Estimated Parameters as Dependent Variables

Judith Lave and Lester Lave, in their article in this Review, have proposed two imaginative, if simple, techniques for estimating cost functions in multiproduct, multiservice industries. Both techniques rest on the assumption that output mix varies among firms but is constant over time within any one firm. The first proposed technique involves two stages of analysis and allows a simple cost function to differ among firms. In the first stage a cost function is estimated for each firm. In the second stage a search is made for the factors causing variation among the estimated individual firm cost function parameters. The second proposed technique is based on the more limited assumption that, apart from the intercept term, all firms have the same cost function. Here the data are pooled and a single cost function estimated. Lave and Lave, in the course of analysis, conclude that their dichotomous first technique is too inflexible, given their particular problems and opt for the second method. Despite their discarding of the first technique, others have been tempted to try it out and I think some discussion of this proposed estimation procedure is in order.1 I think this discussion will also have some bearing on the Laves' second method. The Laves' search for the causes of variation among individual cost functions involves taking firm cost function parameters and regressing them on various sets of independent variables in an effort to find significant relationships. Research workers using such a procedure should be aware that it is unusual in that before these regressions are ever run, an estimate of the variance of each dependent variable is available. Such knowledge may well be unpleasant. If on testing the hypothesis of homogeneity of variance using likelihood ratio methods, the null hvpothesis is rejected, then the problem of heteroscedasticity in regressions using estimated parameters as dependent variables must be confronted. The now classical results indicate that applying ordinary least squares to an equation with a heteroscedastic error structure, while giving unbiased estimates of the coefficients of the independent variables, is nonetheless an inefficient method. Also, using the usual formula to calculate sampling variances may well involve an overstatement of the true sampling variances. Given the comparatively low I-statistics reported in the Laves' Tables 1, 2, and 3, the possibility exists that many of the variables reported as significant were really not at all significant. What can be done about this problem of heteroscedasticity? Since estimates of the variance of each observation on the dependent variable are available, a straightforward procedure would imply weighting each observation on all variables used in the secondstage equation by the inverse of the estimated standard error of the dependent variable. This weighting technique, of course, is nothing more than an application of generalized least squares, where an estimate of the variance-covariance matrix is used in place of the true variance-covariance matrix. One may well wonder whether the twostage procedure is necessary at all. After all in the Laves' case, the linear second-stage equation (4) can easily be substituted into the linear first-stage equation (3). Unfortunately, to the extent that the variances of the stochastic terms in the first-stage cost equations are hospital specific, the fundamental equation resulting from this substitution will still contain a heteroscedastic error structure. Interestingly enough, * The University of Michigan. I would like to thank Takeshi Amemiya for good advice and Judith Lave and Lester Lave for providing the data used in the analysis in this paper. I See for example, J. G. Williamson. With increased interest in the last ten years in models where it is assumed that the parameters have a stochastic component, it is not surprising that there also be increased interest in models where the parameters are subject to systematic variation.