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

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.

Number of Pages in PDF File: 85

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

JEL Classification: E32, E43, E44, G12