# Serial Correlation

Maxwell L. King

1 Introduction

In its most general form, serial correlation involves the correlation of successive time series observations. It has been the subject of much research in econometrics over the last half century, particularly with respect to serial correlation in regres­sion disturbances. The seminal work of Cochrane and Orcutt (1949) did much to alert econometricians to the difficulties of assuming independent regression errors in time series applications of the standard linear regression model. It is now well known that the neglect of disturbance correlation can lead to inefficient parameter estimates, misleading inferences from hypothesis tests and inefficient predictions. This led to a vast literature on testing for serial correlation in linear regression models and other models, see for example King (1987) for a review of this literature.

Early econometric models were often built using a static representation of the economic forces at work. Such static models were generally estimated using annual data and typically any problems resulting from the poor specification of the dynamics of the economic situation being modeled were swept into the error term. The availability of quarterly data in the 1970s brought the realization that better modeling of the dynamics was needed. This led to a greater interest by econometricians in a class of univariate time series models proposed by Box and Jenkins (1970). It also led to greater use of dynamic linear regression models in which the lagged dependent variable is included as a regressor. In summary, modeling serial correlation really involves taking care of the dynamic part of a model specification.

The class of Box-Jenkins models provides important building blocks for a wide range of model specifications that handle the dynamics of the process. In this chapter we provide a summary of these different types of models and consider related issues of estimation and testing. We constrain our attention purely to univariate models. These models can all be generalized to linear simultaneous equations and vector autoregressive models.

The plan of this chapter is as follows. A range of models are introduced in Section 2. These include Box-Jenkins time series models, regression disturbance models of serial correlation and dynamic linear regression models. Section 3 discusses the problem of estimation of the linear regression model with serial correlation in the disturbances. Hypothesis testing is the topic of Section 4. Particular emphasis is placed on the Durbin-Watson and related tests in the context of the standard linear regression model and the dynamic linear regression model. The chapter ends with a short discussion on model selection.

2 Models