Cointegration and Representation of Cointegrated Autoregressive Processes in Banach Spaces

Abstract

We extend the notion of cointegration for time series taking values in a potentially infinite dimensional Banach space. Examples of such time series include stochastic processes in $C[0,1]$ equipped with the supremum distance and those in a finite dimensional vector space equipped with a non-Euclidean distance. We then develop versions of the Granger–Johansen representation theorems for I(1) and I(2) autoregressive (AR) processes taking values in such a space. To achieve this goal, we first note that an AR(p) law of motion can be characterized by a linear operator pencil (an operator-valued map with certain properties) via the companion form representation, and then study the spectral properties of a linear operator pencil to obtain a necessary and sufficient condition for a given AR(p) law of motion to admit I(1) or I(2) solutions. These operator-theoretic results form a fundamental basis for our representation theorems. Furthermore, it is shown that our operator-theoretic approach is in fact a closely related extension of the conventional approach taken in a Euclidean space setting. Our theoretical results may be especially relevant in a recently growing literature on functional time series analysis in Banach spaces.

Functional principal component analysis for cointegrated functional time series

“…Finiteness of 𝜑 seems to be reasonable in many empirical examples; see Chang et al (2016, Section 5) and Nielsen et al (2022, Section 5). Moreover, it is known that functional autoregressive (AR) processes with unit roots (including functional ARMA processes that are considered in for example, Klepsch et al, 2017) with compact AR operators always satisfy Assumption M(ii); see Beare and Seo (2020), Franchi and Paruolo (2020) and Seo (2022). It is common to assume compactness of AR operators in statistical analysis of such time series, and thus Assumption M(ii) does not seem to be restrictive in practice.…”