C O M P A R I S O N O F F I T T I N G T E C H N I Q U E S F O R S Y S T E M S OF F O R E S T R Y E Q U A T I O N S By Valerie LeMay B. Sc. (Forestry). University of Alberta, 1981 M . Sc. (Forestry). University of Alberta, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF FORESTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1988 © Valerie LeMay, 1988 in p resen t ing this thesis in partial fu l f i lment of the requ i remen ts for an a d v a n c e d d e g r e e at the Univers i ty of Brit ish C o l u m b i a , I agree that t he Library shall make it f reely avai lable fo r re fe rence and s tudy. I fur ther agree that p e r m i s s i o n for ex tens ive c o p y i n g of this thesis fo r scholar ly p u r p o s e s may be g ran ted by the h e a d of m y depa r tmen t o r by his o r her representat ives. It is u n d e r s t o o d that c o p y i n g o r pub l i ca t i on of this thesis for f inancial ga in shal l no t b e a l l o w e d w i thou t my wr i t ten p e r m i s s i o n . D e p a r t m e n t of The Univers i ty of Brit ish C o l u m b i a V a n c o u v e r , C a n a d a Da te DE-6 (2/88) Abstract 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 ( M S L S ) . The M S L S 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 i i 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 M S L S technique resulted in goodness-of-fit measures which were close to the OLS goodness-of-fit measures. Of most importance, however, is that the M S L S 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 M S L S technique were similar to the O L S fit of individual equations. The main disadvantage to using the M S L S 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 M S L S fit were suggested. This would result in some loss of efficiency, but estimated coefficients and their variances would be consistent. i i i Table of Contents Abstract ii List of Tables ix List of Figures xi Acknowledgement xii 1 Introduction 1 2 Previous Attempts at Fitting Forestry Equation Systems 6 2.1 Ordinary Least Squares for Individual Equations 6 2.2 Compatible Systems using Substitution 15 2.3 M i n i m u m Loss Function 17 2.4 Econometric Methods Based on the Assumption of i id Error Terms . . . 19 2.5 Econometric Methods without the Assumption of i id Error Terms . . . . 25 2.6 Discussion of Previous Approaches 29 3 An Alternative Simultaneous Fitting Procedure 33 3.1 Extension of Econometric Methods to Multistage Least Squares 33 3.2 Estimation of the Error Covariance Matr ix 41 3.2.1 Autocorrelation and Contemporaneous Correlation 41 3.2.2 Heteroskedasticity and Contemporaneous Correlation 47 iv 3.2.3 Autocorrelation, Heteroskedasticity, and Contemporaneous Corre-lation 50 3.3 Confidence Limits , Hypothesis Testing, and Prediction 54 4 Procedures for Comparison of Fitting Techniques 59 4.1 Selection of Equations for the Systems . 59 4.2 Obtaining the O L S , Composite, and M S L S Fits 60 4.3 Tests for Heteroskedasticity and Serial Correlation 62 4.4 Criteria for Comparison of the Three Techniques 64 4.4.1 Goodness-of-fit Measures 64 4.4.2 Trace of the Estimated Coefficient Covariance Matr ix 66 4.4.3 Table of Estimated Coefficients and Standard Deviations 67 4.4.4 Ranking for Other Features : . . . . 67 5 Application 1: Tree Volume Estimation 69 5.1 Introduction 69 5.2 Preparation of Data 69 5.3 Model Selection 72 5.4 Ordinary Least Squares Fit 80 5.4.1 Unweighted Simple or Mult iple Linear Regression 80 5.4.2 Testing for i id Error Terms 80 5.4.3 Estimating the Error Covariance Matrix of Each Equation . . . . 82 5.4.4 Appropriate OLS Fit Based on Error Structure 84 5.5 Composite Model Fit 84 5.5.1 Derivation of the Composite Model 84 5.5.2 Unweighted Regression of Composite Model 85 5.5.3 Testing for i id Error Terms 86 v 5.5.4 Estimating the Variance of the Error Terms 86 5.5.5 Weighted Regression of the Composite Model 87 5.6 M S L S Fit 88 5.6.1 First Stage Equations 88 5.6.2 Second Stage Equations 89 5.6.3 Testing for i id Error Terms 89 5.6.4 Estimation of the Error Covariance Matr ix . 90 5.6.5 E G L S to Fit the System of Equations 92 5.7 Comparison of the Three Fi t t ing Techniques 93 5.7.1 Goodness-of-fit Measures 93 5.7.2 Relative Variances 97 5.7.3 Table of Estimated Coefficients and Standard Deviations 98 5.7.4 Ranking for Other Features 100 5.8 Conclusion 102 6 Application 2: Estimation of Tree Diameter Distribution 105 6.1 Introduction 105 6.2 Preparation of Data 107 6.3 Model Selection 110 6.4 Ordinary Least Squares Fi t . . 113 6.4.1 Unweighted Simple or Mult iple Linear Regression 113 6.4.2 Testing for i id Error Terms 113 6.4.3 Estimating the Error Covariance Matr ix of Each Equation . . . . 114 6.4.4 Appropriate O L S Fit Based on Error Structure 116 6.5 Composite Model Fit 116 6.5.1 Derivation of the Composite Model 116 vi 6.5.2 Unweighted Regression of Composite Model 117 6.5.3 Testing for i id Error Terms and Weighted Regression 118 6.6 M S L S Fit 119 6.6.1 First Stage Equations 119 6.6.2 Second Stage Equations 119 6.6.3 Testing for i id Error Terms 120 6.6.4 Estimation of the Error Covariance Matr ix 121 6.6.5 E G L S to Fit the System of Equations 123 6.7 Comparison of the Three Fit t ing Techniques 124 6.7.1 Goodness-of-fit Measures 124 6.7.2 Relative Variances 129 6.7.3 Table of Estimated Coefficients and Standard Deviations 130 6.7.4 Ranking for Other Features 132 6.8 Conclusion 133 7 Application 3: Volume Growth and Yield 136 7.1 Introduction 136 7.2 Data Preparation 137 7.3 Model Selection 138 7.4 Ordinary Least Squares Fit 139 7.4.1 Unweighted Simple or Mult iple Linear Regression 139 7.4.2 Testing for i id Error Terms 139 7.4.3 Appropriate O L S Fit Based on Error Structure 140 7.5 Composite Model Fit . 140 7.5.1 Derivation of the Composite Model 140 7.5.2 Unweighted Regression of the Composite Model 141 vii 7.5.3 Testing for i id Error Terms 141 7.5.4 Weighted Fi t of the Composite Model 142 7.6 M S L S Fit 142 7.6.1 First Stage Equations 142 7.6.2 Second Stage Equations 143 7.6.3 Testing for i id Error Terms 143 7.6.4 Estimation of the Error Covariance Matr ix 144 7.6.5 E G L S to Fit the System of Equations 144 7.7 Comparison of the Three Fit t ing Techniques 145 7.7.1 Goodness-of-fit Measures 145 7.7.2 Relative Variances 148 7.7.3 Table of Estimated Coefficients and Standard Deviations 150 7.7.4 Ranking for the Three Techniques 150 7.8 Conclusion 152 8 Overall Discussion and Conclusions 155 9 References Cited 160 A Glossary of Terms and Abbreviations 170 vm List of Tables 2.1 Simultaneous Estimation Assuming i id Error Terms 21 5.2 Distribution of Selected Trees b}' Height and Dbh Classes 71 5.3 D i s t r i b u t i o n of 100 S a m p l e Trees 73 5.4 Fit Indices for OLS and M S L S Fits of the Volume Equation System . . . 93 5.5 M . A . D . for Five Classes for the O L S Fit of the Volume Equation System 94 5.6 M . A . D . for Five Classes for the M S L S Fit of the Volume Equation System 95 5.7 M . A . D . for Merchantable Volume 95 5.8 M . D . for Five Classes for the O L S Fit of the Volume Equation System . 96 5.9 M . D . for Five Classes for the M S L S Fit of the Volume Equation System . 96 5.10 M . D . for Merchantable Volume 97 5.11 Trace of the Coefficient Covariance Matr ix for Each Equation of the Vol-ume Equation System 98 5.12 Estimated Coefficients and Standard Deviations of Coefficients for the Volume Equation System 99 5.13 Ranks for the Three Techniques 101 6.14 Distribution of Selected Psps by Age and Stems per Hectare Classes . . . I l l 6.15 Fit Indices for OLS and M S L S Fits of the Dbh Distribution System . . . 124 6.16 M . A . D . for Five Classes for the O L S Fi t of the D b h Distribution System 126 6.17 M . A . D . for Five Classes for the M S L S Fit of the D b h Distribution System 126 6.18 M . D . for Five Classes for the O L S Fit of the Dbh Distribution System . . 128 6.19 M . D . for Five Classes for the M S L S Fi t of the Dbh Distribution System . 128 ix 6.20 Trace of the Coefficient Covariance Matr ix for Each Equation of the Dbh Distribution System 130 6.21 Estimated Coefficients and Standard Deviations of Coefficients for the Dbh Distribution System 131 6.22 Ranks for the Three Techniques for the F i t of the Dbh Distribution Systeml32 7.23 Distribution of Selected Psps for the T h i r d Application 137 7.24 Fit Indices for O L S , Composite Model , M S L S and M S L S Fits of the Yield Equation System 145 7.25 M . A . D . for Five Classes for the O L S Fi t of the Yield Equation System . 146 7.26 M . A . D . for Five Classes for the M S L S Fi t of the Yield Equation System . 146 7.27 M . A . D . for Five Classes for the Yield Composite Model F i t 147 7.28 M . D . for Five Classes for the OLS Fit of the Yield Equation System . . . 147 7.29 M . D . for Five Classes for the M S L S Fit of the Yield Equation System . . 148 7.30 M . D . for Five Classes for the Yie ld Composite Model Fit 148 7.31 Trace of the Coefficient Covariance Matr ix for Each Equation of the Yie ld Equation System 149 7.32 Estimated Coefficients and Standard Deviations of Coefficients for the Yie ld Equation System 150 7.33 Ranks for the Three Techniques 151 x List of Figures Extension of Econometric Least Squares Methods for Fitt ing Systems to M S L S xi Acknowledgement M y greatest thanks go to the professors at University of British Columbia. In particular, Dr. A . Kozak, as well as the other members of my thesis committee, Drs. J . H . G . Smith, M . Bonner, P. Marshall , and D. T a i l , of the Faculty of Forestry, and Dr. J . G . Cragg of the Department of Economics provided great assistance in completing my research. In addition, I would like to thank Dr. J . Wilson and Ms. N . Bertrams for their efforts in securing funding and for processing the necessary information. Data were provided by the Timber Management Branch of the Alberta Forest Service. Special thanks go to M r . D. J . Morgan, Forester-in-charge, Forest Measurement Section, and his staff members for their prompt response to my data needs and questions. I would also like to thank Patrick Pelletier, my husband, for his patience and encour-agement . xii Chapter 1 Introduction In order to describe forestry problems, a system of equations is frequently used. For instance, for the prediction of tree volume, tree height is first estimated, and then tree volume is estimated from measured diameter outside bark at breast height (dbh) and the estimated tree height. The desired parameter, tree volume, is therefore estimated through the use of a system of two equations. The error structure for a system of equations will affect the results of any technique used to fit the system. The error terms among equations of the system may be correlated (contemporaneous correlation)1 or variables appearing as dependent variables in one equation may appear as an independent variable in another equation of the system. For example, the system of equations for the prediction of tree volume may be chosen as follows: where volume is the volume of the main bole of the tree from ground to tree top; height is the height of the main stem from ground to tree top; dbh is the diameter outside bark measured at breast height, 1.3 metres above ground; log is the logarithm, base 10 or base e; •"•For definitions of terms and abbreviations used in this thesis, see A p p e n d i x A . height = /3Q + flidbh2 + ei (1.1) log volume = log 02 + 83 log dbh + 84 log height + e 2 (1.2) 1 Chapter 1. Introduction 2 0o, 0x, 02, 03, 04 axe coefficients to be estimated; ei, £2 are error terms. Height is a stochastic variable which occurs as both a dependent variable on the left hand side (LHS) of an equation and also as an independent variable on the right hand side (RHS) of an equation. Also, a measure of tree taper such as form factor has been excluded from the system. Taper would likely affect both height and volume, hence both Ci and include the error due to the exclusion of a measure of taper, and are therefore probably correlated (contemporaneous correlation). Further complications arise if the variances of the error terms for any of the equations in the system vary over the range of independent variables (heteroskedasticity 2 ) , or if the error terms for a given equation are correlated with the previous error terms (serially correlation). The error structure for a system of equations can therefore have any or all of the following characteristics. 1. Dependent variables may appear on the LHS of an equation in the system and also as a RHS variable in another equation so that the OLS assumption that the RHS variables are uncorrelated with the error term is not met for every equation. Systems with this characteristic are termed simultaneous equations. 2. The error terms among equations may be correlated indicating contemporaneous correlation. 3. Within individual equations, the error terms may be serially correlated (not inde-pendent), or the variances of the error terms may be heterogenous (not identically distributed), or both (neither independent nor identically distributed (non-iid)). The fitting approach for systems of equations for forestry applications should ideally meet all of the following criteria. 2 This may also be spelled as heteroscedasiicity. See McCulloch (1985) for discussion. Chapter 1. Introduction 3 1. The routine should be simple in that few fitting steps are required. 2. Estimates of the coefficients and their associated variances should be unbiased or at least consistent. The estimates should have low variance (high efficiency). 3. Reported information on asymptotic and small sample properties of the estimated coefficients and their variances should be available. 4. Estimates should result in a compatible system of equations in that logical rela-tionships among variables in the system should be maintained in the fitted system. Criteria one through four should be met regardless of the error structure of the systems of equations (characteristics 1, 2, and 3, page 2). The main objectives of this research were as follows: 1. To review forestry, econometrics, biometrics, and statistics literature and to choose, from this literature, a technique which satisfies all of the above criteria for fitting simultaneous, contemporaneously correlated systems of forestry equations, in which the error terms of individual equations are non-iid. 2. To compare this alternative technique to the most commonly used methods of (1) an appropriate OLS fit to each of the equations and (2) an appropriate O L S fit to a composite model created by substituting the equations of the system into one composite equation. The central hypotheses of this thesis are: first, a fitting technique exists which satisfies the desired criteria for simultaneous, contemporaneously correlated systems of equations in which individual equations have non-iid error terms; and second, that any additional computational burden in using the technique is compensated by the benefits of meeting the desired criteria. Chapter 1. Introduction 4 To meet the first objective, a literature search was conducted, first by examining forestry literature, and then by extending the search to econometrics, biometrics, and statistics literature. In order to restrict the scope of this thesis, only techniques based on least squares methodology were considered. The main alternative to the least squares approach is a maximum likelihood approach; however, the maximum likelihood approach requires that an assumption about the distribution of the dependent variables be made, and is more difficult to calculate. Also, the maximum likelihood approach is more sen-sitive to model specification error, to the presence of outliers, and to the presence of multicollinearity (Cragg, 1967; Summers, 1965). Least squares methodology was there-fore selected as a more desirable method. Objective one was not met, because the search of the literature failed to provide a fitting technique which satisfies all of the desirable properties for fitting systems with all three of the characteristics listed for the error structure. However, the information in the econometrics literature was used as the basis for the derivation of a new technique as part of this research. Results from several authors were combined into a comprehensive technique and labelled multistage least squares (MSLS) for this thesis. 3 The M S L S technique is restricted to fitting systems of equations in which one sample set is used to fit the equations and, therefore, the number of samples is the same for every equation of the system. A n iterated procedure was also derived and labelled iterated multistage least squares ( IMSLS) . To meet the second objective, the derived M S L S technique, and the two most com-monly used techniques were used to fit a system of equations for each the following forestry problems. 3 T h e use of the terms multistage least squares in this thesis refers only to the technique derived by this author. Pienaar and Shiver (1986) use the term multistage least squares in the summary of their paper; however, it is likely that they are referring to established econometric techniques for systems of equations where error terms for individual equations are iid, which can be considered to be a subset of the MSLS comprehensive technique described in this thesis. Chapter 1. Introduction 5 1. The estimation of tree volume. 2. The estimation of diameter distributions from stand measurements, such as stand age and number of stems per unit area. 3. The estimation of volume and basal area growth and yield. The systems of equations for these problems were expected to have different error struc-tures. For the first problem, the error terms were expected to be heteroskedastic within equations and contemporaneously correlated among equations. The second problem in-volved the prediction of parameters of a probability distribution; each equation of the system estimated one of the parameters. For the third problem, the error terms were expected to be serially correlated and perhaps heteroskedastic within equations, and con-temporaneously correlated among equations. These estimation problems are commonly encountered in forest management and endogenous variables tend to be estimates which are later expanded. For instance, the predicted volume on a given tree may be expanded to a per hectare estimate of volume. The evaluation of the three fitting procedures, O L S , composite model, and M S L S , was based on comparing goodness-of fit measures, comparing the traces of the estimated coefficient covariance matrices, and calculating a summed rank, based on the presence or absence of desired properties of the estimates. In order to distinguish between variable types, the terms used in econometrics are used. Stochastic variables which appear on the L H S of equations and may also appear on the R H S are termed endogenous variables derived from a Greek word meaning "generated from the inside" (Hu, 1973, page 121). Variables which appear only on the R H S are termed exogenous variables meaning "generated from the outside". These exogenous variables were assumed to have very little error and could be considered to be fixed variables (nonstochastic). Chapter 2 Previous Attempts at Fitting Forestry Equation Systems 2 . 1 Ordinary Least Squares for Individual Equations For systems of equations in forestry, the most common fitting method has been the independent fitting of each equation in the system using an appropriate least squares pro-cedure, such as simple linear regression, multiple linear regression, weighted regression, or nonlinear least squares (Burkhart, 1986: Furnival and Wilson, Jr., 1971). The O L S approach has appeal in that the method is well known and calculations to obtain esti-mates of coefficients and variances of coefficients and of dependent variables are relatively simple. The standard O L S procedure of simple or multiple linear regression applied to indi-vidual equations yields estimates of coefficients which are best linear unbiased estimates ( B L U E ) for linear equations if the following assumptions are met. 1. The error terms for an equation are i id ; serial correlation and heteroskedasticity are not present within individual equations. 2. The variables on the R H S of each equation are uncorrelated with the error term of the equation (nonstochastic). 3. The error terms among equations are not correlated, meaning that contemporane-ous correlation is not present. 6 Chapter 2. Previous Attempts ai Fitting Forestry Equation Systems A system of three equations appears as follows: y 2 = X2/32 + £2 y3 = X3/33 — £3 (2 .3 ) (2 .4 ) (2 .5) where y; is an n by 1 matrix of the sample values for the ith endogenous variable in the system of equations; X2- is an n by k{ matrix of the sample values for all of the exogenous variables which affect the ith endogenous variable in this equation of the system; fli is a ki by 1 matrix of the true coefficients associated with the exogenous variables of this equation; €i is an n by 1 matrix of the error terms associated with each sample of the endogenous variable; n is the number of samples. These equations can also be expressed as follows: ( 2 .6 ) where 0 n is an n by n submatrix of zeros. If all of these assumptions are met for the system of equations, the covariance matrix of the error terms of the system, is a diagonal matrix as follows: y i x On On A y = y2 On x 2 On ft + £2 = XB + E ys 0 n On x 3 03 £3 2T On On fl = On 2T On o n On C/3An ( 2 .7 ) where ft is the error covariance matrix for the system of equations; Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 8 erf In is a diagonal submatrix representing iid error terms within the equation i. Each of the diagonal elements is equal to erf for the ith equation. I n is the identity matrix of size n by n; 0 n is an n by n submatrix of zeros, indicating no contemporaneous correlation. The covariance matrix then "falls apart" into separate matrices for each equation and simple or multiple linear regression can be used to fit each equation separately. The estimated coefficients will be BLUE, and also maximum likelihood estimates (MLE) if the error terms are normally distributed. The estimated coefficient covariance matrix also will be unbiased. For individual equations, if the error terms are not identically distributed, in that their variances are not homogeneous, the error covariance matrix remains a diagonal matrix, but the diagonal terms for each equation are unequal, as shown below for three equations and three samples. 0 0 0 0 0 0 0 0 0 „2 a 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 „2 a 2 \ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 _2 ^23 0 0 0 0 0 0 0 0 0 _2 0 0 0 0 0 0 0 0 0 „ 2 a Z 2 0 0 0 0 0 0 0 0 0 _2 ° 3 3 (2.8) where tr?. is the variance for the ith equation and the jth sample. Again, the covariance matrix can be divided into a separate matrix for each equation of the system. In this case, a specialized OLS technique such as a weighted regression Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 9 procedure or another transformation of the variables in each equation can be used so that the variance of the transformed regression is as the matrix, in equation 2.7 (iid error terms), and a least squares procedure can be used to obtain estimates which are asymptotically BLUE for the untransformed variables (Judge et al., 1985, pages 420 and 421). Alternatively, the elements of the error covariance matrix can be estimated, and an estimated generalized least squares method (EGLS) could be used to fit each equation separately as follows: y; = Xifc + ei (2.9) fit GLS = ( X ^ . n x ^ ' x ^ r i y , (2.10) where y; is an n by 1 matrix of the sample values for the ith endogenous variable in the system of equations; Xj is an n by k{ matrix of the sample values for all of the exogenous variables which affect the ith endogenous variable in this equation of the system; fii is a ki by 1 matrix of the true coefficients associated with the exogenous variables of this equation; ej is an n by 1 matrix of the error terms associated with each sample of the endogenous variable; fii GLS is an estimate of the fii matrix; <£; is the estimated error covariance matrix for ith equation of the system. Generally, for EGLS, if the estimated error covariance matrix is consistent, the esti-mated coefficients and variances of these coefficients are consistent, and the distribution of each of the estimated coefficients is asymptotically normal. Several estimators for the error covariance matrix were presented by Judge and others (1985, pages 419 to 464) depending on the assumptions made concerning the heteroskedasticity. The procedure may be iterated by using the results of the EGLS fit to obtain a new estimate of the Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 10 error covariance matrix and from this obtain a new 0i GLS , and so on until the estimated coefficients converge. These iterations will result in what Malinvaud (1980. page 285) termed a quasi-maximum likelihood estimator of the coefficients and the covariance ma-trix. If normality of the error terms is assumed, then the estimators become M L E and the asymptotic covariance matrix will reach the Cramer-Rao lower bound for efficiency (minimum variance). If serial correlation is present, the covariance matrix is no longer diagonal, because relationships within the sample appear. In this case, if the relationship can be defined such as a first order autoregressive process, in which the correlation between error terms declines geometrically as the time between disturbances increases, a term may be added to the regression equation or a transformation of the variables may be used (Cochrane and Orcutt, 1949; Kadiyala, 1968). This technique was used by Monserud (1984) for a single equation to describe height growth using stem analysis data. Alternatively, as with the case of heteroskedasticity, a consistent estimator of the error covariance matrix may be found, and used in an EGLS procedure (Judge et al, 1985, pages 283 to 286). Gregoire (1987) demonstrated this procedure for a single equation where permanent sample plots were correlated over time. If endogenous variables appear on the RHS of one equation of the system and on the LHS of another equation of the system (simultaneous equations), the assumption that the variables on the RHS of each equation are not correlated with the error is not met. The independent OLS fit using simple or multiple linear regression will result in biased estimates of coefficients. To prove that this bias exists, equation 2.3 can be extended to include endogenous variables on the right hand side as follows: yi = Y i 7 i + Xif3i + ez (2.11) where y; is an n by 1 matrix of the sample values for the ith endogenous variable in the Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 11 system of equations; Yj is an n by g — 1 matrix of the sample values for all of the endogenous variables excluding the itn endogenous variable; 7i is a g — 1 by 1 matrix of coefficients associated with the endogenous variables on the RHS of the equation. Coefficients are set to zero if the associated endogenous variables do not affect the endogenous variable of the system; Xj is an n by k matrix of the sample values for all of the exogenous variables of the system; /3j is a k by 1 matrix of coefficients associated with the exogenous variables. Coefficients are set to zero if the associated exogenous variables do not affect the itn endogenous variable of the system; 6j is an n by 1 matrix of the errors associated with the itn endogenous variable of the system. The RHS variables and coefficients may be combined as follows: Yi = [Yj Xj] Pi + £j = Zj£j + £j (2.12) For the g equations of the system, using g — 3 equations, yi On On " s1' = on z2 on + On 0„ Z 3 s3 (2.13) which may be restated as follows: y'= ZA + E (2.14) where y is a gn by 1 matrix of n samples for each of g endogenous variables; Z is a gn by g({g — 1) + k) matrix of the RHS variables; Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 12 A is a g([g — 1) - f k) by 1 matrix of the coefficients associated with the RHS variables; E is a gn by 1 matrix of the residual error for n samples for each of g endogenous variables. For a single equation, the OLS estimator of 5,; is therefore: The second term does not disappear as the Zj are not independent of the error term and so the OLS fit is biased (Judge et al, 1985, page 571). This simultaneity bias does not disappear if the sample size is increased. The estimates are also inconsistent (see proof in Judge et ai, 1985, page 571). Also, since the estimated coefficients are biased and inconsistent, the estimated coefficient covariance matrix cannot be used to calculate confidence limits for the true coefficients. If confidence limits are calculated using these estimated variances of the coefficients, the limits would be incorrectly narrow. If endogenous variables do not appear on the RHS, but contemporaneous correlation is present, the resulting OLS estimates of coefficients will not be most efficient as this correlation among equations is not included in the OLS fit of single equations. Zellner (1962) demonstrated this problem and called the system of equations with this contem-poraneous correlation, seemingly unrelated regression (SUR) equations. If endogenous variables appear on the RHS and contemporaneous correlation is present, OLS estimates will again be biased and inconsistent, and, also, a loss in efficiency would result, because the information concerning correlation among equations would not be used in the OLS 5i = (z^ r 1 z;Yt (2.15) The expectation of the OLS estimator is: (2.16) (2.17) Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 13 fit. Finally, if contemporaneous correlation is present, endogenous variables appear on the R H S , and the i id assumption is not met for one or more equations, the standard O L S estimates using simple or multiple linear regression for individual equations wil l result in biased and inconsistent estimates of the coefficients and their variances. In addition, predictions of endogenous variables are not constrained to be logically compatible if an O L S fit is used. The practical implications of using the O L S fitting method for individual equations are as follows: 1. For simultaneous systems, even though any bias between an estimated and the actual coefficient, or between a predicted and the actual value of the dependent variable may be small, these small biases may be magnified over a forest inventory. For instance, if volume per tree was in error by 0.10 cubic metres, and this tree represented 100 trees in the stand, the volume for the stand would be in error by 10 cubic metres. Also, biases in certain coefficients may have a dramatic effect in the resulting estimates of dependent variables over the entire range or over partial ranges of the independent variables. 2. For simultaneous systems, estimates of coefficients based on the O L S fit of individ-ual equations are not only biased, they are inconsistent. Consequently, no matter how many samples are collected, there is no assurance that the sample estimates will be close to the population values for coefficients and dependent variables. 3. The estimated coefficient covariance matrix from the O L S fit of simultaneous equa-tions cannot be used to calculate confidence intervals. If these estimated variances are used, the confidence intervals wil l be incorrectly narrow, and there wil l be a higher chance that the true coefficients are not in the confidence interval. Also, hypotheses cannot be tested. Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 14 4. If contemporaneous correlation is present, O L S estimates wil l be less efficient, be-cause the information concerning this correlation is not used in determining the O L S fit of individual equations. The resulting confidence limits will therefore be wider if O L S is used, and more samples wi l l be required to obtain a desired preci-sion. 5. Because the equations are fitted independently, compatibility of estimates is not assured. For instance, one equation may be used to estimate tree height and another equation may be used to estimate site index. If site index is defined as the height of the tree (or average tree) at 50 years measured at breast height, a desirable trait of this system of two equations would be that the height at 50 years from breast height predicted from the first equation, is equal to the site index of that tree as predicted from the second equation. The O L S fit of individual equations does not assure this logical compatibility. The independent O L S approach has been termed the "naive" approach by Intriliga-tor (1978, page 373), because information concerning the error structure of the system of equations is ignored. However, the method is still useful in that it is the easiest to calculate and computer programs are the most widely available. In addition, for pre-liminary work to define the system of equations, or where the system of equations is very large such as in systems for forestry growth and yield modelling, an independent O L S fit is probably the most practical method. However, compatibility is not assured, and estimates can be biased, inconsistent and not most efficient, depending on the error structure of the system. Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 15 2.2 Compatible Systems using Substitution Another frequently used method for fitting systems of forestry equations is to use differentiation or substitution of variables and equations within the system. For instance, in an attempt to ensure logical compatibility of growth and yield estimates, Clutter (1963) suggested that growth equations should be obtained by taking the derivatives of the yield equations. Resulting estimates were not considered efficient. Bailey and Ware (1983) produced a compatible basal area growth and yield model by using the growth model to calculate yield. Matney and Sullivan (1982) described a system of substitutions to obtain compatible stock and stand tables. They estimated the parameters of the Weibull distribution by relating the integration of the distribution for volume and basal area to the predicted volume and basal area, thereby ensuring compatibility. Other applications of the method of substitution include Ramirez-Maldonado and others (1987). who developed a system of equations for predicting height growth and yield. A model to predict height at time one was first fitted, and then used to predict one of the coefficients of the model to estimate height at time two, so that estimates for growth or yield of height are compatible. McTague and Bailey (1987a) developed a compatible system of equations for basal area and diameter distribution by recovering the parameters of a Weibull distribution from predicted stand variables. First, the 10th, and 63rd percentiles, present and future, were predicted from site index, age and stems per hectare. The basal area was then predicted from these current percentiles. The "a" parameter of the Weibull distribution was predicted from age, number of stems, and the 10th percentile. The 90th percentile was then predicted from the 10th and 63rd percentiles, and site index and age. The "b" and "c" parameters were then calculated mathematically. Substitution of all equations into one composite model has perhaps been even more widely used. Sullivan and Clutter (1972) developed a single linear model by substituting Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 16 the basal area growth equation into the volume yield equation to obtain a composite model. A maximum likelihood procedure was used to obtain unbiased estimates of the regression coefficients, because of serial correlation. The problem with this approach was that some of the variables from the original models were not significant in the composite model and disappeared from the resulting fit. The original biologically based models were lost; the result of changes in the variables was difficult to interpret. For example, the original model for predicting the log of volume at time two was as follows: lnV2 = h (SI, 1/A2, lnBA2) (2.18) where lnV2 is the natural logarithm of volume a.t time 2; SI is the site index; A2 is the age at time 2; BA2 is the basal area at time 2. A substitution for basal area at time two resulted in the following equation. ' n V i = h (SI- h %x l n B A i ' 1 -1? I 1 - x S I ) ( 2 I 9 ) where Ai is the age at time 1; BAi is the basal area at time 1. The composite model no longer retained the expected biological relationship between volume and basal area. Also, even though the authors suggested that the combined linear model was compatible for prediction of growth and yield, the growth was obtained indirectly by subtracting yield at time two from yield at time one, rather than by an unique equation for growth and for yield. A composite model for height growth based on the Chapman-Richard's model (Pien-aar and Turnbull , 1973) is frequently proposed. The coefficients of the model are es-timated for each tree or plot and then these coefficients are related to site index or Chapter 2. Previous Attempts ai Fitting Forestry Equation Systems 17 other sample observations such as hab i ta t or region. T h e resul t ing equations to describe changes i n the coefficients are then placed back in to the C h a p m a n - R i c h a x d ' s equat ion to o b t a i n a composite m o d e l (e.g. Beck , 1971; G r a n e y a n d B u r k h a r t , 1973; L u n d g r e n a n d D o l i d , 1970; T r o u s d e l l et ai, 1974). A n o t h e r e x a m p l e of the use of a composi te m o d e l is V a n D e u s e n a n d others (1982) w h o showed that a sys tem of equations for merchantable v o l u m e to any height or to any diameter , t o t a l v o l u m e , a n d merchantab le height c o u l d be m a d e compat ib le by e s t i m a t i n g one of the equat ions , a n d us ing this est imate for the other equations. O n l y one es t imated coefficient was required. M c T a g u e and B a i l e y (1987b) proposed a more c o m p l e x composi te equat ion to estimate merchantable v o l u m e . T h e equat ion c o u l d be rearranged to estimate t o t a l v o l u m e , a n d for taper . T h e compos-ite equat ion was f i t , and the f i t ted coefficients were then used for each of the equations o b t a i n e d by rearranging the composi te m o d e l . F o r taper m o d e l l i n g , K o z a k (in press) sub-s t i tu ted an equat ion for the diameter at the in f lec t ion p o i n t , a n d another equat ion for the exponent of the equat ion , to o b t a i n a composi te m o d e l w h i c h descr ibed the diameter for a given height above g r o u n d . T h e s u b s t i t u t i o n m e t h o d has been wide ly used to ensure c o m p a t i b i l i t y of systems of forestry equations. However , the o r i g i n a l b io log ica l ly based models m a y be changed i n the s u b s t i t u t i o n , and neither efficiency nor unbiasedness of the es t imated coefficients or their variances is assured. A l s o , o n l y one var iable o n the L H S is es t imated . 2.3 M i n i m u m Loss Function A s imultaneous fit of two equations was o b t a i n e d b y B u r k h a r t a n d S p r i n z (1984) b y m i n i m i z i n g a loss f u n c t i o n w h i c h combined squared errors for the first equat ion w i t h the squared errors for the second equat ion as fol lows: F = £ {Vi - V - ) 2 / 4 B.)2/*B (2-20) Chapter 2. Previous Attempts at Fitting Forestry Equation Systems 18 where V;, V; is volume, actual and predicted; Bi, Bi is basal area, actual and predicted; u\ is the mean square error from the O L S fit of the volume equation; In) • The estimated auto-correlation coefficients (pi) are then used to obtain the P i matrices. This technique results in a consistent estimate of $7 which is required so that the proper-ties of G L S can be applied to E G L S asymptotical!}'. The P ; matrix can also be extended Chapter 3. An Alternative Simultaneous Fitting Procedure 47 to higher order correlations if these are found to be significant (see Ameniya, 1985, pages 164 to 170). 3.2.2 Heteroskedasticity and Contemporaneous Correlation The error covariance matrix for heteroskedastic, contemporaneously correlated systems of equations without serial correlation (independent, not identically distributed) is block diagonal as shown in equation 2.22. The variance for an individual equation can be shown as $ = where is a diagonal matrix with unequal diagonal elements. To obtain a consistant estimate of the error covariance matrix, a consistent estimate of a\ and a consistent estimate of ^ are required. A method for estimating the error covariance matrix for heteroskedastic, contempora-neously correlated systems of equations was not found in literature. Parks (1967) method for autocorrelated systems was therefore extended to heteroskedastic, contemporaneously correlated systems. The following steps were derived to obtain a consistent estimate of the error covariance matrix. 1. For each second stage equation, simple or multiple linear regression is used to obtain consistent estimates of the error terms for each sample as in step 1 for autocorrelated errors. The estimated coefficients wi l l be consistent. If the system of equations is not simultaneous (stage 1 of M S L S procedure is not required), the estimated coefficients will be also unbiased. 2. The error terms, squared, are then graphed against the predicted values for the L H S endogenous variables using coefficients from step 1 for each equation as follows: Vim = ZimCli (3.44) ^ i m ^-im Vim Vim (3.45) Chapter 3. An Alternative Simultaneous Fitting Procedure 48 where u ; m is e;m for the i equation and the m observation; yim is the predicted endogenous variable from step 1 for the ith equation and the m.th observation; Z ; m is the vector of predicted R H S endogenous variables and exogenous variables for the ith equation and the mth sample; a ; is the coefficient matrix from the step 1 simple or multiple linear regression fit of each equation. 3. Using the graph of estimated error squared versus predicted endogenous variables, an estimated functional form of the variance for each given sample (represented by the squared error) against the expected value of the endogenous variable given the R H S variables is chosen. «L=/(fcm) ( 3 - 4 6 ) This equation may be fit using simple or multiple linear regression for each equation, resulting in a consistent estimate of the variances (Judge et al, 1985, p. 437) or all equations may be fit simultaneously using S U R as with step 2a of the process to obtain the error covariance matrix for autocorrelated systems. 4. The fitted equations are then used to obtain crfm€ = u\m to yield the following matrix. Vile 0 0 . . 0 0 0 . . 0 0 0 038) 039 a r e coefficients to be estimated; error i 5 is the error term. Chapter 5. Application 1: Tree Volume Estimation 79 Because top dib ranged from 0.00 through 15.0 centimetres, for small trees, the specified top dib was sometimes larger than dbh. To restrict the top dib/dbh ratio, the ratio was allowed a maximum value of stump dib/'dbh where stump dib is the diameter inside bark measured at stump height. The R2 value for this model was 0.9722. Using a stepwise regression procedure, other transformations of the top dib/dbh ratio, such as the first and third power, entered into the equation before hs; however, when these other transformations of the diameter ratio were included in the equation, the inclusion of hs in the model resulted in an R2 value change of less than 0.005. Since the effect of changes in the stump height on the height ratio was considered important, the equation presented above was selected for the system of equations. The chosen system of equations for estimating tree volume was the following: height = Su + 8i2dbh + 8izdbh2 + ex (5.85) merch. volume total volume (5.86) (5.87) (5.88) merch. volume = total volume x VR (5.89) ml — height x HR (5.90) merch. height = hm = ml + hs (5.91) The system is therefore composed of seven equations, and estimates of coefficients are needed for four of these equations. Chapter 5. Application 1: Tree Volume Esiimaiion 80 5.4 Ordinary Least Squares Fit 5.4.1 Unweighted Simple or Multiple Linear Regression The estimated coefficients for the chosen equations using unweighted multiple linear regression to fit each equation were as follows: pred. height = 0.584579 + 1.07123922 -0.912563 -0.898900 0.021266 0.018179 <$23 -0.226424 -0.217870 0.039766 0.024722 total volume 831 0.0116150 0.003757 0.003155 0.003851 832 3.552805 x I O " 5 3.738763 x 10~ 5 4.505443 x 10~ 7 5.887156 x I O " 7 V R <541 0.990691 0.985298 0.001643 0.006660 842 -0.986292 -0.947024 0.006209 0.023751 Chapter 5. Application 1: Tree Volume Estimation 100 The estimated coefficients for the M S L S and for the O L S fits were very similar, except for the intercepts of the height and volume equations, and the slope of the volume ratio equation. The standard deviations for the estimated coefficients of the height model were lower from the M S L S fit than from the O L S fit. This was reflected in the trace calculated for this equation, and indicates that the inclusion of the contemporaneous variances in the system fit using M S L S resulted in a lower variance. Results were similar for the fits of the height ratio equation. For the volume and volume ratio equations, the standard deviations for the coefficients were higher for the M S L S fit; again, this was reflected in the trace values for these equations, and indicates that O L S underestimates the confidence interval for the true value, because of simultaneity bias. If the O L S estimates for the volume and volume ratio equations were used to test hypothesis statements, results would not be correct. For instance, <5>41 would be expected to be equal to 1.0, so that a, volume ratio of one is predicted, if the merchantable length is equal to the total height. Using an alpha level of 0.01, the confidence limits using the biased estimates from O L S would be 0.94923 to 0.98646. Using the consistent M S L S estimates, the confidence limits would be 0.96830 to 1.00230. The hypothesis that this coefficient is equal to 1.0 would be rejected with the O L S estimates, but not rejected with the M S L S consistent estimates. 5.7.4 Ranking for Other Features The ranks assigned to each fitting technique are shown in Table 5.13. 4 The composite model was assigned a rank of one for information as only one endogenous variable is estimated. For consistency, both the M S L S and the composite model for this system of equations provided consistent estimates. The fit of the composite model was unbiased 4See Chapter 4 for an explanation of the ranking system used. Chapter 5. Application 1: Tree Volume Estimation 101 Table 5.13: Ranks for the Three Techniques Feature O L S Composite M S L S Information 2 1 2 Consistency 1 2 2 Confidence Limits 1 2 2 Asymptotic Efficiency 1 1 2 Compatibility 1 2 2 Ease of fit 2 3 1 Total 8 11 11 as all R H S variables were exogenous. Confidence limits for both the M S L S fit and the fit of the composite model can be calculated. M S L S was considered to have the highest asymptotic efficiency as the equations appear to be correlated. Both the M S L S and the composite model fits provide compatible estimates. The number of steps required to obtain the O L S fit was four for the unweighted -fit, eight to check for i id errors, one to estimate the error covariance matrix, and one to obtain the weighted fit of the total volume model, for a total of 14'step's. For the composite model, three steps were required to derive the model, two steps to check for i id error terms, and three steps to obtain the unweighted fit, the estimated error covariance matrix, and the weighted fit, for a total of eight steps. For the M S L S fit, two steps were required to fit the first stage models, four steps for the second stage models (actually used O L S estimates for two of the second stage models), eight to check for i id errors (actually only four as results from O L S used for two of the models), one to estimate the weights for the volume equation, one to estimate the covariances among equations, and one to fit the system of equations, simultaneously, for a total of 17 steps. Chapter 5. Application 1: Tree Volume Estimation 102 5.8 Conclusion The chosen system of equations to estimate tree volume was composed of seven equa-tions. Four of these equations had coefficients which were to be estimated. Using the O L S technique, two of the equations were found to have i id error terms, and were fitted using multiple linear regression. The error terms for the height ratio equation were considered i i d , although a lack-of-fit was noted. The total volume equation had heteroskedastic error terms. Many models of this heteroskedasticity were tested and found to be inadequate. The variances of the error terms were therefore estimated by calculating the average of the estimated error terms, squared, from the unweighted fit, for classes of the predicted total volume values, also from the unweighted fit. The total volume equation was then refitted using E G L S . A composite model was derived by combining the seven equations into one to estimate merchantable volume. The unweighted fit, using a criterion of a minimum R2 change of 0.005, included only three of the 23 terms of the composite model. Error terms were found to be heteroskedastic and so the variances of the error terms were estimated. Mult iple linear regression was then used to fit the weighted equation. The first stage equations for the M S L S fit of the volume equation did not reflect the quadratic nature of the height ratio variable; M S L S estimates were therefore not aymptotically efficient for this system. The error terms for the second stage equations of the M S L S fit were found to be similar to the error terms from the unweighted O L S fit. However, a lack-of-fit was noted for the estimated error terms of the volume ratio equation; no attempt was made to explain the cause of this lack-of-fit. A second set of second stage equations were therefore derived in which the second stage total volume equation was weighted and fit using E G L S . M S L S was used to fit the system using the estimated error covariance matrix. I M S L S was not used to fit the equations, because the Chapter 5. Application 1: Tree Volume Estimation 103 number of samples was large. The best goodness-of-fit measures (high F.I. and low M . A . D . and M . D . values) were obtained for the O L S fit. However, the goodness-of-fit measures for the M S L S fit were very similar. The composite model resulted in the worst goodness-of-fit measures, likely because some of the important variables were excluded from the composite model fit. The trace of the estimated coefficient covariance matrix was lower for M S L S than for OLS . The trace values of the submatrices of this matrix, corresponding to individual equations, showed that the O L S fit resulted in lower trace values than the M S L S fit for the total volume and volume ratio equations. Each of these equations had R H S endoge-nous variables. The trace values for the O L S fit were higher for the height and height ratio equations than for the M S L S fit; these equations did not have R H S endogenous variables. The M S L S fit therefore resulted in an increase in efficiency as shown by the lower trace values for the submatrices estimated for the height and height ratio equations. Higher trace values for the submatrices of the total volume and volume ratio equations resulted from the M S L S fit. However, the O L S estimates of the variances cannot be used to calculate confidence intervals or test hypotheses concerning the true coefficients. The M S L S fit results in a consistent estimate of the coefficients and of the coefficient co-variance matrix, and, therefore, confidence limits for true coefficients can be calculated. Hypotheses can also be tested using the M S L S fit. The table of estimated 'coefficients and their standard deviations further supported the evidence given by the traces. The O L S fit results in larger standard deviations than the M S L S fit for the height and height ratio equations, and results in underestimates of confidence limits for the volume and volume ratio equations. The summed rank for other features was the same for the M S L S fit as for the compos-ite model fit. The M S L S technique has the advantage of all of the endogenous variables being predicted, whereas the composite model fit is easier to compute. The O L S fit Chapter 5. Application 1: Tree Volume Estimation 104 had the lowest summed rank, largely because estimates of the coefficients are biased and inconsistent. Also, the O L S fit is a single equation approach which does not re-sult in compatible estimates, and efficiency is lost because the information concerning contemporaneous correlation is not utilized in the O L S fit. The M S L S technique is therefore preferable for this application, because the goodness-of-fit measures are close to the O L S fit and higher than the composite model fit. Estimates of the coefficients and their standard deviations are consistent, and more efficient. This was indicated by a high summed rank for other features. The composite model fit resulted in unbiased estimates of the coefficients and their standard deviations; however, only one variable was estimated. In addition, the M S L S fit required only three more steps than the O L S fit. The main disadvantage of the M S L S fit was that the number of samples had to be reduced from 500 to 100. This was due to a l imitation of the computer memory to seven megabytes for this research. If a very large number of samples were used, and larger computer memory was not available, a subsample could be used, as was used for this apphcation. Alternatively, the M S L S fit could be modified by eliminating the last step, which is E G L S applied to the system using the estimated error covariance matrix. Instead, consistent estimates of the coefficients could be obtained by using the weighted second stage coefficients (equations 5.116, 5.117, 5.118, and 5.119). The gain in efficiency by accounting for the contemporaneous variances in the final system fit would be lost, but simultaneity bias would be removed, which is an improvement over the O L S fit. Chapter 6 Application 2: Estimation of Tree Diameter Distribution 6.1 Introduction Diameter distribution information is required in order to choose stands for harvest and to assess the expected financial return. The estimation of diameter distributions from stand measures can be done by first selecting a known probability density function (pdf), finding the parameters, and then by relating the parameters of the selected pdf to current stand measures. These equations can then be combined with a stand level growth model where stand attributes are first predicted and parameters of the pdf are then "recovered" from these predicted stand attributes. Hyink and Moser (1983) referred to this diameter distribution modelling technique as the parameter recovery method. Stand attributes from forest inventories can also be used as inputs to predict the parameters of the selected pdf, once the relationships between each parameter and the stand attributes are established. The alternative to estimating the parameters of a known pdf is to relate the stand attributes (measured or predicted) to percentiles of the diameter distribution (Anon. , 1987; Bailey et al., 1981; Borders et al., 1987b). To limit the scope of this thesis, the estimation of diameter distribution was restricted to the first method. The most commonly selected pdf is the Weibull distribution (Clutter et ai, 1983). The probability density function of this distribution is as follows: (a < X < co) im = o otherwise (6.127) 105 Chapter 6. Application 2: Estimation of Tree Diameter Distribution 106 where b > 0 c > 0 The parameter a is the location parameter, b is the scale parameter, and c is the shape parameter (Clutter et al., 1983). Although a is sometimes negative for the distribution, a must be nonnegative for diameter distributions. The Weibull distribution is attractive for representing diameter distributions in that the equation for the pdf is relatively simple with only three parameters, and the shape of the distribution is flexible (Bailey and Dell , 1973). The cumulative form of the Weibull distribution, the cumulative density function (cdf), is as follows: F(X.) = l - e [ _ ( ^ ) C ] (a* + 0.242275 qdiamwted + 0.001292 stemswted -0.030403 totvolwted + 1.302598 pred. a u t w t e d (6.169) pred.c2nd = 5.177297 - 0.300508 i o p f a + 0.331522 pred. a l r t +0.064493 pred. blst (6.170) where wt is the inverse of the square root of the estimated variance of the error term; pred.b2ndwied is the b parameter times the weight; qdiamwted is quadratic mean diameter times the weight; stems w t e d is stems per hectare times the weight; totvol w U d is volume per hectare times the weight; pred. a l B t w l e d is the predicted a parameter times the weight. Contemporaneous variances were calculated using equation 3.28, resulting in the following S matrix. 5.2812 -1.1863 0.4059 -1.1863 0.8688 0.2898 (6.171) 0.4059 0.2898 2.8225 where diagonal elements are the variances for an equation and off-diagonal elements are covariances between equations. The error terms between the a and b parmeter equations appeared negatively correlated, whereas the c parameter Avas positively correlated with the other two equations of the system. Based on equation 3.51, the estimated error covariance matrix was therefore the Chapter 6. Apphcation 2: Estimation of Tree Diameter Distribution 123 following matrix. 5 . 2 8 1 2 W 1 W i - 1 . 1 8 6 3 W ! W 2 0 . 4 0 5 9 W ! W 3 fi = - 1 . 1 8 6 3 W 2 W i 0 . 8 6 8 8 W 2 W 2 0 . 2 8 9 8 W 2 W 3 (6.172) 0 . 4 0 5 9 W 3 W ; 0 . 2 8 9 8 W 3 W 2 2 . 8 2 2 5 W 3 W 3 Because the a and b parameter equations have i id error terms, W j and W 3 are equal to the identity matrix of size n by n. The W 2 matrix is an n by n matrix with the diagonal elements equal to the square root of the estimated variances of the error terms for the second stage b parameter equation. The simplified error covariance matrix was therefore as follows: r 5.2812In - 1 . 1 8 6 3 W 2 . 0.4059In fi = - 1 . 1 8 6 3 W 2 0 . 8 6 8 8 W 2 W 2 0 .2898W 2 (6.173) 0.4059In 0 .2898W 2 2.8225L, 6.6.5 E G L S to F i t the S y s t e m of E q u a t i o n s The last step of the M S L S fitting technique is to use the estimated error covariance matrix, and obtain an E G L S fit of the system of equations simultaneously. The resulting M S L S fit was as follows: pred.a MSLS pred. b MSLS = 6.572920 + 0.076069 topht - 9.476689 x 1 0 - 4 stems +0.025394 totvol - 0.403223 pred. blst (6.174) = -6.095781 + 0.235510 qdiam + 0.001234 stems -0.028857 totvol + 1.235273pred. a u t (6.175) = 5.204059 - 0.299683 topht + 0.330314pred. a u t +0.058220 pred. but (6.176) The I M S L S technique was not used, because the number of samples was considered large enough for the asymptotic properties of the M S L S technique to be assumed. pred. c MSLS Chapter 6. Application 2: Estimation of Tree Diameter Distribution 124 6.7 Comparison of the Three Fitting Techniques 6.7.1 Goodness-of-fit Measures Fit Index The Fit Indices for each of the four equations for the O L S and the M S L S fits are presented in Table 6.15. The Fi t Indices for the M S L S fit were lower than those of the Table 6.15: Fit Indices for O L S and M S L S Fits of the D b h Distribution System Endogenous Variable a parameter b parameter c parameter OLS 0.7383 0.6744 0.1563 M S L S 0.6398 0.5383 0.1057 O L S fit. The result was expected as the O L S fit minimizes the squared difference between the endogenous variable and the corresponding predicted value, whereas the M S L S fit minimizes the squared difference between weighted values. The differences between the O L S and the M S L S Fi t Indices are larger for this second application than the differences shown for Application 1. For the composite model, the Fi t Index was calculated using the following equation. F I = ffUi (0-24 - F (DI))2 + (0.63 - F (D2))2 + (0.93 - F (DZ))2 S ^ = 1 (0.24 - 0.3333)2 + (0.63 - 0.3333)2 + (0.93 - 0.3333)2 where F (DI), F (D2), and F (DZ) are the estimated cumulative probabilities from the estimated Weibull distribution up to the Dl, D2, and D 3 diameter limits, respectively; 0.3333 is the mean of all three percentiles for the 121 plots; 0.24, 0.63, and 0.93 were the percentiles selected to represent each plot. Chapter 6. Apphcation 2: Estimation of Tree Diameter Distribution 125 In order to calculate F(D1), F(D2), and F(D3), the a parameter was reset to 0.0 if a negative value was predicted. The Fit Index using this equation for the composite model fit was 0.6442. To compare to the OLS technique, the predicted values for the a, b, and c parameter using the final OLS equations were used to obtain the estimated cumulative probabilities. One of the plots had an estimated a parameter which was reset to 0.0, another had a negative b parameter which was reset to 1.0 X 1 0 - 7 , and another had a negative c parameter which was not reset. The overall Fit Index for the O L S technique, using the above equation, was 0.6679. Similarly, an overall F i t Index for the M S L S technique was calculated using predicted a, b, and c parmeters from the final M S L S fit. For one of the 121 plots, the predicted value of the a parameter was negative, and was reset to 0.0. The overall Fit Index for the M S L S technique was 0.7125. The overall F i t Index was lowest for the composite model fit which may have resulted because a local rather than a global minimum may have been found using N L P . Also, in the composite model, the parameters of the Weibull distribution are a function of the stand parameters alone and are not a function of the other parameters of the Weibull distribution. The M S L S overall F i t Index was slightly higher than the overall O L S Fit Index, which may be due the simultaneous fit of the three parameters using the M S L S technique. The presence of negative b and c parameters from the O L S fit may also have caused the lower overall Fit Index. Mean Absolute Deviation The mean absolute deviations ( M . A . D . ) were calculated by class for each of the L H S endogenous variables. The classes were created in a similar manner as for Apphcation 1, with each class having 24 observations except the last class which had 25 observations. The M . A . D . values for the OLS fit and for the M S L S fit are presented in Table 6.16 and Table 6.17. As with the Application 1, the OLS technique resulted in lower M . A . D . Chapter 6. Application 2: Estimation of Tree Diameter Distribution 126 Table 6.16: M . A . D . for Five Classes for the O L S Fi t of the D b h Distribution System Endogenous Variable Classes, from Low to High Values 1 2 3 4 5 a parameter 1.884 0.706 0.841 1.083 1.951 b parameter 0.407 0.489 0.344 0.521 0.910 c parameter 1.174 0.924 0.551 0.356 2.445 Table 6.17: M . A . D . for Five Classes for the M S L S Fit of the D b h Distribution System Endogenous Variable Classes, from Low to High Values 1 2 3 4 5 a parameter 2.901 1.257 0.701 0.798 2.221 b parameter 0.557 0.524 0.395 0.638 1.149 c parameter 1.349 1.009 0.617 0.431 2.466 Chapter 6. Application 2: Estimation of Tree Diameter Distribution 127 values across the range of the endogenous variables, although the differences between the O L S and M S L S M . A . D . values by class, were small. For the composite model fit, an M . A . D . value was calculated for each percentile as shown below for the first percentile. Negative values for estimated parameters were reset as for the calculation of Fit Index. The M . A . D . for the 24thpercentile was 0.190, for the 63 r d percentile was 0.276, and for the 93 r d percentile was 0.075. Using the estimated parameters from the final O L S fit to obtain F(D1), F(D2), and F(D3), the M . A . D . for the 24th percentile was 0.236, for the 63 r d percentile was 0.239, and for the 93 r d percentile was 0.067. For the M S L S fit, the M . A . D . for the 24th percentile was 0.220, for the 63 r d percentile was 0.223, and for the 93 r d percentile was 0.070. The M . A . D . values were much the same for the three percentiles. Mean Deviation The mean deviations (M.D.) were calculated for the same classes as with the M . A . D . values. The results for the OLS fit are presented in Table 6.18, and for the M S L S fit in Table 6.19. As with the M . A . D . values by class, the M . D . values by class were lower with the OLS fit; however, the differences between the O L S and the M S L S fits are generally small. Also, the trends of over- and underestimation across the range of the endogenous variables are similar with the two fits. For the composite model fit, an M . D . value was calculated for each percentile as shown for the first percentile. M.A.D. = 5 & = 1 1 0 . 2 4 ( 0 1 ) (6.178) n M.D. = E » = 1 ( 0 . 2 4 - F p l ) ) (6.179) n Chapter 6. Application 2: Estimation of Tree Diameter Distribution 128 Table 6.18: M . D . for Five Classes for the O L S Fit of the D b h Distribution System Classes, from Low to High Values Endogenous Variable 1 2 0 o 4 5 a parameter -0.727 -0.110 -0.457 -0.219 1.452 b parameter -0.293 -0.448 -0.268 0.224 0.752 c parameter -1.079 -0.896 -0.531 0.073 2.335 Table 6.19: M . D . for Five Classes for the M S L S Fi t of the Dbh Distribution System Endogenous Variable Classes, from Low to High Values 1 2 3 4 5 a parameter -1.421 -0.324 -0.420 0.262 1.832 b parameter -0.529 -0.484 -0.329 0.242 1.064 c parameter -1.349 -0.844 -0.536 0.159 2.466 Chapter 6. Application 2: Estimation of Tree Diameter Distribution 129 Negative estimated parameters were reset as for the calculation of F i t Index. The M . D . for the 24th percentile was 0.175, for the 63 r d percentile was 0.197, and for the 93 r d percentile was -0.030. For the O L S fit, using the estimated Weibull parameters, the M . D . for the 24 t h percentile was 0.185, for the 63 r d percentile was 0.033, and for the 93 r d percentile was -0.054. The 24th and 63 r d percentiles were therefore underestimated and the 93 r d percentile was slightly overestimated. For the M S L S fit, the M . D . for the 24 t h percentile was 0.182, for the 63 r d percentile was 0.035, and for the 93 r d percentile was -0.064, much the same as for the O L S fit. 6.7.2 Relative Variances The trace of the estimated coefficient covariance matrix for the M S L S fit was 49.64200, whereas the trace of the OLS fit was 4.7411 for the system of equations. Because the system of parameter equations was simultaneous, the O L S estimate of the coefficient covariance matrix cannot be used to calculate confidence limits for the true parame-ter. However, because the simultaneity bias was removed in the first step of the M S L S technique, the estimated coefficient covariance matrix from the M S L S fit is consistent. Confidence limits using these O L S variance estimates would appear, incorrectly, to be much narrower than the those using the consistently estimated coefficient covariance matrix from the M S L S fit. The traces for the submatrices of the estimated coefficient covariance matrix corre-sponding to each individual equation are given in Table 6.20. Unlike Application 1, all of the equations of the dbh distribution system have endogenous variables on the R H S . The O L S estimates of the variance of the coefficients appear lower for all of the equations of the system, similar to the trace for the overall matrix. The comparison of the coefficient covariance matrix for the composite model with those from the other fitting techniques was not possible, because the L H S variable of the Chapter 6. Apphcation 2: Estimation of Tree Diameter Distribution 130 Table 6.20: Trace of the Coefficient Covariance Ma.trix for Each Equation of the D b h Distribution System Endogenous Variable a parameter b parameter c parameter O L S 3.690011 0.170987 0.880147 M S L S 5.056920 42.410241 1.174840 composite model does not appear on the L H S of equations of the system fitted by the O L S or the M S L S procedure. Also, the variances were not calculated for the composite model. 6.7.3 Table of Estimated Coefficients and Standard Deviations A summary of the estimated coefficients and their associated standard deviations from the OLS and M S L S fits is shown in Table 6.21. The estimated standard deviations for the coefficients were higher for the M S L S fit than for the O L S fit, but the O L S estimates of the coefficients are inconsistent, and so these standard deviations would result in underestimated confidence intervals for all of the coefficients in the system. The coefficients for the a parameter and c parameter equations were similar for the two techniques, but, for the b parameter equation, the coefficients were quite different. Since the M S L S estimated coefficients are similar to the unweighted multiple least squares fit of each of the second stage equations (equations 6.164, 6.165, and 6.166), the difference in coefficients must be due to simultaneity bias. Because the OLS results in biased coefficients, hypothesis statements tested using the results from the O L S fit wi l l be incorrect. For instance, using a value of 1.96 from the normal distribution, confidence intervals for the <515 coefficient which was associated with the b parameter in the first equation, were calculated as -1.58437 to -0.86721 from Chapter 6. Apphcation 2: Estimation of Tree Diameter Distribution 131 Table 6.21: Estimated Coefficients and Standard Deviations of Coefficients for the Dbh Distribution System Endogenous Coefficients Standard Deviations Variable O L S M S L S O L S M S L S a parameter on 7.214783 6.572920 1.908552 2.199947 012 0\260097 0.076069 0.118161 0.160974 013 -0.000956 -0.000948 0.000133 0.000152 014 0.026935 0.025394 0.003106 0.003645 015 -1.225793 -0.403223 0.182949 0.437296 b parmeter 021 3.574736 -6.095781 0.411770 6.516691 £22 0.162501 0.235510 0.020156 0.057110 023 -0.000180 0.001234 0.000047 0.000960 024 0.007432 -0.028857 0.001329 0.024576 25 -0.212128 1.235273 0.031998 0.969079 c parameter 031 4.436816 5.204059 0.917986 1.027642 032 -0.349270 -0.299683 0.090431 0.118933 033 0.237814 0.330314 0.056247 0.088472 0~34 0.540540 0.058220 0.161578 0.311160 Chapter 6. Apphcation 2: Estimation of Tree Diameter Distribution 132 the O L S fit, and as -1.26032 to 0.45388 from the M S L S fit. Using the results from the M S L S fit, this coefficient would be considered to be zero, whereas for the O L S fit, the coefficient is nonzero. Similar differences in hypothesis testing were noted for the 825 and 834 coefficients. 6.7.4 R a n k i n g f o r O t h e r F e a t u r e s The ranks assigned to each fitting technique are shown in Table 6.22. For the Infor-Table 6.22: Ranks for the Three Techniques for the Fi t of the D b h Distribution System Feature O L S Composite M S L S Information 2 1 2 Consistency 1 2 2 Confidence Limits 1 2 - 2 Asymptotic Efficiency 1 .1 2 Compatibility 1 2 2 Ease of fit 3 1 2 T o t a l 9 9 12 mation feature, the composite model was assigned a rank of one as only one endogenous variable was estimated. The only consistent estimates were the M S L S coefficient esti-mates, as the nonlinear least squares procedure used to fit the composite model was restricted to obtain positive values for the predicted b and for the c parameter values. Confidence limits can be calculated for M S L S as the sample size was quite large and the estimated coefficients are asymptotically normally distributed. For the composite model, confidence limits could be calculated if generalized nonlinear least squares were used to fit the model, because estimates of the variances of the coefficients could be obtained. In terms of efficiency, the M S L S approach should be more efficient than the O L S approach; Chapter 6. Application 2: Estimation of Tree Diameter Distribution 133 however, since the O L S estimates of all of the coefficients of this system were biased and inconsistent, this expected increase in efficiency using the M S L S approach was difficult to witness. The M S L S and composite model fits both result in compatibility, whereas the O L S fit does not. The number of steps for the M S L S fit is greater than that for the O L S fit; for the composite model, the number of steps was not calculated as no attempt was made to estimate the variances of the coefficients of the nonlinear composite model. Because the fit of the nonlinear composite model would require simultaneous estimation of the error covariance matrix and the coefficients if the error terms were non-iid, the composite model was considered to be the most difficult to fit. Also, the number of coef-ficients to be estimated in the nonlinear composite model was 15; the chance of obtaining a local minimum rather than a global minimum is probably high with this large number. The composite model was therefore given a low "ease of fit" ranking. The M S L S technique was therefore assigned the highest sum of ranks for this second application. 6.8 Conclusion The chosen system of parameter prediction equations was simultaneous. However, the L H S variable of the third equation, the c parameter, did not appear on the R H S . One of the equations, the 6 parameter equation, was found to have heteroskedastic error terms using the O L S technique. The error terms for the c parameter equation indicated some lack-of-fit, but the error terms for this equation were assumed to be i id , as were the error terms for the a parameter equation. E G L S was used to fit the b parameter equation, whereas multiple linear regression was used to fit the other two equations. For the composite model, a nonlinear model was derived with 15 coefficients to be Chapter 6. Application 2: Estimation of Tree Diameter Distribution 134 estimated. The equation was fitted using restricted nonlinear least squares. No attempt was made to check for serial correlation or heteroskedasticity. The unweighted fit of the second stage equations, for the M S L S fit, indicated that the characteristics of the error terms were similar to those of the error terms from the unweighted fit for the O L S technique. The variances of the error terms for the second stage b parameter equation were estimated and the equation was refitted using E G L S . The estimated error covariance matrix was then used to obtain the final M S L S fit. The goodness-of-fit measures for the individual parameter prediction equations were best for the O L S fit, as expected. Unlike Application 1, the goodness-of-fit measures for the M S L S fit were somewhat worse than for the OLS fit. The goodness-of-fit measures for the composite model were compared to the simulated composite model using the O L S fit, and, separately, the M S L S fit. The goodness-of-fit measures, in this case, were generally best for the M S L S fit, except for the M . D . values; however, since the O L S fit resulted in one negative predicted b parameter and one negative predicted c parameter, the goodness-of-fit measures may be somewhat misleading for the simulated composite model. The goodness-of-fit measures for the composite model were the worst. The trace of the estimated coefficient covariance matrix was lowest with the O L S fit. The trace values for the submatrices of this matrix, corresponding to each equation of the system, were lower for O L S than for M S L S . Since all three of the parameter prediction equations have R H S endogenous variables, these results are similar to those found for the total volume and volume ratio equation of Application 1 which also had R H S endogenous variables. Since the O L S fit coefficients are biased and inconsistent, these estimated variances can not be used to obtain confidence limits for the true coefficients. The summed rank for other features was highest for the M S L S fit. The composite model was given a low ease-of-fit rank because the simultaneous estimation of the 15 coefficients was difficult, and if estimates of the variances of these coefficients is also Chapter 6. Application 2: Estimation of Tree Diameter Distribution 135 desired, the estimation would be even more difficult. The O L S fit resulted in biased and inconsistent estimates as all of the equations have R H S endogenous variables. In terms of goodness-of-fit, the O L S fit was better; if a fit of the sample data were required and this fit was not to be used for other samples of the population, the O L S fit would be appropriate. However, the O L S fit results in inconsistent estimates of the coefficients, and the results could not be used to test hypothesis statements and should not be used for other sample data. The composite model fit would require a difficult fitting procedure to obtain estimates of the coefficient covariance matrix, and the goodness-of-fit measures were lower than those of the M S L S fit for the simulated composite model. Because tests of hypothesis statements are useful, and the fitted equation may be used for other samples of the population, the M S L S approach was considered the best technique for fitting this system of equations; the coefficients and the estimated variances of the coefficients are consistent. Chapter 7 Application 3: Volume Growth and Yield 7.1 Introduction The management of forest resources requires accurate information about the current and future wood supply. Systems of equations have been widely used to represent for-est growth and yield. Simultaneous fitting techniques such as 2SLS, 3SLS, restricted 3SLS, and minimizing loss functions for the system have been used to fit these equations (Borders and Bailey, 1986; Burkhart and Sprinz, 1984; Furnival and Wilson, Jr., 1971; Hans, 1986; Murphy and Beltz, 1981; Murphy and Sternitzke, 1979; Reed et al, 1986). However, for each of these studies, the assumption was made that individual equations of the system have i id error terms. Permanent sample plots are often used in fitting growth and yield systems. Because psps are measured repeatedly over time, error terms may be serially correlated, and also, the variance of the error terms may differ among plots. The presence of serial correlation between error terms wil l depend on the length of time between measurements and whether overlapping intervals are used in estimation (Borders et al, 1987a). The use of simultaneous fitting techniques for systems of growth and yield equations assuming that the error terms are i id may therefore be less efficient than the M S L S technique, depending on the degree of serial correlation or heteroskedasticity of the error terms. 136 Chapter 7. Apphcation 3: Volume Growth and Yield 137 7.2 Data Preparation The psp data used for Application 2 were also selected for this apphcation. Plots with more than 80 percent pine by volume, breast height age recorded, and no treat-ment applied were selected. Data from Northeastern Alberta were deleted so that only lodgepole pine was represented in the data. One plot was selected from each cluster of four plots at a location. Summary information from the establishment measurement and the subsequent measurement was selected; any plot having only the establishment measurement was deleted from the data. Data were graphed and no outliers were noted. The distribution of-the remaining 28 plots is given in Table 7.23. Table 7.23: Distribution of Selected Psps for the Third Application Age at Breast Height Stems 21 41 61 81 101 121 per to to to to to to Total Hectare 40 60 80 10.0 120 140 0 to 1000 1 1 1 3 1001 to 2000 4 2 4 1 11 2001 to 3000 1 3 1 1 6 3001 to 4000 3 1 1 5 4001 to 5000 1 1 5001 to 6000 1 1 6001 to 7000 1 1 Total 7 5 5 2 6 3 28 Chapter 7. Apphcation 3: Volume Growth and Yield 138 7.3 Model Selection Two equations of the growth and yield system developed by Clutter (1963) were selected for analysis. These equations were as follows: lnBA2 = InBA^+aAl-^f] +a2(l-^f] SI+ error1 (7.180) A2 V A2J \ A21 lnV2 = 0o+ 0i SI + / 3 2 - J - + 03lnBA2 + error2 (7.181) where InBAi and lnBA2 are the natural logarithms of the basal area per hectare measured at times 1 and 2, respectively; Ai and A2 are the number of years counted at breast height at times 1 and 2, respectively; SI is the site index for a reference age of 50 years measured at breast height; In V2 is the natural logarithm of volume per hectare at time 2; ai and c t 2 , and 0O through 03 are coefficients to be estimated; errori and error2 are the error terms. These equations were fitted simultaneously by Burkhart and Sprinz (1984) using a min-imum loss function for the two equations. Borders and Bailey (1986), Murphy and Ster-nitzke (1979), and Hans (1986) used simultaneous fitting techniques from econometrics, assuming that error terms of individual equations are i i d , to fit modifications of these equations. The equations can be rearranged as follows: (in BA2 - In BAX ^ ) = * u + 612 (l - ^ ) + S13 (l - ^ ) SI + e1 (7.182) lnV2 = $21 + 8 22 SI + 823 — + 824 In BA2 + e2 (7.183) ^2 The term InBAij^ is a combined term where BAi is a lagged endogenous variable which can be treated as a predetermined variable, and the term j+ is an exogenous Chapter 7. Apphcation 3: Volume Growth and Yield 139 variable. This combined term can be treated as a constant which is subtracted from the endogenous variable, lnBA2, to obtain the L H S endogenous variable shown in the basal area equation above. A n intercept coefficient was added to this basal area equation, although this coefficient is expected to be close to zero. 7.4 Ordinary Least Squares Fit 7.4.1 Unweighted Simple or Multiple Linear Regression The estimated coefficients using multiple linear regression to fit each equation of the system individually were as follows: pred. (in BA2 - In BAj = -0.017785 + 4.458907 ( l --0.011270 ( l - ^ ) Si" (7.184) pred. In V2 = 2.315771 -f 0.066766 S J - 47.758153-^j-A2 -(-1.014053 In BA2 (7.185) The estimated error terms from the unweighted fit were then used to test for serial correlation and heteroskedasticity. 7.4.2 Testing for iid Error Terms To check for serial correlation in each equation, the estimated error terms were sorted by the predicted endogenous variables, and a graph of the current error term versus the previous error term was obtained. The R2 value for the simple linear regression of the current error term with the previous error term was 0.00965 for the basal area equation and 0.011511 for the volume equation. The Durbin and Watson (1951) test statistic for each model was not significant for either positive or negative serial correlation using an Chapter 7. Application 3: V o l u m e Growth and Yield 140 alpha of 0.10 (0.05 for positive and 0.05 for negative serial correlation). Each equation of the system therefore had independent error terms. To test for heteroskedasticity, a graph of the estimated error versus the predicted L H S endogenous variable was done. The regression of the estimated error, squared, with the predicted variable on the L H S of the basal area equation resulted in an R2 value of 0.00270. The error terms for this equation were therefore considered indentically distributed. The regression of the estimated error, squared, with the predicted logarithm of volume from the second equation, resulted in an R2 value of 0.01425. The test statistic for the Goldfeld and Quandt (1965) test was 1.9149 for nonincreasing variance with increasing predicted logarithm of volume. The critical F value for 6 degrees of freedom for the numerator and the denominator, and an alpha level of 0.05 was 4.28. The error terms of the second equation of the system were therefore considered identically distributed. 7.4.3 Appropriate O L S Fit Based on Error Structure Because each of the equations of the system had iid error terms, the unweighted O L S fit was considered appropriate. This unweighted fit was therefore used in the comparison with other methods. 7.5 Composite M o d e l Fit 7.5.1 Derivation of the Composite M o d e l A composite model was derived by Sullivan and Clutter (1972), by substituting the equation for the logarithm of basal area at time 2 into the equation to estimate yield at time two. The resulting equation was as follows: lnV2 = 7o + 7 i SI + 7 2 +^/3lnBA1 ^ + 7 4 (l -Chapter 7. Application 3: Volume Growth and Yield 141 +75 ( l ~ ^ ) SI + e3 (7.186) where 7 0 through 75 are coefficients to be estimated; e3 is the error term. 7.5.2 Unweighted Regression of the Composite Model A l l of the R H S variables of the composite model can be considered predetermined variables and are uncorrelated with the error term. The fit using multiple linear regression results in unbiased estimates of the coefficients and their variances. The fitted equation was as follows: pred. In V2 = 2.513266 + 0.063926 SI - 39.294313 -j- + 0.961204 A-i In BA-i ^ + 3.655157 ( x ~ ^ ) ~ 0.003875 ( l - ^ 5/(7.187) 7.5.3 Testing for iid Error Terms Data were ordered by the predicted logarithm of volume using the unweighted fit of the composite model, and a graph of the estimated error with the previous estimated error term was examined for serial correlation. The regression of the current error term with the previous error term resulted in an R2 value of 0.0082, and the Durbin and Watson test statistics for positive and for negative serial correlation were not significant for an alpha of 0.10 for both tests. The error terms for the composite model were therefore independent. To test for heteroskedasticity. a graph of the estimated error term with the predicted logarithm of volume was obtained, and the simple linear regression of the estimated error term, squared, with the predicted logarithm of volume was performed. The R2 value was 0.03734. The Goldfeld and Quandt (1965) test statistic was 1.0567 assuming Chapter 7. Application 3: Volume Growth and Yield 142 that the variance is nonincreasing with increasing predicted logarithm of volume. This test statistic was less than the critical F value of 5.05 for 5 degrees of freedom for the numerator and for the denominator, and for an alpha of 0.05. The error terms were therefore considered homoskedastic. 7.5.4 Weighted Fit of the Composite Model The error terms for the composite model were found to be i id . The unweighted fit was therefore used to compare to other fitting techniques. 7.6 MSLS Fit 7.6.1 First Stage Equations The two equation of the system met the rank and order conditions and were identified. For the second equation of the system, In BA2 appears on the R H S and is the part of the complex variable which appears as the endogenous variable on the L H S of the first equation. To remove the simultaneity bias for the second equation, the following first stage model was obtained. pred. (lnBA2 - lnBA1 ^) = 0.163469 - 0.013701 SI - 1.026070 4~ + 4.071904 V A j l B t A2 (l - -^j + 0.026270 ( l - -^j SI (7.188) 12 / V -12 < The predicted In BA2 values were then recovered from this first stage equation using the following equation. / 4 \ A pred.lnBA2lst = pred. [InBA2 - I n B A 1 ^ ) +lnBA1-± (7.189) Chapter 7. Application 3: Volume Growth and Yield 143 7.6.2 Second Stage Equations The second stage equations were derived by substituting the predicted value for the logarithm of basal area at time 2 from the first stage equation into the R H S of the In V2 equation. Because the second stage equation for lnBA2 — InBAi ^ has only exogenous variables on the R H S , the unweighted O L S fit of this equation was the same as the fit of the second stage equation. pred. (lnBA2 - lnBA1 ^f] = -0.017785 + 4.458907 ( l - - 1 ^ ^ 2 / 2nd ^ A _ -0.011270 ( l - SI (7.190) pred.lnV22nd = 2.550358 + 0.066408 5 / - 47.37629 - j -A2 +0.9503109 pred. lnBA2ut (7.191) 7.6.3 Testing for iid Error Terms The basal area second stage equation was the same as the basal area equation for the unweighted OLS fit; therefore, the error terms for this second stage equation were i i d . To test for serial correlation in the second stage volume model, the estimated error terms were ordered by the predicted logarithm of volume, and a graph of the estimated error term with the previous error term was obtained. The R2 value for the simple linear regression of the current with the previous error term was 0.01856. The Durbin and Watson (1951) test statistics for positive and negative serial correlation were not significant for an alpha of 0.10 for the two sided test. The error terms were therefore considered independent for the second stage logarithm of volume equation. To test for heteroskedasticity, a graph of the estimated error with the predicted log-arithm of volume using the second stage model was obtained. The regression of the estimated error, squared, with predicted logarithm of volume resulted in an R2 value Chapter 7. Application 3: Volume Growth and Yield 144 of 0.0950. The Goldfeld and Quandt (1965) test for an alpha of 0.05 was not signifi-cant. The error terms of the second stage logarithm of volume equation were therefore identically distributed. 7.6.4 E s t i m a t i o n of the E r r o r C o v a r i a n c e M a t r i x Because the equations of the system have i id error terms, the M S L S procedure simplifies to the 3SLS procedure. Estimates of contemporaneous variances using equation 3.28 were calculated from the estimated error terms of the second stage models, resulting in the following S matrix. r 0.0067 0.0060 0.0060 0.0078 The error covariance matrix for the system was therefore as follows: (7.192) fi (7,193) 0.0067In 0.0060In 0.0060L, 0.00781,, The covariance of the error terms between equations 0.0060 was quite high relative to the variance of the error terms within each equation which were 0.0067 and 0.0078. The equations therefore appear to be correlated. 7.6.5 E G L S t o F i t the S y s t e m of E q u a t i o n s The final step of the M S L S fit was to fit the system simultaneously using the estimated error covariance matrix, fl. The resulting M S L S fit of the system of equations was as follows: pred. (lnBA2 - I n B A ^ ) = -0.0480136 + 4.588017 f l - ^ ) V A2/ MSLS \ A2J -0.008839 - 5"! (7.194) Chapter 7. Apphcation 3: Volume Growth and Yield 145 pred. In V 2 M S L S = 2.264741 + 0.072275 SI - 47.47299 — ^±2 + lMA2Upred.lnBA2ut (7.195) The I M S L S technique in this case is the same as the I3SLS technique. The estimated coefficients from the M S L S fit were used to estimate new values for the contemporaneous correlation by using equation 3.32, and a new M S L S fit was calculated. This process was repeated for eight iterations, until the criterion specified in equation 4.65 was met. Coefficients were similar to those for the M S L S fit. 7.7 Comparison of the Three Fitting Techniques 7.7.1 Goodness-of-fit Measures Fit Index The Fit Indices for each of the two equations for the O L S , M S L S , and I M S L S fits are presented in Table 7.24. Fit Indices for the composite model fit are presented for the logarithm of volume only. The Fit Indices for the M S L S fit were marginally lower than those for the O L S fit, for the basal area equation. For the volume equation, the Fit Index for the O L S fit was the highest, followed by the composite model, and the lowest F i t Index was for the I M S L S fit: Table 7.24: Fit Indices for O L S , Composite Model , M S L S and I M S L S Fits of the Yie ld Equation System Endogenous Variable O L S Composite Model M S L S I M S L S In BA2 - In BAi £ 0.9536 none 0.9519 0.9496 In V2 0.9802 0.9129 0.9024 0.8974 Chapter 7. Application 3: Volume Growth and Yield 146 Mean Absolute Deviation The mean absolute deviations ( M A . D . ) were calculated by class for each of the L H S endogenous variables. The classes were created by sorting the 28 samples by the endoge-nous variable, and then dividing the sorted data into four classes of six samples each, with the fifth class having only four samples. The M . A . D . values for the O L S fit, for the M S L S fit, and for the composite model fit for the logarithm of volume only, are presented in Table 7.25, Table 7.26, and Table 7.27. No values are shown for the I M S L S fit; the values were within 0.005 units from the M S L S fit. Table 7.25: M . A . D . for Five Classes for the O L S Fit of the Yie ld Equation System Classes, from Low to High Values Endogenous Variable 1 2 3 4 5 In BA2 - In BA1 % 0.043 0.081 0.074 0.035 0.044 lnV2 0.043 0.037 0.032 0.021 0.035 Table 7.26: M . A . D . for Five Classes for the M S L S Fit of the Yie ld Equation System Classes, from Low to High Values Endogenous Variable 1 2 3 4 5 lnBA2 - lnBA1 0.039 0.087 0.078 0.034 0.046 lnV2 0.090 0.070 0.052 0.078 0.054 For the basal area equation, the M . A . D . values were only marginally lower for the O L S fit compared to the M S L S fit. For the volume equation, the M . A . D . values were somewhat higher for the M S L S and composite model fits, relative to the O L S fit. The Chapter 7. Application 3: Volume Growth and Yield 147 Table 7.27: M . A . D . for Five Classes for the Yie ld Composite Model Fit Classes, from Low to High Values Endogenous Variable 1 2 3 4 5 In Vi 0.089 0.073 0.039 0.080 0.057 composite model fit had similar M . A . D . values to the M S L S fit. M e a n D e v i a t i o n The mean deviations (M.D.) were calculated for each equation for the same five classes as for M . A . D . Results for the O L S fit are presented in Table 7.28, for the M S L S fit in Table 7.29, and for the Composite Model fit in Table 7.30. The M . D . values for the I M S L S fit were similar to the M S L S fit for the volume equation. For the basal area equation, values for the I M S L S fit were close the those for the O L S fit. Table 7.28: M . D . for Five Classes for the OLS Fit of the Yie ld Equation System Classes, from Low to High Values Endogenous Variable 1 2 3 4 5 In BA2 - In BAX ^ -0.024 -0.010 0.008 -0.004 0.044 lnV2 -0.003 -0.017 0.003 0.002 0.023 For the basal area equation, the M . D . values were similar for the O L S and M S L S fits. The directions of the deviations between the observed and predicted values were similar also. For the volume equation, the M . D . values were krwer for the O L S fit than for the M S L S fit. The M . D . values for the composite model fit were the highest. Chapter 7, Apphcation 3: Volume Growth and Yield 148 Table 7.29: M . D . for Five Classes for the M S L S Fi t of the Yield Equation System Endogenous Variable Classes, from Low to High Values 1 2 3 4 5 In BA2 - In BA1 ^ -0.006 0.001 0.009 -0.015 0.019 In V2 0.006 -0.070 0.004 0.023 0.054 Table 7.30: M . D . for Five Classes for the Yie ld Composite Model Fit Classes, from Low to High Values Endogenous Variable l . | 2 3 4 5 In V2 -0.007 | -0.068 0.008 0.030 0.057 7.7.2 Relative Variances The trace of the estimated coefficient covariance matrix for the M S L S fit was 12.35046 whereas the trace of the O L S fit was 7.500340 for the system of equations. However, the O L S fit results in coefficients which are inconsistent, because the endogenous variable on the L H S of the first equation was modified and included as an endogenous variable on the R H S of the second equation of the system. The traces for the submatrices of the estimated coefficient covariance matrix corre-sponding to each individual equation using the O L S and M S L S fitting techniques are given in Table 7.31. The trace of the coefficient covariance matrix for the composite model fit is shown in the table; however, the composite model had six coefficients in the volume equation, whereas for the O L S and M S L S fit of the system of equations, only four coefficients were estimated for the volume equation. Chapter 7. Application 3: Volume Growth and Yield 149 Table 7.31: Trace of the Coefficient Covaria,nce Matr ix for Each Equation of the Yie ld Equation System Endogenous Variable O L S M S L S Composite Model lnBA2 - InBA^ 0.59786 0.21083 none In V2 6.9025 12.1396 98.1090 The high trace value for the composite model was attributable mostly to the high variance of the coefficient associated with the -~ variable. Also, the mean squared error for the composite model was 0.00933, whereas for the O L S fit of the volume equation, the mean squared error was only 0.00194. The higher mean squared error and larger number of variables likely caused the high value for the trace of the estimated coefficient covariance matrix from the composite model. The OLS fit of the basal area equation resulted in unbiased estimates of the coefficients and their variances, because all R H S variables of this equation were exogenous. The M S L S fit resulted in a lower trace value for the estimated coefficient covariance matrix, which was the result of using the information from the volume equation in fitting the basal area equation (contemporaneous variances). The O L S fit of the volume equation results in biased and inconsistent coefficients, because basal area appears as a L H S and as a R H S variable in the system of equations. The trace of the estimated coefficient covariance matrix was therefore lower than for the M S L S fit. Because the number of samples used to fit this system of equations was small, the large sample properties of consistency and asymptotic efficiency may not apply for the 3SLS fit. The O L S fit has been shown to be more biased than 3SLS, and since the M S L S fit for this system was simply a 3SLS fit, the O L S results may be more biased than those of the M S L S fit, for the volume equation. Chapter 7. Apphcation 3: Volume Growth and Yield 150 7.7.3 Table of Estimated Coefficients and Standard Deviations A summary of the estimated coefficients and their associated standard deviations from the O L S and M S L S fits is shown in Table 7.32. Table 7.32: Estimated Coefficients and Standard Deviations of Coefficients for the Yield Equation System Endogenous Variable Coefficients Standard Deviation O L S M S L S I M S L S O L S M S L S I M S L S biBA2-lnBA,± on 012 013 -0.017785 4.458907 -0.011270 -0.048014 4/588017 -0.008839 -0.059548 4.561855 -0.003256 0.048789 0.770864 0.035310 0.034473 0.453306 0.022136 0.032591 0.394548 0.019709 lnV2 8 21 $22 023 023 2.315771 0.066766 -47.758153 1.104053 2.264741 0.072275 .-47.47299 1.004244 2.202168 0.074283 -46.69858 1.010136 0.134221 0.005269 2.623611 0.033243 0.163504 0.006985 3.480158 0.036704 0.145804 0.006259 3.119119 0.031243 The estimated coefficients were similar for the three techniques. Because the I M S L S technique was simply I3SLS for this problem, and the I3SLS should converge to M L E if the error terms are normally distributed, the I3SLS estimates of the coefficients may be more appealing. 7.7.4 Ranking for the Three Techniques The ranks assigned to each fitting technique are shown in Table 7.33. Because of the small number of samples, the assumption that the M S L S estimates are normally distributed may not be applicable: the calculation of confidence limits using the results from this small number of samples would likely be incorrect. The M S L S estimators are Chapter 7. Application 3: Volume Growth and Yield 151 Table 7.33: Ranks for the Three Techniques Feature O L S Composite M S L S Information 2 1 2 Consistency 1 2 2 Confidence Limits 1 2 1 Asymptotic Efficiency 1 1 2 Compatibility 1 2 2 Ease of fit 2 3 1 Total 8 11 10 consistent, but, because of the small number of samples, the estimates obtained may be quite different from the true values. The summed rank for these features was highest for the composite model, because estimates of the coefficients are unbiased and so are the estimated variances of these coef-ficients. However, the composite model only estimates the coefficients for the endogenous variable of the second equation of the system. For the first equation, the basal area equa-tion, the coefficients from the fitted volume equation may be used, as demonstrated by Sullivan and Clutter (1972), but these estimated coefficients are not necessarily unbiased or consistent estimates. The O L S fit resulted in inconsistent estimates of the coefficients for the volume model, as one of the R H S variables is endogenous. Also, the ease-of-fit rank was lower than for the composite model fit, as only one equation is fitted for the composite model. Finally, the OLS fit does not result in compatible equations. Chapter 7. Application 3: Volume Growth and Yield 152 7.8 Conclusion The system of equations proposed by Clutter (1963) was selected for this application. This system is simultaneous, although the first equation has only exogenous variables on the R H S . The estimated error terms from the unweighted OLS fit were i i d , even though two measurements from psps were used to analyze the two equation yield system, and serial correlation was expected to be significant. This may have resulted because the plot data were pooled and serial correlation within measurements of a plot was masked by this grouping of data. Also, the measurement period between the first and second varied from five to 14 years, and the longer periods result in a reduction i n the correlation between measurements. The unweighted fit was used for comparison with the other fitting techniques. The composite model was derived by Sullivan and Clutter (1972). The estimated error terms from the multiple linear regression of the composite model were i i d . The estimated error terms from the unweighted fit of the second stage equations for the M S L S fit, were i i d . The M S L S technique therefore was reduced to the 3SLS technique. The estimated contemporaneous variance between the two equations was high relative to the estimated variances with each equation and the error terms between equations appeared to be highly correlated. The goodness-of-fit measures were generally best for the O L S fit. For the basal area equation, the M S L S goodness-of-fit measures were close to those of the O L S fit, and sometimes better. For the volume equations, the goodness-of-fit measures for the composite model fit and for the M S L S fit were similar, and somewhat worse than those of the OLS fit. The trace of the estimated coefficient covariance matrix was lower for the O L S fit than Chapter 7. Apphcation 3: Volume Growth and Yield 153 for the M S L S fit. A n examination of the trace values for the submatrices indicated that, like Apphcation 1, the trace value for the equation with no R H S endogenous variables (basal area equation) was lower for the M S L S fit than for the O L S indicating an increase in efficiency by accounting for contemporaneous correlation. Like Applications 1 and 2, the trace value for the submatrix of the estimated coefficient covariance matrix for the equation with R H S endogenous variables, the volume equation, was lower for the OLS fit. If the variances of the coefficients from the O L S fit were used to calculate confidence limits, the confidence interval would be underestimated. The composite model fit was given the highest summed rank for other features. This was largely due to the small number of samples used to test this application. The M S L S technique was given a lower rank because some of the properties of E G L S could not be applied to the M S L S fit for only 28 samples. If the number of samples was larger, the summed rank for the M S L S fit would be equal to that of the composite model fit. The composite model only estimated the L H S variable; however, unbiased estimates of the coefficients were obtained. In terms of goodness-of-fit measures, the O L S fit was best, but the estimates of the coefficients of the volume model were biased and inconsistent. Also, information concerning the contemporaneous correlation was not used for the OLS fit, and so the estimates are less efficient than with the M S L S technique. The M S L S fit was not necessarily the best for this apphcation, because of the small number of samples used. The large sample properties of E G L S , applicable to M S L S could not be assumed. If more two-measurement psps were available, the large sample properties of the M S L S technique could be assumed and so this technique would be the most favorable. Alternatively, all possible pairs of the measurements from the selected psps could have been used to fit the model, but serial correlation between the measurements of each plot may have been significant. Using all possible pairs of measurements, the error structure would likely be more complex than that given for the serially and contemporaneously Chapter 7. Application 3: Volume Growth and Yield 154 correlated systems of equations in Chapter 3 of this thesis. If compatibilitjr and estimates of each of the L H S variables of this system were re-quired, the I M S L S procedure would be the most appropriate, because these estimates woidd be equal to the M L E estimates if error terms were normally distributed. The standard deviations shown for the I M S L S fit were the lowest for the basal area equation, and were lower than those of the M S L S fit for the volume equation. Chapter 8 Overall Discussion and Conclusions The first hypothesis of this thesis was that a fitting technique exists which satisfies the desired criteria for simultaneous, contemporaneously correlated systems of equations, in which individual equations have non-iid error terms. The second hypothesis was that any additional computational burden in using the technique is compensated by the benefits of meeting the desired criteria. The first objective of this research, related to the first hypothesis, was to find a technique from the literature which meets the desired criteria for simultaneous, con-temporaneously correlated systems of equations, in which the error terms for individual equations are non-iid. This objective was not met, because no technique was found which satisfied these criteria for systems of equations with this error structure. However, infor-mation from the literature was used to derive a new fitting technique, labelled multistage least squares ( M S L S ) , which is an extension of the 3SLS technique to systems in which the error terms of individual equations are non-iid. The estimated coefficients from the M S L S technique are consistent and asymptotically efficient, if the estimated error covari-ance matrix is consistent and the error structure has been correctly determined for the system. Confidence limits can be calculated for large sample sizes, and compatibility is maintained. The second objective, related to the second hypothesis, was to compare the chosen technique to the common techniques of O L S applied to each equation, and O L S applied 155 Chapter 8. Overall Discussion and Conclusions 156 to a composite model. Since no technique was found in literature, the M S L S technique was used for this comparison. The three techniques were applied to three forestry prob-lems for which systems of equations are used. The criteria for examining the results of the three techniques included the comparison of goodness-of-fit measures (Fit Index, Mean Absolute Deviation, Mean Deviation), the comparison of the trace of the estimated coefficient covariance matrix, and the calculation of a summed rank based on the amount of information given, the consistency of estimates using information from literature to assess the system, the ability to calculate confidence intervals, the efficiency using in-formation from literature to assess the system, the compatibility, and the ease of fit in terms of the number and difficulty of steps required. The OLS fit of individual equations is simple to calculate and algorithms are readily available. The O L S fit of each of the systems of equations for the three applications, resulted in better goodness-of-fit measures than did the M S L S fit, as expected, because the OLS fit minimizes the sum of squared differences. Also, the estimated coefficients from the O L S fit were generally close to those from the M S L S fit. The OLS fit requires less computer memory than the M S L S fit; large forestry problems with many equations, variables, and samples can be fitted. However, for simultaneous systems of equations, the estimated coefficients are biased and inconsistent. The estimates do not converge to the true estimates, with increasing sample size. Also, confidence limits cannot be calculated and compatibility within the system is not assured. The OLS fit of a composite model, created by substituting all of the equations into one equation, was simple to perform for the applications tested. Also, for the two l in-ear composite models, the estimated coefficients were unbiased, because all of the R H S variables were exogenous. For the nonlinear composite model, coefficients were restricted and are therefore likely biased. For the first apphcation, the volume system, the derived Chapter 8. Overall Discussion and Conclusions 157 composite model did not appear to have all of the important variables. For the dbh dis-tribution, the composite model did not show the relationships among the parameters of the Weibull distribution. Because the composite model derived for the growth and yield system is useful for predicting only the volume yield, the basal area yield was not pre-dicted. The technique of deriving and fitting a composite model meets all of the desired criteria for some systems of forestry equations. However, if endogenous variables remain on the R H S , estimated, coefficients are biased and consistent. Since only one endogenous variable is- predicted, the composite model fit may be undesirable. Also, the original biological relationships may be lost, and important variables may not be retained in the derived model. Finally, for large problems with many variables and many equations, this technique is impractical. For all three of the applications tested, the goodness-of-fit measures for the M S L S fit were close to those for the O L S fit, and were sometimes better than the composite •model fit. The number of steps required was similar to the OLS fit, also. In addition, the estimated coefficients of the M S L S fit were consistent and asymptotically efficient, except for the first application for which some efficiency was lost. Compatibil i ty was also obtained with the M S L S fit. Hypothesis statements can also be tested. For the first and second applications, the use of the O L S fit to incorrectly test hypotheses about coefficients resulted in different conclusions than if the consistent estimates from the M S L S fit were used. The selected applications did not demonstrate the use of the M S L S technique for serially correlated error terms in individual equations, or for heteroskedastic, serially correlated error terms. However, the desired criteria would still be met for these error structures, and the difficulty in obtaining the M S L S fit would likely be similar to the applications tested in this thesis. The main disadvantage of the M S L S fit was that more computer memory is required, Chapter 8. Overall Discussion and Conclusions 158 than for the OLS or composite model fits. For large forestry problems with many equa-tions, variables, and samples, a more efficient computer program or enough computer memory would be required. 1 Alternatively, a modified M S L S technique could be used, in which the final step of the M S L S technique is not performed. This modified M S L S technique is simply an extension of the 2SLS technique to non-iid error terms. Esti-mates of coefficients and their variances would remain consistent for this modified M S L S technique. A loss in efficiency would be incurred, because the information about contem-poraneous correlation used in the last step of the M S L S technique would not be utilized. The final alternative, which was used for Apphcation 1, is to fit the system using a subsample of the data. Again, a loss of efficiency would result. Another disadvantage of the M S L S technique is that estimates are not unbiased. For small samples, a Monte Carlo study to examine the degree of bias for small sample sizes for the M S L S and I M S L S techniques should be conducted. The contemporaneous correlation, correlation of the R H S variables with the error term, serial correlation, and heteroskedasticity should be varied to examine the effects on bias. However, studies using 3SLS for small samples indicated that the bias is often less for this technique than for the O L S technique. Similar results may occur for the M S L S technique. In summary, the first hypothesis of this thesis was refuted, because no technique was found in literature which meets the desired criteria for the error structure described. However, a fitting technique was derived as part of this research. In terms of the number of steps required, the second hypothesis was met using M S L S for the applications tested. 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Estimators of seemingly unrelated regressions: some exact finite sample results. J . Amer. Stat. Assn. 58: 977-992. Zellner, A . and D . S. Huang. 1962. Further properties of efficient estimators for seemingly unrelated regression equations. Int. Econ. Rev. 3: 300-313. Zellner, A . and H . Theil . 1962. Three-stage least squares: simultaneous estimation of simultaneous equations. Econometrica 30: 54-78. Appendix A Glossary of Terms and Abbreviations 2SLS Two Stage Least Squares; a technique to fit systems of equations which are simul-taneous but not contemporaneously correlated. 3SLS Three Stage Least Squares; a technique to fit systems of equations which are simultaneous and contemporaneously correlated. A F S Alberta Forest Service B L U E Best Linear Unbiased Estimator compatibility Logical relationships between equations of a system are retained in the fitted system of equations. composite A l l equations of the system are combined into one equation. consistent The probability that the estimate is within a small deviation from the pop-ulation value approaches one as the number of samples is increased; the estimate converges to the true value as the sample size is increased. contemporaneous correlation Correlation of error terms between equations of a sys-tem. contemporaneous variances The variances corresponding to contemporaneous corre-lation. dbh diameter at breast height (1.3 metres above ground) efficient The Cramer-Rao lower bound for efficiency is met. More efficient means that the variance of the estimate is lower whereas less efficient means that the variance of the estimate is higher. E G L S Estimated Generalized Least Squares; a technique developed for single equations to fit a linear model using an estimated error covariance matrix. endogenous variables Variables generated by the system of equations, stochastic. 170 Appendix A. Glossary of Terms and Abbreviations 171 exogenous variables Variables generated outside the system of equations, nonstochas-tic. F .I. F i t Index GLS Generalized Least Squares; a technique developed for single equations to fit a linear model using a known error covariance matrix. heteroskedasticity Variances of the error terms are not equal across the range of the sample data; error terms are not identically distributed. homoskedasticity Variances of the error terms are equal across the range of the sample data; error terms are identically distributed. I3SLS Iterated Three Stage Least Squares; the estimated error covariance matrix from the first 3SLS fit is used to fit the system of equations again. The process is repeated unti l convergence occurs. iid independent and identically distributed IMSLS Iterated Multistage Least Squares; the estimated error covariance matrix from the first M S L S fit is used to fit the system of equations again. The process is repeated until convergence occurs. inconsistent The probability that the estimate is within a small deviation from the population value does not approach one as the number of samples is increased. LHS Left Hand Side M.A.D. Mean Absolute Deviation M.D. Mean Deviation MLE M a x i m u m Likelihood Estimator MSLS Multistage Least Squares; derived in this thesis for fitting systems of equations which are simultaneous and contemporaneously correlated and have with non-iid error terms. non-iid Either not independent or not identically distibuted or both. OLS Ordinary Least Squares; techniques include simple linear regression, multiple linear regression, nonlinear least squares, and regression of weighted models. One of the assumptions for O L S is that the error terms are. i i d . Appendix A. Glossary of Terms and Abbreviations 172 psps permanent sample plots; plots that are established and marked so that future measurements from the same trees can be taken. RHS Right Hand Side serial correlation Dependence between the current error term and the previous one(s). simultaneity bias The bias in the estimated coefficients which is results if an equation with endogenous variables on the R H S is fit using O L S , G L S , or E G L S . simultaneous equations A system of equations wherein the L H S variable of one or more equations also appears on the R H S of one or more equations in the system. SUR Seemingly Unrelated Regression; a technique to fit systems of equations which are contemporaneously correlated but not simultaneous.