Robust Estimation of Double Autoregressive Models via Normal Mixture QMLE*

This paper investigates the estimation of the double autoregressive (DAR) model in the presence of skewed and heavy-tailed innovations. We propose a novel Normal Mixture Quasi-Maximum Likelihood Estimation (NM-QMLE) method to address the limitations of conventional quasi-maximum likelihood estimation (QMLE) under non-Gaussian conditions. By incorporating a normal mixture distribution into the quasi-likelihood framework, NM-QMLE effectively captures both heavy-tailed behavior and skewness. A critical contribution of this paper is addressing the often-overlooked challenge of selecting the appropriate number of mixture components, $K$, a key parameter that significantly impacts model performance. We systematically evaluate the effectiveness of different model selection criteria. Under regularity conditions, we establish the consistency and asymptotic normality of the NM-QMLE estimator for DAR($p$) models. Numerical simulations demonstrate that NM-QMLE outperforms commonly adopted QMLE methods in terms of estimation accuracy, particularly when the innovation distribution deviates from normality. Our results also show that while criteria like BIC and ICL improve parameter estimation of $K$, fixing a small order of components provides comparable accuracy. To further validate its practical applicability, we apply NM-QMLE to empirical data from the S\&P 500 index and assess its performance through Value at Risk (VaR) estimation. The empirical findings highlight the effectiveness of NM-QMLE in modeling real-world financial data and improving risk assessment. By providing a robust and flexible estimation approach, NM-QMLE enhances the analysis of time series models with complex innovation structures, making it a valuable tool in econometrics and financial modeling.