Henri Theil

Three-Stage Least Squares: Simultaneous Estimation of Simultaneous Equations

In simple though approximate terms, the two-stage least squares method of estimating a structural equation consists of two steps, the first of which serves to estimate the moment matrix of the reduced-form disturbances and the second to estimate the coefficients of one single structural equation after its jointly dependent variables are “purified” by means of the moment matrix just mentioned. The three-stage least squares method, which is developed in this paper, goes one step further by using the two-stage least squares estimated moment matrix of the structural disturbances to estimate all coefficients of the entire system simultaneously. The method has full-information characteristics to the extent that, if the moment matrix of the structural disturbances is not diagonal (that is, if the structural disturbances have nonzero “contemporaneous” covariances), the estimation of the coefficients of any identifiable equation gains in efficiency as soon as there are other equations that are over-identified. Further, the method can take account of restrictions on parameters in different structural equations. And it is very simple computationally, apart from the inversion of one big matrix.

On Pure and Mixed Statistical Estimation in Economics

A basic problem in the statistical estimation of economic relations is the discrepancy between the statistician’s claim when he formulates his maintained hypothesis and the procedure which he actually follows. He claims that he includes all his relevant a priori information in the maintained hypothesis, and it is on the maintenance of this hypothesis that his probability statements are based. Yet it is not uncommon to find statisticians discarding their maintained hypotheses after the estimation has been undertaken.

Specification Errors and the Estimation of Economic Relationships

The usual first step in the estimation and testing procedures of economic relationships is the formulation of a “maintained hypothesis”. As is well-known1, this amounts to a specification of the general framework within which the estimation or testing is carried out. Most of the results in this field have been derived under the assumption that the “maintained hypothesis” is correct. This is indeed in accordance with its name: the hypothesis is not subject to test, but “maintained.”

The Analysis of Disturbances in Regression Analysis

Classical regression analysis is concerned with the estimation of the parameter vector\(beta \)of the equation\(y = X\beta + u\)(1.1)where y is the T-element column vector of values taken by the dependent variable, X the \(TxA\) matrix of values taken by the A independent variables, \(beta \) a column of A parameters, and u a column of T disturbance.

Testing the Independence of Regression Disturbances

Abstract This article deals with the distribution of the Von Neumann ratio of least-squares estimated regression disturbances. This distribution is approximated by a beta distribution under the condition that the behaviour of explanatory variables of the regression over time is sufficiently smooth. Two examples are presented, together with a table containing 1 and 5 per cent significance limits for a number of observations ranging from 15 to 100 and a number of coefficients adjusted ranging from 2 to 6.

The Final Form of Econometric Equation Systems

An important problem in econometrics is the pattern of dependence of endogenous variables on exogenous variables in a series of consecutive periods. It has recently been stressed again by the work on macroeconomic decision problems done by Van den Bogaard, Theil and others (1959, 1961); and it is related to the classical Keynesian multiplier concept, particularly when this is formulated in a dynamic manner as has recently been done by Goldberger (1959).

RECENT EXPERIENCES WITH THE MUNICH BUSINESS TEST An Expository Article

Since many of the difficulties connected with the analysis of entrepreneurial behavior must be ascribed to a lack of empirical data, it is a fortunate thing that in recent years large, regular, and detailed surveys in the field of industry and trade have been established. One of these, that of the IFO-Institut fur Wirtschaftsforschung, Munich, is the subject of the present paper. An important feature of this survey, which started in the beginning of 1950, is that it is concerned both with actual and anticipated phenomena. These and other aspects are described in section 2; section 3 deals with the relationship of the results of this survey to conventional statistical data; and the remaining sections are devoted to the interrelationships of the survey results themselves.

The Aggregation Implications of Identifiable Structural Macrorelations

This article is concerned with the interpretation of econometric equation systems as systems of macrorelations. If an aggregate consumption function, say, is part of such a system and if it is also regarded as the macroeconomic analogue of a set of consumption functions of individual households, what is then the relationship between the macroeconomic consumption function and the underlying microeconomic consumption functions? The analysis, which is partly statistical, is carried out for linear equation systems.

A Reconsideration of the Keynes-Tinbergen Discussion on Econometric Techniques

It is now about twenty-five years ago when Professor Tinbergen started his work in eneva, under the auspices of the League of Nations, to verify empirically the validity f a number of competing business cycle theories. The League had not been a success in olitical affairs, but there were other areas in which it took happier and more successful nitiatives. This project was, I think, one of them. It was a double project and Professor aberler was in charge of the other half. His task was to survey and to analyze existing heories as such, that is, at the theoretical rather than the empirical level and his results ere laid down in the now-famous book Prosperity and Depression. Tinbergen’s results ere published under the title Statistical Testing of Business Cycle Theories in two olumes, the first of which was called A Method and Its Application to Investment Activity and the second Business Cycles in the United States, 1919–1932.

Some Developments of Economic Thought in the Netherlands

Contrary to several other small European countries, such as Sweden and Austria, the Netherlands cannot claim to have a great tradition in economic theory which goes back during many decades. Originally, economics was a minor field that could be studied in the departments of law of the universities; the first change in this respect took place in 1913 when the Netherlands School of Economics was established at Rotterdam. In the next decade this move was followed in two other places, leading to the Department of Economics of the (Municipal) University of Amsterdam and the Catholic School of Economics at Tilburg. After the second World War two other universities established economics departments, the State University of Groningen and the Free (Calvinist) University of Amsterdam.