This paper develops the asymptotic theory of a Fully Modified Generalized Least Squares estimator for multivariate cointegrating polynomial regressions. Such regressions allow for deterministic trends, stochastic trends and integer powers of stochastic trends to enter the cointegrating relations. Our fully modified estimator incorporates: (1) the direct estimation of the inverse autocovariance matrix of the multidimensional errors, and (2) second order bias corrections. The resulting estimator has the intuitive interpretation of applying a weighted least squares objective function to filtered data series. Moreover, the required second order bias corrections are convenient byproducts of our approach and lead to standard asymptotic inference. We also study several multivariate KPSS-type of tests for the null of cointegration. A comprehensive simulation study shows good performance of the FM-GLS estimator and the related tests. As a practical illustration, we reinvestigate the Environmental Kuznets Curve (EKC) hypothesis for six early industrialized countries as in Wagner et al. (2020).
Cointegrating Polynomial Regression, Cointegration Testing, Environmental Kuznets Curve, Fully Modified Estimation, Generalized Least Squares
We develop a Feasible Generalized Least Squares (FGLS) estimator of structural break date in models that allow level and/or trend breaks. The proposed FGLS estimator reduces the mean squared errors and improves the testing power compared to existing estimators, leading to more informative confidence sets. This estimator is based on a consistent estimation of the inverse autocovariance matrix of errors that are stacked over time. We find a monotone and continuous cubic polynomial transformation of the break date estimates can be approximated by a nonstandard yet nuisance parameter free distribution asymptotically. The transformation itself contains some unknown parameters which can be estimated consistently. The asymptotic results can be applied for inference with a single, known, set of critical values, requiring no simulation. The asymptotic distribution captures the asymmetry and bimodality observed in finite samples. We obtain the confidence sets/intervals by inverting multiple likelihood ratio (LR) tests for break dates. An extensive simulation study shows good coverage and short length using the LR-based constructions even in models with highly persistent errors and small break sizes.
Level Break, Trend Break, Generalized Least Squares, Confidence Sets
The Environment Kuznets Curve (EKC) predicts an inverted U-shaped relationship between economic growth and environmental pollution. Current analyses frequently employ models which restrict the nonlinearities in the data to be explained by the economic growth variable only. We propose a Generalized Cointegrating Polynomial Regression (GCPR) with flexible time trends to proxy time eects such as technological progress and/or environmental awareness. More specifically, a GCPR includes flexible powers of deterministic trends and integer powers of stochastic trends. We estimate the GCPR by nonlinear least squares and derive its asymptotic distribution. Endogeneity of the regressors can introduce nuisance parameters into this limiting distribution but a simulated approach nevertheless enables us to conduct valid inference. Moreover, a subsampling KPSS test can be used to check the stationarity of the errors. A comprehensive simulation study shows good performance of the simulated inference approach and the subsampling KPSS test. We illustrate the GCPR approach on a dataset of 18 industrialised countries containing GDP and CO2 emissions. We conclude that: (1) the evidence for an EKC is significantly reduced when a nonlinear time trend is included, and (2) a linear cointegrating relation between GDP and CO2 around a power law trend also provides an accurate description of the data.
Cointegration Testing, Environmental Kuznets Curve, Generalized Cointegrating Polynomial Regression, Nonlinear Least Squares, Power Law Trends