Autocovariance Functions of Series and of Their Transforms

We derive a method to link exactly the autocovariance functions of two arbitrary instantaneous transformations of a time series. This is useful, for example, when one wishes to describe the time-series effect of applying a nonlinear transformation to a series whose properties are known. As an illustration, we provide two corollaries and three examples. The first corollary is on the commonly-used logarithmic transformation, and is applied to a geometric Auto-Regressive (AR) process, as well as to a positive Moving-Average (MA) process. The second corollary is on the tan⁻¹(.) transformation which will turn possibly unstable series into stable ones. As an illustration, we obtain the autocovariance function of the tan⁻¹(.) of an arithmetic AR process. This filter, while always producing a bounded process, preserves the stability/instability distinction of the original series, a feature that can be turned to an advantage in the design of tests. We then present a probabilistic interpretation of the main features of the new autocovariance function. We also provide a mathematical lemma on a general integral which is of independent interest