The Estimation of Spectra and Cross-Spectra on Short Time Series Data

Abstract

IN HIS RECENT BOOK [20, (Chapters 4, 5)] Professor Hannan has mnade an extensive theoretical study on the asymptotic properties of the multivariate-analytic estimators composed of the direct Fourier Transforms of time series data as contrasted to the estimators obtained through the autocovariances.2 The present paper is concerned with these estimators for the case in which short time series covering, say, 30 to 200 time points have to be dealt with. The direct Fourier transforms of data are adopted as the basic material, not because of speedy computations (Fast Fourier transform) to obtain them, which does not matter much in the presenit case, but because of the ease which they introduce in devising and understanding the estimators in the framework of standard multivariate analysis. We exploit, in particular, the closeness to diagonality of the covariance matrix of the transforms over the multi-frequency points.3 Thus a solution is proposed for a well known problem in the field, that is, how to recover the loss of degrees of freedom caused by a simple data window which has to be used to eliminate the leakages from enormous peaks located at frequencies distant from the one of interest. An empirical study demonstrates that the proposed method is extremely effective in estimating spectra and cross-spectra of economic time series data that are frequently used in econometric studies, say, 15 years, quarterly data. It frees us fromn bothering too much with the prewhitening, and, perhaps it does away with the seasonal adjustment. The present author's conclusion as to the required length of data is contrary to the view expressed in Hannan [20], and, in fact, the present study might contribute to enlarging the applicability of econometric, spectral methods to the development of which