278 Investigating Economic Trends and Cycles
filter has only a short span, it matters little which assumptions are made about the
length of the data sequence. Only at the ends of the data sequence are there liable
to be problems.
The assumption of a double-infinite data sequence also sustains the theory of
time-invariant infinite impulse response (IIR) rational filters, such as the Butter-
worth and HP filters of section 6.6.2, which correspond to moving averages of
infinite order. These are not so easily applied to short sequences. Nevertheless, if
the data sequence is sufficiently lengthy to allow the transient effects of the arbi-
trary start-up values to disappear, then such filters can be implemented successfully
via bidirectional feedback procedures which comprise only a handful of recent data
values. (In effect the start-up values purport to summarize the history of the infinite
data sequence, insofar as it affects the IIR filter.)
In econometric applications, attention is often focused upon the most recent
observations at the upper end of a short data sequence. In such cases, a theory of
filtering is called for that fully recognizes the finite nature of the data sequence.
Also, in cases where the data are trended, it becomes essential to supply appropriate
non-zero initial conditions to the filter, and these should be the products of a
finite-sample theory.
The theory that we shall expound here depends upon replacing the symbolz
within the various polynomial operators by a matrix lag operator. However, it is
immediately apparent that this replacement alone is insufficient for the purpose
of creating adequate finite-sample filters.
To demonstrate the effects of the replacement, letLT=[e 1 ,e 2 ,...,eT− 1 ,0]be
the matrix version of the lag operator, which is formed from the identity matrix
IT=[e 0 ,e 1 ,e 2 ,...,eT− 1 ]of orderTby deleting the leading column and by append-
ing a column of zeros to the end of the array. Then, the matrix of orderTthat
corresponds to thepth difference operator∇p(z)=( 1 −z)pis:
∇pT=(I−LT)p. (6.111)We may partition this matrix so that∇pT=[Q∗,Q]′, whereQ∗′hasprows. Ifyis a
vector ofTelements, then:
∇Tpy=[
Q∗′
Q′]
y=[
g∗
g]
, (6.112)andg∗is liable to be discarded, whereasgwill be regarded as the vector of thepth
differences of the data.
The inverse matrix is partitioned conformably to give∇−Tp=[S∗,S]. It follows
that:
[
S∗ S][Q′
∗
Q′]
=S∗Q∗′+SQ′=IT, (6.113)