Speed of Adjustment in Cointegrated Systems

This paper considers the speed of adjustment to long-run equilibria, in the context of cointegrated Vector Autoregressive Processes (VAR). We discuss the definition of multivariate p-lives for any indicator of predictive ability, concentrating on cumulated interim multipliers which converge to impact factor for increasing forecasting horizon. Interim multipliers are related to autoregressive Granger-causality coefficients, structural or generalized cumulative impulse responses. We discuss the relation of the present definition of multivariate p-lives with existing definitions for univariate time series and for nonlinear multivariate stationary processes. For multivariate (possibly cointegrated) VAR systems, p-lives are functions of the dynamics of the system only,and do not depend on the history path on which the forecast is based. Hence one can discuss inference on p-lives as (discrete) functions of parameters in the VAR model. We discuss a likelihood-based approach, both for point estimation and for confidence regions. An illustrative application to adjustment to purchasing-power parity (PPP) is presented.