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Summary
Summary
Taking the topics of a quantitative methodology course and illustrating them through Monte Carlo simulation, this book examines abstract principles, such as bias, efficiency, and measures of uncertainty in an intuitive, visual way. Instead of thinking in the abstract about what would happen to a particular estimator "in repeated samples," the book uses simulation to actually create those repeated samples and summarize the results. The book includes basic examples appropriate for readers learning the material for the first time, as well as more advanced examples that a researcher might use to evaluate an estimator he or she was using in an actual research project. The book also covers a wide range of topics related to Monte Carlo simulation, such as resampling methods, simulations of substantive theory, simulation of quantities of interest (QI) from model results, and cross-validation. Complete R code from all examples is provided so readers can replicate every analysis presented using R.
Table of Contents
Acknowledgments | p. ix |
1 Introduction | p. 1 |
1.1 Can You Repeat That Please? | p. 2 |
1.2 Simulation and Resampling Methods | p. 4 |
1.2.1 Simulations as Experiments | p. 4 |
1.2.2 Simulations Help Develop Intuition | p. 5 |
1.2.3 An Overview of Simulation | p. 6 |
1.2.4 Resampling Methods as Simulation | p. 7 |
1.3 OLS as a Motivating Example | p. 8 |
1.4 Two Brief Examples | p. 12 |
1.4.1 Example 1: A Statistical Simulation | p. 13 |
1.4.2 Example 2: A Substantive Theory Simulation | p. 15 |
1.5 Looking Ahead | p. 15 |
1.5.1 Assumed Knowledge | p. 16 |
1.5.2 A Preview of the Book | p. 16 |
1.6 R Packages | p. 17 |
2 Probability | p. 19 |
2.1 Introduction | p. 19 |
2.2 Some Basic Rules of Probability | p. 20 |
2.2.1 Introduction to Set Theory | p. 20 |
2.2.2 Properties of Probability | p. 22 |
2.2.3 Conditional Probability | p. 22 |
2.2.4 Simple Math With Probabilities | p. 23 |
2.3 Random Variables and Probability Distributions | p. 24 |
2.4 Discrete Random Variables | p. 29 |
2.4.1 Some Common Discrete Distributions | p. 30 |
2.5 Continuous Random Variables | p. 33 |
2.5.1 Two Common Continuous Distributions | p. 36 |
2.5.2 Other Continuous Distributions | p. 39 |
2.6 Conclusions | p. 43 |
3 Introduction to R | p. 45 |
3.1 Introduction | p. 45 |
3.2 What Is R? | p. 45 |
3.2.1 Resources | p. 46 |
3.3 Using R With a Text Editor | p. 46 |
3.4 First Steps | p. 47 |
3.4.1 Creating Objects | p. 47 |
3.5 Basic Manipulation of Objects | p. 48 |
3.5.1 Vectors and Sequences | p. 48 |
3.5.2 Matrices | p. 49 |
3.6 Functions | p. 50 |
3.6.1 Matrix Algebra Functions | p. 51 |
3.6.2 Creating New Functions | p. 51 |
3.7 Working With Data | p. 52 |
3.7.1 Loading Data | p. 52 |
3.7.2 Exploring the Data | p. 53 |
3.7.3 Statistical Models | p. 54 |
3.7.4 Generalized Linear Models | p. 57 |
3.8 Basic Graphics | p. 59 |
3.9 Conclusions | p. 61 |
4 Random Number Generation | p. 63 |
4.1 Introduction | p. 63 |
4.2 Probability Distributions | p. 63 |
4.2.1 Drawing Random Numbers | p. 65 |
4.2.2 Creating Your Own Distribution Functions | p. 67 |
4.3 Systematic and Stochastic | p. 68 |
4.3.1 The Systematic Component | p. 69 |
4.3.2 The Stochastic Component | p. 70 |
4.3.3 Repeating the Process | p. 71 |
4.4 Programming in R | p. 72 |
4.4.1 for Loops | p. 73 |
4.4.2 Efficient Programming | p. 74 |
4.4.3 If-Else | p. 76 |
4.5 Completing the OLS Simulation | p. 77 |
4.5.1 Anatomy of a Script File | p. 80 |
5 Statistical Simulation of the Linear Model | p. 83 |
5.1 Introduction | p. 83 |
5.2 Evaluating Statistical Estimators | p. 84 |
5.2.1 Bias, Efficiency, and Consistency | p. 84 |
5.2.2 Measuring Estimator Performance in R | p. 87 |
5.3 Simulations as Experiments | p. 96 |
5.3.1 Heteroskedasticity | p. 96 |
5.3.2 Multicollinearity | p. 103 |
5.3.3 Measurement Error | p. 105 |
5.3.4 Omitted Variable | p. 109 |
5.3.5 Serial Correlation | p. 112 |
5.3.6 Clustered Data | p. 114 |
5.3.7 Heavy-Tailed Errors | p. 118 |
5.4 Conclusions | p. 125 |
6 Simulating Generalized Linear Models | p. 127 |
6.1 Introduction | p. 127 |
6.2 Simulating OLS as a Probability Model | p. 128 |
6.3 Simulating GLMs | p. 130 |
6.3.1 Binary Models | p. 130 |
6.3.2 Ordered Models | p. 135 |
6.3.3 Multinomial Models | p. 141 |
6.4 Extended Examples | p. 145 |
6.4.1 Ordered or Multinomial? | p. 145 |
6.4.2 Count Models | p. 150 |
6.4.3 Duration Models | p. 157 |
6.5 Computational Issues for Simulations | p. 162 |
6.5.1 Research Computing | p. 162 |
6.5.2 Parallel Processing | p. 163 |
6.6 Conclusions | p. 167 |
7 Testing Theory Using Simulation | p. 169 |
7.1 Introduction | p. 169 |
7.2 What Is a Theory? | p. 169 |
7.3 Zipf's Law | p. 171 |
7.3.1 Testing Zipf's Law With Frankenstein | p. 171 |
7.3.2 From Patterns to Explanations | p. 174 |
7.4 Punctuated Equilibrium and Policy Responsiveness | p. 181 |
7.4.1 Testing Punctuated Equilibrium Theory | p. 183 |
7.4.2 From Patterns to Explanations | p. 185 |
7.5 Dynamic Learning | p. 190 |
7.5.1 Reward and Punishment | p. 193 |
7.5.2 Damned If You Do, Damned If You Don't | p. 195 |
7.5.3 The Midas Touch | p. 197 |
7.6 Conclusions | p. 200 |
8 Resampling Methods | p. 201 |
8.1 Introduction | p. 201 |
8.2 Permutation and Randomization Tests | p. 202 |
8.2.1 A Basic Permutation Test | p. 203 |
8.2.2 Randomization Tests | p. 205 |
8.2.3 Permutation/Randomization and Multiple Regression Models | p. 208 |
8.3 Jackknifing | p. 209 |
8.3.1 An Example | p. 210 |
8.3.2 An Application: Simulating Heteroskedasticity | p. 213 |
8.3.3 Pros and Cons of Jackknifing | p. 214 |
8.4 Bootstrapping | p. 215 |
8.4.1 Bootstrapping Basics | p. 217 |
8.4.2 Bootstrapping With Multiple Regression Models | p. 220 |
8.4.3 Adding Complexity: Clustered Bootstrapping | p. 225 |
8.5 Conclusions | p. 228 |
9 Other Simulation-Based Methods | p. 231 |
9.1 Introduction | p. 231 |
9.2 QI Simulation | p. 232 |
9.2.1 Statistical Overview | p. 232 |
9.2.2 Examples | p. 235 |
9.2.3 Simulating QI With Zelig | p. 245 |
9.2.4 Average Case Versus Observed Values | p. 249 |
9.2.5 The Benefits of QI Simulation | p. 254 |
9.3 Cross-Validation | p. 255 |
9.3.1 How CV Can Help | p. 257 |
9.3.2 An Example | p. 258 |
9.3.3 Using R Functions for CV | p. 266 |
9.4 Conclusions | p. 267 |
10 Final Thoughts | p. 269 |
10.1 A Summary of the Book | p. 270 |
10.2 Going Forward | p. 271 |
10.3 Conclusions | p. 272 |
References | p. 275 |
Index | p. 283 |