A Persistence-Based Wold-Type Decomposition for Stationary Time Series

85 Pages Posted: 15 Dec 2011 Last revised: 13 Aug 2015

Fulvio Ortu

Bocconi University - Department of Finance

Federico Severino

Bocconi University

Andrea Tamoni

London School of Economics & Political Science (LSE)

Claudio Tebaldi

Bocconi University, IGIER and CAREFIN

Date Written: August 12, 2015

Abstract

The Classical Wold Decomposition Theorem allows to split a weakly stationary time series x into a non-deterministic component, driven by uncorrelated innovations, and a deterministic term. This decomposition is a special case of the Abstract Wold Theorem, which deals with isometric operators defined on Hilbert spaces. As the lag operator is isometric on the Hilbert space H_t(x) spanned by the sequence {x_{t-k}_k}, the Classical Wold Decomposition for time series obtains. Moreover, the \emph{scaling operator} is isometric on the Hilbert space H_t(e), spanned by the classical Wold innovations of x, and it provides an Extended Wold Decomposition. Thus, the process x may be seen as a sum, across scales, of uncorrelated components that explain different layers of persistence, from temporary fluctuations to low-frequency shocks. Multiscale impulse response functions are, then, defined. Conversely, the sum of suitable uncorrelated components delivers a weakly stationary process. This decomposition fruitfully applies to ARMA and fractional ARIMA processes.

Keywords: Wold decomposition, Abstract Wold Theorem, persistence heterogeneity, impulse response functions, forecasting

JEL Classification: E32, E43, E44, G12

Suggested Citation

Ortu, Fulvio and Severino, Federico and Tamoni, Andrea and Tebaldi, Claudio, A Persistence-Based Wold-Type Decomposition for Stationary Time Series (August 12, 2015). Available at SSRN: https://ssrn.com/abstract=1973049 or http://dx.doi.org/10.2139/ssrn.1973049

Fulvio Ortu

Bocconi University - Department of Finance ( email )

Via Roentgen 1
Milano, MI 20136
Italy

Federico Severino

Bocconi University ( email )

Via Sarfatti, 25
Milan, MI 20136
Italy

Andrea Tamoni

London School of Economics & Political Science (LSE) ( email )

Houghton Street
London, WC2A 2AE
United Kingdom
02079557303 (Phone)

Claudio Tebaldi (Contact Author)

Bocconi University, IGIER and CAREFIN ( email )

Via Roentgen 1
Milan, 20136
Italy

Paper statistics

Downloads
161
Rank
147,728
Abstract Views
888