["Oxford Bulletin of Economics and Statistics, Volume 88, Issue 3, Page 473-483, June 2026. ", "\nABSTRACT\nA new feasible generalized least squares estimator is proposed. Our estimator incorporates (1) the inverse autocovariance matrix of 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 conventional asymptotic inference. Based on the proposed fully modified (FM) estimator, a multivariate KPSS‐type test for the null of cointegration is constructed. We subsequently undertake a comprehensive Monte Carlo study to compare the performance of the FM estimators and the related tests. The proposed estimator and the implied test statistics for linear hypotheses and cointegration show good performance in finite samples. We illustrate our methods by estimating long‐run fiscal reaction functions for Austria, Germany, Norway, Portugal, and Switzerland.\n"]