Autocovariance Functions of Series and of Their Transforms

27 Pages Posted: 15 Jan 2012

See all articles by Karim M. Abadir

Karim M. Abadir

Imperial College Business School

Gabriel Talmain

University of York - Department of Economics and Related Studies

Date Written: 2005

Abstract

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.

Suggested Citation

Abadir, Karim M. and Talmain, Gabriel, Autocovariance Functions of Series and of Their Transforms (2005). Journal of Econometrics, Vol. 124, No. 2, p. 227, 2005. Available at SSRN: https://ssrn.com/abstract=1985286

Karim M. Abadir (Contact Author)

Imperial College Business School ( email )

South Kensington Campus
Exhibition Road
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/portal/page?_pageid=61,629646&_dad=portallive&_schema=PORTALLIVE

Gabriel Talmain

University of York - Department of Economics and Related Studies ( email )

Heslington
York, YO1 5DD
United Kingdom

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