Mathematical and Statistical Methods for Actuarial Sciences and Finance

Nonlinear cointegration in financial time series

Claudio Pizzi

Abstract.In this paper, the concept of linear cointegration as introduced by Engle and

Granger [5] is merged into the local paradigm. Adopting a local approach enables the achieve-

ment of a local error correction model characterised by dynamic parameters. Another important

result obtained using the local paradigm is that the mechanism that leads the dynamic system

back to a steady state is no longer a constant: it is a function not defined a priori but estimated

point by point.

Key words:nonlinearity, cointegration, local polynomial model

1 Introduction

One of the aims of the statistical analysis of a time series is to enable the researcher

to build a simplified representation of the data-generating process (DGP) and/or the

relationship amongst the different phenomena under study. The methods for identi-

fying and estimating these models are based on the assumption of the stationarity of

the DGP. Nevertheless, this assumption is often violated when considering financial

phenomena, for example stock price, interest rates, exchange rates and so on. The

financial time series usually present a non-stationarity of the first order if not higher.

In the case of the construction of a regressive model, the presence of unit roots in

the time series means attention should be paid to the possible cointegration amongst

the variables.

The cointegration idea, which characterises the long–run relationship between

two (or several) time series, can be represented by estimating a vector of parameters

and can be used to build a dynamic model that enables both long-run relationships

and also some transitional short-run information to be highlighted. This enables the

representation of an error correction model that can be considered as a dynamic system

characterised by the fact that any departure from the steady state generates a short-run

dynamic.

The linear cointegration concept introduced by Engle and Granger [5] has been

broadly debated in the literature and much has been published on this topic. The

M. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
© Springer-Verlag Italia 2010