Periodograms of unit root time series: Distributions and tests

Periodograms are often used to characterize time series. They decompose the variation in the data into periodic components and their statistical properties for stationary series are well understood. The periodogram can be computed for any sequence of numbers and we are interested in studying its statistical properties when the underlying data have a time series structure with a unit root. Knowing these properties gives us the ability to look at nonstationarity on a frequency by frequency basis. The pervasive use of periodogram ordinates in applied work and the frequent occurrence of apparently nonstationary data in practice are sufficient alone to motivate interest in these distributions, however we also suggest a way to use the results in a formal test for unit roots. The test has the advantage of using nonseasonal ordinates of the periodogram, thus being invariant to regular periodicities in the data. Section 1 of the paper is introductory. Section 2 develops the periodogram and a convenient normalization far an autoregressive process of order 1, leading to distributional results and a table of percentiles. In section 3, an extension is made to general processes in which the first difference is a stationary autoregressive moving average. The effect of a deterministic trend or drift on the periodogram is considered in Section 4 and an adjustment for trend is given in Section 5. Interestingly, it will be seen that the adjustment has an advantage over the usual time domain tests far unit roots in that standard chi(2) distributions result. Tests based on the periodogram and their power properties are displayed with examples in Section 6. An advantage of these tests is their invariance to deterministic seasonal components which, for the example series, is a reasonably likely scenario. In the second example on retail sales, it is seen that breaking trends can also be accommodated with a minor modification of this technique and the advantage of having standard distributions will become apparent.