[This paper looks at some aspects of the three stage least squares approach to estimating simultaneous econometric models. Three stage least squares is derived along the lines of best linear unbiased estimators in classical regression, whereby it is indicated that the usual assumption of non-singularity of the disturbance covariance matrix is unnecessary. Consistency of the estimator is shown as is the irrelevance of exactly identified equations to the estimation of other equations in a model when three stage least squares is used. Also included are an easily computed test for the validity of all specified overidentifying restrictions, and a method of efficiently estimating the reduced form using only the information contained in structural equations that are thought to be well specified.]

In the first section of this paper the overidentifying restrictions on a system of linear simultaneous equations are expressed in terms of restrictions on the reduced form parameters. These restrictions provide the basis of a test of the structure using only the unrestricted reduced form parameter estimates. Under Ho the test proposed is asymptotically equivalent to a likelihood ratio test. The test may be applied as a single equation or complete system procedure and it may be presented as either a x2 or an F statistic. The case is also made here for system overidentification tests rather than single equation procedures, the arguments being drawn from the statistical literature on hypothesis testing by induction. The computational advantages of the present proposals are substantial when compared to FIML based likelihood -ratio tests and Monte Carlo experiments confirm that a system version of the test performs well in large samples. The system version of the test behaves like the FIML likelihood ratio test in large sample situations both under Ho and H1. However, the Monte Carlo studies indicate that both the single equation and system versions of the test perform poorly in small samples. THE AIM OF THIS paper is to investigate the possibility of deciding on the specification of a simultaneous equation model prior to the estimation of the structure. A well established test procedure is suggested which uses OLS estimates of the reduced form parameters; it enables the null hypothesis, that the model specified is not significantly different from the model which generated the sample, to be tested. Because of the one-to-one correspondence between the overidentified structure and the restricted reduced form, it is possible to make inferences about the structure from the observed compatibility of the reduced form restrictions with the sample information. In addition, the reduced form restrictions resulting from a particular equation may be isolated and tested separately, if desired. The principle underlying the test would appear to be due to Wald [17J; namely, that if the null hypothesis is correct and the structure postulated as the maintained hypothesis was responsible for the generation of the observed sample, then the unrestricted reduced form parameter estimates will tend, if the sample size is large enough, to satisfy the reduced form restrictions advanced under the maintained hypothesis. A number of problems relating to identification of linear simultaneous equations make life a little difficult and are discussed subsequently. Now, take the linear structure

[In this paper we study the properties of the ordinary least squares estimator ofthe coefficients of linear macor models when the coefficients of linear micro relations are random with identical mean and variance. It is shown that the ordinary least squares estimator is equal to a minimum variance linear unbiased estimator of the coefficients of a linear macro equation obtained by aggregating over all micro relations. Futher, some problems associated with estimating the covariance matrix of the ordinary least squares estimator, using only aggregate data, are indicated.]

[The core is the set of all unblocked allocations. Implicit in this definition is the idea that if an allocation is proposed which could be blocked, some coalition will form and issue a counterproposal which it can enforce. A process of successive counterproposals based on this idea is shown to converge in a finite period of time (amost surely) to the core.]

R. L. Basmann, D. H. Richardson, R. J. Rohr, An Experimental Study of Structural Estimators and Test Statistics Associated with Dynamical Econometric Models, Econometrica, Vol. 42, No. 4 (Jul., 1974), pp. 717-730

R. L. Basmann, D. H. Richardson, R. J. Rohr, Finite Sample Distributions Associated with Stochastic Difference Equations--Some Experimental Evidence, Econometrica, Vol. 42, No. 5 (Sep., 1974), pp. 825-839

Phoebus J. Dhrymes, R. Berner, D. Cummins, A Comparison of Some Limited Information Estimators for Dynamic Simultaneous Equations Models with Autocorrelated Errors, Econometrica, Vol. 42, No. 2 (Mar., 1974), pp. 311-332

Decentralized decision making is consistent if it is executed without cost (i.e., without a loss of output or utility). Consistency requires that the objective function be appropriately structured. In this paper, a hierarchical decision making structure is rationalized by an objective function which combines some of the properties of homothetic separability and asymmetric separability. THIS PAPER EXAMINES consistent decentralized decision making in a hierarchical structure. The decision making process is rationalized by a class of objective functions which combines some of the properties of homothetic separability [1, 3] and recursive, or asymmetric, separability [5,8]. We refer to these functions as homothetically recursive. After introducing our basic notation and definitions, we prove a representation theorem for homothetically recursive functions in Section 1. In Section 2 we prove two duality theorems for homothetically recursive structures. A recursively decentralized decision making process, described in Section 3, is made manifest in the structure of the associated cost function. The analysis is carried out in the context of the theory of the firm but other applications are discussed briefly in Section 4.