Estimating moving average parameters: Classical pileups and bayesian posteriors

We analyze posterior distributions of the moving average parameter in the first-order case and sampling distributions of the corresponding maximum likelihood estimator. Sampling distributions “pile up” at unity when the true parameter is near unity; hence if one were to difference such a process, estimates of the moving average component of the resulting series would spuriously tend to indicate that the process was overdifferenced. Flat-prior posterior distributions do not pile up, however, regardless of the parameter’s proximity to unity; hence caution should be taken in dismissing evidence that a series has been overdifferenced.