Knowledge that Transforms

FOR VARIOUS REASONS, data availability being not the least of these, empirical studies of production processes can often be carried out more conveniently in terms of cost functions instead of production functions. Assuming cost minimizing behavior by entrepreneurs, cost function studies can, in principle, reveal the same information [23, 26]. Such dual cost functions have an empirical difficulty in common with production functions, however. They must be concave and linear homogeneous in input prices so that, when limitations of known estimation techniques are considered, the choice of appropriate algebraic structure is severely restricted. Even when only economies of scale information is desired, the preferred time-series studies [27, p. 39ff] are placed under this handicap since construction of a proper index to deflate price variation is equivalent to specification and parameter estimation of a production function [16]. For this reason, attention is directed to the sometimes criticized cross-section studies where input price variation may be negligible.

[The parameters of a system of demand equations are estimated subject to the prior information of classical demand theory. The equations are estimated as a system using a variant of generalized least squares, the parametric restrictions being imposed by Lagrange multipliers. Tests of significance are given, both for individual restrictions and for the restrictions applied collectively. The method is applied to Barten's sixteen commodity consumer expenditure data for Holland. The work was done independently of R. H. Court's [6] similar treatment; however, there are significant differences in the method which warrant further discussion and the application is itself of some interest.]

[The problem to be considered in this paper is that in a linear regression model, y = Xβ + ε (where X is n × k of rank r ≤ k), the disturbance vector ε′ = ( extlesstex-math extgreater$ extbackslashvarepsilon _\1\, extbackslashvarepsilon _\2\,..., extbackslashvarepsilon _ $ extless/tex-math extgreater) is distributed according to the null hypothesis, H0, as multivariate normal with mean vector 0 and variance-covariance matrix proportional to extlesstex-math extgreater$ extbackslashSigma _\0$ extless/tex-math extgreater, against the alternative hypothesis, H_1, that it is distributed as multivariate normal with mean vector 0 and variance-covariance matrix proportional to Σ _1. Three test statistics, extlesstex-math extgreaters_\1\,s_\2\ extless/tex-math extgreater, and s_3, all functions of estimated disturbances from the fitted regression are proposed to test the hypothesis H_0. It is shown (in Section 3) that all three tests based on extlesstex-math extgreaters_\1\,s_\2\ extless/tex-math extgreater, and s_3 are unbiased and that the test T(1) based on s_1 is most powerful. In Section 4, ε is assumed to have a special covariance structure, namely, a first order stationary Markov process, uniform covariance structure, and moving average of order one, and the general results obtained in Section 3 are simplified. It is also shown that the hypothesis H_0, in general, cannot be tested and that only an implication of it can be tested. Section 5 contains three numerical illustrations comparing the results proposed in this study with the Durbin-Watson procedure.]

1. IN THIS NOTE we consider the estimation of linear models when the coefficients of the structural form are not the same for all observations for which the model is postulated to be valid. An example of such a model is given in [3], where some structural relations have a piecewise linear form. Another example is the water melon market model of Suits [2] where there are two alternative harvest supply schedules. Also discussed here is the case where for one part of the sample period one or more variables are endogenously determined while for another part they are exogenous, for instance, the wage rate or the rate of exchange. Such a change in the nature of the model can also be interpreted as a change in the coefficients of the structural form. It is assumed throughout that it is known a priori for what observations each specification holds. 2. A shift in the value of the coefficients of a predetermined variable does not cause special problems. If, say, only one shift occurs, one defines two predetermined variables to replace the original one. The vector of observations for the first of these consists of the observations on the original variable with the exception of those observations for which the second value is supposed to hold. These latter observations are replaced by zero's. The vector of observations on the second variable is simply the difference between the vector of observations for the original variable and the one for the first variable. 3. Next, consider the case where there is a shift in the value of one or more structural coefficients associated with an endogenous variable. Let the model in structural form be (1) Yt = y'B + x'C + ur where yt is an M-element vector of endogenous or jointly dependent variables, xt an L-element vector of predetermined variables, while ut is the M-element vector of structural disturbances. The matrix B is the M x M matrix of coefficients associated with the endogenous variables and C is the L x M matrix of coefficients associated with the predetermined variables. It is assumed that one or more elements of B take for some observations a different value than for others. The superscripts a and b are used to distinguish between the two situations. The following partitioning of the sets of T observations in two subsets of T7 and Tb observations, respectively, are introduced:

[Average cost functions for the life insurance industry, all of which show increasing and then constant returns, are estimated from cross section data for 237 companies. The special problems of measuring output, controlling for product mix, and accounting for the effect of rate of growth in output are examined and dealt with. The article concludes that average costs are constant beyond $100 million of premiums.]

In this paper four alternative quarterly econometric models of investment behavior are compared with regard to predictive performance. Predictive performance may be assessed in two ways: (i) We compare prediction errors for a period of prediction with errors for a period of fit. (ii) We fit investment functions for both periods and test for structural change. These two procedures may be viewed as alternative tests of the hypothesis of structural change; the second is more powerful from the statistical point of view. Tests of predictive performance supplement the comparisons of alternative models given in our preceding paper [17]. Goodness of fit may be exaggerated by consideration of a wide range of alternatives and selection of the one that fits best. If goodness of fit is exaggerated, a predictive test should produce evidence of structural change between the period of fit and the period of prediction. Of course, the better an econometric model fits the data, the more stringent this criterion for predictive performance. The econometric models included in our study are those of Anderson [1], Eisner [7], Jorgenson and Stephenson [19], and Meyer and Glauber [21]. On the basis of predictive performance the ranking of the alternative models is as follows: (1) Eisner, (2) JorgensonStephenson, (3) Meyer-Glauber, and (4) Anderson. This ranking is similar to that resulting from comparisons based on goodness of fit presented in our preceding paper [17]. For econometric models of quarterly investment behavior, the models that fit the best also have the best predictive performance.

[The object of this paper is to provide a systematic treatment of aggregation of individual welfare as a basis for social preference. Two polar cases of interpersonal comparability seem to have received all the attention in the literature so far. Either it is assumed that individual welfare measures are fully comparable, e.g., in Marshall [12], or that they are not comparable at all, e.g., in Robbins [15]. It is clear, however, that we frequently make judgments that are not consistent with noncomparability but which do not require full comparability. Part of the object of this paper is to examine the formal basis of such judgements and to develop a continuum of intermediate assumptions.]