Testing for Homogeneous Variations in Periodic Data

When analyzing periodic data, there are not one but two intervals to be considered. Often these intervals correspond to seasons and years. Specifically, one expects relationships to exist (a) between observations for successive seasons in a particular year and (b) between the observations for the same season in successive years. Periodic data of this type may be tabulated in $s$ rows due to $s$ seasons and $n$ columns due to $n$ years, where $s$ is the periodicity of the data. When the column variances are equal, the periodic pattern of the data generally remains stable over the years. On the other hand, when the variances are unequal, the periodic pattern changes over the years. The usual Bartlett (1937) test for the equality of $n$ variances may be highly misleading because of the correlation among observations within rows as well as columns. In the present paper, we develop an asymptotically most powerful test for testing the homogeneity of column variances in a two-way layout assuming that observations follow a seasonal ARMA process. A modification to the standard Bartlett test is also proposed which accounts for the autocorrelations. The modified Bartlett test is simple to compute and it has asymptotically $\chi^2$ distribution. The tests are illustrated by a numerical example.