FILTERING MACROECONOMIC DATA
By D.S.G. Pollock
University of Leicester
Email:
stephen
[email protected]
This chapter sets forth the theory of linear filtering together with an ac
companying frequencydomain analysis. It employs the classical Wiener–
Kolmogorov theory in describing some of the filters that are used by econo
metricians.
This theory, which was developed originally in reference to
stationary stochastic processes defined on a doublyinfinite index set, is
adapted to cater to short nonstationary sequences. An alternative method
ology of filtering is also described. This operates in the frequency domain,
by altering the amplitudes of the trigonometrical functions that are the
elements of the Fourier decomposition of the detrended data.
1. Introduction
The purpose of a filter is to remove unwanted components from a stream of data
so as to enhance the clarity of the components of interest. In many engineering
applications and in some econometric applications, there is a single component
of interest, described as the signal, to which a component has been added that
can be described as the noise.
A complete separation of the signal and the noise is possible only if they
reside in separate frequency bands. It they reside in overlapping frequency
bands, then their separation is bound to be tentative. The signal typically
comprises elements of low frequency and the noise comprises elements of higher
frequencies. Filters are, therefore, designed by engineers with reference to their
frequencyselective properties.
In econometric applications, some additional components must be taken
into account. The foremost of these is the trend, which may be defined as an
underlying trajectory of the data that cannot be synthesised from trigonomet
rical functions alone. It is diﬃcult to give a more specific definition, which
may account for the wide variety of procedures that have been proposed for
extracting trends from the economic data. A business cycle component might
also be extracted from the data; but this is often found in combination with
the trend.
Another component that is commonly present, if it has not been removed
already by the providers of the economic data, is a pattern of seasonal ﬂuc
tuations. In this case, given that the ﬂuctuations reside in limited frequency
bands, it is easier to provide a specific definition of the seasonal component,
albeit that there is still scope for alternative definitions.
Notwithstanding the illdefined nature of these components, econometri
cians have tended to adopt particular models for the trend and for the seasonal
1
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D.S.G. POLLOCK: Filtering Macroeconomic Data
ﬂuctuations. The trend is commonly modelled by a firstorder random walk
with drift, which is an accumulation of a whitenoise sequence of independently
and identically distributed random variables. The drift occurs when the vari
ables have a nonzero mean—a positive mean giving rise to an upward drift.
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 Fall '11
 D.S.G.Pollock
 Signal Processing, LTI system theory, Stationary process, D.S.G. POLLOCK, Ωη

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