Summability of stochastic processes: a generalization of integration for non-linear processes

The order of integration is valid to characterize linear processes; but it is not appropriate for non-linear worlds. We propose the concept of summability (a re-scaled partial sum of the process being O-p(1)) to handle non-linearities. The paper shows that this new concept, S (delta): (i) generalizes I (delta); (ii) measures the degree of persistence as well as of the evolution of the variance; (iii) controls the balancedness of non-linear relationships; (iv) opens the door to the concept of co-summability which represents a generalization of co-integration for non-linear processes. To make this concept empirically applicable, an estimator for delta and its asymptotic properties are provided. The finite sample performance of subsampling confidence intervals is analyzed via a Monte Carlo experiment. The paper finishes with the estimation of the degree of summability of the macroeconomic variables in an extended version of the Nelson-Plosser database. Financial support from SEJ-2007-63098, ECO-2010-19357, CONSOLIDER 2010 (CSD 2006-00016), and EXCELECON S-2007/HUM-044 grants is gratefully acknowledged.