The SPECTRA Procedure

White Noise Test

PROC SPECTRA prints two test statistics for white noise when the WHITETEST option is specified: Fisher’s Kappa (Davis, 1941; Fuller, 1976) and Bartlett’s Kolmogorov-Smirnov statistic (Bartlett, 1966; Fuller, 1976; Durbin, 1967).

If the time series is a sequence of independent random variables with mean 0 and variance ${{\sigma }^{2}}$, then the periodogram, ${J_{k}}$, will have the same expected value for all ${k}$. For a time series with nonzero autocorrelation, each ordinate of the periodogram, ${J_{k}}$, will have different expected values. The Fisher’s Kappa statistic tests whether the largest ${J_{k}}$ can be considered different from the mean of the ${J_{k}}$. Critical values for the Fisher’s Kappa test can be found in Fuller 1976.

The Kolmogorov-Smirnov statistic reported by PROC SPECTRA has the same asymptotic distribution as Bartlett’s test (Durbin, 1967). The Kolmogorov-Smirnov statistic compares the normalized cumulative periodogram with the cumulative distribution function of a uniform(0,1) random variable. The normalized cumulative periodogram, ${F_{j}}$, of the series is

\[  F_{j} = \frac{\sum _{k=1}^{j}{J_{k}}}{\sum _{k=1}^{m}{J_{k}}}, j = 1, 2 \ldots , m-1  \]

where ${m=\frac{n}{2}}$ if n is even or ${m=\frac{n-1}{2}}$ if n is odd. The test statistic is the maximum absolute difference of the normalized cumulative periodogram and the uniform cumulative distribution function. Approximate p-values for Bartlett’s Kolmogorov-Smirnov test statistics are provided with the test statistics. Small p-values cause you to reject the null-hypothesis that the series is white noise.