On the efficiency of regression analysis with AR(p) errors

Abstract

In this paper we will consider a linear regression model with the sequence of error terms following an autoregressive stationary process. The statistical properties of the maximum likelihood and least squares estimators of the regression parameters will be summarized. Then, it will be proved that, for some typical cases of the design matrix, both methods produce asymptotically equivalent estimators. These estimators are also asymptotically efficient. Such cases include the most commonly used models to describe trend and seasonality like polynomial trends, dummy variables and trigonometric polynomials. Further, a very convenient asymptotic formula for the covariance matrix will be derived. It will be illustrated through a brief simulation study that, for the simple linear trend model, the result applies even for sample sizes as small as 20.

Keywords:

• linear regression
• autoregressive stationary process
• maximum likelihood
• least squares
• trend
• seasonality
• linear difference equation

Acknowledgements

One of the authors was partially supported by FCT (Fundação para a Ciência e Tecnologia) through the program POCTI (Portugal/FEDER-EU).