Cointegration, Root Functions and Minimal Bases

This paper discusses the notion of cointegrating space for linear processes integrated of any order. It first shows that the notions of (polynomial) cointegrating vectors and of root functions coincide. Second, it discusses how the cointegrating space can be defined (i) as a vector space of polynomial vectors over complex scalars, (ii) as a free module of polynomial vectors over scalar polynomials, or finally (iii) as a vector space of rational vectors over rational scalars. Third, it shows that a canonical set of root functions can be used as a basis of the various notions of cointegrating space. Fourth, it reviews results on how to reduce polynomial bases to minimal order—i.e., minimal bases. The application of these results to Vector AutoRegressive processes integrated of order 2 is found to imply the separation of polynomial cointegrating vectors from non-polynomial ones.

Dynamic Pricing Recognition on E-Commerce Platforms with VAR Processes

In an environment such as e-commerce, characterized by the presence of numerous agents, competition based on product characteristics is a very important aspect. This paper proposes a model based on vector autoregressive processes (VAR) and Lasso penalization to detect and examine the dynamics that govern real-time price competition in electronic marketplaces. Employing this model, an empirical study was performed on the price trends of smartphone models on the major electronic sales platforms of the Italian market. The proposed model detects real-time price variations in single vendors, based on the variations of their direct competitors. The statistical method adopted in this analysis may be useful for e-commerce companies that conduct market analyses of competitors’ pricing strategies.

REPRESENTATION OF I(1) AND I(2) AUTOREGRESSIVE HILBERTIAN PROCESSES

We develop versions of the Granger–Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.