Growth Curve Modeling: Theory and Applications

450 Pages·2014·4.209 MB·English

by  Michael J. Panik

GROWTH CURVE MODELING
GROWTH CURVE MODELING
Theory and Applications
MICHAEL J. PANIK
Department of Economics
University of Hartford
West Hartford, Connecticut
Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
Panik, Michael J.
Growth curve modeling : theory and applications / Michael J. Panik.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-76404-6 (cloth)
1. Mathematical statistics. 2. Time series analysis. 3. Regression analysis.
4. Multivariate analysis. I. Title.
QA276.P2243 2013
519.5–dc23
2013026535
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
In memory of Michael Christopher Duffy
CONTENTS
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
vii
viii CONTENTS
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
CONTENTS ix
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
x CONTENTS
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
C urve, 258
Appendix 7.C Derivation of the Bleasdale and Nelder Yield–Density
C urve, 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
CONTENTS xi
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
xii CONTENTS
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 t), 407
v
Table A.3 Quantiles of the Chi-Square Distribution (X Is (cid:70)2), 408
v
Table A.4 Quantiles of Snedecor’s F Distribution (F Is F ), 410
v1,v2
Table A.5 Durbin–Watson DW Statistic—5% Significance Points d and d
L U
(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