The structure of simultaneous equation estimators - Hallett, A (1976)
The thesis is concerned with developing a coherent theory of estimation suitable for the standard simultaneous equation econometric model. Particular attention is paid to the case of large models and many of the estimators considered may be used when the sample size is inadequate, i.e. there are more predetermined variables than observations in the model. However, the analysis is not restricted to this case. A great many new estimators have been proposed in recent years but little is known of their properties or performance. I consider iterative estimators (under a classical model specification) from the repeated least squares and instrumental variable groups, as well as conventional and maximum likelihood estimators. Included at the same time are four separate methods of obtaining maximum likelihood estimates. The questions to be answered for each estimator are: Have they the properties of consistency and efficiency? Are their statistical properties affected in any way by the numerical algorithm used to obtain them? Under what parameters of the model and data does each estimator exist? What is their suitability or ease of use from a computational view point? Which estimator entails the smallest numerical burden? What are the relations between them and to conventional maximum likelihood estimators? Is a family structure of estimators available? Finally a comparison is made to provide, in given circumstances, a preference ordering over the available estimators. This is done from the answers to the above questions, after a discussion using known results to show the impossibility of making any general comparisons on small sample properties. I can view each estimator as an iterative process with a pre- specified or infinite number of iterates and a particular starting point. I therefore begin by reviewing the consistency efficiency, optimality, availability and starts of the estimators. They are then classified according to iterative algorithms; and a new algorithm is suggested which is computationally superior. There is a substantial analysis of the iterative solution of equation systems. New convergence theorms, theorems on optimal acceleration para- meters, orderings, etc. and comparison theorems are developed here. Both linear and nonlinear iterations are considered, but for large systems linear iterations appear adequate and are generally superior computationally. Thus the analysis of the thesis extends to nonlinear estimation techniques and iterative estimators involving nonlinear iteration. Next relations with maximum likelihood estimators are developed and a complete classification and ranking by asymptotic efficiency is provided. The iterative schemes involved in the estimators may then be analysed for convergence conditions and optimal convergence conditions for their existence and availability are established and it is possible to make recommendations on the best way of using any iterative estimator, and, with some numerical analysis, to provide a preference ordering among them. Maximum likelihood estimators are also included in the computational analysis. The general conclusion is that iterative estimators, appropriately used, are always competitive with conventional methods and except for very small models are superior from the computational point of view. A number of numerical examples are presented illustrating the effectiveness of these results. A sample of nine well known models of various sizes has been used for this purpose. The main contributions are intended to be: the classification and statistical properties of the estimators considered; several new theorems on the convergence, comparison and optimal use of various iterative solutions to systems of equations; the first computational analysis of alternative estimators; a coherent estimation theory for the simultaneous equation model; a preference ordering over the estimators. The theory is important in itself and because most of the large, well known models have so far been estimated inconsistently and inefficiently for lack of such a theory. They remain however the typical models in econometrics. Finally the whole structure and theory of the estimators considered is summarised in an estimator generating function.