Sample selection bias as a specification error This paper discusses the bias that results from using non-randomly selected samples to estimate behavioral relationships as an ordinary specification error or «omitted variables» bias. A simple consistent two stage estimator is considered that enables analysts to utilize simple regression methods to estimate behavioral functions by least squares methods. The asymptotic distribution of the estimator is derived.

Sample Selection Bias as a Specification Error

In this paper a method of estimating the parameters of a set of regression equations is reported which involves application of Aitken's generalized least-squares [1] to the whole system of equations. Under conditions generally encountered in practice, it is found that the regression coefficient estimators so obtained are at least asymptotically more efficient than those obtained by an equation-by-equation application of least squares. This gain in efficiency can be quite large if “independent” variables in different equations are not highly correlated and if disturbance terms in different equations are highly correlated. Further, tests of the hypothesis that all regression equation coefficient vectors are equal, based on “micro” and “macro” data, are described. If this hypothesis is accepted, there will be no aggregation bias. Finally, the estimation procedure and the “micro-test” for aggregation bias are applied in the analysis of annual investment data, 1935–1954, for two firms.

An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias

Structural Equations with Latent Variables.

Methods for studying the stability over time of regression relationships are considered. Recursive residuals, defined to be uncorrelated with zero means and constant variance, are introduced and tests based on the cusum and cusum of squares of recursive residuals are developed. Further techniques based on moving regressions, in which the regression model is fitted from a segment of data which is moved along the series, and on regression models whose coefficients are polynomials in time are studied. The Quandt log-likelihood ratio statistic is considered. Emphasis is placed on the use of graphical methods. The techniques proposed have been embodied in a comprehensive computer program, TIMVAR. Use of the techniques is illustrated by applying them to three sets of data.

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1975.tb01532.x

Techniques for Testing the Constancy of Regression Relationships Over Time

Preface MATRICES: Basic Properties of Vectors and Matrices Kronecker Products, the Vec-Operator and the Moore- Penrose Inverse Miscellaneous Matrix Results DIFFERENTIALS: THE THEORY: Mathematical Preliminaries Differentials and Differentiability The Second Differential Static Optimization DIFFERENTIALS: THE PRACTICE: Some Important Differentials First- Order Differentials and Jacobian Matrices Second-Order Differentials and Hessian Matrices INEQUALITIES: Inequalities THE LINEAR MODEL: Statistical Preliminaries The Linear Regression Model Further Topics in the Linear Model APPLICATIONS TO MAXIMUM LIKELIHOOD ESTIMATION: Maximum Likelihood Estimation Simultaneous Equations Topics in Psychometrics Subject Index Bibliography.

Matrix Differential Calculus with Applications in Statistics and Econometrics

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.

Three-Stage Least Squares: Simultaneous Estimation of Simultaneous Equations

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.

On Pure and Mixed Statistical Estimation in Economics

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.”

Specification Errors and the Estimation of Economic Relationships

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.

The Analysis of Disturbances in Regression Analysis

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.

Henri Theil

Papers

Three-Stage Least Squares: Simultaneous Estimation of Simultaneous Equations

On Pure and Mixed Statistical Estimation in Economics

Specification Errors and the Estimation of Economic Relationships

The Analysis of Disturbances in Regression Analysis

Testing the Independence of Regression Disturbances