An analysis and comparison of estimation methods for self-referencing equations

Abstract

Self-referencing equations (SREs) play an important role in modeling stand and individual-tree growth and yield. Over the decades, forest modelers have applied ordinary least-squares (OLS) or generalized least-squares to fit SREs (namely, the SRE method). In this article, we discuss the statistical properties of the SRE method via theoretical and empirical analyses. The SRE method has its disadvantages: (i) the parameter estimates are not the OLS estimates; (ii) the standard errors of the parameters are underestimated; (iii) the model mean squared error is overestimated; and (iv) the model random errors are always correlated and have heterogeneous variances. Thus, statistical inferences based on these model statistics may not be valid. In addition, there is no simple way to overcome these problems, because they arise from the data structures used for model fitting. This study demonstrates that the disadvantages of the SRE method can be circumvented by fitting the corresponding base model, rather than the transformed model, using two alternative methods: dummy variable regression (DVR method) and mixed effect models (MIX method). The DVR and MIX methods can efficiently account for serial autocorrelation and variance heterogeneity and, thus, produce valid model statistics for hypothesis testing and confidence intervals.