Statistical demography and forecasting

Sustainability of pension systems, intergeneration fiscal equity under population aging, and accounting for health care benefits for future retirees are examples of problems that cannot be solved without understanding the nature of population forecasts and their uncertainty. Similarly, the accuracy of population estimates directly affects both the distributions of formula-based government allocations to sub-national units and the apportionment of political representation. The book develops the statistical foundation for addressing such issues. Areas covered include classical mathematical demography, event history methods, multi-state methods, stochastic population forecasting, sampling and census coverage, and decision theory. The methods are illustrated with empirical applications from Europe and the U.S.

For statisticians the book provides a unique introduction to demographic problems in a familiar language. For demographers, actuaries, epidemiologists, and professionals in related fields, the book presents a unified statistical outlook on both classical methods of demography and recent developments. To facilitate its classroom use, exercises are included. Over half of the book is readily accessible to undergraduates, but more maturity may be required to benefit fully from the complete text. Knowledge of differential and integral calculus, matrix algebra, basic probability theory, and regression analysis is assumed.

Juha M. Alho is Professor of Statistics, University of Joensuu, Finland, and Bruce D. Spencer is Professor of Statistics and Faculty Fellow at the Institute for Policy Research, Northwestern University. Both have contributed extensively to statistical demography and served in advisory roles and as statistical consultants in the field.

Preface	p. vii
Acknowledgments	p. ix
List of Examples	p. xix
List of Figures	p. xxv
Chapter 1 Introduction	p. 1
1 Role of Statistical Demography	p. 1
2 Guide for the Reader	p. 4
3 Statistical Notation and Preliminaries	p. 4
Chapter 2 Sources of Demographic Data	p. 9
1 Populations: Open and Closed	p. 9
2 De Facto and De Jure Populations	p. 11
3 Censuses and Population Registers	p. 15
4 Lexis Diagram and Classification of Events	p. 16
5 Register Data and Epidemiologic Studies	p. 19
5.1 Event Histories from Registers	p. 19
5.2 Cohort and Case-Control Studies	p. 19
5.3 Advantages and Disadvantages	p. 20
5.4 Confounding	p. 22
6 Sampling in Censuses and Dual System Estimation	p. 24
Exercises and Complements	p. 27
Chapter 3 Sampling Designs and Inference	p. 31
1 Simple Random Sampling	p. 32
2 Subgroups and Ratios	p. 35
3 Stratified Sampling	p. 36
3.1 Introduction	p. 36
3.2 Stratified Simple Random Sampling	p. 37
3.3 Design Effect for Stratified Simple Random Sampling	p. 38
3.4 Poststratification	p. 39
4 Sampling Weights	p. 40
4.1 Why Weight?	p. 40
4.2 Forming Weights	p. 41
4.3 Non-Response Adjustments	p. 43
4.4 Effect of Weighting on Precision	p. 45
5 Cluster Sampling	p. 46
5.1 Introduction	p. 46
5.2 Single Stage Sampling with Replacement	p. 47
5.3 Single Stage Sampling without Replacement	p. 47
5.4 Multi-Stage Sampling	p. 49
5.5 Stratified Samples	p. 50
6 Systematic Sampling	p. 52
7 Distribution Theory for Sampling	p. 53
7.1 Central Limit Theorems	p. 53
7.2 The Delta Method	p. 55
7.3 Estimating Equations	p. 56
8 Replication Estimates of Variance	p. 61
8.1 Jackknife Estimates	p. 61
8.2 Bootstrap Estimates	p. 62
8.3 Replication Weights	p. 63
Exercises and Complements	p. 64
Chapter 4 Waiting Times and Their Statistical Estimation	p. 71
1 Exponential Distribution	p. 71
2 General Waiting Time	p. 76
2.1 Hazards and Survival Probabilities	p. 76
2.2 Life Expectancies and Stable Populations	p. 79
2.2.1 Life Expectancy	p. 79
2.2.2 Life Table Populations and Stable Populations	p. 81
2.2.3 Changing Mortality	p. 82
2.2.4 Basics of Pension Funding	p. 84
2.2.5 Effect of Heterogeneity	p. 85
2.3 Kaplan-Meier and Nelson-Aalen Estimators	p. 85
2.4 Estimation Based on Occurrence-Exposure Rates	p. 88
3 Estimating Survival Proportions	p. 91
4 Childbearing as a Repeatable Event	p. 93
4.1 Poisson Process Model of Childbearing	p. 93
4.2 Summary Measures of Fertility and Reproduction	p. 96
4.3 Period and Cohort Fertility	p. 101
4.3.1 Cohort Fertility is Smoother	p. 101
4.3.2 Adjusting for Timing	p. 103
4.3.3 Effect of Parity on Pure Period Measures	p. 104
4.4 Multiple Births and Effect of Pregnancy on Exposure Time	p. 106
5 Poisson Character of Demographic Events	p. 107
6 Simulation of Waiting Times and Counts	p. 109
Exercises and Complements	p. 110
Chapter 5 Regression Models for Counts and Survival	p. 117
1 Generalized Linear Models	p. 118
1.1 Exponential Family	p. 118
1.2 Use of Explanatory Variables	p. 119
1.3 Maximum Likelihood Estimation	p. 119
1.4 Numerical Solution	p. 120
1.5 Inferences	p. 121
1.6 Diagnostic Checks	p. 122
2 Binary Regression	p. 123
2.1 Interpretation of Parameters and Goodness of Fit	p. 123
2.2 Examples of Logistic Regression	p. 124
2.3 Applicability in Case-Control Studies	p. 129
3 Poisson Regression	p. 130
3.1 Interpretation of Parameters	p. 130
3.2 Examples of Poisson Regression	p. 131
3.3 Standardization	p. 133
3.4 Loglinear Models for Capture-Recapture Data	p. 136
4 Overdispersion and Random Effects	p. 138
4.1 Direct Estimation of Overdispersion	p. 139
4.2 Marginal Models for Overdispersion	p. 139
4.3 Random Effect Models	p. 140
5 Observable Heterogeneity in Capture-Recapture Studies	p. 143
6 Bilinear Models	p. 146
7 Proportional Hazards Models for Survival	p. 150
8 Heterogeneity and Selection by Survival	p. 154
9 Estimation of Population Density	p. 156
10 Simulation of the Regression Models	p. 158
Exercises and Complements	p. 159
Chapter 6 Multistate Models and Cohort-Component Book-Keeping	p. 166
1 Multistate Life-Tables	p. 167
1.1 Numerical Solution Using Runge-Kutta Algorithm	p. 167
1.2 Extension to Multistate Case	p. 168
1.3 Duration-Dependent Life-Tables	p. 172
1.3.1 Heterogeneity Attributable to Duration	p. 172
1.3.2 Forms of Duration-Dependence	p. 173
1.3.3 Aspects of Computer Implementation	p. 174
1.3.4 Policy Significance of Duration-Dependence	p. 175
1.4 Nonparametric Intensity Estimation	p. 175
1.5 Analysis of Nuptiality	p. 177
1.6 A Model for Disability Insurance	p. 179
2 Linear Growth Model	p. 180
2.1 Matrix Formulation	p. 180
2.2 Stable Populations	p. 183
2.3 Weak Ergodicity	p. 185
3 Open Populations and Parametrization of Migration	p. 186
3.1 Open Population Systems	p. 186
3.2 Parametric Models	p. 186
3.2.1 Migrant Pool Model	p. 187
3.2.2 Bilinear Models	p. 187
4 Demographic Functionals	p. 189
5 Elementwise Aspects of the Matrix Formulation	p. 191
6 Markov Chain Models	p. 191
Exercises and Complements	p. 193
Chapter 7 Approaches to Forecasting Demographic Rates	p. 198
1 Trends, Random Walks, and Volatility	p. 198
2 Linear Stationary Processes	p. 201
2.1 Properties and Modeling	p. 202
2.1.1 Definition and Basic Properties	p. 202
2.1.2 ARIMA Models	p. 203
2.1.3 Practical Modeling	p. 206
2.2 Characterization of Predictions and Prediction Errors	p. 210
2.2.1 Stationary Processes	p. 210
2.2.2 Integrated Processes	p. 211
2.2.3 Cross-Correlations	p. 216
3 Handling of Nonconstant Mean	p. 216
3.1 Differencing	p. 216
3.2 Regression	p. 218
3.3 Structural Models	p. 219
4 Heteroscedastic Innovations	p. 220
4.1 Deterministic Models of Volatility	p. 221
4.2 Stochastic Volatility	p. 222
Exercises and Complements	p. 223
Chapter 8 Uncertainty in Demographic Forecasts: Concepts, Issues, and Evidence	p. 226
1 Historical Aspects of Cohort-Component Forecasting	p. 228
1.1 Adoption of the Cohort-Component Approach	p. 228
1.2 Whelpton's Legacy	p. 228
1.3 Do We Know Better Now?	p. 231
2 Dimensionality Reduction for Mortality	p. 234
2.1 Age-Specific Mortality	p. 234
2.2 Cause-Specific Mortality	p. 236
3 Conceptual Aspects of Error Analysis	p. 238
3.1 Expected Error and Empirical Error	p. 238
3.2 Decomposing Errors	p. 238
3.2.1 Error Classifications	p. 238
3.2.2 Alternative Decompositions	p. 240
3.3 Acknowledging Model Error	p. 240
3.3.1 Classes of Parametric Models	p. 240
3.3.2 Data Period Bias	p. 241
3.4 Feedback Effects of Forecasts	p. 242
3.5 Interpretation of Prediction Intervals	p. 244
3.5.1 Uncertainty in Terms of Subjective Probabilities	p. 244
3.5.2 Frequency Properties of Prediction Intervals	p. 248
3.6 Role of Judgment	p. 249
3.6.1 Expert Arguments	p. 249
3.6.2 Scenarios	p. 250
3.6.3 Conditional Forecasts	p. 251
4 Practical Error Assessment	p. 251
4.1 Error Measures	p. 252
4.2 Baseline Forecasts	p. 253
4.3 Modeling Errors in World Forecasts	p. 256
4.3.1 An Error Model for Growth Rates	p. 256
4.3.2 Second Moments	p. 257
4.3.3 Predictive Distributions for Countries and the World	p. 259
4.4 Random Jump-off Values	p. 261
4.4.1 Jump-off Population	p. 262
4.4.2 Mortality	p. 263
5 Measuring Correlatedness	p. 264
Exercises and Complements	p. 267
Chapter 9 Statistical Propagation of Error in Forecasting	p. 269
1 Törnqvist's Contribution	p. 269
2 Predictive Distributions	p. 271
2.1 Regression with a Known Covariance Structure	p. 271
2.2 Random Walks	p. 274
2.3 ARIMA(1,1,0) Models	p. 276
3 Forecast as a Database and Its Uses	p. 277
4 Parametrizations of Covariance Structure	p. 278
4.1 Effect of Correlations on the Variance of a Sum	p. 279
4.2 Scaled Model for Error	p. 280
4.3 Structure of Error in Migration Forecasts	p. 283
5 Analytical Propagation of Error	p. 284
5.1 Births	p. 284
5.2 General Linear Growth	p. 285
6 Simulation Approach and Computer Implementation	p. 287
7 Post Processing	p. 289
7.1 Altering a Distributional Form	p. 289
7.2 Creating Correlated Populations	p. 292
7.2.1 Use of Seeds	p. 292
7.2.2 Sorting Techniques	p. 293
Exercises and Complements	p. 294
Chapter 10 Errors in Census Numbers	p. 296
1 Introduction	p. 296
2 Effects of Errors on Estimates and Forecasts	p. 297
2.1 Effects on Mortality Rates	p. 297
2.2 Effects on Forecasts	p. 298
2.3 Effects on Evaluation of Past Population Forecasts	p. 298
3 Use of Demographic Analysis to Assess Error in U.S. Censuses	p. 299
4 Assessment of Dual System Estimates of Population Size	p. 300
5 Decomposition of Error in the Dual System Estimator	p. 303
5.1 A Probability Model for the Census	p. 303
5.2 Poststratification	p. 304
5.3 Overview of Error Components	p. 305
5.4 Data Error Bias	p. 308
5.5 Decomposition of Model Bias	p. 309
5.5.1 Synthetic Estimation Bias and Correlation Bias	p. 309
5.5.2 Poststratified Estimator	p. 310
5.6 Estimation of Correlation Bias in a Poststratified Dual System Estimator	p. 312
5.7 Estimation of Synthetic Estimation Bias in a Poststratified Dual System Estimator	p. 314
6 Assessment of Error in Functions of Dual System Estimators and Functions of Census Counts	p. 316
6.1 Overview	p. 316
6.2 Computation	p. 317
Exercises and Complements	p. 319
Chapter 11 Financial Applications	p. 327
1 Predictive Distribution of Adjustment for Life Expectancy Change	p. 327
1.1 Adjustment Factor for Mortality Change	p. 327
1.2 Sampling Variation in Pension Adjustment Factors	p. 329
1.3 The Predictive Distribution of the Pension Adjustment Factor	p. 330
2 Fertility Dependent Pension Benefits	p. 332
3 Measuring Sustainability	p. 335
4 State Aid to Municipalities	p. 337
5 Public Liabilities	p. 339
5.1 Economic Series	p. 340
5.2 Wealth in Terms of Random Returns and Discounting	p. 340
5.3 Random Public Liability	p. 341
Exercises and Complements	p. 342
Chapter 12 Decision Analysis and Small Area Estimates	p. 344
1 Introduction	p. 344
2 Small Area Analysis	p. 345
3 Formula-Based Allocations	p. 346
3.1 Theoretical Construction	p. 346
3.1.1 Apportionment of the U.S. House of Representatives	p. 347
3.1.2 Rationale Behind Allocation Formulas	p. 348
3.2 Effect of Inaccurate Demographic Statistics	p. 349
3.3 Beyond Accuracy	p. 350
4 Decision Theory and Loss Functions	p. 351
4.1 Introduction	p. 351
4.2 Decision Theory for Statistical Agencies	p. 353
4.3 Loss Functions for Small Area Estimates	p. 357
4.4 Loss Functions for Apportionment and Redistricting	p. 359
4.1.1 Apportionment	p. 359
4.1.2 Redistricting	p. 360
4.5 Loss Functions and Allocation of Funds	p. 361
4.5.1 Effects of Over- and Under-Allocation	p. 361
4.5.2 Formula Nonoptimality	p. 362
4.5.3 Optimal Data Quality with Multiple Statistics and Uses	p. 363
5 Comparing Risks of Adjusted and Unadjusted Census Estimates	p. 363
5.1 Accounting for Variances of Bias Estimates	p. 364
5.2 Effect of Unmeasured Biases on Comparisons of Accuracy	p. 365
6 Decision Analysis of Adjustment for Census Undercount	p. 365
7 Cost-Benefit Analysis of Demographic Data	p. 367
Exercises and Complements	p. 368
References	p. 371
Author Index	p. 397
Subject Index	p. 405

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