Asymptotic Distribution of the Bias Corrected Least Squares Estimators in Measurement Error Linear Regression Models Under Long Memory

This article derives the consistency and asymptotic distribution of the bias corrected least squares estimators (LSEs) of the regression parameters in linear regression models when covariates have measurement error (ME) and errors and covariates form mutually independent long memory moving average processes. In the structural ME linear regression model, the nature of the asymptotic distribution of suitably standardized bias corrected LSEs depends on the range of the values ofD max = max { d X + d ε , d X + d u , d u + d ε , 2 d u } , where d X , d u , and d ε are the LM parameters of the covariate, ME and regression error processes respectively. This limiting distribution is Gaussian whenD max < 1 / 2 and non‐Gaussian in the caseD max > 1 / 2 . In the former case some consistent estimators of the asymptotic variances of these estimators and a log( n )‐consistent estimator of an underlying LM parameter are also provided. They are useful in the construction of the large sample confidence intervals for regression parameters. The article also discusses the asymptotic distribution of these estimators in some functional ME linear regression models, where the unobservable covariate is non‐random. In these models, the limiting distribution of the bias corrected LSEs is always a Gaussian distribution determined by the range of the values of d ε  −  d u .