Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 281

IfQ′is thepth difference operator, thenQ′φ=μι, withι=[1, 1,...,1]′, will
contain a constant sequence of values, which will be zeros if the degree ofφ
is less thanp. Also,Q′ζwill be a vector sampled from a mean-zero stationary
process. Therefore,δ=Q′ξis from a stationary process with a constant mean. Thus,
there is:

Q′y=Q′ξ+Q′η

=δ+κ=g,

(6.126)

where:

E(δ)=μι, D(δ)=,

E(κ)=0, D(κ)=Q′Q.

(6.127)

Now consider the conditional expectation ofηgiveng =Q′y, which is also
its minimum mean square error estimator on the assumption that the various
stochastic processes are normally distributed. This is:

E(η|g)=E(η)+C(η,g)D−^1 (g){g−E(g)}

=Q(+Q′Q)−^1 {Q′y−μι}.

(6.128)

If the vectorE(g)=μιis non-zero it will, nevertheless, be virtually nullified by the

matrixQ(+Q′Q)−^1 , which is a matrix version of a highpass filter. Therefore, it
may be deleted from the expressions of (6.128). Next, sinceξ=y−η, the estimate
of the trend isx=E(ξ|g)=y−E(η|g), which is exactly equation (6.123).
The HP filter may be derived by specializing the statistical assumptions of (6.125)
and (6.127). It is assumed that:

D(η)==ση^2 I, D(δ)==σδ^2 I and λ=

ση^2

σδ^2

. (6.129)

Putting these details into equation (6.123) gives equation (6.124).
It is straightforward to derive the dispersion matrices that are found within the
formulae for the finite-sample estimators from the corresponding autocovariance
generating functions. Letγ(z)={γ 0 +γ 1 (z+z−^1 )+γ 2 (z^2 +z−^2 )+···}denote
the autocovariance generating function of a stationary stochastic process. Then,
the corresponding dispersion matrix for a sample ofTconsecutive elements drawn
from the process is:

=γ 0 IT+

T∑− 1

τ= 1

γτ(LτT+FTτ), (6.130)

whereFT=L′Tis in place ofz−^1. SinceLTandFTare nilpotent of degreeT, such
thatLqT,FTq=0 whenq≥T, the index of summation has an upper limit ofT−1.