Knowledge that Transforms

[This paper is concerned with the so-called k-class estimators of structural parameters in a simultaneous system. The structural equation being estimated is assumed, as is common in other small-sample investigations, to consist of two endogenous variables; and the number of the exogenous variables (included or excluded) as well as the number of equations in the system are arbitrary so long as the identifiability condition of the estimated equation is satisfied. Moreover, we assume that the system contains no lagged endogenous variables and disturbance terms of each period are independently distributed as multivariate normal. The exact finite-sample moments of the k-class estimators are evaluated for 0 @ extless k extless 1. For k extgreater 1 it is proved that the estimator does not possess even the first-order moment. The exact moment functions are expanded in terms of the inverse of the noncentrality (or concentration) parameter. This expansion sheds more light on the comparative study of alternative k-class estimators. Numerical calculations of the mean square error and the bias for some specific cases are also given for illustrative purposes.]

Gramlich and Galper (hereafter referred to as GG) indeed have performed a useful service by pointing out that an erroneous equation was included in my paper [3]. The corrected inventory change equation, the equation with all lagged orders included, was almost identical with the results obtained by applying GG's method (ii) to the order-inventory stock equation presented in Column (2), Table IV of my paper [3, p. 161]. The results are consistent when GG's methods (ii) and (iii) are applied on the correct equations. In any case, GG's methods (ii) and (iii) are well known and deserve no debate here. However, the same cannot be said of their method (i). GG's method (i) is based on two crucial assumptions concerning the pattern of the production process and inventories of defense products. The validity of using the method to derive the coefficients of an inventory change equation from an order-shipment relationship depends directly on these assumptions. If the assumptions describe exactly the actual patterns of the production process and of inventory build-up, then method (i) can be used for GG's purpose. Unfortunately, both of these assumptions are very unrealistic, and the use of method (i) yields misleading results. In their derivation of method (i), GG assume that the production process is rectangular. Furthermore, the authors claim that the method is valid regardless of what is assumed about the shape of the production process-a rectangular assumption serves as well as any.' This claim is faulty, as the following example demonstrates.

[Consider a sequence of markets for goods and securities at successive dates, with no market at any date complete in the Arrow-Debreu sense. A concept of common expectations is proposed that requires traders to associate the same future prices to the same future exogenous events, but does not require them to agree on the (subjective) probabilities associated with those events. An equilibrium is a set of prices at the first date, a set of common price expectations for the future, and a consistent set of individual plans for consumers and producers such that, given the current prices and price expectations, each individual agent's plan is optimal for him, subject to an appropriate sequence of budget constraints. The existence of such an equilibrium is demonstrated under assumptions about technology and consumer preferences similar to those used in the typical Arrow-Debreu theory of complete markets. However, an equilibrium can fail to exist if some provision is not made for the elimination of "unprofitable" enterprises. The usual assumptions of "rationality" imply, in this model, that agents learn from experience and modify their expectations as Bayesians.]

On the basic assumption that individual consumption of a good is a stochastic phenomenon, the first part of this article shows that under general conditions market quantity demanded is asymptotically (as n, the number of individuals in the market, increases) distributed as normal with and variance a function of own price given all other prices and individual incomes. Next, by the use of integral transforms, it is shown that the unknown market demand function can be approximated by a specific functional form. The estimation problems involved with such a model are discussed in the last part of the paper. Two BASIC, but fundamental, problems facing any econometrician attempting to estimate market demand curves are the choice of functional form and the justification of the normal form for the distribution of the disturbance terms. This paper goes some little way toward meeting both problems. The first step is to regard quantity demanded as a random variable. It is assumed that the axioms of choice of modern demand theory refer to the mean quantities demand curves one can derive the normal distribution as a limiting form for it is assumed that the consumer in determining his preferences determines the parameters of the distribution function of quantity demanded. Market stochastic demand curves are obtained from individual stochastic demand curves by taking the sum of the quantities demanded over all individuals in the market. It is shown that under certain weak assumptions about the characteristics of individual demand curves one can derive the normal distribution as a limiting form for stochastic market demand curves. The limits are taken as n, the number of individuals in the market, approaches infinity. It is shown that under the assumptions of the problem both the and variance of the market stochastic demand function are decreasing functions of own price. The second step in the argument is to obtain approximations for the functional relationships between the and own price and between the variance and own price for the market curve. This is achieved by stating those conditions under which upper and lower bound functions can be defined. The approximations to the actual, but unknown, functions are obtained by considering the convex combination of both bound functions. The last section of the paper discusses the problems involved in estimating the parameters of the limiting form of the market stochastic demand function.