Summary
As is well known, least squares estimates of regression coefficients are inconsistent if the variables are measured with random errors. In the classical case of known variances and covariances for these error variables, consistent estimates can be derived. It is shown that these estimators generally have a joint asymptotic normal distribution, the covariance matrix of which is derived. No use is made of normality assumptions, but knowledge of the third and fourth moments of error variables is utilized.
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Literature
Anderson, T.W.: An Introduction to Multivariate Statistical Analysis. New York-London-Sydney 1958.
Cochran, W.G.: Errors of Measurement in Statistics. Technometrics,10, pp. 637–666, 1968.
Dorff, M., andJ. Gurland: Estimation of the parameters of a Linear Functional Relation. Journal of the Royal Statistical Society, B23, pp. 160–170, 1961.
Grubbs, F.E.: On Estimating Precision of Measuring Instruments and Product Variability. Journal of American Statistical Association,43, pp. 243–264, 1948.
Halperin, M., andJ. Gurian: A Note on Estimation in Straight Line Regression when Both Variables are Subject to Error. Journal of the American Statistical Association,66, pp. 587–589, 1971.
Johnston, J.: Econometric Methods. New York-San Francisco-Toronto-London (a) 1963. (b) 2nd ed. 1972.
Kendall, M.G., andA. Stuart.: The Advanced Theory of Statistics. Vol. 2, London 1961.
Lindley, D.V.: Regression Lines and the Linear Functional Relationship. Journal of the Royal Statistical Society, B (Supplement)9, 2 pp. 218–244, 1947.
Madansky, A.: The Fitting of Straight Lines when Both Variables are Subject to Error. Journal of the American Statistical Association,54, pp. 173–205, 1959.
Malinvaud, E.: Statistical Methods of Econometrics. Amsterdam 1966.
Morgenstern, O.: Über die Genauigkeit wirtschaftlicher Beobachtungen. Wien-New York 1965.
Nowak, E.: Konsistente Schätzung von Parametern in autoregressiven Systemen mit Fehlern in den Variablen, bei gegebener Kovarianzmatrix der Fehler. Forschungsberichte aus dem Institut für Statistik und Wissenschaftstheorie der Universität München, Serie Oe. Nr. 1, 1975.
Nowak, E.: Eine einheitliche Methode zur konsistenten Schätzung von Parametern in ökonometrischen Systemen mit Fehlern in den Variablen. Forschungsberichte aus dem Institut für Statistik und Wissenschaftstheorie der Universität München, Serie Oe. Nr. 2, 1975.
Richardson, D.H, andDe-min Wu: Least Squares and Grouping Method Estimators in the Errors in Variables Models. Journal of the American Statistical Association,65, pp. 724–748, 1970.
Schneeweiß, H.: Ökonometrie. Würzburg 1971, 1974.
Stroud, W.F.: Comparing Conditional Means and Variances in a Regression Model with Measurement Errors of Known Variances. Journal of the American Statistical Association, pp. 407–412, 1972.
Tawil, J.J.: The Linear Structural Relationship. Unpublished paper 1972.
Theil, H.: Principles of Econometrics. Amsterdam-London 1971.
Villegas, C.: On the Asymptotic Efficiency of Least Squares Estimators. Annals of Mathematical Statistics37, pp. 1676–1683, 1966.
Ware, J.H. The Fitting of Straight Lines when Both Variables are Subject to Error and the Ranks. of the Means are Known. Journal of the American Statistical Association,67, pp. 891–897, 1972.
Zarkovich, S.S.: Quality of Statistical Data. (Food and Agricultural Organization of the United Nations), Rome 1966.
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Schneeweiß, H. Consistent estimation of a regression with errors in the variables. Metrika 23, 101–115 (1976). https://doi.org/10.1007/BF01902854
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DOI: https://doi.org/10.1007/BF01902854