Statistica Sinica 23 (2013), 251-270

doi:http://dx.doi.org/10.5705/ss.2011.086

QUASI-MAXIMUM EXPONENTIAL LIKELIHOOD

ESTIMATORS FOR A DOUBLE AR(p) MODEL

Ke Zhu and Shiqing Ling

Chinese Academy of Sciences and

Hong Kong University of Science and Technology

Abstract: The paper studies the quasi-maximum exponential likelihood estimator (QMELE) for the double AR(p) (DAR(p)) model:

$$y_t=\sum_{i=1}^{p}\phi_{i} y_{t-i}+\eta_t\sqrt{w+\sum_{i=1}^{p}\alpha_{i} y_{t-i}^2},$$

where $\{\eta_{t}\}$ is a white noise sequence. Under a fractional moment of $y_{t}$ with $E\eta_{t}^{2}<\infty$, strong consistency and asymptotic normality of the global QMELE are established. A formal comparison is given with the QMLE in Ling (2007) and WLADE in Chan and Peng (2005). A simulation study is carried out to compare the performance of these estimators in finite samples. An example on the exchange rate is given.

Key words and phrases: Asymptotic normality, double AR(p) model, QMELE and strong consistency.