Edwin Kuh

How Extraneous are Extraneous Estimates

IN order to overcome the harmful effects on regression and correlation estimates of using highly collinear time series observations, and in order to obtain structurally more accurate estimates of income elasticities of demand, economic statisticians have turned increasingly to the device of extraneous estimators. This technique has been most commonly employed in demand studies but also could be used in other applications. In the case of demand functions the procedure has been to obtain the income coefficient from cross-section budget data for which the price variables are presumably constant. Thus an estimate of the partial regression of quantity on income, with price given, is obtained. This cross-section estimate of the income regression-coefficient is then multiplied by the time-series aggregate of income, and the product is in turn subtracted from the annual time series of quantity demanded, to form a new dependent variable. This new dependent variable, as a possibly unintended consequence, usually has a larger variance than the original dependent series, which often displays little variation beyond the simplest trend component. Having been thus corrected, the dependent series is then regressed against the time series of the price variables to obtain an estimate of the price elasticity of demand. It should be fairly clear that when the purpose is to make short-run forecasts, the described techniques often may be unnecessary and in many instances could actually prove harmful.2 Specifically, someone making forecasts need not be especially worried about multicollinearity. If some of the explanatory variables are multicollinear, the prediction interval obtained from such a set of observations will be quite large. By eliminating a number of the collinear variables it will usually be possible to substantially reduce the prediction interval for given values of included independent variables. Of course, while the elimination of collinear explanatory variables will tend to reduce the prediction interval, the actual prediction, by hypothesis, will change very little. Hence the pragmatic forecaster might be indifferent to the extent of collinearity, while the more sophisticated forecaster will not be indifferent; both will make similar forecasts and the actual errors of the forecast will be approximately the same. The combined use of cross-section and timeseries data is therefore intended to overcome multicollinearity (which entails the arbitrary splitting up of the influence of the explanatory variables) in order to obtain structurally mnore accurate estimates of the various coefficients. The question, however, can legitimately be asked: Exactly what structure does the statistician seek to estimate? Insofar as demand studies are concerned, it is quite possible, as will shortly be argued at length, that the kind of behavior measured from cross-section * For helpful comments and discussions on an earlier draft of this paper, we are indebted to John S. Chipman, Gregory Chow, James S. Duesenberry, John Lintner, Guy H. Orcutt, Robert Solow, and Charles Zwick. The authors were aided in preparing this paper by research grants from the School of Industrial Management, Massachusetts Institute of Technology, and Division of Research, Harvard Business School (under a Rockefeller Foundation grant for a Study of Profits and the Functioning of the Economy). While the techniques used differ in some important respects, the rationale is fully explained in each of the following sources: Richard Stone, The Measurement of Consumers Expenditure and Behavior in the United Kingdom 1920-1938 (Cambridge, England, I954); and particularly J. Durbin, A Note on Regression When There is Extraneous Information About One of the Coefficients, Journal of the American Statistical Association, xLvm (December I953), 799-808; Herman Wold and Lars Jureen, Demand Analysis (New York, I953). This method has also been used by J. Tobin, A Statistical Demand Function for Food in the U.S.A., Journal of the Royal Statistical Society, Series A, cxiII (Part II I950), II3-4I. A comprehensive review article, William C. Hood, Empirical Studies of Demand, Canadian Journal of Economics and Political Science, xxi (August I955), 309-27, provides a worthwhile reference on the subject. The distinction between longand short-run estimates is to be found in the useful paper by Richard J. Foote, Price Elasticity of Demand for Nondurable Goods with Emphasis on Food, Agricultural Marketing Service Bulletin 96, USDA (Washington, D.C., I956). Although none of the people who have used the combined techniques has had short-run forecasting as his immediate goal, it seems useful to indicate how such prediction fits into the scheme of possible objectives.

Structural Sensitivity in Econometric Models.

Essentials of Model Structure Procedures for Model Understanding Some Properties of the MQEM Preliminary Complete Model Dynamics Multiplier and Parameter Perturbations in the MQEM Reduced Models and the MQEM Bibliography Index.