The definitions of fractional Gaussian noise and integrated (or fractionally differenced) series are generalized, and it is shown that the two concepts are equivalent. A new estimator of the long memory parameter in these models is proposed, based on the simple linear regression of the log periodogram on a deterministic regressor. The estimator is the ordinary least squares estimator of the slope parameter in this regression, formed using only the lowest frequency ordinates of the log periodogrqm. Its asymptotic distribution is derived, from which it is evident that the conventional interpretation of these least squares statistics is justified in large samples. Using synthetic data the asymptotic theory proves to be reliable in samples of 50 observations or more. For three postwar monthly economic time series, the estimated integrated series model provides more reliable out-of-sample forecasts than do more conventional procedures.

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This article develops practical methods for Bayesian inference in the autoregressive fractionally integrated moving average (ARFIMA) model using the exact likelihood function, any proper prior distribution, and time series that may have thousands of observations. These methods utilize sequentially adaptive Bayesian learning, a sequential Monte Carlo algorithm that can exploit massively parallel desktop computing with graphics processing units (GPUs). The article identifies and solves several problems in the computation of the likelihood function that apparently have not been addressed in the literature. Four applications illustrate the utility of the approach. The most ambitious is an ARFIMA(2,d,2) model for the Campito tree ring time series (length 5405), for which the methods developed in the article provide an essentially uncorrelated sample of size 16,384 from the exact posterior distribution in under four hours. Less ambitious applications take as little as 4 minutes without exploiting GPUs.