Multivariate Regression with Errors in Variables: Issues on Asymptotic Robustness

  • Albert Satorra
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 120)


Estimation and testing of functional or structural multivariate regression with errors in variables, with possibly unbalanced design for replicates, and not necessarily normal data, is developed using only the sample cross-product moments of the data. We give conditions under which normal theory standard errors and an asymptotic chi-square goodness-of-fit test statistic retain their validity despite non-normality of constituents of the model. Assymptotic optimality for a subvector of parameter estimates is also investigated. The results developed apply to methods that are widely available in standard software for structural equation models, such as LISREL or EQS.


Functional Model Normality Assumption Covariance Structure Analysis Moment Structure Linear Latent Variable Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amemiya, Y. 1985. “On the goodness-of-fit tests for linear statistical relationships.”Technical Report No. 10.Econometric workshop, Stanford University.Google Scholar
  2. Amemiya, Y.&Anderson, T.W. 1990. “Asymptotic chi-square tests for a large class of factor analysis models.”The Annals of Statistics 3:1453–1463.MathSciNetCrossRefGoogle Scholar
  3. Anderson, T.W. 1989. “Linear latent variable models and covariance structures”. Journal of Econometrics41:91–119.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Anderson, T. W. 1987. “Multivariate linear relations.” Pp. 9–36 inProceedings of the Second International Conference in Statistics editedby T. Pukkila&S. Puntanen. Tampere, Finland: University of Tampere.Google Scholar
  5. Anderson, T.W. 1989. “Linear Latent Variable Models and Covariance Structures.”Journal of Econometrics 41:91–119.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Anderson, T.W. & Amemiya, Y. 1988. “The asymptotic normal distribution of estimates in factor analysis under general conditions.”The Annals of Statistics16:759–771.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Arminger, G. & Sobel, M.E., 1990. “Pseudo-maximum likelihood estimation of mean and covariance structures with missing data”Journal of the American Statistical Association 85:195–203.MathSciNetCrossRefGoogle Scholar
  8. Bentler, P.M. 1983. “Simultaneous equation systems as moment structure models”Journal of Econometrics 22:13–42.MathSciNetCrossRefGoogle Scholar
  9. Bentler, P.M. 1989.EQS Structural Equations Program Manual.Los Angeles: BMDP Statistical Software, Inc.Google Scholar
  10. Bollen, K.A., 1989.Structural equations with latent variables.New York: Wiley.zbMATHGoogle Scholar
  11. Browne, M.W. 1984. “Asymptotically distribution-free methods for the analysis of covariance structures.”British Journal of Mathematical and Statistical Psychology 37:62–83.MathSciNetzbMATHGoogle Scholar
  12. Browne, M.W. 1987. “Robustness in statistical inference in factor analysis and related models.”Biometrika 74:375–384.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Browne, M.W. 1990. “Asymptotic Robustness of Normal Theory Methods for the Analysis of Latent Curves.” Pp. 211–225 inStatistical Analysis of Measurement Errors and Applicationsedited by P. J. Brown and W.A. Fuller. Rhode Island: American Mathematical Society.Google Scholar
  14. Browne, M. W.&Shapiro, A. 1988. “Robustness of normal theory methods in the analysis of linear latent variable models.”British Journal of Mathematical and Statistical Psychology 41:193–208.MathSciNetzbMATHGoogle Scholar
  15. Chamberlain, G. 1982. “Multivariate regression models for panel data.”Journal of Econometrics 18:5–46MathSciNetzbMATHCrossRefGoogle Scholar
  16. Dham, P.F.&Fuller, W.A. (1986).Generalized Least Squares Estimation of the Functional Multivariate Linear Errors-in-variables Model. Journal of Multivariate Analysis 19132–141.MathSciNetCrossRefGoogle Scholar
  17. Hasabelnaby, N.A., Ware, J.H., and Fuller, W.A. (1989). “Indoor Air Pollution and Pulmonary Performance Investigating Errors in Exposure Assessment” (with comments)Statistics in Medicine 81109–1126.CrossRefGoogle Scholar
  18. Gleser, L. J. (1992). The importance of Assessing Measurement Reliability in Multivariate RegressionJournal of the American Statisticl Association 87696–707.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Jöreskog, K. 1971. “Simultaneous factor analysis in several populations”Psychometrika57: 409–426.CrossRefGoogle Scholar
  20. Jöreskog, K.&Sörbom, D. 1989.LISREL 7 A Guide to the Program and Applications.(2nd ed.) Chicago: SPSS IncGoogle Scholar
  21. Kano, Y. (1993) “Asymptotic Properties of Statistical Inference Based on Fisher-Consistent Estimates in the Analysis of Covariance Structures” inStatistical Modelling and Latent Variablesedited by K. Haagen, D.J. Bartholomew and M. Deistler. Elsevier: Amsterdam.Google Scholar
  22. Magnus J. & Neudecker, H. 1988.Matrix differential calculus.New York: Wiley.zbMATHGoogle Scholar
  23. Mooijaart, A.&Bentler, P. M. 1991. “Robustness of normal theory statistics in structural equation models.”Statistica Neerlandica 45:159–171.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Muthén, B. 1987.LISCOMP: Analysis of linear structural equations with a comprehensive measurement model(User’s Guide)., Mooresville IN: Scientific Software.Google Scholar
  25. Muthén, B. Kaplan, D., & Hollis, M. (1987). On structural equation modeling with data that are not missing completely at random.Psychometrika 52431–462.zbMATHCrossRefGoogle Scholar
  26. Newey, W.K. 1988. “Asymptotic equivalence of closest moments and GMM estimators.” Econometric Theory4:336–340.MathSciNetCrossRefGoogle Scholar
  27. Satorra, A. 1989a. “Alternative test criteria in covariance structure analysis: a unified approach.”Psychometrika 54:131–151.MathSciNetCrossRefGoogle Scholar
  28. Satorra, A. 1992b. “Asymptotic robust inferences in the analysis of mean and covariance structures.”Sociological Methodology 1992P.V. Marsden (edt.) pp. 249–278. Basil Blackwell: Oxford & Cambridge, MAGoogle Scholar
  29. Satorra, A. 1993a. “Multi-Sample Analysis of Moment-Structures: Asymptotic Validity of Inferences Based on Second-Order Moments”, inStatistical Modelling and Latent Variablesedited by K. Haagen, D.J. Bartholomew and M. Deistler. Elsevier: Amsterdam.Google Scholar
  30. Satorra, A. 1993b. “Asymptotic robust inferences in multiple-group analysis of augmented-moment structures”, in pp. 211–229Multivariate Analysis: Future Directions 2edited by C.M. Cuadras and C.R. Rao. Elsevier: Amsterdam.Google Scholar
  31. Satorra, A. & Bentler, P.M. 1990. “Model conditions for asymptotic robustness in the analysis of linear relations.”Computational Statistics 8f Data Analysis 10:235–249.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Satorra, A.&Neudecker, H. 1994. On the asympotic optimality of alternative minimum-distance estimators in linear latent-variable models.Econometric Theory 10:867–883.MathSciNetCrossRefGoogle Scholar
  33. Shapiro, A. 1985. “Asymptotic equivalence of minimum discrepancy function estimators to G.L.S.estimators.”South African Statistical Journal19:73–81.MathSciNetzbMATHGoogle Scholar
  34. Shapiro, A. 1986. “Asymptotic theory of overparameterized models.”Journal of the American Statistical Association 81:142–149.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1997

Authors and Affiliations

  • Albert Satorra
    • 1
  1. 1.Departament d’EconomiaUniversitat Pompeu FabraBarcelonaSpain

Personalised recommendations