Linear Models and Time-Series Analysis

5Regression Extensions: AR(1) Errors and Time‐varying Parameters

The place of econometrics at the centre of economics is now confirmed.

(The Economist, Oct. 11, 2003, p. 84)

A popular univariate time‐series model that is useful in itself and also serving as a baseline for advanced models in econometrics is the regression framework of Chapter 1 combined with the AR(1) model for the regression error term from Chapter 4,. This chapter considers this model in detail.

After discussing the likelihood in Section 5.1 , we develop point and interval estimators for the AR(1) parameter amid regressor covariates in Section 5.2 . Section 5.3 discusses methods for testing the null hypothesis of the AR(1) coefficient being zero. Section 5.4 builds on the methods from Section 4.6 for bias‐adjusted point estimation of the AR(1) parameter. Section 5.5 details some basic methods for unit‐root testing. Finally, we turn to the regression model with time‐varying coefficients in Section 5.6 .

5.1 The AR(1) Regression Model and the Likelihood

Let , , be a set of vectors of non‐stochastic, known constants, ...