Introduction. Measurement error and latent variables. About this book. Bibliographical notes. Regression and Measurement Error. The model. Asymptotic properties of the OLS estimators. Attenuation. Errors in a single regressor. Various additional results. Bibliographical notes. Bounds on the Parameters. Reverse regression. Reverse regression and the analysis of discrimination. Bounds with multiple regression. Bounds on the measurement error. Uncorrelated measurement error. Bibliographical notes. Identification. Structural versus functional models. Maximum likelihood estimation in the structural model. Maximum likelihood estimation in the functional model. General identification theory. Identification of the measurement error model under normality. A general identification condition in the structural model. Bibliographical notes. Consistent Adjusted Least Squares. The CALS estimator. Measurement error variance known. Weighted regression. Orthogonal regression. Bibliographical notes. Instrumental Variables. Assumptions and estimation. Application to the measurement error model. Heteroskedasticity. Combining data from various sources. Limited information maximum likelihood. LIML and weak instruments. Grouping. Instrumental variables and nonnormality. Measurement error in panel data. Bibliographical notes. Factor Analysis and Related Methods. Towards factor analysis. Estimation in the one-factor FA model. Multiple factor analysis. An example of factor analysis. Principal relations and principal factors. A taxonomy of eigenvalue-based methods. Bibliographical notes. Structural Equation Models. Confirmatory factor analysis. Multiple causes and the MIMIC model. The LISREL model. Other important general parameterizations. Scaling of the variables. Extensions of the model. Equivalent models. Bibliographical notes. Generalized Method of Moments. The method of moments. Definition and notation. Basic properties of GMM estimators. Estimation of the covariance matrix of the sample moments. Covariance structures. Asymptotic efficiency and additional information. Conditional moments. Simulated GMM. The efficiency of GMM and ML. Bibliographical notes. Model evaluation. Specification tests. Comparison of the three tests. Test of overidentifying restrictions. Robustness. Model fit and model selection. Bibliographical notes. Nonlinear Latent Variable Models. A simple nonlinear model. Polynomial models. Models for qualitative and limited-dependent variables. The LISCOMP model. General parametric nonlinear regression. Bibliographical notes. Appendix A. Matrices, Statistics, and Calculus. Some results from matrix algebra. Some specific results. Definite matrices. 0-1 matrices. On the normal distribution. Slutsky's theorem. The implicit function theorem. Bibliographical notes. Appendix B. The Chi-Square Distribution. Mean and variance. The distribution of quadratic forms in general. The idempotent case. Robustness characterizations. Bibliographical notes. References. Author Index. Subject Index.
The book first discusses in depth various aspects of the well-known inconsistency that arises when explanatory variables in a linear regression model are measured with error. Despite this inconsistency, the region where the true regression coeffecients lies can sometimes be characterized in a useful way, especially when bounds are known on the measurement error variance but also when such information is absent. Wage discrimination with imperfect productivity measurement is discussed as an important special case.
Next, it is shown that the inconsistency is not accidental but fundamental. Due to an identification problem, no consistent estimators may exist at all. Additional information is desirable. This information can be of various types. One type is exact prior knowledge about functions of the parameters. This leads to the CALS estimator. Another major type is in the form of instrumental variables. Many aspects of this are discussed, including heteroskedasticity, combination of data from different sources, construction of instruments from the available data, and the LIML estimator, which is especially relevant when the instruments are weak.
The scope is then widened to an embedding of the regression equation with measurement error in a multiple equations setting, leading to the exploratory factor analysis (EFA) model. This marks the step from measurement error to latent variables. Estimation of the EFA model leads to an eigenvalue problem. A variety of models is reviewed that involve eignevalue problems as their common characteristic.
EFA is extended to confirmatory factor analysis (CFA) by including restrictions on the parameters of the factor analysis model, and next by relating the factors to background variables.
These models are all structural equation models (SEMs), a very general and important class of models, with the LISREL model as its best-known representation, encompassing almost all linear equation systems with latent variables.
Estimation of SEMs can be viewed as an application of the generalized method of moments (GMM). GMM in general and for SEM in particular is discussed at great length, including the generality of GMM, optimal weighting, conditional moments, continuous updating, simulation estimation, the link with the method of maximum likelihood, and in particular testing and model evaluation for GMM.
The discussion concludes with nonlinear models. The emphasis is on polynomial models and models that are nonlinear due to a filter on the dependent variables, like discrete choice models or models with ordered categorical variables.
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- © North Holland 2000
- 8th December 2000
- North Holland
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"...Graduate - or advanced undergraduate - level textbook deals with measurement error and latent variables in econometrics." --Journal of Economic Literature
"...for all researchers who are concerned with measurement error and latent variables the book is highly recommended." --Zentralblatt Fur Mathematik
"Represents an important addition to the literature and will be a useful reference for both graduate students and researchers working in this area. The authors should also be complimented on the style of presentation. Finally, most results are derived formally which, although adding to the level of technical sophistication required by the reader, is to be applauded." --Journal of Applied Econometrics
"...This book will be very useful for applied researchers who are interested in data contaminations. The treatment of measurement errors is very comprehensive and rigorous. Most of the mathematical derivations are given in detail. This feature will make the book particularly appealing to graduate students studying econometrics. It will be a welcome addition to any library collection." --Mathematical Reviews
"...I strongly encourage using this book as a basis for an advanced course on latent variable models and/or structural equation models for an Economics or Management audience. ....One of the strongest points of the book is its courageous integration of the Econometrics, Statistics and Psychometrics literatures. ...All in all, this is an excellent book which deserves a careful reading.." --Psychometrika
"...After reading the book, I believe that it will help econometricians to appreciate psychometrics and, hopefully, it will inspire pyschometricians to consider work in Econometrics. ...This is an excellent book which I strongly recommend reading carefully. Needless to say, this book is no light reading. Given the amount of material covered in just over 400 pages the writing is often terse and demanding on the reader, particularly if s/he has not been exposed previously to the Econometrics literature. But the efforts are well worth it, the book is a gem." --Psychometrika
Rijksuniversiteit Groningen, The Netherlands