  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).
