Historical Misconceptions in Autocorrelation Estimation

The sample autocovariance function, which is estimated as mean lagged products (LPs) of random observations, can also be obtained as the inverse Fourier transform of the periodogram of the data that are augmented with zeros. Hence, the quality of the sample autocovariance as a representation of stochastic data is the same as that of a raw periodogram. The LP estimate is not based on any efficient estimation principle for random data. The spectral density and the autocovariance function can be estimated much more accurately with parametric time series models. A recent development in time series analysis gives the possibility to select automatically the type and the order of the best time series model for any set of random observations. The spectral accuracy of the selected model is better than the accuracy of modified periodograms. In addition, the accuracy of the parametric estimate of the autocovariance function is the same or better for every individual lag than what can be achieved by the nonparametric mean-LP estimates. Perhaps even more important, the estimated time series parameters define the autocovariance as a complete function, for all lags together.