Windowed periodograms and moving average models

A windowed and tapered periodogram can be computed as the Fourier transform of an estimated covariance function of tapered data, multiplied by a lag window. Covariances of finite length can also be modeled as moving average (MA) time series models. The direct equivalence between periodograms and MA models is shown in the method of moments for MA estimation. A better MA representation for the covariance and the spectral density is found with Durbin's improved MA method. That uses the parameters of a long autoregressive (AR) model to find MA models, followed by automatic selection of the MA order. A comparison is made between the two MA model types. The best of many MA models from windowed periodograms is compared to the single selected MA model obtained with Durbin's method. The latter typically has a better quality.