Chapter 5: Filtering macroeconomic data
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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. If 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 synthesized from trigonometrical 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.

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