Knowledge that Transforms

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.]

[The Ramsey model of optimal capital accumulation is reconsidered under the additional restriction that gross investment must be nonnegative. An effective characterization of the optimal solution in open-loop form is obtained. It is shown, however, that in general no restriction can be placed on the number of intervals in which the non negativity constraint is binding.]

In this paper decision systems are structured so that topological concepts can be applied to formulate and help solve existence problems. Existence of stationary states and orbits is established. The analysis is then applied to programs, a special class of decision systems in which the decision operator is a mathematical program. The theory is extended to of many decision makers with rolling schedules of future actions. RECURSIVE DECISION SYSTEMS (RDS's) are dynamic systems based on discrete time that represent the positive behavior of decision makers. They have been put to three basic uses, (1) to describe the behavior of various economic sectors, (2) to show how indirect policies can in some particular way improve the performance of the economic system under investigation, and (3) to formulate and analyze a variety of dynamic economic theories. In this paper we define RDS's so that topological concepts and theorems can be used to study existence questions. Existence theorems are then given for stationary states and compact orbit sets. Special attention is given to the class of RDS's called programs (RP) of which various recursive program- ming models are special cases. The paper concludes with some brief comments on the assumptions used in the analysis. Before proceeding to the formal definitions, we briefly review in a nontechnical manner the basic concepts underlying RDS's and their use in economic research. RDS's as defined here are mathematical of socioeconomic processes having two basic components: (1) a decision operator that describes the manner in which final decisions or actions are derived from a given amount of information about the decision maker's environment ;2 and (2) afeedback operator that describes how decisions once acted on, or once scheduled for the future, interact with the decision maker's environment to produce new information upon which succeeding plans can be based.3 A given decision operator may represent the decision process not only of a single decision maker, but also of a group of decision makers who make their decisions independently-or collusively-during the same time period. Further- more, the decision at a given time may represent not only an immediate choice,