Semiparametric estimation for seasonal long-memory time series using generalized exponential models

We consider a generalized exponential (GEXP) model in the frequency domain for modeling seasonal long-memory time series. This model generalizes the fractional exponential (FEXP) model [Beran, J., 1993. Fitting long-memory models by generalized linear regression. Biometrika 80, 817–822] to allow the singularity in the spectral density occurring at an arbitrary frequency for modeling persistent seasonality and business cycles. Moreover, the short-memory structure of this model is characterized by the Bloomfield [1973. An exponential model for the spectrum of a scalar time series. Biometrika 60, 217–226] model, which has a fairly flexible semiparametric form. The proposed model includes fractionally integrated processes, Bloomfield models, FEXP models as well as GARMA models [Gray, H.L., Zhang, N.-F., Woodward, W.A., 1989. On generalized fractional processes. J. Time Ser. Anal. 10, 233–257] as special cases. We develop a simple regression method for estimating the seasonal long-memory parameter. The asymptotic bias and variance of the corresponding long-memory estimator are derived. Our methodology is applied to a sunspot data set and an Internet traffic data set for illustration.