Nonlinear cointegrating regressions with nonstationary nonlinear heteroskedasticity

This article considers the parametric estimation of nonlinear cointegration with nonstationary nonlinear heteroskedasticity (NNH). Due to the presence of NNH error, only a subset of nonlinear least squares (NLS) estimators can be shown to be consistent, with the aid of our newly proposed sequential proof strategy. To achieve estimation consistency, the weighted NLS (WNLS) estimators are further entertained, and their asymptotic properties are found to depend on the integrability of the heterogeneity-generating function. To our surprise, the WNLS estimators are not only consistent but can even enjoy accelerated convergence rates compared with their NLS counterparts under certain scenarios. We then consider efficient estimation by removing the asymptotic bias induced by the dependence between the regression errors and the innovations of integrated regressors. The resulting estimators become asymptotically mixed normal and are more efficient compared with the corresponding NLS and WNLS estimators. Finally, the Wald tests based on the above estimators are investigated, with a bootstrap procedure proposed to approximate their finite sample distributions. Simulation results suggest that the proposed estimators and tests enjoy satisfactory finite sample performance. Finally, an empirical application to the U.S. macroeconomic data demonstrates the conditional heteroskedasticity for the cointegration between the per capita real personal consumption expenditure and disposable personal income, which can be modeled as a quadratic function of the logarithm of gross domestic product.