Dedicated to the memory of E. J. Hannan. A central limit theorem is established for time series regression estimates which include generalized least squares, ...
The Annals of Statistics 1997, Vol. 25, No. 1, 77 ] 104
TIME SERIES REGRESSION WITH LONG-RANGE DEPENDENCE1 BY P. M. ROBINSON AND F. J. HIDALGO London School of Economics Dedicated to the memory of E. J. Hannan A central limit theorem is established for time series regression estimates which include generalized least squares, in the presence of long-range dependence in both errors and stochastic regressors. The setting and results differ significantly from earlier work on regression with long-range-dependent errors. Spectral singularities are permitted at any frequency. When sufficiently strong spectral singularities in the error and a regressor coincide at the same frequency, least squares need no longer be n1r 2 -consistent, where n is the sample size. However, we show that our class of estimates is n1r 2 -consistent and asymptotically normal. In the generalized least squares case, we show that efficient estimation is still possible when the error autocorrelation is known only up to finitely many parameters. We include a Monte Carlo study of finite-sample performance and provide an extension to nonlinear least squares.
1. Introduction. This paper derives central limit theorems for estimates of the slope coefficient vector b in the multiple linear regression.
Ž 1.1.
yt s a q b X x t q u t ,
t s 1, 2, . . . ,
where both the K-dimensional column vector of regressors x t and the unobservable scalar error u t are permitted to exhibit long-range dependence, a is an unknown intercept and the prime denotes transposition. Two widely used methods of estimating b are ordinary least squares and generalized least squares. Both are known to be asymptotically normal under a wide variety of regularity conditions. However, it was pointed out by Robinson Ž1994a. that, when x t and u t collectively exhibit long-range dependence of a sufficiently high order, the least squares estimate is not asymptotically normal. We show that a class of weighted least squares estimates, which includes generalized least squares as a special case, is asymptotically normal under rather general forms of long-range dependence in both x t and u t . Given observations yt , x t , t s 1, . . . , n, consider estimates of the following type w indexed by the function f Ž l., which is real-valued, even, integrable and
Received February 1995; revised March 1996. 1 Research supported by ESRC Grant R000235892. AMS 1991 subject classifications. Primary 62M10, 60G18; secondary 62F12, 62J05, 62J02. Key words and phrases. Long-range dependence, linear regression, generalized least squares, nonlinear regression.
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P. M. ROBINSON AND F. J. HIDALGO
periodic of period 2p x :
ˆy1 ˆ bˆf s A f bf ,
Ž 1.2. where, with Ý t denoting Ý nts1 ,
1
H I Ž l. w Ž l. d l , 2p yp
ˆw s A
p
x
1
H I 2p yp
ˆbw s
p
xy
Ž l. w Ž l. d l ,
I x Ž l . s wx Ž l . wx Ž yl . , X
I x y Ž l . s wx Ž l . wy Ž yl . , wx Ž l . s
Ž 1.3.
wy Ž l . s xs
1
Ž 2p n .
1r2
t
1
Ž 2p n .
Ý Ž x t y x . e itl ,
1r2
Ý Ž yt y y . e i t l , t
1
Ý xt , n
ys
t
1 n
Ý yt , t
ˆf is nonsingular. Here A ˆf and ˆbf can equivalently be written as and A ˆf s A Ž 1.4.
ˆbf s
1 n 1 n
Ý Ý Ž x t y x . Ž x s y x . ftys , X
s
t
Ý Ý Ž x t y x . Ž ys y y . ftys , s
t
where
Ž 1.5.
fj s
1
Hypf Ž l. cos jl d l . p
Ž 2p .
2
In case f Ž l. ' 1, so that f 0 s 1r2p and f j s 0 for j / 0, bˆf s bˆ1 , the least squares estimate,
Ž 1.6.
bˆ1 s
½ Ý Ž x y x. x 5 X t
t
t
y1
Ý Ž x t y x . yt . t
We assume throughout that u t is covariance stationary, having mean that is Žwithout loss of generality. 0, and absolutely continuous spectral distribution function, so that it has spectral density, denoted f Ž l., satisfying
Ž 1.7.
g j sdef E Ž u1 u1qj . s
Hyp f Ž l. cos jl d l , p
j s 0, 1, . . . .
Suppose f Ž l. ) 0, yp - l F p . Then taking f Ž l. s f Ž l.y1 gives a generalized least squares estimate bˆfy1 . In case f is not known up to scale, but only
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LONG-RANGE-DEPENDENT REGRESSION
up to a finite-dimensional vector of parameters u , so that f Ž l. s f Ž l; u . is a given function of l and u , a feasible generalized least squares estimate is bˆfˆy1 , obtained by taking f Ž l. s f y1 Ž l; uˆ., where uˆ is an estimate of u . Now suppose that x t is a stochastic sequence, independent of the u t sequence, and covariance stationary with autocovariance matrix Gj s EŽ x 1 y Ex 1 .Ž x 1qj y Ex 1 .X . Then
Ž 1.8.
Hype p
Gj s
i jl
dF Ž l . ,
where the matrix F has Hermitian nonnegative definite increments and is uniquely defined by the requirement that it is continuous from the right. Under suitable conditions we then have the central limit theorem Žclt.
Ž 1.9.
ž
/
n1r2 bˆf y b ªd N Ž 0, Sfy1 Sc Sfy1 . ,
where c Ž l. s f 2 Ž l. f Ž l. and
Ž 1.10.
Sx s
1
H x Ž l. dF Ž l. 2p yp p
for x Ž l. such that Sx is finite and nonsingular. In particular,
Ž 1.11.
ž
/
n1r2 Ž bˆ1 y b . ªd N 0, Ž 2p . G0y1 S f G0y1 ,
ž ž bˆˆ
/ yb/ ª
2
Ž 1.12.
n1r2 bˆfy1 y b ªd N Ž 0, Sy1 fy 1 . ,
Ž 1.13.
n1r2
fy1
d
N Ž 0, Sy1 fy 1 . .
Some conditions for Ž1.9. and Ž1.11. ] Ž1.13. have already been laid down in the literature. In particular, the case when f is at least bounded,
Ž 1.14.
sup f Ž l . - `, l
has effectively been covered in a large literature, some relatively complete results appearing in Hannan Ž1979.. There is also interest in cases where Ž1.14. does not hold, when we can say that u t has long-range dependence: f Ž l. has a singularity at one or more l, such as l s 0. Here the relevant literature is much less extensive, and it has stressed the least squares estimate in case of nonstochastic x t satisfying what have come to be called ‘‘Grenander’s conditions.’’ Eicker Ž1967. gave a clt under the assumption that
Ž 1.15.
ut s
Ý t j « tyj ,
Ý t j2 - `,
where Ý will always denote a sum over 0, " 1, . . . of an obvious index, and the « t are independent with finite variance, and Hannan Ž1979. relaxed the latter assumption to square-integrable martingale differences. The squaresummability condition in Ž1.15. is equivalent only to covariance stationarity of u t given the other conditions Ži.e., to integrability of f .. Such stronger conditions on the t j as absolute summability would rule out long-range dependence, and imply Ž1.14.. Mention must also be made of important early,
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P. M. ROBINSON AND F. J. HIDALGO
contributions in the simple location model special case. Ibragimov and Linnik Ž1971., Theorem 18.6.5, considered this under essentially the same conditions on u t as Eicker Ž1967., while Taqqu Ž1975. initiated a new avenue of research by assuming u t is a nonlinear function of a Gaussian process. The work of Yajima Ž1988, 1991. contained some central limit theory Žunder conditions on cumulants of all orders. but stressed other aspects of regression with nonstochastic regressors and long-range-dependent errors. He assumed that
Ž 1.16.
f Ž l . s f U Ž l . < l
