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 squared, then the periodogram, upper J Subscript k, will have the same expected value for all k. For a time series with nonzero autocorrelation, each ordinate of the periodogram, upper J Subscript k, will have different expected values. The Fisher’s Kappa statistic tests whether the largest upper J Subscript k can be considered different from the mean of the upper J Subscript 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, upper F Subscript j, of the series is

upper F Subscript j Baseline equals StartFraction sigma-summation Underscript k equals 1 Overscript j Endscripts upper J Subscript k Baseline Over sigma-summation Underscript k equals 1 Overscript m Endscripts upper J Subscript k Baseline EndFraction comma j equals 1 comma 2 ellipsis comma m minus 1

where m equals StartFraction n Over 2 EndFraction if n is even or m equals StartFraction n minus 1 Over 2 EndFraction 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.

Last updated: June 19, 2025