Detection of periodic autocorrelation in time series data via zero-crossings (Q2703245)

When analyzing a seasonal time series \(\{Y(t);\;t=1,2,\dots,N\}\) it is frequently assumed that \(Y(t)\) may be written as NEWLINE\[NEWLINEY(t)=\mu(t)+\sigma(t)Z(t),NEWLINE\]NEWLINE where \(\mu(t)\) and \(\sigma^{2}(t)\) are the periodic mean and variance functions, respectively, and \(Z(t)\) is wide-sense stationary with mean zero, variance one, and autocorrelation function NEWLINE\[NEWLINE\text{corr} \{Z(t), Z(t-k)\}\equiv \rho_{k}(t)NEWLINE\]NEWLINE which does not depend on \(t.\) The time series \(Y(t)\) is than adjusted by subtracting an estimate of \(\mu(t)\) and dividing by an estimate of \(\sigma(t)\) (or, if only the mean is assumed periodic, by differencing). However, in fields such as hydrology, meteorology, climatology and economics, there are examples of seasonal time series with apparent periodicities in the autocorrelation function at various lags \(k.\)NEWLINENEWLINENEWLINEIn this paper, a statistical procedure based on zero-crossing counts for detection of periodic autocorrelations in time series such as \(Z(t)\) which has been adjusted for periodic mean and variance functions is developed. A strong point of this method is its robustness. It is shown that percentiles of the test statistic are relatively stable for a variety of ARMA processes.