplace ($\square$), or $t_3$ ($+$) distribution.
• Figure 2: Time plot for centered weekly log returns in percentage (black line) of BTC from July 18, 2010, to August 16, 2020, with one-week negative VaR forecasts at the level of 5% based on the G-QMLE (dotted line) and the E-QMLE (dashed line) from March 26, 2017, to August 16, 2020.
• Figure 3: QQ plots of the residuals $\{\widehat{\eta}_t\}$ against the Student's $t_{2}$ (left panel), $t_{3}$ (middle panel), and $t_{4}$ (right panel) distributions.
• Figure 4: Residual ACF plots for $\widehat{\rho}_l$ (left panel) and $\widehat{\gamma}_l$ (right panel), where the dashed lines are the corresponding 95% pointwise confidence intervals.
• Figure 5: The empirical size of $Q(M)$ (a) and $Q^G(M)$ (b) with respect to the lag order $M$ at the significance level 0.05, where the line with color blue or red presents $N=500$ or 2000, the symbol $\times$ or $\triangle$ represents $\eta_t$ being a standard normal or Laplace random variable; the empirical size of $Q(M)$ (c) and $Q^G(M)$ (d) with respect to the sample size $n$ at the significance level 0.05, where the line with color blue or red presents $M=6$ or 12, the symbol $\times$ or $\triangle$ represents $\eta_t$ being a standard normal or Laplace random variable.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Remark 1
• Theorem 3
• Theorem 4
• Remark 2
• Theorem 5
• Remark 3
• Remark 4
• Lemma 1