Tests for periodic components in multiple time series

Abstract

SUMMARY The problem of testing for periodicities in multiple time series is considered. The test statistic proposed is based on the Euclidean norm of the matrix of periodograms. We first obtain the exact distribution of our test statistic under the hypothesis that observations are taken from an n-vector time series consisting of independent series of independent normal variables. We extend the theory to normal linear series with a specified spectral density matrix and then show that a suitable estimate can be used to replace this matrix in our test statistic. These results are asymptotic in nature. Extreme value theorems are stated for the case of normal variates and it is shown that the distributions for nonnormal series can be approximated adequately by those for normal series. An exact test for periodic components in an univariate time series was first proposed by Fisher (1929) who considered the null hypothesis of normal white noise. The test was based upon the maximum of a set of periodogram ordinates each divided by their sum, the division making it unnecessary to know in advance the variance of the components of the series. The use of the rth largest of these ordinates, r > 1, in testing for periodicities was also discussed by Fisher (1940). Whittle (1951, p. 101) showed how one could apply the Fisher tests to normal linear series provided the number of observations was large and one could specify the spectral density for the series. Hannan (1961) proposed a modification to this test that removed the necessity of knowing beforehand the spectral density function. Nicholls (1967) proposed a variation on Hannan's test statistic that improves its power under certain alternatives. Some extreme value results for periodogram ordinates were given by Walker (1965), who also demonstrated that dropping the normality condition has little effect on the large sample distributions, at least in the upper tails. In the sequel we consider the problem of testing for periodicities in multiple time series and pursue a course that parallels the historical development for univariate series as outlined in the paragraph above. Our tests are based on the Euclidean norm of the matrix of periodo- grams. In this connection Nicholls (1969) has discussed a test for discrete mass in a cross- spectrum. We consider real, discrete time parameter, n-vector, time series, Z(t) (t = 0, + 1, + 2, ...), where Z(t) = It+m(t) +X(t), with ,u+m(t) being a mean value function and X( ) being a weakly stationary, zero mean, time series possessing a spectral density matrix fxx(A) with tr{fxx(A)} <1 M < oo (JAI < 7r). We also assume the existence of a moving average representation 00 wo,t) , CO v - )