Wiener–kolmogorov Filtering, Frequency-selective Filtering and Polynomial Regression

The classical theory of statistical signal extraction presupposes lengthy data sequences, which are assumed, in theory, to be doubly infinite or semi-infinite— see Whittle (1983), for example. In many practical cases, and in most econometric applications, the available data are, to the contrary, both strongly trended and of a limited duration. This paper is concerned with the theory of finite-sample signal extraction; and it shows how the classical Wiener–Kolmogorov theory of signal extraction can be adapted to cater to short sequences generated by processes that may be nonstationary. Alternative methods of processing finite samples, which work in the frequency domain, are also described; and their relation to the time-domain implementations of the Wiener–Kolmogorov methods are demonstrated. The frequency-domain methods have the advantage that they are able to achieve clear separations of components that reside in adjacent frequency bands in a way that the time-domain methods cannot.