Tests of an Adaptive Regression Model

A NY econometric equation representing a complex behavioral or technical relationship is, of necessity, an approximation of reality. As such, it is subject to errors in specification and structural change over time. This problem is well recognized by econometricians. Duesenberry and Klein (1965) point out that *'. . . as technology, institutional arrangements, tastes and managerial techniques change over time, the relationships represented by our equations inevitably change. Furthermore, when statistical tests are applied to econometric relationships, the hypothesis of structural stability is frequently rejected.' Some methods for dealing with structural change have evolved. Quandt (1957) has developed a maximum likelihood technique for estimating a point of structural change within a sample.2 Klein and Evans (1967) adjust the intercepts of the Wharton Model to account for structural change.3 The purpose of this paper is to test the robustness of Adaptive Regression (1973) to specification errors causing structural change over time, relative to ordinary least squares analysis with and without the autoregressive correction.4 Since econometricians are inevitably faced with structural change and errors in specification, they should use a technique which is robust rela-tive to such problems. The device most commonly used is to assume that the disturbances are subject to an autoregressive process. The autoregressive correction may frequently ameliorate the effects of misspecification and structural change, but it is doubtful whether such processes, except in rare instances, describe the true distribution of the disturbances. The economics literature seldom gives any justification for this scheme except that omitted variables may be subject to an autoregressive process or the structure of the model may be changing.5 We suspect the reasons for the widespread use of the autoregressive correction are that it is a simple hypothesis, explains serial correlation in the disturbances, and can be dealt with efficiently. The adaptive regression model considered in this paper is equally simple but more general, explains serial correlation, and can also be dealt with efficiently.6 In the next section the adaptive regression model is presented and the Bayesian estimators are developed. In section II the results of a Monte Carlo Study are presented. Two models are considered for which data are generated by eleven different schemes. The estimation and forecasting efficiency of adaptive regression, and ordinary least squares with and without the autoregressive correction are compared. Section III contains an analysis of the role of time trends in econometric relationships. In section IV the relative forecasting ability of the three estimation techniques is tested on real data. The three models suggested by Received for publication February 10, 1972. Revision accepted for publication November 30, 1972. *The authors acknowledge helpful comments of Professors F. G. Adams, R. Roll and R. Summers and the participants of the NBER conference on Bayesian Statistical Inference in Economics. Computations were executed on the University of Pennsylvania computer. 'Examples of such tests include Brown (1966), Goldfield (1969) and Howrey (19-70). One of the most extensive studies was done by Duffy (1969). 'The Quandt technique is limited by the fact that it is mainly useful for finding stable subsamples. If structural change occurs often, it is not very useful. Rosenberg (1968) has used stepwise composition to develop the computationally efficient Aitken estimates of a model subject to structural change over time. His procedure, however, requires that the true covariance matrix of the disturbances be known up to a constant scale factor. 'Adjusting the intercepts is an ad hoc method for keeping the model on track for ex ante forecasting. The intercepts are not assumed to change over the sample period which is always much longer than the forecasting period. 'The autoregressive correction assumes the error is subject to a first or second order autoregressive scheme. See Dhrymes (1969) for the maximum likelihood approach and Zellner and Tiao (1965) for the Bayesian development. The latter approach is used in this paper. 'In fact, if omitted variables are subject to an autoregressive process, the disturbances will, in general, be subject to a more complicated process. 6 A test with sufficient power to differentiate betweer these two models (or others which result in serial correlation) using sample sizes generally available to econometricians does not appear to exist. Further, if one did, its usefulness would be limited as neither structure is likely to be an exact representation of reality. That one structure is more likely on the basis of the data does not imply thai it will forecast better if, in fact, a third structure is generating the data.