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

publication date

  • January 2014

start page

  • 331

end page

  • 341


  • Part 2


  • 178

International Standard Serial Number (ISSN)

  • 0304-4076

Electronic International Standard Serial Number (EISSN)

  • 1872-6895


  • 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.


  • Economics


  • co-integration; co-summability; integrated processes; non-linear balanced relationships; non-linear processes; summability; macroeconomic time-series; asymptotics; cointegration; regressions; convergence; root