Factor analysis, path analysis, structural equation modeling, and related multivariate statistical methods are based on maximum likelihood or generalized least squares estimation developed for covariance structure models. Large-sample theory provides a chi-square goodness-of-fit test for comparing a model against a general alternative model based on correlated variables. This model comparison is insufficient for model evaluation: In large samples virtually any model tends to be rejected as inadequate, and in small samples various competing models, if evaluated, might be equally acceptable. A general null model based on modified independence among variables is proposed to provide an additional reference point for the statistical and scientific evaluation of covariance structure models. Use of the null model in the context of a procedure that sequentially evaluates the statistical necessity of various sets of parameters places statistical methods in covariance structure analysis into a more complete framework. The concepts of ideal models and pseudo chi-square tests are introduced, and their roles in hypothesis testing are developed. The importance of supplementing statistical evaluation with incremental fit indices associated with the comparison of hierarchical models is also emphasized. Normed and nonnormed fit indices are developed and illustrated.

Significance tests and goodness of fit in the analysis of covariance structures

Using the result that under the null hypothesis of no misspecification an asymptotically efficient estimator must have zero asymptotic covariance with its difference from a consistent but asymptotically inefficient estimator, specification tests are devised for a number of model specifications in econometrics. Local power is calculated for small departures from the null hypothesis. An instrumental variable test as well as tests for a time series cross section model and the simultaneous equation model are presented. An empirical model provides evidence that unobserved individual factors are present which are not orthogonal to the included right-hand-side variable in a common econometric specification of an individual wage equation.

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Specification Tests in Econometrics

Structural equation modeling (SEM) is a vast field and widely used by many applied researchers in the social and behavioral sciences. Over the years, many software packages for structural equation modeling have been developed, both free and commercial. However, perhaps the best state-of-the-art software packages in this field are still closed-source and/or commercial. The R package lavaan has been developed to provide applied researchers, teachers, and statisticians, a free, fully open-source, but commercial-quality package for latent variable modeling. This paper explains the aims behind the development of the package, gives an overview of its most important features, and provides some examples to illustrate how lavaan works in practice.

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lavaan: An R Package for Structural Equation Modeling

In Acemoglu, Johnson, and Robinson, henceforth AJR, (2001), we advanced the hypothesis that the mortality rates faced by Europeans in different parts of the world after 1500 affected their willingness to establish settlements and choice of colonization strategy. Places that were relatively healthy (for Europeans) were—when they fell under European control—more likely to receive better economic and political institutions. In contrast, places where European settlers were less likely to go were more likely to have “extractive” institutions imposed. We also posited that this early pattern of institutions has persisted over time and influences the extent and nature of institutions in the modern world. On this basis, we proposed using estimates of potential European settler mortality as an instrument for institutional variation in former European colonies today. Data on settlers themselves are unfortunately patchy—particularly because not many went to places they believed, with good reason, to be most unhealthy. We therefore followed the lead of Curtin (1989 and 1998) who compiled data on the death rates faced by European soldiers in various overseas postings. 1 Curtin’s data were based on pathbreaking data collection and statistical work initiated by the British military in the mid-nineteenth century. These data became part of the foundation of both contemporary thinking about public health (for soldiers and for civilians) and the life insurance industry (as actuaries and executives considered the

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The Colonial Origins of Comparative Development: An Empirical Investigation

Structural Equations with Latent Variables.

A method is given for estimating the coefficients of a single equation in a complete system of linear stochastic equations (see expression (2.1)), provided that a number of the coefficients of the selected equation are known to be zero. Under the assumption of the knowledge of all variables in the system and the assumption that the disturbances in the equations of the system are normally distributed, point estimates are derived from the regressions of the jointly dependent variables on the predetermined variables (Theorem 1). The vector of the estimates of the coefficients of the jointly dependent variables is the characteristic vector of a matrix involving the regression coefficients and the estimate of the covariance matrix of the residuals from the regression functions. The vector corresponding to the smallest characteristic root is taken. An efficient method of computing these estimates is given in section 7. The asymptotic theory of these estimates is given in a following paper [2]. When the predetermined variables can be considered as fixed, confidence regions for the coefficients can be obtained on the basis of small sample theory (Theorem 3). A statistical test for the hypothesis of over-identification of the single equation can be based on the characteristic root associated with the vector of point estimates (Theorem 2) or on the expression for the small sample confidence region (Theorem 4). This hypothesis is equivalent to the hypothesis that the coefficients assumed to be zero actually are zero. The asymptotic distribution of the criterion is shown in a following paper [2] to be that of $\chi^2$.

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Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations

In this paper we discuss some methods of factor analysis. The entire discussion is centered around one general probability model. We consider some mathematical problems of the model, such as whether certain kinds of observed data determine the model uniquely. We treat the statistical problems of estimation and tests of certain hypotheses. For these purposes the asymptotic distribution theory of some statistics is treated. The primary aim of this paper is to give a unified exposition of this part of factor analysis from the viewpoint of the mathematical statistician. The literature on factor analysis is scattered; moreover, the many papers and books have been written from many different points of view. By confining ourselves to one model and by emphasizing statistical inferences for this model we hope to present a clear picture to the statistician. The development given here is expected to point up features of model-building and statistical inference that occur in other areas where statistical theories are being developed. For example, nearly all of the problems met in factor analysis are met in latent structure analysis. There are also some new results given in this paper. The proofs of these are mainly given in a technical Part II of the paper. In confining.ourselves to the mathematical and statistical aspects of one model, we are leaving out of consideration many important and interesting topics. We shall not consider how useful this model may be nor in what substantive areas one may expect to find data (and problems) that fit the model. We also do not consider methods based on other models. In doing this, we do not mean to imply that the model considered here is the most useful or important. It seems that this model has some usefulness and importance, it has been studied considerably, and one can give a fairly unified exposition of it. Extensive discussion of the purposes and applications (as well as other developments) of factor analysis is given in books by psychologists (for example, Holzinger and Harmon [10], Thomson [23], Thurstone [24]). Some general discussion of statistical inference has been given in papers by Bartlett [9] and Kendall [12].

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Statistical Inference in Factor Analysis

This paper deals with a class of statistical quality control procedures and continuous inspection procedures which are optimum for a specified income function and a production model which can only be in one of four states, two of which are states of repair, with known transition probabilities. The Markov process, generated by the model and the class of decision procedures, approaches a limiting distribution and the integral equations from which the optimum procedures can be derived are given.

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A bayes approach to a quality control model

A Constant-Utility Index of the Cost of Living

In a previous paper [2] the authors have given a method for estimating the coefficients of a single equation in a complete system of linear stochastic equations. In the present paper the consistency of the estimates and the asymptotic distributions of the estimates and the test criteria are studied under conditions more general than those used in the derivation of these estimates and criteria. The point estimates, which can be obtained as maximum likelihood estimates under certain assumptions including that of normality of disturbances, are consistent even if the disturbances are not normally distributed and (a) some predetermined variables are neglected (Theorem 1) or (b) the single equation is in a non-linear system with certain properties (Theorem 2). Under certain general conditions (normality of the disturbances not being required) the estimates are asymptotically normally distributed (Theorems 3 and 4). The asymptotic covariance matrix is given for several cases. The criteria derived in [2] for testing the hypothesis of over-identification have, asymptotically, $\chi^2$-distributions (Theorem 5). The exact confidence regions developed in [2] for the case that all predetermined variables are exogenous (that is, that the difference equations are of zero order) are shown to be consistent and to hold asymptotically even when this assumption is not true (Theorem 6).

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Herman Rubin

Papers

Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations

Statistical Inference in Factor Analysis

A bayes approach to a quality control model

A Constant-Utility Index of the Cost of Living

The Asymptotic Properties of Estimates of the Parameters of a Single Equation in a Complete System of Stochastic Equations