Asymptotic properties of the corrected score estimator in the autoregressive model with measurement errors

Abstract

The autoregressive model with errors in variables with normally distributed control se- quence is considered. For the main sequence, two cases are dealt with: (a) main sequence has station- ary distribution, and (b) initial distribution is arbitrary, independent of the control sequence and has finite fourth moment. Here the elements of the main sequence are not observed directly, but surrogate data that include a normally distributed additive error are observed. Errors and main sequence are assumed to be mutually independent. We estimate unknown parameter using the Corrected Score method and in both cases prove strict consistency and asymptotic normality of the estimator. To prove asymptotic normality we apply the theory of strong mixing sequences. Finally, we compare the efficiency of the Least Squares (naive) estimator and the Corrected Score estimator in the forecasting problem and conclude that the naive estimator gives better forecast.