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 (<(beta)over bar>(MC)(GLS), an idealized GLS estimator (<(beta)over bar>(E)(GLS)) based on the simplifying assumptions of <(beta)over bar>(MC)(GLS) with true underlying statistics in a region, the best possible GLS estimator (<(beta)over bar>(T)(GLS)) obtained using the true residual error covariance matrix, and the ordinary least squares estimator (<(beta)over bar>(T)(OLS)). Useful analytic expressions are developed for the variance of <(beta)over bar>(T)(GLS), <(beta)over bar>(E)(GLS), and <(beta)over bar>(T)(OLS). For previously examined cases the average sampling mean square error (mse(s)) of <(beta)over bar>(T)(GLS), and the mse(s) of <(beta)over bar>(MC)(GLS) usually was larger than the mse(s) of both <(beta)over bar>(E)(GLS) and <(beta)over bar>(T)(GLS). The loss in efficiency of <(beta)over bar>(MC)(GLS) 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 oil the estimators' relative performance. For a more heteroscedastic model error variance and larger return periods, some increase in the mse(s) of <(beta)over bar>(E)(GLS) relative to the mse(s) <(beta)over bar>(T)(GLS) was observed.