Prediction in polynomial errors-in-variables models

TL;DR

This paper analyzes polynomial errors-in-variables models with normally distributed covariates and errors, showing that classical estimators are inconsistent for prediction, and provides insights into their asymptotic behavior in a homoskedastic setting.

Contribution

It extends the understanding of errors-in-variables models to polynomial cases, demonstrating the limitations of classical estimators for prediction in these models.

Findings

OLS estimators approximate the best predictor almost surely

Consistent estimators are ineffective for prediction under certain error conditions

Results apply to both linear and polynomial EIV models

Abstract

A multivariate errors-in-variables (EIV) model with an intercept term, and a polynomial EIV model are considered. Focus is made on a structural homoskedastic case, where vectors of covariates are i.i.d. and measurement errors are i.i.d. as well. The covariates contaminated with errors are normally distributed and the corresponding classical errors are also assumed normal. In both models, it is shown that (inconsistent) ordinary least squares estimators of regression parameters yield an a.s. approximation to the best prediction of response given the values of observable covariates. Thus, not only in the linear EIV, but in the polynomial EIV models as well, consistent estimators of regression parameters are useless in the prediction problem, provided the size and covariance structure of observation errors for the predicted subject do not differ from those in the data used for the model fitting.