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Autocovariance functions of series and of their transforms. (English) Zbl 1335.62123

Summary: 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 \(\tan^{-1}(\cdot)\) transformation which will turn possibly unstable series into stable ones. As an illustration, we obtain the autocovariance function of the \(\tan^{-1}(\cdot)\) 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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