Examples of Regression with Serially Correlated Errors

Abstract

Three data sets are analysed to illustrate methods of modelling regression errors which are serially correlated. An autoregressive-moving average error process is used in fitting a regression equation to the energy demands of a mechanical model of a suckler cow. Drug-induced currents in ion- channels are represented by a realisation of a stochastic compartment system. First-order linear stochastic difference equations are used to model milk yield of cows. It is concluded that error models should be used with caution. situations it is assumed that the function is deficient, and it is changed. But there are cases where the assumption of independent errors is not wholly plausible. For example, some sources of error will persist over several observations when repeated measurements are made on a single experimental unit. Systematic departures may then be modelled either by another regression function, or by correlated errors. The modelling objective determines the choice: for a simple summary it may be preferable for the regression function to explain all systematic variability, whereas a correlated stochastic component may be of more assistance in understanding the data generating mechanism. A succinct summary of data is often achieved by using the regression function to describe the long-term trends and the correlations the short-term fluctua- tions. In the presence of correlated errors, ordinary least squares regression parameter estimators may be inefficient and the conventional estimators of the variances of these estimators are usually biased. The simplest way round these problems is to discard the biased standard errors; the argument being that least-squares estimation is often not very inefficient, and is intuitively appealing because of its simplicity. This approach is most useful when no estimate of precision is required, for example when data are available from independent units and within-unit variability is of little importance. (See, for example, Rowell & Walters, 1976.) Alternatively, if it can be assumed that the errors arose from a particular stochastic model, any parameters can be estimated jointly with the regression ones by maximising the likelihood. Empirical and mechanistic approaches to modelling errors will be considered in the following two sections. In essence, the mechanistic approach requires knowledge of the processes by which the data were generated, whereas the empirical method is purely data-based (Thornley, 1976, pp. 4-6).