An Exploratory Study of the Bi-Spectrum of Economic Time Series

The main motivation of the analysis presented here is to investigate certain non-linear properties of the mechanism which may be supposed to generate a time series. I investigate specifically quadratic terms in the generating model. For a wide-sense stationary process the spectrum describes the linear mechanism. For a third moment stationary process the bi-spectrum describes the quadratic terms in the mechanism. I will discuss the derivation of the bi-spectrum. The interpretation of the bi-spectrum in terms of certain non-linear transformations will also be discussed. As I have chosen to compute bi-spectra by means of complexdemodulation, there will be a major digression into the computation and analysis of complex-demodulates. For background information concerning the estimation of spectra the reader is referred to Blackman and Tukey (1959). Analysis of certain non-linear systems is discussed in Wiener (1958), while the statistical theory of higher order spectra (polyspectra) is treated in Brillinger (1964). An extremely interesting application of the bispectrum is given in Hasselman et al. (1963).