Conditional maximum Lq-likelihood estimation for regression model with autoregressive error terms

Abstract

In this article, we consider the parameter estimation of regression model with pth-order autoregressive (AR(p)) error term. We use the maximum Lq-likelihood (MLq) estimation method proposed by Ferrari and Yang (Ann Stat 38(2):753–783, 2010), as a robust alternative to the classical maximum likelihood (ML) estimation method to handle the outliers in the data. After exploring the MLq estimators for the parameters of interest, we provide some asymptotic properties of the resulting MLq estimators. We give a simulation study and three real data examples to illustrate the performance of the proposed estimators over the ML estimators and observe that the MLq estimators have superiority over the ML estimators when some outliers are present in the data.

Appendices

Appendix A

Let $\underline{\theta } = \left( { \beta_{1} , \beta_{2} , \ldots ,\beta_{M} ,\;\phi_{1} ,\phi_{2} , \ldots ,\phi_{p} ,\sigma^{2} } \right).$ The elements of $U\left( {a_{t} ;\underline{\theta } } \right)$ are

$$ \begin{aligned} \frac{\partial lnf}{{\partial \beta_{k} }} & = \frac{1}{{\sigma^{2} }}\left( {\Phi \left( B \right)y_{t} - \mathop \sum \limits_{i = 1}^{M} \beta_{i}\Phi \left( B \right)x_{t,i} } \right)\Phi \left( B \right)x_{t,k} ,\quad k = 1,2, \ldots ,M \\ \frac{\partial lnf}{{\partial \phi_{r} }} & = \frac{1}{{\sigma^{2} }}\left( {\Phi \left( B \right)y_{t} - \mathop \sum \limits_{i = 1}^{M} \beta_{i}\Phi \left( B \right)x_{t,i} } \right)\left( {y_{t - r} - \mathop \sum \limits_{i = 1}^{M} \beta_{i} x_{t - r,i} } \right) ,\quad r = 1,2, \ldots ,p, \\ \frac{\partial lnf}{{\partial \sigma^{2} }} & = - \frac{1}{{2\sigma^{2} }} + \frac{1}{{2\sigma^{4} }}\left( {\Phi \left( B \right)y_{t} - \mathop \sum \limits_{i = 1}^{M} \beta_{i}\Phi \left( B \right)x_{t,i} } \right)^{2} , \\ \end{aligned} $$

where $t = p + 1, \ldots ,N.$

The second partial derivatives are

$$ \begin{aligned} \frac{{\partial^{2} \ln f}}{{\partial \beta_{j} \partial \beta_{k} }} & = - \frac{1}{{\sigma^{2} }}\Phi \left( B \right)x_{t,j}\Phi \left( B \right)x_{t,k} , \frac{{\partial^{2} \ln f}}{{\partial \beta_{j} \partial \phi_{r} }} = - \frac{1}{{\sigma^{2} }}\left[ {e_{t - r}\Phi \left( B \right)x_{t,j} + a_{t} x_{t - r,j} } \right], \\ \frac{{\partial^{2} \ln f}}{{\partial \beta_{j} \partial \sigma^{2} }} & = - \frac{1}{{\sigma^{4} }}a_{t}\Phi \left( B \right)x_{t,j} , \frac{{\partial^{2} \ln f}}{{\partial \phi_{r} \partial \phi_{s} }} = - \frac{1}{{\sigma^{2} }}e_{t - r} e_{t - s} , \\ \frac{{\partial^{2} \ln f}}{{\partial \phi_{r} \partial \sigma^{2} }} & = - \frac{2}{{\sigma^{4} }}a_{t} e_{t - r} , \frac{{\partial^{2} \ln f}}{{\partial \left( {\sigma^{2} } \right)^{2} }} = \frac{1}{{2\sigma^{4} }} - \frac{1}{{\sigma^{6} }}a_{t}^{2} , \\ \end{aligned} $$

where $t = p + 1, \ldots ,N, j,k = 1,2, \ldots ,M$ and $r,s = 1,2, \ldots ,p.$

Appendix B

The elements of $U^{*} \left( {a_{t} ;\underline{\theta } ,q} \right)$ are

$$ \begin{aligned} \frac{\partial Lq}{{\partial \beta_{k} }} & = \frac{1}{{\sigma^{2} }}\omega_{t} \left( {\Phi \left( B \right)y_{t} - \mathop \sum \limits_{i = 1}^{M} \beta_{i}\Phi \left( B \right)x_{t,i} } \right)\Phi \left( B \right)x_{t,k} , \quad k = 1,2, \ldots ,M \\ \frac{\partial Lq}{{\partial \phi_{r} }} & = \frac{1}{{\sigma^{2} }}\omega_{t} \left( {\Phi \left( B \right)y_{t} - \mathop \sum \limits_{i = 1}^{M} \beta_{i}\Phi \left( B \right)x_{t,i} } \right)\left( {y_{t - r} - \mathop \sum \limits_{i = 1}^{M} \beta_{i} x_{t - r,i} } \right),\quad r = 1,2, \ldots ,p, \\ \frac{\partial Lq}{{\partial \sigma^{2} }} & = \omega_{t} \left[ { - \frac{1}{{2\sigma^{2} }} + \frac{1}{{2\sigma^{4} }}\left( {\Phi \left( B \right)y_{t} - \mathop \sum \limits_{i = 1}^{M} \beta_{i}\Phi \left( B \right)x_{t,i} } \right)^{2} } \right], \\ \end{aligned} $$

where

$$ \omega_{t} = \left( {\frac{1}{{\sqrt {2\pi \sigma^{2} } }}exp\left\{ { - \frac{{\left( {\Phi \left( B \right)y_{t} - \mathop \sum \nolimits_{i = 1}^{M} \beta_{i}\Phi \left( B \right)x_{t,i} } \right)^{2} }}{{2\sigma^{2} }}} \right\} } \right)^{1 - q} , \quad t = p + 1, \ldots N. $$

The second partial derivatives of conditional Lq-likelihood function can be obtained by substituting the derivatives given in “Appendix A” in Eq. (25).