Co-summability from linear to non-linear cointegration
While co-integration theory is an ideal framework to study linear relationships among persistent economic time series, the intrinsic linearity in the concepts of integration and co-integration makes it unsuitable to study non-linear long run relations among persistent processes. This drawback hinders the empirical analysis of modern macroeconomics, which often addresses asymmetric responses to policy interventions, multiplicity of equilibria, transitions between regimes or polynomial approximations to unknown functions. In this paper, to cope with non-linear relations and consequently to generalise co-integration, we formalise the idea of co-summability. It is built upon the concept order of summability developed by Berenguer-Rico and Gonzalo (2013), which, in turn, was conceived to address non-linear transformations of persistent processes. Theoretically, a co-summable relationship is balanced -in terms of the variables involved having the same order of summability- and describes a long run equilibrium that can be non-linear -in the sense that the errors have a lower order of summability. To test for these types of equilibria, inference tools for balancedness and cosummability are designed and their asymptotic properties are analysed. Their finite sample performance is studied via Monte Carlo experiments. The practical strength of co-summability theory is shown through two empirical applications. Specifically, asymmetric preferences of central bankers and the environmental Kuznets curve hypothesis are studied through the lens of co-summability.
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