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# ELGARTEXT - FILTERING MACROECONOMIC DATA By D.S.G Pollock...

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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 frequency-domain 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 doubly-infinite 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 frequency-selective 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 ill-defined 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 first-order random walk with drift, which is an accumulation of a white-noise 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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ELGARTEXT - FILTERING MACROECONOMIC DATA By D.S.G Pollock...

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