Whittle Pseudo-Maximum Likelihood Estimation of Nonstationary Time Series Carlos Velasco Universidad Carlos III de Madrid, Departamento de Estadstica y Econometra Calle Madrid 126 28903 Getafe (Madrid), Spain
[email protected] Peter M. Robinson London School of Economics, Department of Economics Houhgton Street London WC2A 2AE, UK
[email protected]
1. Nonstationary Time Series Exact and approximate maximum likelihood estimation for the parameters of stationary time series has been justi ed under various sets of conditions, including spectral densities with a peak at the origin due to a persistent behaviour (e.g. Fox and Taqqu (1986)). Thus it is often assumed that the spectral density f () of an observed covariance stationary sequence satis es, for 0 < G < 1, (1)
f () Gjj,2d
as ! 0;
where d 2 (, 21 ; 12 ) is the parameter that governs the degree of memory of the series. This is the interval of values of d for which the process is stationary and invertible. If d 2 (0; 12 ) then we say that the series exhibits long memory or long range dependence. When the observations are nonstationary, they are usually di erenced an integer number of times to achieve stationarity. If an observed processes fXtg has covariance stationary increments Xt , with spectral density satisfying fX () G jj,2(d,1) as ! 0, d 12 , it is possible to de ne a generalized or pseudo spectral density (psd) which also displays power-law behaviour at the origin: f () = j1 , exp(i)j,2 fX () G jj,2d
as ! 0:
This psd f () so de ned is not integrable and cannot represent a decomposition of the (in nite) variance of the nonstationary time series. Hurvich and Ray (1995) suggested that the psd f () has equivalent properties to the stationary case, including its interpretation as the limit of the expectation of the sample periodogram. This was used in Velasco (1998) to show that semiparametric estimates of d for stationary time series (see Robinson (1995)), are consistent also for nonstationary sequences. In this paper we exploit these properties of the nonstationary periodogram to approximate parametric (pseudo) spectral densities f () for all frequencies and to construct estimates of the parameters in the frequency domain, including the memory d, without a priori assumptions about the possible nonstationarity. When the memory is too high, tapering might be necessary to reduce the bias in the estimation or to implicitly eliminate stochastic and polynomial trends.
We assume that Xt is a linear process and that f () satis es (1) with parameterization f (; ; ) = k (; )=(2 ); for > 0 and 2 Ra ; such that is compact and d is a R component of inside a closed interval. We assume that the normalization , log k(; )d = 0 holds, which indicates that the free parameter is the variance of the one-step-ahead best linear predictor if d < . It has been often employed for the analysis of Gaussian estimates of stationary series and holds for standard parameterizations of fractional ARIMA models. 2
2
2
2
1 2
2. Whittle Estimates The tapered periodogram of Xt for n observations, t = 1; : : : ; n, and integer, and a taper sequence fhtgnt of order p = 1; 2; : : : ; is de ned as
j
= 2j=n,
j
=1
Ip (j )
= 2
X
n 1 X t t=1
h
2
,
2
ht Xt exp(ij t) :
Though we do not require stationarity or Gaussianity, we use the discrete frequencydomain version of the Whittle or Gaussian log-likelihood for stationary series to estimate , 2p X Ip(j ) ; Q( ) = n j p k (j ; ) where Pj p is a sum over j = p; 2p; : : : ; n , p, n=p is an integer, and cannot be replaced by an integral. We omit zero frequency, for mean-correction purpouses in the stationary case, while the exclusion of frequencies j between p; p; : : : ; n,p is for trend correction in the nonstationary case. The omission of frequencies when p > 1 could be avoided in some circumstances to achieve greater eciency, for example if it is known that d < and E [Xt ] = 0. We de ne the estimates b = arg min Q(), and b = 2Q(b): If the data is not tapered (p = 1) and d < , or if we employ a taper of order p > 1, such that p bd + c +1, we show that the Whittle estimates are consistent. Furthermore they retain the same asymptotic distribution and convergence rate as in the stationary case. The only modi cation is the standard taper correlation contribution to the variance, together with the use of only n=p frequencies. ( )
( )
2
2
3 4
2 3
1 2
REFERENCES Fox, R. and Taqqu, M.S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian times series. Annals of Statistics 14, 517-532. Hurvich, C.M. and Ray, B.K. (1995). Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes. Journal of Time Series Analysis 16, 17-42. Robinson, P.M. (1995). Gaussian semiparametric estimation of long range dependence. Annals of Statistics 23, 1630-1661. Velasco, C. (1998). Gaussian semiparametric estimation for non-stationary time series. Journal of Time Series Analysis 87-127, 1999.
RESUME
Nous presentons quelques results concernant le comportement asymptotique de estimateurs de Whittle par des processus no stationnaire a dependance longue.
