Simultaneous Confidence Regions for the Frequency Analysis of Multiple Time Series

Abstract

Abstract In the frequency analysis of several time series observed simultaneously, it is often necessary to adapt classical results of linear models and multiple regression to situations involving complex-valued variates. Given a real-valued stochastic time series, Y(t), and a suitable vector-valued fixed series, X(t), with components Xi (t) (j = 1, 2, …, r, t = 0, ± 1, …), it may be of interest to assess the effect, in the frequency domain, of each component series of X(t) on the behavior of Y(t). If values of Y(t) and X(t) are made available for t = 0, 1, …, T − 1, then under certain conditions and for large T at frequency λ the finite Fourier transform of Y(t) is approximately linearly related to the finite Fourier transform of X(t) through a complex-valued vector A(λ). In an asymptotic sense, A(λ), with entries Aj (λ) (j = 1, 2, …, r), may be interpreted as a vector of complex regression coefficients of Y(t) on X(t) at frequency λ The amount of amplification and phase shift by which the Xi (t) series must be modified in order to give an optimum fit, on average, to the Y(t) series at frequency λ is, therefore, determined by the real-valued gain, Gj (λ) = |Aj (λ)|, and phase, φ j (λ) = arg Aj (λ), respectively, of Y(t) over Xj (t). After estimating Aj (λ), and also Gj (λ) and φ j (λ), using results from multiple regression analysis adapted to complex-valued variates, a gain plot and a phase plot can be made for each j = 1, 2, …, r, with appropriate confidence regions drawn on the plot at each frequency λ to assess the degree of association between Y(t) and the Xj (t) series. In this article new asymptotic simultaneous confidence regions for gains and phases are derived, and we show theoretically that they are shorter than similar confidence regions currently available in the statistical literature. The tools used here for constructing these improved regions include a differential identity for the complex multivariate Gaussian distribution and a multivariate probability inequality for the amplitudes of components of a complex-valued Gaussian random vector. Numerical computations covering a variety of possible situations show the actual amount of asymptotic improvement. Empirical investigations of the finite sample behavior of the proposed new simultaneous confidence regions indicate good overall performance.