Elsevier

Journal of Econometrics

Volume 91, Issue 2, August 1999, Pages 325-371
Journal of Econometrics

Non-stationary log-periodogram regression

https://doi.org/10.1016/S0304-4076(98)00080-3Get rights and content

Abstract

We study asymptotic properties of the log-periodogram semiparametric estimate of the memory parameter d for non-stationary (d⩾12) time series with Gaussian increments, extending the results of Robinson (1995) for stationary and invertible Gaussian processes. We generalize the definition of the memory parameter d for non-stationary processes in terms of the (successively) differentiated series. We obtain that the log-periodogram estimate is asymptotically normal for d∈[12,34) and still consistent for d∈[12,1). We show that with adequate data tapers, a modified estimate is consistent and asymptotically normal distributed for any d, including both non-stationary and non-invertible processes. The estimates are invariant to the presence of certain deterministic trends, without any need of estimation.

Introduction

Statistical inference for stationary long range dependent time series is often based on semiparametric estimates that avoid parameterization of the short run behaviour. One of most popular semiparametric estimates in the frequency domain is the log-periodogram regression, proposed initially by Geweke and Porter-Hudak (1983). Robinson (1995) showed the consistency and asymptotic normality of a version of that estimate for stationary and invertible Gaussian vector time series. He assumed that the spectral density f(λ) of the observed stationary sequence satisfies for one constant 0<G<∞, f(λ)∼Gλ−2dasλ→0+,where d∈(−12,12) is the parameter that governs the memory of the series. This is the interval of values of d for which the process is stationary and invertible. If d∈(0,12) then we say that the series exhibits long memory or long range dependence. Expression (1) reflects a linear relationship between the spectral density and the frequency in log–log coordinates, with slope −2d. This, together with the fact that the periodogram ordinates at Fourier frequencies around the origin are still approximately independent and unbiased for the spectral density f in the long memory case (1), constitute the basis for the log-periodogram estimate.

There have been proposals to extend the applicability of the log-periodogram estimate for non-stationary (d⩾12) or non-invertible (d⩽−12) time series, and indeed log-periodogram regressions have been applied to non-stationary observations (e.g. Agiakloglou et al., 1993; Bloomfield, 1991), opening the question of analysing the properties of the estimates when they give values outside the interval of allowed values of d. For d⩾12, a function f(λ) behaving like Eq. (1)can be defined in terms of the differenced series, but it is no longer a spectral density, since it is not integrable and the time series is non-stationary with infinite variance. Hassler (1992) used the log-periodogram estimate to construct a unit root test (d=1), but he gave no theoretical justification 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 d∈[12,32) 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.

Robinson (1995) advocated an initial differentiation (integration) of the observed time series when non-stationarity (non-invertibility) is suspected to obtain a value of d in the stationary and invertible interval (−12,12) and then perform the periodogram regression on the transformed series, adjusting the estimate with the number of differences (integrations) taken. However, the simulation work of Hassler (1992) and Hurvich and Ray (1995) suggests that, at least for values d∈[12,1), the estimation procedure using the original series can be consistent, although it will not coincide in general with the pre-differenced estimate.

Using Hurvich and Ray’s definitions we extend Robinson’s (1995) results to cover the non-stationary case. We find that in the Gaussian case the log-periodogram estimate is asymptotically normal for d<34 and still consistent for d<1. Here we are trying to approximate a different function than in the stationary situation, explaining the discrepancy with respect to the estimates which use previously differentiated 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<32.

We also consider a general non-stationary model for any d⩾12, where the presence of deterministic time trends is allowed, and show that it is possible to design data tapers which deliver asymptotic normally distributed estimates of d under Gaussianity. The main idea is the same as in, e.g. Zhurbenko (1979), Zhurbenko 1980, Zhurbenko 1982), Robinson (1986) or Dahlhaus (1988), who showed that certain tapers or data windows allow statistical inference in the presence of non-stationary properties at certain frequencies. Their analyses used the improved convergence properties of the spectral window of some tapers and we will require those and some other special features to deal with the stochastic trends of non-stationary processes. The same principle will make the estimates robust to deterministic time trends up to certain order, avoiding any trend specification, testing or estimation as in most of non-stationary inference literature, both with the autoregressive approach (e.g. Durlauf and Phillips, 1988) or in the fractional differencing framework (Robinson, 1994b). Related ideas allow also the estimation of d⩽−12 for Gaussian non-invertible processes that may arise in overdifferencing to eliminate stochastic and deterministic trends. These properties enable us to abstract from deterministic behaviours and concentrate on the stochastic trends and their implications on the non-invertibility (d⩽−12), non-stationarity (d⩾12), mean reversion (d<1), etc., of the observed time series.

Finally, we analyse empirically the performance of the estimates for finite sample sizes. We show how to base a choice of the degree of tapering, identifying when it produces biased estimates for all possible choices of a bandwidth parameter.

The paper is organized as follows. First we give the main assumptions and definitions. In Section 3we study the non-tapered situation and in Section 4we analyse the cosine bell window taper. Then we consider in Section 5a general model for non-stationary time series and suitable data windows for their analysis and in Section 6we apply the same methods to non-invertible processes. In Section 7we analyse the performance of the estimates proposed for simulated data. Then we conclude and give some proofs and technical lemmas in three appendices.

Section snippets

Assumptions and definitions

Following Hurvich and Ray (1995), we say that the non-stationary process {Xt} has memory parameter d (12⩽d<32) if the zero mean stationary process εtXt has spectral density fε(λ)=|1−exp(iλ)|−2(d−1)f(λ),where f(λ) is a positive, integrable, even function on [−π,π] which is bounded above and away from zero and is continuous at λ=0. We will relax this assumption later, and consider a more general non-stationary process. Then, we can write, for any t⩾1, Xt=X0+k=1tεk,where X0 is a random

Non-tapered periodogram

In this section we analyse the asymptotic properties of d̂ as defined previously in Eq. (8), in terms of the raw (non-tapered) periodogram for non-stationary time series. We analyse the univariate case for simplicity, but the multivariate model does not involve new ideas and can be dealt with as in Robinson (1995), since the relationships between the elements of the spectral density matrix of εt go through for a matrix function f(λ), although the interpretation is different.

Under Assumptions 1

Cosine bell tapered periodogram

We consider in this section the full cosine bell taper, as suggested by Hurvich and Ray (1995). The tapered discrete Fourier transform for any taper sequence {ht}t=1n is defined as wTj)=1∑ht2t=1nhtXtexp(iλjt).For the full cosine bell ht=12(1−cos[2πt/n]), and the sum of the squared taper weights is ∑ht2=3n/8. This is called the asymmetric version of the cosine bell by (Percival and Walden (1993), p. 325). The usual discrete Fourier transform w(λ) is obtained setting ht≡1,∀t.

The benefits of

General non-stationary processes and data tapers

In this section we propose a general model for non-stationary time series d⩾12 and show how to extend the previous ideas to the estimation of the memory parameter d when we use appropriate data tapers. The consideration of processes with, e.g. one or two unit roots in the classical sense will also lead us to the discussion of polynomic trends of time and how to discriminate between these deterministic trends and the stochastic trends produced by the integration of (zero mean) processes.

We say

Non-invertible processes

Differencing the observed time series is an effective way of reducing the magnitude of the memory parameter d and the maximum order of any polynomial deterministic trend. However, differencing to remove deterministic or stochastic trends may lead to non-invertible stationary time series satisfying (Eq. (1)) with d⩽−12. Otherwise we will not find the non-invertible (d⩽−12) situation very often in practical applications.

Hurvich and Ray (1995) considered the limit of the expectation of the

Simulation results

In this section we describe briefly the practical implementation of the previous estimates of the memory parameter d, with simulated non-stationary data. We will concentrate on Zhurbenko–Kolmogorov tapers with different values of p. We have plotted these data tapers for p=1,2,3 and 4 and n=96 on the first row of Fig. 1. We can observe that the larger the order p, the smoother is the transition in the extremes of the taper weights in the observed interval 1,…,n. The (logarithms of the) spectral

Conclusions

We have given a unified asymptotic theory for the log-periodogram estimate of the memory parameter d for Gaussian processes, including non-stationary (d⩾12) and non-invertible (d⩽−12) time series, with possibly deterministic trends, making of this semiparametric estimate a convenient tool for the analysis of the memory structure of a general class of processes under weak assumptions.

We have described the effects of tapering in terms of bias reduction, trend removal and estimation of

Acknowledgements

I am grateful to P.M. Robinson for helpful discussions and suggestions. I also thank an Associate Editor and two Referees whose comments led to a great improvement of the paper. I also wish to thank P. Zaffaroni, C. Michelacci, F.J. Hidalgo, L. Giraitis and L.A. Gil Alaña for valuable comments. The first version of this paper was written while the author was at the London School of Economics and Political Science. Financial support from the Fundación Ramón Areces (Spain), the Economic and

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