UBC Theses and Dissertations
Comparison of fitting techniques for systems of forestry equations LeMay, Valerie
In order to describe forestry problems, a system of equations is commonly used. The chosen system may be simultaneous, in that a variable which appears on the left hand side of an equation also appears on the right hand side of another equation in the system. Also, the error terms among equations of the system may be contemporaneously correlated, and error terms within individual equations may be non-iid in that they may be dependent (serially correlated) or not identically distributed (heteroskedastic) or both. Ideally, the fitting technique used to fit systems of equations should be simple; estimates of coefficients and their associated variances should be unbiased, or at least consistent, and efficient: small and large sample properties of the estimates should be known; and logical compatibility should be present in the fitted system. The first objective of this research was to find a fitting technique from the literature which meets the desired criteria for simultaneous, contemporaneously correlated systems of equations, in which the error terms for individual equations are non-iid. This objective was not met in that no technique was found in the literature which satisfies the desired criteria for a system of equations with this error structure. However, information from the literature was used to derive a new fitting technique as part of this research project, and labelled multistage least squares (MSLS). The MSLS technique is an extension of three stage least squares from econometrics research, and can be used to find consistent and asymptotically efficient estimates of coefficients, and confidence limits can also be calculated for large sample sizes. For small sample sizes, an iterative routine labelled iterated multistage least squares (IMSLS) was derived. The second objective was to compare this technique to the commonly used techniques of using ordinary least squares (simple or multiple linear regression and nonlinear least squares regresion), and of substituting all of the equations into a composite model and using ordinary least squares to fit the composite model. The three techniques were applied to three forestry problems for which a system of equations is used. The criteria for comparing the results included comparing goodness-of-fit measures (Fit Index, Mean Absolute Deviation, Mean Deviation), comparing the traces of the estimated coefficient co variance matrices, and calculating a summed rank, based on the presence or absence of desired properties of the estimates. The comparison indicated that OLS results in the best goodness-of-fit measures for all three forestry- problems; however, estimates of coefficients are biased and inconsistent for simultaneous systems. Also, the estimated coefficient covariance matrix cannot be used to calculate confidence intervals for the true parameters, or to test hypothesis statements. Finally, compatibility among equations is not assured. The fit of the composite model was attractive for the systems tested; however, only one left hand side variable was estimated, and, for larger systems with more variables and more equations, this technique may not be appropriate. The MSLS technique resulted in goodness-of-fit measures which were close to the OLS goodness-of-fit measures. Of most importance, however, is that the MSLS fit ensures compatibility among equations, estimates of coefficients and their variances are consistent, estimates are asymptotically efficient, and confidence limits can be calculated for large sample sizes using the estimated variances and probabilities from the normal distribution. Also, the number and difficulty of steps required for the MSLS technique were similar to the OLS fit of individual equations. The main disadvantage to using the MSLS technique is that a large amount of computer memory is required; for some forestry problems with very large sample sizes, the use of a subsample or the exclusion of the final step of the MSLS fit were suggested. This would result in some loss of efficiency, but estimated coefficients and their variances would be consistent.
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