equation, yet estimates of the income elasticity of housing demand derived from micro data are systematically lower than estimates obtained from grouped (metropolitan-wide) data. The explanation proposed was based on the effects of common specification errors related to the housing price term. It was demonstrated theoretically that, given certain assumptions, the omission of the price term biases the ungrouped income elasticity estimate downward (whether stratified by metropolitan area or not) and the grouped estimate upward. It was also shown that the inclusion of a metropolitan-wide housing price index in the micro equation worsens the downward bias of the income elasticity estimate (and biases the corresponding price elasticity estimate toward zero). At the time the paper was written, the only empirical evidence directly relevant to this price misspecification hypothesis was contained in a study by Maisel, Burnham, and Austin [1]. It was shown that their income elasticity estimates were entirely consistent with the proposed explanation. Although their price elasticity results were not consistent, it was suggested that this could easily be attributed to sampling error. The purpose of this note is to discuss the relevance of some new empirical estimates of housing demand for the price misspecification hypothesis. These results are part of a broader study of housing demand undertaken by the author with D. T. Ellwood [3]. Since that study discusses the data and the specification of the equations in detail, the focus here is on those aspects of the data and results which are especially relevant to the price misspecification explanation.

[In large and complicated management systems, solutions with the same indices, but with different degrees of aggregation, are used and coordinated. For example, in hierarchical systems, those in higher management levels make decisions with more aggregated indices than those in lower management levels. Managers of an individual subsystem within a large and complicated system use detailed information about their own subsystem and aggregated information (in some degree or other) about other subsystems. The principal idea of the iterative aggregation method is to consecutively recompute the aggregated indices characterizing the activities of the whole system, followed by a recomputation of the detailed indices characterizing each of its subsystems. From a theoretical point of view these methods are generalizations of some classes of iterative and decomposition methods.]

A simple test for heteroscedastic disturbances in a linear regression model is developed using the framework of the Lagrangian multiplier test. For a wide range of heteroscedastic and random coefficient specifications, the criterion is given as a readily computed function of the OLS residuals. Some finite sample evidence is presented to supplement the general asymptotic properties of Lagrangian multiplier tests.

[Conventionally labor supply modeling has been dichotomized from consumption expenditure allocation. We estimate a model unifying both aspects of the consumer's decision problem, and we test for the two-stage decision implied by the conventional dichotomy. We investigate the gains from joint modeling. We use a version of the Rotterdam model recently shown by Barnett [6] to be derivable at the aggregate level under weaker assumptions than those needed to acquire empirically usable theoretical results at the aggregate level with other models; our results are not subject to the restrictiveness imputed to earlier uses of versions of the Rotterdam model.]

THIS PAPER IS a preliminary investigation of dynamics under uncertainty. We attempt to develop a general approach to the continuous time stochastic processes that arise in dynamic economics from the maximizing behavior of agents. The analysis builds on recent results of Bismut [2, 3] concerning the characterization of the extrema of stochastic variational problems over a finite horizon and on our own investigations [6, 7, 20, 21] of the stability properties of the equations of dynamic economics.2 We consider a class of discounted infinite horizon maximum problems. While it is convenient to pose the basic economic problem as a stochastic control problem, to obtain the full benefit of Bismut's elegant characterization of a maximizing process it is convenient to transform this problem into an equivalent stochastic variational problem along the lines indicated by Rockafellar [27] in the deterministic case and generalized by Bismut [2] to the stochastic case. Within this framework we show that the idea of a competitive path introduced in the continuous time deterministic case in [21] generalizes in a natural way in the case of uncertainty to a competitive process. We show, under a concavity assumption on the basic integrand of the problem, that a competitive process which satisfies a transversality condition is optimal under a discounted catching up criterion (Section 2). In Section 3 we examine the sample path properties of a competitive process. If for almost every realization of a competitive process the associated dual price process generates a path of subgradients for the value function, we call the process McKenzie competitive, since it was McKenzie [22] who first recognized the importance of this property in the deterministic case. We show that two McKenzie competitive processes starting from distinct nonrandom initial conditions converge almost surely if the processes are bounded almost surely and if a certain curvature condition is satisfied by the Hamiltonian of the system. The earlier convergence result extensively studied in the deterministic case thus continues to hold in the stochastic case. The problem of finding sufficient conditions for the existence of a McKenzie competitive process remains an open problem. Section 4 examines the long-run behavior of the probability measure associated with a competitive process. We give conditions under which a McKenzie competitive process is a Markov process with an invariant probability measure and

[This paper is concerned with the sources of variation in the earnings of American scientists over the decade 1960-70. It focuses on sources of variation over time as well as at a point in time. Earnings variation over time is decomposed into sources due to measurable variables representing an earnings function, a random effect individual variance component in the level of earnings, a random effect individual component in earnings growth, and a serially correlated transitory component. Maximum likelihood estimates of the implied parameters are presented and a method for obtaining GLS estimates of the earnings regression function, when the residual variance components are given, is explored.]