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On the Asymptotic Optimality of Alternative Minimum-Distance Estimators in Linear Latent-Variable Models

Published online by Cambridge University Press:  11 February 2009

Albert Satorra  and
Heinz Neudecker
Albert Satorra
Affiliation:
Universitat Pompeu Fabra, Barcelona
Heinz Neudecker
Affiliation:
Universiteit van Amsterdam
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Abstract

In the context of linear latent-variable models, and a general type of distribution of the data, the asymptotic optimality of a subvector of minimum-distance estimators whose weight matrix uses only second-order moments is investigated. The asymptotic optimality extends to the whole vector of parameter estimators, if additional restrictions on the third-order moments of the variables are imposed. Results related to the optimality of normal (pseudo) maximum likelihood methods are also encompassed. The results derived concern a wide class of latent-variable models and estimation methods used routinely in software for the analysis of latent-variable models such as LISREL, EQS, and CALIS. The general results are specialized to the context of multivariate regression and simultaneous equations with errors in variables.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 5 , December 1994 , pp. 867 - 883
Copyright
Copyright © Cambridge University Press 1994

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References

1Anderson, T.W.Linear latent variable models and covariance structures. Journal of Econometrics 41 (1989): 91–119.CrossRefGoogle Scholar
2Anderson, T.W. & Amemiya, Y.. The asymptotic normal distribution of estimators in factor analysis under general conditions. The Annals of Statistics 16 (1988): 759–771.CrossRefGoogle Scholar
3Bentler, P.M.Simultaneous equation systems as moment structure models. Journal of Econometrics 22 (1983): 13–42.CrossRefGoogle Scholar
4Bentler, P.M.EQS Structural Equations Program Manual. Los Angeles: BMDP Statistical Software, 1989.Google Scholar
5Bentler, P.M. & Dijkstra, T.. Efficient estimation via linearization in structural models. In Krishnaiah, P.R. (ed.), Multivariate Analysis–VI, pp. 9–42. Amsterdam: North-Holland, 1985.Google Scholar
6Browne, M.W.Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology 37 (1984): 62–83.CrossRefGoogle ScholarPubMed
7Browne, M.W. Asymptotic robustness of normal theory methods for the analysis of latent curves. In Brown, P. J. & Fuller, W.A. (eds.), Statistical Analysis of Measurement Errors and Applications, pp. 211–225. Providence, Rhode Island: American Mathematical Society, 1990.Google Scholar
8Browne, M.W. & Shapiro, A.. Robustness of normal theory methods in the analysis of linear latent variable models. British Journal of Mathematical and Statistical Psychology 41 (1988): 193–208.CrossRefGoogle Scholar
9Chamberlain, G.Multivariate regression models for panel data. Journal of Econometrics 18 (1982): 5–46.CrossRefGoogle Scholar
10Chiang, C.L.On regular best asymptotically normal estimates. Annals of Mathematical Statistics 27 (1956): 336–351.CrossRefGoogle Scholar
11Fuller, W.A.Measurement Error Models. New York: John Wiley & Sons, 1987.CrossRefGoogle Scholar
12Jöreskog, K. & Sörbom, D.. LISREL 7: A Guide to the Program and Applications, 2nd ed.Chicago: SPSS, 1989.Google Scholar
13Magnus, J.R. & Neudecker, H.. Symmetry, 0–1 matrices and Jacobians. Econometric Theory 2 (1986): 157–190.CrossRefGoogle Scholar
14Magnus, J.R. & Neudecker, H.. Matrix Differential Calculus, 2nd ed.Chichester: Wiley, 1991.Google Scholar
15Mooijaart, A. & Bentler, P.M.. Robustness of normal theory statistics in structural equation models. Statistica Neerlandica 45 (1991): 159–171.CrossRefGoogle Scholar
16Muthén, B.LISCOMP: Analysis of Linear Structural Equations with a Comprehensive Measurement Model (User's Guide). Mooresville, Indiana: Scientific Software, 1987.Google Scholar
17Neudecker, H. & Satorra, A.. Linear structural relations: Gradient and hessian of the fitting function. Statistics & Probability Letters 11 (1991): 57–61.CrossRefGoogle Scholar
18Neudecker, H. & Satorra, A.. Simple proof of a general matrix equality South African Statistical Journal 25 (1991): 79–82.Google Scholar
19Neudecker, H. & Satorra, A.. A matrix invariance problem. Econometric Theory 8 (1992): 310.CrossRefGoogle Scholar
20Newey, W.K.Asymptotic equivalence of closest moments and OMM estimators. Econometric Theory 4 (1988): 336–340.CrossRefGoogle Scholar
21Rao, C.R. & Mitra, S.K.. Generalized Inverse of Matrices and Its Applications. New York: Wiley, 1971.Google Scholar
22SAS Institute. SAS/STAT Software: CALIS and LOGISTIC Procedures. SAS technical report P-200, SAS Institute, Cary, North Carolina, 1990.Google Scholar
23Satorra, A.Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika 54 (1989): 131–151.CrossRefGoogle Scholar
24Satorra, A. Asymptotic robust inferences in the analysis of mean and covariance structures. In Marsden, P.V. (ed.), Sociological Methodology 1992, pp. 249–278. Oxford and Cambridge, Massachusetts: Basil Blackwell, 1992.Google Scholar
25Satorra, A.The variance matrix of sample second-order moments in multivariate linear relations. Statistics & Probability Letters 15 (1992): 63–69.CrossRefGoogle Scholar
26Satorra, A. & Bender, P.M.. Model conditions for asymptotic robustness in the analysis of linear relations. Computational Statistics & Data Analysis 10 (1990): 235–249.CrossRefGoogle Scholar
27Satorra, A. & Neudecker, H.. A matrix equality applicable in the analysis of mean-andcovariance structures. Problem section. Econometric Theory 8 (1993): 581.CrossRefGoogle Scholar
28Schoenberg, R.J.LINCS: Linear Covariance Structure Analysis. User's Guide. Kent, Washington: RJS Software, 1989.Google Scholar
29Shapiro, A.Asymptotic equivalence of minimum discrepancy function estimators to G.L.S. estimators. South African Statistical Journal 19 (1985): 73–81.Google Scholar
30Shapiro, A.Asymptotic theory of overparameterized models. Journal of the American Statistical Association 81 (1986): 142–149.CrossRefGoogle Scholar
31Shapiro, A.Robustness properties of the MDF analysis of moment structures. South African Statistical Journal 21 (1987): 39–62.Google Scholar
32White, H.Maximum likelihood estimation of misspecified models. Econometrica 50 (1982): 1–25.CrossRefGoogle Scholar