THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS

TLDR

The paper generalizes the definitions of fractional Gaussian noise and integrated series, showing their equivalence. The study proposes a new estimator for the long memory parameter using linear regression of the log periodogram on a deterministic regressor. The estimator is the ordinary least squares slope from a regression using only the lowest frequency log periodogram ordinates. The asymptotic distribution of the estimator is derived, confirming conventional interpretation, and simulations show reliable performance with samples of 50 or more observations, while application to three postwar monthly economic series demonstrates that the integrated model yields more reliable out‑of‑sample forecasts than conventional methods. Abstract.

Abstract

Abstract. 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 periodogram. 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.