ESTIMATION OF THE MEMORY PARAMETER FOR NONSTATIONARY OR NONINVERTIBLE FRACTIONALLY INTEGRATED PROCESSES

Clifford Hurvich and Bonnie K. Ray

Journal of Time Series Analysis, 1995, vol. 16, issue 1, 17-41

Abstract: Abstract. We consider the asymptotic characteristics of the periodogram ordinates of a fractionally integrated process having memory parameter d≥ 0.5, for which the process is nonstationary, or d≤ ‐.5, for which the process is noninvertible. Series having d outside the range (‐.5,.5) may arise in practice when a raw series is modeled without preliminary consideration of the stationarity and invertibility of the series or when a wrong decision is made concerning the stationarity and invertibility of the series. We find that the periodogram of a nonstationary or noninvertible fractionally integrated process at the jth Fourier frequency ωj= 2πj/n, where n is the sample size, suffers from an asymptotic relative bias which depends on j. We also examine the impact of periodogram bias on the regression estimator of d proposed by Geweke and Porter‐Hudak (1983) in finite samples. The results indicate that the bias in the periodogram ordinates can strongly affect the GPH estimator even when the number of Fourier frequencies used in the regression is allowed to depend on the length of the series. We find that data tapering and elimination of the first periodogram ordinate in the regression can reduce this bias, at the cost of an increase in variance for nonstationary series. Additionally, we find for nonstationary series that the GPH estimator is more nearly invariant to first‐differencing when a data taper is used.

Date: 1995
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