Methodology for Trend Estimation

The problem of trend estimation in econometrics has had a long history, and the techniques which can be deployed have been evolving gradually over many years. The forces of evolution have been twofold. On one hand is the gradual improvement in statistical and computational techniques which has been accompanied by improvements in the processing power of computers and in the accessibility of software programs. On the other hand are the methodological developments within the discipline of econometrics. The econometric approach to trend estimation is based upon the notion that a time series is composed of several components of independent origin which are combined by addition or by multiplication. Usually, a multiplicative combination can be reduced to an additive one by the simple expedient of taking logarithms. The components of the time series can be regarded as Fourier combinations of trigonometrical functions—i.e. of sines and cosines—whose frequencies fall within specified ranges. Over the range of the frequencies which pertain to a particular component, one can define a spectral density function which represents the squared amplitudes of the constituent trigonometrical functions. If the frequency ranges of the components are completely segregated, then it is possible, in principle, to achieve a definitive separation of the time series into its independent components. If the frequency ranges of the components overlap, then it is possible to achieve a tentative separation in which trigonometrical functions of the same frequencies are present in two or more components of the time series. In each estimated component, the functions acquire the amplitudes which are indicated by the appropriate spectral density function. Since the estimates of such overlapping components originate from the same empirical data sequence, they are bound to be correlated with each other—which contradicts the theoretical assumptions regarding the components. In recent years, the mainstay of econometric trend estimation has been the Wiener–Kolmogorov theory of signal extraction (See [10] and [17]). The theory