D.S.G. Pollock 279and that:
[
Q′∗
Q′
][
S∗ S]
=[
Q∗′S∗ Q∗′S
Q′S∗ Q′S]
=[
Ip 0
0 IT−p]. (6.114)
Ifg∗is available, thenycan be recovered fromgvia:
y=S∗g∗+Sg. (6.115)The lower-triangular Toeplitz matrix∇T−p=[S∗,S]is completely characterized by
its leading column. The elements of that column are the ordinates of a polynomial
of degreep−1, of which the argument is the row indext=0, 1,...,T−1. More-
over, the leadingpcolumns of the matrix∇
−p
T , which constitute the submatrixS∗,
provide a basis for all polynomials of degreep−1 that are defined on the integer
pointst=0, 1,...,T−1.
It follows thatS∗g∗=S∗Q∗′ycontains the ordinates of a polynomial of degreep−1,
which is interpolated through the firstpelements ofy, indexed byt=0, 1,...,p−1,
and which is extrapolated over the remaining integerst=p,p+1,...,T−1.
6.7.1 Polynomial regression and HP filtering
A polynomial that is designed to fit the data should take account of all of the
observations iny. Imagine, therefore, thaty=φ+η, whereφcontains the ordinates
of a polynomial of degreep−1 andηis a disturbance term withE(η)=0 and
D(η)=. Then, in forming an estimatef=S∗r∗ofφ, we should minimize the
sum of squaresη′−^1 η. Since the polynomial is fully determined by the elements
of a starting value vectorr∗, this is a matter of minimizing:
(y−φ)′−^1 (y−φ)=(y−S∗r∗)′−^1 (y−S∗r∗), (6.116)with respect tor∗. The resulting values are:
r∗=(S′∗−^1 S∗)−^1 S∗′−^1 y and φ=S∗(S∗′−^1 S∗)−^1 S′∗−^1 y. (6.117)An alternative representation of the estimated polynomial is available, which
avoids the inversion of. This is provided by the identity:
P∗=S∗(S′∗−^1 S∗)−^1 S∗′−^1=I−Q(Q′Q)−^1 Q′=I−PQ,(6.118)which gives two representations of the projection matrixP∗. The equality follows
from the fact that, if Rank[R,S∗]=Tand ifS′∗−^1 R=0, then:
S∗(S′∗−^1 S∗)−^1 S′∗−^1 =I−R(R′−^1 R)−^1 R′−^1. (6.119)SettingR=Qgives the result. It follows that the ordinates of the polynomial
fitted to the data by generalized least-squares regression can be represented by:
φ=y−Q(Q′Q)−^1 Q′y. (6.120)