Palgrave Handbook of Econometrics: Applied Econometrics

272 Investigating Economic Trends and Cycles

which is obtained fromε(t)by differencingm−dtimes. Then, the filter that is
applied toy(t)to estimateξ(t), which is thed-fold integral ofδ(t), takes the form:

β(z)=

σζ^2 ( 1 +z−^1 )n( 1 +z)n

σζ^2 ( 1 +z−^1 )n( 1 +z)n+σε^2 ( 1 −z−^1 )m( 1 −z)m

, (6.93)

regardless of the degreedof differencing that would be necessary to reducey(t)to
stationarity.
Two special cases are of interest. By settingd=m=2 andn=0 in (6.92),
a model is obtained of a second-order random walkξ(t)affected by white-noise
errors of observationη(t)=ε(t). The resulting lowpass WK filter, in the form:

β(z)=

1

1 +λ( 1 −z−^1 )^2 ( 1 −z)^2

with λ=

ση^2

σδ^2

, (6.94)

is the HP filter. The complementary highpass filter, which generates the residue, is:

βc(z)=

( 1 −z−^1 )^2 ( 1 −z)^2

λ−^1 +( 1 −z−^1 )^2 ( 1 −z)^2

. (6.95)

Here,λ, which is described as the smoothing parameter, is the single adjustable
parameter of the filter.
By settingm=n, a filter for estimatingξ(t)is obtained that takes the form:

β(z)=

σζ^2 ( 1 +z−^1 )n( 1 +z)n

σζ^2 ( 1 +z−^1 )n( 1 +z)n+σε^2 ( 1 −z−^1 )n( 1 −z)n

=

1

1 +λ

(

i

1 −z

1 +z

) 2 n with λ=

σε^2

σζ^2

.

(6.96)

This is the formula for the Butterworth lowpass digital filter. The filter has two
adjustable parameters and, therefore, is a more flexible device than the HP filter.
First, there is the parameterλ. This can be expressed as:

λ={ 1 /tan(ωd)}^2 n, (6.97)

whereωdis the nominal cut-off point of the filter, which is the mid-point in the
transition of the filter’s frequency response from its pass band to its stop band. The
second of the adjustable parameters isn, which denotes the order of the filter. Asn
increases, the transition between the pass band and the stop band becomes more
abrupt.
These filters can be applied to the non-stationary data sequencey(t)in the man-
ner indicated by equation (6.91), provided that the appropriate initial conditions
are supplied with which to start the recursions. However, by concentrating on the
estimation of the residual sequenceη(t), which corresponds to a stationary process,