Limiting Distribution of Least Squares in an Errors-in-Variables Linear Regression Model

摘要

There is a substantial literature concerning linear regression when some of the predictors (independent variables) are measured with error. Such models are of importance in econometrics (instrumental variables models), psychometrics (correction for attenuation, models of change), and in instrumental calibration studies in medicine and industry. Recent theoretical work concerning maximum likelihood estimation in such models appears in Healy (1980), Fuller (1980), and Anderson (1984), while Reilly and Patino-Leal (1981) take a Bayesian approach. In an applied context, an investigator may either overlook the measurement errors in the predictors, or choose the classical ordinary least squares (OLS) estimator of the parameters because of its familiarity and ease of use. Certainly, the methodology of classical least squares theory (confidence intervals, multiple comparisons, tests of hypotheses, residual analysis) is considerably more developed than the corresponding errors-in-variables methodology, particularly in samples of moderate size. In this paper, it is shown that under reasonable regularity conditions such linear combinations are (jointly) asymptotically normally distributed.