Measurement Error in Linear Autoregressive Models

Time series data are often subject to measurement error, usually the result of needing to estimate the variable of interest. While it is often reasonable to assume the measurement error is additive, that is, 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 paper we address estimation of the parameters in linear autoregressive models in the presence of additive and uncorrelated measurement errors, allowing heteroskedasticity in the measurement error variances. The asymptotic properties of naive estimators that ignoremeasurement error are established, andwe propose an estimator based on correcting the Yule-Walker estimating equations. We also examine a pseudolikelihood method, which is based on normality assumptions and computed using the Kalman filter. Other techniques that have been proposed, including two that require no information about the measurement error variances are reviewed as well. The various estimators are compared both theoretically and via simulations. The estimator based on corrected estimating equations is easy to obtain, readily accommodates (and is robust to) unequal measurement error variances, and asymptotic calculations and finite sample simulations show that it is often relatively efficient.