Summability of stochastic processes—A generalization of integration for non-linear processes

Abstract

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 Op(1)) to handle non-linearities. The paper shows that this new concept, S(δ): (i) generalizes I(δ); (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 δ 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.