CFDP 2303

Discrete Fourier Transforms of Fractional Processes with Econometric Applications


Publication Date: October 2021

Pages: 66


The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d ≥ 1 2: Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1; the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases. 

Keywords: Discrete Fourier transform, Fractional Brownian motion, Fractional integration, Log periodogram regression, Nonstationarity, Operator decomposition, Semiparametric estimation, Whittle likelihood

JEL Classification Codes: C22

JEL Classification Codes: C22