[This paper deals with the question of appropriately specifying the error structure in equations nonlinear in the parameters. An approach is presented which nests various disparate hypotheses (including those of additive normal and multiplicative lognormal error distributions) and suggests an approximate testing procedure. An example is given in which the method is applied in the context of estimating an aggregate production function for the U.S..R.]

A nonparametric framework for deriving the asymptotic MSE-optimal predictor for a multiplicative model is presented. The resulting predictor is compared to several known competitors in a limited Monte Carlo experiment. RECENT PAPERS BY Zellner [10] and Teekens and Koerts [7] address themselves to the problem of minimum-MSE prediction in a Cobb-Douglas-type multiplicative model under a lognormal distribution assumption for the disturbance term. Each derives the finite sample predictor (which turns out to be a function of the familiar least-squares predictor) for the model based on the assumption that ?2, the variance of the lognormally distributed disturbance, is known. An approximately optimal finite sample predictor is then suggested, where an estimate of w2 is utilized. Under certain conditions the approximately optimal predictor poses a computational burden. Under others, the predictor is easily computed, but no longer are small sample properties guaranteed. Our purpose in this note is to present a general framework for deriving the asymptotically optimal-MSE predictor for this multiplicative model without the imposition of a distributional assumption at the outset. Not only does this exercise provide us with a convenient vehicle for discussing further the aforementioned contributions, it also yields a viable distribution-free predictor that may

where Yt and Xt are subject to random errors of measurement ut and vt, respectively, with E(u) = E(vt) = 0, E(u2) = u, E(v2) = 2, E(utvt) = , and E(utu')E(vtsv) = 0 for t # t', is applied to a simple system of difference equations. The paper treats two such models, which differ in that one system is specified as deterministic except for measurement error, while the second model includes random in the equations as well. The investigation focuses on the possible consistency of least squares estimates of structural parameters in both cases. Mann and Wald [3] have demonstrated the consistency property of least squares estimates in stochastic difference equations which contain a shock term, and T. W. Anderson [1] has more recently shown such estimates to possess asymptotic normal distributions. One way argue, and perhaps quite legitimately, against the inclusion of measurement errors and shocks as separate entities in systems such as those under consideration. This separation, however, does provide a useful contrast with regard to the consistency property of LS estimates as compared to the case when only shocks (which subsume measurement error) are present in the specification of the system. When measurement errors are separated from shocks, LS yields consistent estimates for the explosive case of cobweb equilibrium and inconsistent estimates under convergence. The above phenomenon rests on the perhaps more interesting results for a recursive model where only measurement errors are present. This change in how the random terms enter the system, as contrasted to the Mann and Wald or Anderson formulations, causes zero correlation between observed variables and inconsistent LS estimates in the equations under convergence. Again, under explosion, LS provides consistent estimators of structural parameters.

Social experiments are characterized by their high cost. A tempting alternative to the establishment of a contemporaneous statistical control group is preexempt observat ion of the treatment group. In this paper, the authors analyze the tr ade-off between these two types of "con-trol" as functions of their relative cost and information content in a multiperiod error compone nts framework, where the allocation of observations across the two gr oups is always done in an optimal manner. Solutions for the optimal p roportion of the sample to be devoted to the contemporaneous control group are presented and their behavior as a function of relevant para meters is studied.

Results of the Koopmans-Beckmann (K-B) analysis of the quadratic assignment problem [3] have perplexed many location theorists. K-B hold that indivisibilities of plant, in the presence of minimal interaction between spatially separated plants (namely, the shipment of intermediate goods at positive transportation rates), preclude the existence of a system of rents which will sustain an integral assignment, optimal or otherwise [3, p. 69]. Earlier in their paper K-B show that a sustaining price system does exist when the transportation of intermediate products is excluded; however, it is the presence of such interaction that leads to the more interesting quadratic assignment problem and the pessimistic conclusion reported above. The authors first present the quadratic assignment problem in a permutation search format, and then construct an equivalent linear programming problem which allows fractional assignments to be optimal without forfeiting any integral (one whole plant to each location) solutions that might exist:

[A limiting feature of several theoretically superior "shrinkage" estimators for the linear regression model lies in the fact that there must be a certain degree of orthogonality in regressors in order for them to dominate the ordinary least squares estimator. In this paper we apply variants of pre-test and Stein estimators to data on international trade, and discuss their merits in light of the limitations imposed by the non-orthogonality of these and other sets of economic data.]

The goal of this paper is to provide a comprehensive empirical analysis of majority rule and Tiebout sorting within a system of local jurisdictions. The idea behind the estimation procedure is to investigate whether observed levels of public expenditures satisfy necessary conditions implied by majority rule in a general equilibrium model of residential choice. The estimator controls for observed and unobserved heterogeneity among households, observed and unobserved characteristics of communities, and the potential endogeneity of prices and expenditures, as well as the self-selection of households into communities of their choice. We estimate the structural parameters of the model using data from the Boston Metropolitan Area. The empirical findings reject myopic voting models. More sophisticated voting models based on utility-taking provide a potential explanation of the main empirical regularities.

IN A RECENT PAPER D. R. Capozza and R. Van Order (C.V.) [1] claim to show that there exist conditions under which a competitive firm in industry equilibrium in a spatial environment will charge a higher mill price than a spatial monopolist. This conclusion seems contrary to our intuition and consequently the conditions under which it arises should be made very clear. This paper argues that the C.V. result is not necessarily correct. In particular, there is an apparent error in the three sentences following equation (27). The positive roots of equation (27) are both possible candidates for equilibrium price.2 The difficulty is that there are no grounds for choosing between the two roots within the structure of the model. Capozza has argued, in correspondence, that the smaller root is inappropriate because demand is negative. However, demand is only negative at the border of the market area and one could equally well argue that this is a case of natural monopoly, induced by the cost structure. These remarks are made within the assumptions that C.V. makq in their paper. We can, however, shed further light on the matter by introducing some new elements. In particular, (i) we explicitly impose some simple dynamics on the system, and (ii) we use consumer surplus as a measure cf welfare. (i) Let us suppose that firms are price setters and assume a zero conjectural variation. Under these conditions it is not difficult to show that the larger positive root is stable and the smaller is unstable. (ii) The total of consumer surplus at price m and market radius Do is given by

[Research on the spatial firm has shown that spatial results often differ from non-spatial. This paper considers the relationship between the price of a spatial monopoly and the price of a spatially competitive firm. The conditions under which the competitive price will exceed the monopoly price are outlined.]