D.S.G. Pollock 27700.250.500.751.000 π/ 4 π/ 2 3 π/ 4 πFigure 6.12 The gain function of the trend-extraction filter obtained from the STAMP
program (solid line) together with that of the canonical trend-extraction filter (broken line)
Disaggregated structural time-series models have been treated at length in Harvey
(1989). The methodology has been implemented in the STAMP program, which is
described by Koopmanet al.(2007). A similar approach has been pursued in a pro-
gram within the Captain MATLAB Toolbox, which has been described by Pedregal,
Taylor and Young (2004). A comparative analysis of the STAMP and TRAMO–SEATS
programs has been provided by Pollock (2002b).
Example Figure 6.12 shows the gain of the trend extraction filter that is associated
with a disaggregated structural model that has been applied to the monthly airline
passenger data of Box and Jenkins (1976).
The solid line represents the gain of the ordinary filter and the broken line
represents the gain of the filter that is obtained when the principle of canoni-
cal decomposition is applied to the components of the model. In that case, the
white noise that is contained in the components is removed and reassigned to the
residual component.
The indentations in the gain function at the seasonal frequenciesπj/6;j =
1,..., 6 are due to the zeros of the filter that are to be found on the circumference of
the unit circle and which are effective in removing the seasonal fluctuations from
the trend.
Disregarding these indentations, the gain of the filters is reduced only gradually
as frequency increases. In particular, the ordinary unadjusted filter is liable to trans-
mit a higher proportion of the high-frequency noise of the data. However, given
that such high-frequency noise is largely absent from the airline passenger data,
it transpires that the effect upon the estimated trend of adopting the principle of
canonical decomposition is a minor one.
6.7 Finite-sample signal extraction
The classical theory of linear filtering relies heavily upon the simplifications that
are afforded by the assumption that the data constitute a doubly infinite sequence.
The assumption is an acceptable one in the case of finite impulse response (FIR)
filters that can be realized via low-order moving-average operators. When such a