**Chapter 1. Mathematical Preliminaries**

1.1 Arithmetic Progression

1.2 Geometric Progression

1.3 The Binomial Formula

1.4 The Calculus of Finite Differences

1.5 The Number *e*

1.6 The Natural Logarithm

1.7 The Exponential Function

1.8 Exponential and Logarithmic Functions – Another Look

1.9 Change of Base of a Logarithm

1.10 The Arithmetic (Natural) Scale vs. the Logarithmic Scale

1.11 Compound Interest Arithmetic

**Chapter 2. Fundamentals of Growth**

2.1 Time Series Data

2.2 Relative and Average Rates of Change

2.3 Annual Rates of Change

2.3.A Simple Rates of Change

2.3.B Compounded Rates of Change

2.3.C Comparing Two Time Series: Indexing Data to a Common Starting Point

2.4 Discrete vs. Continuous Growth

2.5 The Growth of a Variable Expressed in Terms of the Growth of its Individual Arguments

2.6 Growth Rate Variability

2.7 Growth in a Mixture of Variables

**Chapter 3. Parametric Growth Curve Modeling**

3.1 Introduction

3.2 The Linear Growth Model

3.3 The Logarithmic Reciprocal Model

3.4 The Logistic Model

3.5 The Gompertz Model

3.6 The Weibull Model

3.7 The Negative Exponential Model

3.8 The von Bertalanffy Model

3.9 The Log-Logistic Model

3.10 The Brody Growth Model

3.11 The Janoschek Model

3.12 The Lundqvist-Korf Growth Model

3.13 The Hossfeld Growth Model

3.14 The Stannard Growth Model

3.15 The Schnute Growth Model

3.16 The Morgan-Mercer-Flodin Growth Model

3.17 The McDill-Amateis Growth Model

3.18 An Assortment of Additional Growth Models

3.18.A Modifications of the Hossfeld Equation

The The Levkovic I Growth Model

The Levkovic III Growth Model

The Yoshida I Growth Model

3.18.B The Sloboda Growth Model

Appendix 3.A The Logistic Model Derived

Appendix 3.B The Gompertz Model Derived

Appendix 3.C The Negative Exponential Model Derived

Appendix 3.D The von Bertalanffy and Richards Model Derived

Appendix 3.E The Schnute Model Derived

Appendix 3.F The McDill-Amateis Model Derived

Appendix 3.G The Slobada Model Derived

Appendix 3.H A Generalized Michaelis-Menton Growth Equation

**Chapter 4. Estimation of Trend**

4.1 Linear Trend Equation

4.2 Ordinary Least Squares (OLS) Estimation

4.3 Maximum Likelihood (ML) Estimation

4.4 The SAS System

4.5 Changing the Unit of Time

4.5.A Annual Totals vs. Monthly Averages vs. Monthly Totals

4.5.B Annual Totals vs. Quarterly Averages vs. Quarterly Totals

4.6 Autocorrelated Errors

4.6.A Properties of the OLS Estimators When is AR (1)

4.6.B Testing for the Absence of Autocorrelation: The Durbin-Watson Test

4.6.C Detection of and Estimation With Autocorrelated Errors

4.7 Polynomial Models in *t*

4.8 Issues Involving Trended Data

4.8.A Stochastic Processes and Time Series

4.8.B Autoregressive Process of Order *p*

4.8.C Random Walk Processes

4.8.D Integrated Processes

4.8.E Testing for Unit Roots

The Structure of the Dickey-Fuller (1979, 1982) Test Equation

Nonstandard Critical Values

The Augmented Dickey-Fuller Test

Testing Downwards for Unit Roots

Appendix 4.A OLS Estimated and Related Growth Rates

The OLS Growth Rate

The Log-Difference (LD) Growth Rate

The Average Annual Growth Rate

Geometric Average Growth Rate

**Chapter 5. Dynamic Site Equations Obtained from Growth Models**

5.1 Introduction

5.2 Base-Age Specific (BAS) Models

5.3 Algebraic Difference Approach (ADA) Models

5.4 Generalized Algebraic Difference Approach (GADA) Models

5.5 A Site Equation Generating Function

5.5.A ADA Derivations

5.5.B GADA Derivations

Linearity

Inverse-Saturation

Quadratic

5.6 The Grounded GADA (g-GADA) Model

Appendix 5.A A Glossary of Selected Forestry Terms

**Chapter 6. Nonlinear Regression**

6.1 Intrinsic Linearity/Nonlinearity

6.2 Estimation of Intrinsically Nonlinear Regression Models

6.2.A Nonlinear Least Squares (NLS)

6.2.B Maximum Likelihood (ML)

Appendix 6.A Gauss-Newton Iteration Scheme: The Single Parameter Case

Appendix 6.B Gauss-Newton Iteration Scheme: The *r *Parameter Case

Appendix 6.C The Newton-Raphson and Scoring Methods

Appendix 6.D The Levenberg-Marquardt Modification/Compromise

**Chapter 7. Yield-Density Curves**

7.1 Introduction

7.2 Structuring Yield-Density Equations

7.3 Reciprocal Yield-Density Equations

7.3.A The Shinozaki and Kira Yield-Density Curves

7.3.B The Holliday Yield-Density Curves

7.3.C The Farazdaghi and Harris Yield-Density Curves

7.3.D The Bleasdale and Nelder Yield-Density Curves

7.4 Weight of a Plant Part and Plant Density

7.5 The Expolinear Growth Equation

7.6 The Beta Growth Function

7.7 Asymmetric Growth Equations (for plant parts)

7.7.1 Model I

7.7.2 Model II

7.7.3 Model III

Appendix 7.A Derivation of the Shinozaki and Kira Yield-Density Curves

Appendix 7.B Derivation of the Farazdaghi and Harris Yield-Density Curves

Appendix 7.C Derivation of the Bleasdale and Nelder Yield-Density Curves

Appendix 7.D Derivation of the Expolinear Growth Curve

Appendix 7.E Derivation of the Beta Growth Function

Appendix 7.F Derivation of Asymmetric Growth Equations

Appendix 7.G Chanter Growth Function

**Chapter 8. Nonlinear Mixed Effects Models for Repeated Measurements Data**

8.1 Some Basic Terminology Concerning Experimental Design

8.2 Model Specification

8.2.1 Model and Data Elements

8.2.2 A Hierarchical (Staged) Model

8.3 Some Special Cases of a Hierarchical Global Model

8.4 The SAS/STAT NLMIXED Procedure for Fitting Nonlinear Mixed Models

**Chapter 9. Modeling the Size and Growth Rate Distributions of Firms**

9.1 Introduction

9.2 Measuring Firm Size and Growth

9.3 Modeling the Size Distribution of Firms

9.4 Gibrat’s Law

9.5 Rationalizing the Firm Size Distribution

9.6 Modeling the Growth Rate Distribution of Firms

9.7 Basic Empirics of Gibrat’s Law (GL)

9.7.1 Firm Size and Expected Growth Rates

9.7.2 Firm Size and Growth Rate Variability

9.7.3 Econometric Issues

9.7.4 Persistence of Growth Rates

9.8 Conclusion

Appendix 9.A Kernel Density Estimation

9.A.1 Motivation

9.A.2 Weighting Functions

9.A.3 Smooth Weighting Functions: Kernel Estimators

Appendix 9.B The Log-Normal and Gibrat Distributions

9.B.1 Derivation of Log-Normal Forms

9.B.2 Generalized Log-Normal Distribution

Appendix 9.C The Theory of Proportionate Effect

Appendix 9.D Classical Laplace Distribution

9.D.1 The Symmetric Case

9.D.2 The Asymmetric Case

9.D.3 The Generalized Laplace Distribution

9.D.4 The Log-Laplace Distribution

Appendix 9.E Power-Law Behavior

9.E.1 Pareto’s Power Law

9.E.2 Generalized Pareto Distribution

9.E.3 Zipf’s Power Law

Appendix 9.F The Yule Distribution

Appendix 9.G Overcoming Sample Selection Bias

9.G.1 Selection and Gibrat’s Law (GL)

9.G.2 Characterizing Selection Bias

9.G.3 Correcting for Selection Bias: The Heckman Two-Step Procedure

9.G.4 The Heckman Two-Step Procedure under Modified Selection

**Chapter10. Fundamentals of Population Dynamics**

10.1 The Concept of a Population

10.2 The Concept of Population Growth

10.3 Modeling Population Growth

10.4 Exponential (Density-Independent) Population Growth

10.4.A The Continuous Case

10.4.B The Discrete Case

10.4.C Malthusian Population Growth Dynamics

10.5 Density-Dependent Population Growth

10.5.1 Logistic Growth Models

10.5.1.A Continuous-Time Logistic

10.5.1.B Discrete-Time Logistic

10.6 Beverton and Holt (B-H) Model

10.7 Ricker Model

10.8 Hassell Model

10.9 Generalized Beverton and Holt (B-H) Model

10.10 Generalized Ricker Model

Appendix 10.A A Glossary of Selected Population Demography/Ecology Terms

Appendix 10.B Equilibrium and Stability Analysis

Appendix 10.C Discretization of the Continuous Time Logistic Growth Equation

Appendix 10.D Derivation of the B-H Stock-Recruitment Relationship

Appendix 10.E Derivation of the Ricker Stock-Recruitment Relationship

Appendix A

Table A.1 Standard Normal Areas (*Z* is *N*(0,1))

Table A.2 Quantiles of the *t* Distribution (*T* is *t*_{v})

Table A.3 Quantiles of the Distribution (*X* is )

Table A.4 Quantiles of the *F* Distribution (*F* is)

Table A.5 Durbin-Watson DW Statistic: 5% Significance Points (*n* is the sample size and is the number of regressors excluding the intercept)

Table A.6 Empirical Cumulative Distribution of for =1

References

Index