Estimating parameters for tree basal area growth with a system of equations and seemingly unrelated regressions

A method was developed for estimating parameters in an individual tree basal area growth model using a system of equations based on dbh rank classes. The estimation method developed is a compromise between an individual tree and a stand level basal area growth model that accounts for the correlation between trees within a plot by using seemingly unrelated regression (SUR) to estimate the restricted parameters. Previously, basal area growth has been modeled on either the stand or the individual tree level. Individual tree models have usually disregarded the regression assumption of independent error terms. Violation of the regression independence assumption may lead to serious underestimation of the mean square error (MSE) and standard error(s) of the parameter estimate(s). The SUR parameter estimation technique has been shown to provide a gain in efficiency for parameter estimation when the error terms for a system of equations are correlated. The data are from an ongoing natural even-aged shortleaf pine growth and yield study being conducted by the USDA Forest Service and Oklahoma State University Department of Forestry for the Ouachita and Ozark National Forests. The basal area growth model based on SUR estimation using a system of four equations (Model 2) corresponding to four dbh rank classes within a plot was compared with a basal area growth model (Model 1) using ordinary least squares (OLS) parameter estimation. The calibration, validation, and complete data set results reveal that Model 2 has a better fit index (FI) and MSE, but that Model 1 has a smaller absolute average error. Model 2 accounts for partial tree interdependency within a plot and consequently should more accurately estimate the parameter standard errors