Efficient use of higher-lag autocorrelations for estimating autoregressive processes

The Yule–Walker estimator is commonly used in time-series analysis, as a simple way to estimate the coefficients of an autoregressive process. Under strong assumptions on the noise process, this estimator possesses the same asymptotic properties as the Gaussian maximum likelihood estimator. However, when the noise is a weak one, other estimators based on higher-order empirical autocorrelations can provide substantial efficiency gains. This is illustrated by means of a first-order autoregressive process with a Markov-switching white noise. We show how to optimally choose a linear combination of a set of estimators based on empirical autocorrelations. The asymptotic variance of the optimal estimator is derived. Empirical experiments based on simulations show that the new estimator performs well on the illustrative model.