Measurement Error in Linear Autoregressive Models

Abstract

Time series data are often subject to measurement error, usually the result of needing to estimate the variable of interest. Although it is often reasonable to assume that the measurement error is additive (i.e., the estimator is conditionally unbiased for the missing true value), the measurement error variances often vary as a result of changes in the population/process over time and/or changes in sampling effort. In this article we address estimation of the parameters in linear autoregressive models in the presence of additive and uncorrelated measurement errors, allowing heteroscedasticity in the measurement error variances. We establish the asymptotic properties of naive estimators that ignore measurement error and propose an estimator based on correcting the Yule–Walker estimating equations. We also examine a pseudo-likelihood method based on normality assumptions and computed using the Kalman filter. We review other techniques that have been proposed, including two that require no information about the measurement error variances, and compare the various estimators both theoretically and via simulations. The estimator based on corrected estimating equations is easy to obtain and readily accommodates (and is robust to) unequal measurement error variances. Asymptotic calculations and finite-sample simulations show that it is often relatively efficient.