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Higher Moment Estimators for Linear Regression Models With Errors in the Variables

  • Denyse L. Dagenais
  • Marcel Dagenais
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    This paper proposes instrumental variable estimators for multiple linear regression models with errors in the explanatory variables, that require no extraneous information. As is very well known, the ordinary least squares estimator (OLS), which is based on the sample moments of order two, is unbiased when there are no errors in the variables, but it becomes biased and inconsistent when there are such errors [Fuller (1987)]. In contrast, the suggested estimators are based on higher sample moments and can be considered as a special type of instrumental variable estimator. They are consistent, under quite reasonable assumptions, when there are measurement errors. While most consistent estimators based on higher moments (HM) proposed previously in the literature [Geary (1942), Drion (1951), Durbin (1954), Pal (1980)] for regressions with errors in the variables seem to be quite erratic [Kendall and Stuart (1963), Malinvaud (1978)], the suggested estimators appear to perform remarkably well in many situations. Although most data do contain errors of measurement, this fact is often ignored by the analysts and statistical procedures designed for data measured without error are applied. It is shown that ignoring the presence of even small measurement errors and using traditonal OLS estimators may lead to performing standard Student t-tests with type I errors of considerably higher sizes than intended, while this is not so with the proposed HM estimators. Our experimental findings suggest also that even if the sample is not very large, when the errors in the variables are non-negligible, our estimators do perform better than the OLS estimators in terms of root mean squared errors, when the explanatory variables are strongly correlated and the multiple correlation of the regression is high. Such situations are typical of many statistical analyses based on aggregate data. When the multiple correlation coefficient is smaller and the explanatory variables are less correlated, our HM estimators will still outperform the OLS estimator if the sample is large, even if the measurement errors are not very important. Such cases are frequently encountered in analyses of survey data. Tests for the presence of errors in the variables are also described, and the power of the tests are assessed in the Monte Carlo experiments. Nous proposons, pour les modèles de régression linéaire où les variables explicatives contiennent des erreurs de mesure, des estimateurs de variables instrumentales d'un type particulier, qui n'exigent aucune information extrinsèque. On sait que l'estimateur des moindres carrés ordinaires (MCO), qui est basé sur les moments échantillonnaux d'ordre deux, est centré lorsqu'il n'y a pas d'erreurs sur les variables0501s qu'il devient biaisé et non convergent en présence de telles erreurs [Fuller (1987)]. Par ailleurs, les estimateurs que nous suggérons sont basés sur des moments d'ordres supérieurs et peuvent être vus comme des estimateurs de variables instrumentales. Sous des hypothèses très raisonnables, ces estimateurs demeurent convergents même lorsqu'il y a des erreurs de mesure. Alors que la plupart des estimateurs convergents basés sur des moments d'ordres supérieurs (MOS) proposés antérieurement [Geary(1942), Drion(1951), Durbin (1954), Pal (1980)] pour les modèles de régression avec erreurs sur les variables, semblent très erratiques [Kendall et Stuart (1963), Malinvaud (1978)], les estimateurs que nous proposons se comportent remarquablement bien, dans un grand nombre de cas. Quoique la plupart des données contiennent des erreurs de mesure, ce fait est souvent ignoré par les analystes qui appliquent, la plupart du temps, des procédures statistiques conçues pour le traitement de données mesurées sans erreur. Nous démontrons que le fait de négliger la présence d'erreurs de mesure même relativement faibles et d'utiliser les estimateurs MCO traditionnels, peut faire en sorte que les tests de Student standards comportent des erreurs de type I dont le niveau est considérablement plus élevé que le niveau désiré, alors que ce n'est pas le cas si on utilise les estimateurs MOS proposés. Même si les échantillons ne sont pas très grands, les résultats de nos expériences suggèrent également que dans les cas où les erreurs sur les variables ne sont pas négligeables, le comportement de nos estimateurs lorsqu'on l'évalue en termes de la racine carrée des écarts quadratiques moyens, est supérieur à celui des MCO, quand les variables explicatives sont fortement corrélées et que le coefficient de corrélation multiple est élevé. Ce genre de situations est typique des analyses statistiques basées sur des données agrégées. Si le coefficient de corrélation multiple est moins élevé et que les variables explicatives sont moins corrélées, nos estimateurs MOS peuvent encore s'avérer supérieurs aux estimateurs MCO lorsque les échantillons sont suffisamment grands, et cela même si les erreurs de mesure ne sont pas aussi importantes. De tels cas se rencontrent fréquemment lorsqu'on a affaire à des données d'enquêtes. Nous décrivons également des tests d'erreurs sur les variables et nous évaluons la puissance de ces tests au moyen d'expériences de Monte-Carlo.

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    Paper provided by CIRANO in its series CIRANO Working Papers with number 95s-13.

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    Length: 51 pages
    Date of creation: 01 Mar 1995
    Handle: RePEc:cir:cirwor:95s-13
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    16. Dagenais, M.G. & Dagenais, D.L., 1994. "GMM Estimators for Linear Regression Models with Errors in the Variables," Cahiers de recherche 9404, Universite de Montreal, Departement de sciences economiques.
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