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. For example, this is useful when one wants to derive the autocovariance of the logarithm of a series from the known autocovariance of the original series and, more generally, when one wishes to describe the time-series effects of applying a nonlinear transformation to a process 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.