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212 Chapter 6 Nonstationary and Seasonal Time Series Models
where Ɣn E WW ′ is the covariance matrix of W.
The generalized least squares (GLS) estimator of β is the value ˆβGLS that
minimizes the weighted sum of squares
(Y − Xβ) ′ Ɣ −1
n (Y − Xβ) . (6.6.6)
Differentiating partially with respect to each component of β and setting the derivatives
equal to zero, we find that
ˆβGLS X ′ Ɣ −1
n X −1 ′ −1
X Ɣn Y. (6.6.7)
If the design matrix X is nonrandom, the GLS estimator is unbiased and has covariance
matrix
Cov ˆβGLS X ′ Ɣ −1
n X −1
. (6.6.8)
It can be shown that the GLS estimator is the best linear unbiased estimator of β, i.e.,
for any k-dimensional vector c and for any unbiased estimator ˆβ of β that is a linear
function of the observations Y1,...,Yn,
Var ≤ Var c ′
ˆβ .
c ′ ˆβGLS
In this sense the GLS estimator is therefore superior to the OLS estimator. However,
it can be computed only if φ and θ are known.
Let V(φ, θ) denote the matrix σ −2 Ɣn and let T(φ, θ) be any square root of V −1
(i.e., a matrix such that T ′ T V −1 ). Then we can multiply each side of (6.6.2) by T
to obtain
T Y TXβ + T W, (6.6.9)
a regression equation with coefficient vector β, data vector T Y, design matrix TX,
and error vector T W. Since the latter has uncorrelated, zero-mean components, each
with variance σ 2 , the best linear estimator of β in terms of T Y (which is clearly the
same as the best linear estimator of β in terms of Y, i.e., ˆβGLS) can be obtained by
applying OLS estimation to the transformed regression equation (6.6.9). This gives
ˆβGLS X ′ T ′ TX −1 X ′ T ′ T Y, (6.6.10)
which is clearly the same as (6.6.7). Cochrane and Orcutt (1949) pointed out that if
{Wt} is an AR(p) process satisfying
φ(B)Wt Zt, {Zt} ∼WN 0,σ 2 ,
then application of φ(B) to each side of the regression equations (6.6.1) transforms
them into regression equations with uncorrelated, zero-mean, constant-variance errors,
so that ordinary least squares can again be used to compute best linear unbiased
estimates of the components of β in terms of Y ∗
t φ(B)Yt, t p + 1,...,n.
This approach eliminates the need to compute the matrix T but suffers from the
drawback that Y∗ does not contain all the information in Y. Cochrane and Orcutt’s