Abstract
For the first order moving average we consider a proposal by Walker (Biometrika, 1961) to use $k$ sample autocorrelations $(1 < k < T, T$ sample size), to estimate the first autocorrelation of the model, and hence its basic parameter. When $k = k_T \rightarrow \infty$ as $T \rightarrow \infty$, the estimator is proved consistent and asymptotically normal and efficient, the latter provided $k_T$ dominates $\log T$ and is dominated by $T^\frac{1}{2}$. An alternative form of the estimator facilitates the calculations and the analysis of the role of $k$, without changing the asymptotic properties.
Citation
Raul Pedro Mentz. "Estimation in the First Order Moving Average Model Based on Sample Autocorrelations." Ann. Statist. 5 (6) 1250 - 1257, November, 1977. https://doi.org/10.1214/aos/1176344012
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