Knowledge that Transforms

[This article considers a two-equation simultaneous equation model in which one of the dependent variables is completely observed and the other is observed only to the extent of whether or not it is positive. A class of generalized least squares estimators are proposed and their asymptotic variance-covariance matrices are obtained. The estimators are based on the principle which is applicable whenever the structural parameters need to be determined from the estimates of the reduced form parameters.]

IF ONE OF THE EXPLANATORY VARIABLES in a linear regression model is measured with error, the ordinary least squares estimator is known to be biased and inconsistent. Given suitable assumptions, an instrumental variables estimator is known to be consistent. In a large sample, the instrumental variables estimator is thus unambiguously preferred, but the choice of an estimator in a small sample remains a puzzle. The method of maximum likelihood sheds light on this puzzle. It will be shown below that the instrumental variables estimate is the maximum likelihood estimate if, and only if, it lies between the ordinary least squares estimate and the reverse least squares estimate, that is, if and only if it satisfies the bounds implied by the simple errors in variables model. The letters Y, x, and z will indicate, respectively, the vector of observations of the dependent variable, the vector of error-ridden measurements of the explanatory variable and the vector of observations of an instrumental variable, each measured around its mean. The ordinary least squares estimate is then

[Our model of a multi-sector infinite horizon economy contains uncertainty about future input-output possibilities and labor supply. Future utilities are not discounted. Even under perfect certainty a competitive program which satisfies some transversality condition (e.g., a bounded price system) does not have to be optimal. We prove that if the technology satisfies some concavity condition, a necessary and sufficient condition for optimality of any competitive program is that the expected present value of the inputs (or outputs) be bounded over time.]

The widely used Cochrane-Orcutt and Hildreth-Lu procedures for estimating the parameters of a linear regression model with first-order autocorrelation typically ignore the first observation. An alternative maximum likelihood procedure which incorporates the first observation and the stationarity condition of the error process is proposed in this paper. It is similar to the Cochrane-Orcutt procedure, and appears to be at least as computationally efficient. This estimator is superior to the conventional ones on theoretical grounds, and sampling experiments suggest that it may yield substantially better estimates in some circumstances.

[The present paper analyzes cropping of a self-reproducible natural resource under competitive conditions. Situations which may lead to extermination of the resource are described. In particular, it is shown that a production function with increasing returns to scale does not exclude the possibility of extermination, as suggested in [1].]

This paper considers the formulation and estimation of simultaneous equation models with both discrete and continuous endogenous variables.The statistical model proposed here is sufficiently rich to encompass the álassjcai simultaneous equation model for continuous endogenous variables and more recent models for purely discrete endogenous variables as special cases of a more general model.Interest in discrete data has been ftsledby a rapid growth in the availability of microeconomic data sets coupled with a growing awareness of the importance of discrete choice models for the analysis of uiicroeconomic problems (see McFadden, 1976).To date, the only available statistical models for the analysis of discrete endogenous variables have been developed for the purely discrete case.The log-linear or logistic model of Goodman (1970) as expanded by Raberman (1974) and Nerlove and Press (1976) is one Vol.II, 1967; Lord and Novick, cbs.16-20, 1967.)It is argued in this paper that this class of statistical models provides a natural framework for generating simultaneous equation models with both discrete and continuous random variables.In contrast, the framework of Goodman, while convenient for formulating descriptive models for discrete data, offers a much less natural apparatus for analyzing econometric structural equation models.This is so primarily because the simultaneous equation model is inherently an unconditional representation of behavioral equations while the model of Goodman is designed to facilitate the analysis of conditional representations, and does not lend itself to the unconditional formulations required in simultaneous equation theory.The structure of this paper is in four parts.in part one general models are discussed.Dummy endogenous variables are introduced in two distinct roles: (1) as proxies for unobserved latent variables and (2) as direct shifters of behavioral equations.Five models incorporating such dummy variables are discussed.Part two, also the longest section, presents a complete analysis of the most novel and most general of the five models presented in part one.This is a model with both continuous and discrete endogenous variables.The issues of identification and estimation are discussed together by proving the existence of consistent estimators.Maximum likelihood estimators and alternative estimators are discussed.In part three, a brief discussion of a multivariate probit model with structural shift is presented.Part four presents a comparison between the models developed in this paper and the models of Goodman and Nerlove and Press.

considered as a graph and the algorithms used are related to graph theory.' The analysis of the qualitative structure defined when the signs of the non-zero elements of the matrix (2) are specified is of particular interest in comparative statics. Each column of the multiplier matrix ay/az' is obtained by solving a linear system of the form:

[The regression relation between regularly sampled Y(t) and X"19t),..., X"N(t) implied by an underlying model in which time enters more generally is studied. The underlying model includes continuous distributed lags, discrete models, and stochastic differential equations as special cases. The relation between parameters identified by regular samplings of Y and X"j and those of the underlying model is characterized. Sufficient conditions for identification of the underlying model in the limit as disaggregation over time proceeds are set forth. Empirical evidence presented suggests that important gains can be realized from temporal disaggregation in the range of conventional measurement frequencies for macroeconomic data.]