Estimation of the Memory Parameter for Nonstationary or Noninvertible Fractionally Integrated Processes

Abstract

We consider the asymptotic characteristics of the periodogram ordinates of a fractionally integrated process having memory parameter d 3 0.5, for which the process is nonstationary, or d -0.5, for which the process is noninvertible. Series having d outside the range (-0.5,O.S) 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 wj = 2xj/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.

Non-stationary log-periodogram regression

“…The covariance matrix of w( H ) can be studied in a similar way as in the stationary framework, extending Hurvich and Ray's (1995) analysis of the expectation of the periodogram. However, due to a bias problem, the same results as in Robinson (1995) can only be obtained for d ( (consistency holds for d(1).…”

“…Following Hurvich and Ray (1995), we say that the non-stationary process +X R , has memory parameter d ( )d( ) if the zero mean stationary process…”

“…, ) and then perform the periodogram regression on the transformed series, adjusting the estimate with the number of di!erences (integrations) taken. However, the simulation work of Hassler (1992) and Hurvich and Ray (1995) suggests that, at least for values d3[ , 1), the estimation procedure using the original series can be consistent, although it will not coincide in general with the pre-di!erenced estimate.…”

“…Here we are trying to approximate a di!erent function than in the stationary situation, explaining the discrepancy with respect to the estimates which use previously di!erentiated observations. When we taper the periodogram with the cosine window, as suggested by Hurvich and Ray (1995), we show that the estimate is asymptotically normal even for d( .…”

“…Hassler (1992) used the log-periodogram estimate to construct a unit root test (d"1), but he gave no theoretical justi"cation for his asymptotic theory in the non-stationary case. Hurvich and Ray (1995) studied the behaviour of the expectation of the periodogram at low Fourier frequencies for Gaussian non-stationary and non-invertible fractionally integrated processes. They showed that the normalized periodogram has bounded expectation for d3 [ , ) but it is biased (for the function f ) in this case, and they proposed to taper the data with the full cosine window in order to reduce this bias.…”