Using simultaneous regression techniques with individual-tree growth models

Individual-tree growth models basically consist of a number of equations (e.g., diameter or basal area increment model, height increment model, crown ratio model) to update tree parameters over time. These equations commonly are assumed to be independent, with parameters of each equation estimated separately rather than simultaneously with linear or nonlinear regression. However, the opposite assumption of interdependence seems more reasonable. A tree is a highly organized system, and assimilation of tissues in the form of diameter increment, height increment, and crown size does not occur independently. Procedures to simultaneously estimate systems of two or more interrelated models are well developed in the econometrics literature and are suggested as a viable alternative. Using more than 7,500 Norway spruce (Picea abies L. Karst) trees from the Austrian National Forest Inventory with remeasured breast height diameters and tree heights (5 yr interval), we compare an individual tree basal area increment model, a height increment model, and a crown ratio model separately using ordinary least squares (OLS) and simultaneously by applying two- and three-stage least squares (2SLS, 3SLS). Results indicate the presence of strong cross-equation correlations, especially between the basal area increment and height increment models. Thus, the 3SLS estimates are more efficient while the separately determined OLS estimates are biased. Two parameters that OLS indicates are significant are found to be nonsignificant when simultaneously estimating all system parameters with 3SLS. Such system simplification reduced the size of the variance-covariance matrix by 100 elements.