Essays on cointegrating polynomial regressions with applications to the EKC

Abstract

We consider cointegrating polynomial regression (CPR) relationships as a special case of nonlinear cointegrating relationships, which steadily gain interest recently in many different areas of applied research, e. g., empirical macroeconomics, empirical finance or environmental economics. CPRs include deterministic variables as well as polynomially transformed integrated variables as explanatory variables and stationary errors. The stochastic regressors are allowed to be endogenous and the errors are allowed to be serially correlated. In this case, the limiting distribution of the OLS estimator is contaminated by second order bias terms. Therefore, various modified OLS estimators are considered that have a zero mean Gaussian mixture limiting distribution and allow for valid asymptotic chi-squared inference. The main motivation for developing estimation and testing techniques for CPRs is the analysis of the environmental Kuznets curve (EKC) hypothesis, which postulates an inverted U-shaped relationship between measures of economic activity, typically proxied by GDP per capita, and pollution. The logarithm of GDP per capita is often found to be integrated of order one. The hypothesized inverted U-shape requires the inclusion of log GDP per capita and its square, or even higher order powers, as explanatory variables. Polynomially transformed integrated processes are not integrated themselves, which is neglected in the large part of the EKC literature. Instead, powers of integrated processes are considered to be integrated and standard unit root and cointegration techniques are applied. We show that the standard fully modified OLS estimator has the same asymptotic distribution in the CPR setting as its tailor-made CPR extension. For cointegration testing, however, the use of standard techniques in CPRs has an effect also asymptotically, since invalid critical values are carried out. A simulation study underlines the theoretical findings in the sense that the difference between standard cointegration methods and their CPR extensions are small in terms of bias and root mean squared error. However, tests based upon the latter outperform tests based upon the former in terms of lower over-rejections under the null and larger (size-corrected) power for hypothesis testing as well as cointegration testing. We also consider an extension of the recently developed integrated modified OLS (IM-OLS) estimator for CPRs, which is shown to have a zero mean Gaussian mixture limiting distribution that forms the basis for asymptotic standard inference. Furthermore, we provide fixed-b asymptotic theory to capture the impact of kernel and bandwidth choices on the sampling distributions of the estimators. A simulation study shows that IM-OLS based tests can lead to substantially smaller size distortions for hypothesis testing under the null at the cost of some minor loss in size-corrected power compared to other modified OLS based tests. We also apply the developed methods to analyze the EKC hypothesis based on a data set containing carbon dioxide emissions and gross domestic product for 19 early industrialized countries over the period 1870-2013. By means of an IM-OLS residual based cointegration test, we find evidence for the existence of a quadratic EKC relationship for six countries and in one additional country for a cubic EKC relationship. Finally, we analyze the EKC hypothesis for carbon dioxide emissions in a multi-country system of equations approach for six early industrialized countries. A seemingly unrelated cointegrating polynomial regressions approach allows to take into account cross-sectional dependence and parameter heterogeneity. Wald-type tests are carried out to test for poolability, i.e. equality of parameters, for subsets of coefficients over potentially different subsets of cross-sections. Subsequent estimation in a group-wise pooled setting reduces the number of estimated parameters about one third in the empirical application, whereas the estimation results are similar to those obtained in unrestricted individual CPRs. We also show that estimation in a panel-type approach including cross-sectional parameter homogeneity, as typically pursued in the empirical EKC literature, is rejected by poolability testing and performs severely worse.