Regional hydrologic analysis: Ordinary and generalized least squares revisited

Generalized least squares (GLS) regional regression procedures have been developed for estimating river flow quantiles. A widely used GLS procedure employs a simplified model error structure and average covariances when constructing an approximate residual error covariance matrix. This paper compares that GLS estimator (b̂GLS MC ), an idealized GLS estimator (b̂GLS E ) based on the simplifying assumptions of b̂GLS MC with true underlying statistics in a region, the best possible GLS estimator (b̂GLS T ) obtained using the true residual error covariance matrix, and the ordinary least squares estimator (b̂OLS T ). Useful analytic expressions are developed for the variance of b̂GLS T , b̂GLS E , and b̂OLS T . For previously examined cases the average sampling mean square error (mses) of b̂GLS E was the same as the mses of b̂GLS T , and the mses of b̂GLS MC usually was larger than the mses of both b̂GLS E and b̂GLS T . The loss in efficiency of b̂GLS MC was mostly due to estimating streamflow statistics employed in the construction of the residual error covariance matrix rather than the simplifying assumptions in presently employed GLS estimators. The new analytic expressions were used to compare the performance of the OLS and GLS estimators for new cases representing greater model variability across sites as well as the effect return period has on the estimators’ relative performance. For a more heteroscedastic model error variance and larger return periods, some increase in the mses of b̂GLS E relative to the mses of b̂GLS T was observed.