Growth Curve Modeling

MICHAEL J. PANIK, PHD, is Professor Emeritus in the Department of Economics at the University of Hartford. He has served as a consultant to the Connecticut Department of Motor Vehicles as well as to a variety of healthcare organizations. In addition, Dr. Panik is the author of numerous books and journal articles in the areas of economics, mathematics, and applied econometrics.

Preface xiii

1 Mathematical Preliminaries 1

1.1 Arithmetic Progression 1

1.2 Geometric Progression 2

1.3 The Binomial Formula 4

1.4 The Calculus of Finite Differences 5

1.5 The Number e 9

1.6 The Natural Logarithm 10

1.7 The Exponential Function 11

1.8 Exponential and Logarithmic Functions: Another Look 13

1.9 Change of Base of a Logarithm 14

1.10 The Arithmetic (Natural) Scale versus the Logarithmic Scale 15

1.11 Compound Interest Arithmetic 17

2 Fundamentals of Growth 21

2.1 Time Series Data 21

2.2 Relative and Average Rates of Change 21

2.3 Annual Rates of Change 25

2.3.1 Simple Rates of Change 25

2.3.2 Compounded Rates of Change 26

2.3.3 Comparing Two Time Series: Indexing Data to a Common Starting Point 30

2.4 Discrete versus Continuous Growth 32

2.5 The Growth of a Variable Expressed in Terms of the Growth of its Individual Arguments 36

2.6 Growth Rate Variability 46

2.7 Growth in a Mixture of Variables 47

3 Parametric Growth Curve Modeling 49

3.1 Introduction 49

3.2 The Linear Growth Model 50

3.3 The Logarithmic Reciprocal Model 51

3.4 The Logistic Model 52

3.5 The Gompertz Model 54

3.6 The Weibull Model 55

3.7 The Negative Exponential Model 56

3.8 The von Bertalanffy Model 57

3.9 The Log-Logistic Model 59

3.10 The Brody Growth Model 61

3.11 The Janoschek Growth Model 62

3.12 The Lundqvist–Korf Growth Model 63

3.13 The Hossfeld Growth Model 63

3.14 The Stannard Growth Model 64

3.15 The Schnute Growth Model 64

3.16 The Morgan–Mercer–Flodin (M–M–F) Growth Model 66

3.17 The McDill–Amateis Growth Model 68

3.18 An Assortment of Additional Growth Models 69

3.18.1 The Sloboda Growth Model 71

Appendix 3.A The Logistic Model Derived 71

Appendix 3.B The Gompertz Model Derived 74

Appendix 3.C The Negative Exponential Model Derived 75

Appendix 3.D The von Bertalanffy and Richards Models Derived 77

Appendix 3.E The Schnute Model Derived 81

Appendix 3.F The McDill–Amateis Model Derived 83

Appendix 3.G The Sloboda Model Derived 85

Appendix 3.H A Generalized Michaelis–Menten Growth Equation 86

4 Estimation of Trend 88

4.1 Linear Trend Equation 88

4.2 Ordinary Least Squares (OLS) Estimation 91

4.3 Maximum Likelihood (ML) Estimation 92

4.4 The SAS System 94

4.5 Changing the Unit of Time 109

4.5.1 Annual Totals versus Monthly Averages versus Monthly Totals 109

4.5.2 Annual Totals versus Quarterly Averages versus Quarterly Totals 110

4.6 Autocorrelated Errors 110

4.6.1 Properties of the OLS Estimators when ε Is AR(1) 111

4.6.2 Testing for the Absence of Autocorrelation: The Durbin–Watson Test 113

4.6.3 Detection of and Estimation with Autocorrelated Errors 115

4.7 Polynomial Models in t 126

4.8 Issues Involving Trended Data 136

4.8.1 Stochastic Processes and Time Series 137

4.8.2 Autoregressive Process of Order p 138

4.8.3 Random Walk Processes 141

4.8.4 Integrated Processes 145

4.8.5 Testing for Unit Roots 146

Appendix 4.A OLS Estimated and Related Growth Rates 158

4.A.1 The OLS Growth Rate 158

4.A.2 The Log-Difference (LD) Growth Rate 161

4.A.3 The Average Annual Growth Rate 161

4.A.4 The Geometric Average Growth Rate 162

5 Dynamic Site Equations Obtained from Growth Models 164

5.1 Introduction 164

5.2 Base-Age-Specific (BAS) Models 164

5.3 Algebraic Difference Approach (ADA) Models 166

5.4 Generalized Algebraic Difference Approach (GADA) Models 169

5.5 A Site Equation Generating Function 179

5.5.1 ADA Derivations 180

5.5.2 GADA Derivations 180

5.6 The Grounded GADA (g-GADA) Model 184

Appendix 5.A Glossary of Selected Forestry Terms 186

6 Nonlinear Regression 188

6.1 Intrinsic Linearity/Nonlinearity 188

6.2 Estimation of Intrinsically Nonlinear Regression Models 190

6.2.1 Nonlinear Least Squares (NLS) 191

6.2.2 Maximum Likelihood (ML) 195

Appendix 6.A Gauss–Newton Iteration Scheme: The Single Parameter Case 214

Appendix 6.B Gauss–Newton Iteration Scheme: The r Parameter Case 217

Appendix 6.C The Newton–Raphson and Scoring Methods 220

Appendix 6.D The Levenberg–Marquardt Modification/Compromise 222

Appendix 6.E Selection of Initial Values 223

6.E.1 Initial Values for the Logistic Curve 224

6.E.2 Initial Values for the Gompertz Curve 224

6.E.3 Initial Values for the Weibull Curve 224

6.E.4 Initial Values for the Chapman–Richards Curve 225

7 Yield–Density Curves 226

7.1 Introduction 226

7.2 Structuring Yield–Density Equations 227

7.3 Reciprocal Yield–Density Equations 228

7.3.1 The Shinozaki and Kira Yield–Density Curve 228

7.3.2 The Holliday Yield–Density Curves 229

7.3.3 The Farazdaghi and Harris Yield–Density Curve 230

7.3.4 The Bleasdale and Nelder Yield–Density Curve 231

7.4 Weight of a Plant Part and Plant Density 239

7.5 The Expolinear Growth Equation 242

7.6 The Beta Growth Function 249

7.7 Asymmetric Growth Equations (for Plant Parts) 253

7.7.1 Model I 254

7.7.2 Model II 255

7.7.3 Model III 256

Appendix 7.A Derivation of the Shinozaki and Kira Yield–Density Curve 257

Appendix 7.B Derivation of the Farazdaghi and Harris Yield–Density Curve 258

Appendix 7.C Derivation of the Bleasdale and Nelder Yield–Density Curve 259

Appendix 7.D Derivation of the Expolinear Growth Curve 261

Appendix 7.E Derivation of the Beta Growth Function 263

Appendix 7.F Derivation of Asymmetric Growth Equations 266

Appendix 7.G Chanter Growth Function 269

8 Nonlinear Mixed-Effects Models for Repeated Measurements Data 270

8.1 Some Basic Terminology Concerning Experimental Design 270

8.2 Model Specification 271

8.2.1 Model and Data Elements 271

8.2.2 A Hierarchical (Staged) Model 272

8.3 Some Special Cases of the Hierarchical Global Model 274

8.4 The SAS/STAT NLMIXED Procedure for Fitting Nonlinear Mixed-Effects Model 276

9 Modeling the Size and Growth Rate Distributions of Firms 293

9.1 Introduction 293

9.2 Measuring Firm Size and Growth 294

9.3 Modeling the Size Distribution of Firms 294

9.4 Gibrat’s Law (GL) 297

9.5 Rationalizing the Pareto Firm Size Distribution 299

9.6 Modeling the Growth Rate Distribution of Firms 300

9.7 Basic Empirics of Gibrat’s Law (GL) 305

9.7.1 Firm Size and Expected Growth Rates 305

9.7.2 Firm Size and Growth Rate Variability 308

9.7.3 Econometric Issues 310

9.7.4 Persistence of Growth Rates 312

9.8 Conclusion 313

Appendix 9.A Kernel Density Estimation 314

9.A.1 Motivation 314

9.A.2 Weighting Functions 315

9.A.3 Smooth Weighting Functions: Kernel Estimators 316

Appendix 9.B The Log-Normal and Gibrat Distributions 322

9.B.1 Derivation of Log-Normal Forms 322

9.B.2 Generalized Log-Normal Distribution 325

Appendix 9.C The Theory of Proportionate Effect 326

Appendix 9.D Classical Laplace Distribution 328

9.D.1 The Symmetric Case 328

9.D.2 The Asymmetric Case 330

9.D.3 The Generalized Laplace Distribution 331

9.D.4 The Log-Laplace Distribution 332

Appendix 9.E Power-Law Behavior 332

9.E.1 Pareto’s Power Law 333

9.E.2 Generalized Pareto Distributions 335

9.E.3 Zipf’s Power Law 337

Appendix 9.F The Yule Distribution 338

Appendix 9.G Overcoming Sample Selection Bias 339

9.G.1 Selection and Gibrat’s Law (GL) 339

9.G.2 Characterizing Selection Bias 339

9.G.3 Correcting for Selection Bias: The Heckman (1976 1979) Two-Step Procedure 342

9.G.4 The Heckman Two-Step Procedure Under Modified Selection 345

10 Fundamentals of Population Dynamics 352

10.1 The Concept of a Population 352

10.2 The Concept of Population Growth 353

10.3 Modeling Population Growth 354

10.4 Exponential (Density-Independent) Population Growth 357

10.4.1 The Continuous Case 357

10.4.2 The Discrete Case 359

10.4.3 Malthusian Population Growth Dynamics 361

10.5 Density-Dependent Population Growth 363

10.5.1 Logistic Growth Model 364

10.6 Beverton–Holt Model 371

10.7 Ricker Model 374

10.8 Hassell Model 377

10.9 Generalized Beverton–Holt (B–H) Model 380

10.10 Generalized Ricker Model 382

Appendix 10.A A Glossary of Selected Population Demography/Ecology Terms 389

Appendix 10.B Equilibrium and Stability Analysis 391

10.B.1 Stable and Unstable Equilibria 391

10.B.2 The Need for a Qualitative Analysis of Equilibria 392

10.B.3 Equilibria and Stability for Continuous-Time Models 392

10.B.4 Equilibria and Stability for Discrete-Time Models 394

Appendix 10.C Discretization of the Continuous-Time Logistic Growth Equation 400

Appendix 10.D Derivation of the B–H S–R Relationship 401

Appendix 10.E Derivation of the Ricker S–R Relationship 403

Appendix A 405

Table A.1 Standard Normal Areas (Z Is N(0, 1)) 405

Table A.2 Quantiles of Student’s t Distribution (T Is tv) 407

Table A.3 Quantiles of the Chi-Square Distribution (X Is 𝛘v 2) 408

Table A.4 Quantiles of Snedecor’s F Distribution (F Is Fv1, v2) 410

Table A.5 Durbin–Watson DW Statistic—5% Significance Points dL and dU (n is the sample size and k′ is the number of regressors excluding the intercept) 415

Table A.6 Empirical Cumulative Distribution of τ for ρ = 1 419

References 420

Index 431