Introduction to the mathematical and statistical foundations of econometrics

Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 17
1.1.1. Introduction......Page 23
1.1.2. Binomial Numbers......Page 24
1.1.4. Algebras and Sigma-Algebras of Events......Page 25
1.1.5. Probability Measure......Page 26
1.2.1. Sampling without Replacement......Page 28
1.2.2. Quality Control in Practice......Page 29
1.2.4. Limits of the Hypergeometric and Binomial Probabilities......Page 30
1.3. Why Do We Need Sigma-Algebras of Events?......Page 32
1.4.1. General Properties......Page 33
1.4.2. Borel Sets......Page 36
1.5. Properties of Probability Measures......Page 37
1.6.1. Introduction......Page 38
1.6.2. Outer Measure......Page 39
1.7.2. Lebesgue Integral......Page 41
1.8.1. Random Variables and Vectors......Page 42
1.8.2. Distribution Functions......Page 45
1.9. Density Functions......Page 47
1.10.2. Bayes’ Rule......Page 49
1.10.3. Independence......Page 50
1.11. Exercises......Page 52
1.B. Extension of an Outer Measure to Probability Measure......Page 54
2.1. Introduction......Page 59
2.2. Borel Measurability......Page 60
2.3. Integrals of Borel-Measurable Functions with Respect to a Probability Measure......Page 64
2.4. General Measurability and Integrals of Random Variables with Respect to Probability Measures......Page 68
2.5. Mathematical Expectation......Page 71
2.6. Some Useful Inequalities Involving Mathematical Expectations......Page 72
2.6.2. Holder’s Inequality......Page 73
2.6.5. Jensen’s Inequality......Page 74
2.7. Expectations of Products of Independent Random Variables......Page 75
2.8.1. Moment-Generating Functions......Page 77
2.9. Exercises......Page 81
2.A. Uniqueness of Characteristic Functions......Page 83
3.1. Introduction......Page 88
3.2. Properties of Conditional Expectations......Page 94
3.3. Conditional Probability Measures and Conditional Independence......Page 101
3.5. Conditional Expectations as the Best Forecast Schemes......Page 102
3.6. Exercises......Page 104
3.A. Proof of Theorem 3.12......Page 105
4.1.1. The Hypergeometric Distribution......Page 108
4.1.2. The Binomial Distribution......Page 109
4.1.4. The Negative Binomial Distribution......Page 110
4.2. Transformations of Discrete Random Variables and Vectors......Page 111
4.3. Transformations of Absolutely Continuous Random Variables......Page 112
4.4.1. The Linear Case......Page 113
4.4.2. The Nonlinear Case......Page 116
4.5.1. The Standard Normal Distribution......Page 118
4.6.1. The Chi-Square Distribution......Page 119
4.6.4. The F Distribution......Page 122
4.7. The Uniform Distribution and Its Relation to the Standard Normal Distribution......Page 123
4.9. Exercises......Page 124
4.A. Tedious Derivations......Page 126
4.B. Proof of Theorem 4.4......Page 128
5.1. Expectation and Variance of Random Vectors......Page 132
5.2. The Multivariate Normal Distribution......Page 133
5.3. Conditional Distributions of Multivariate Normal Random Variables......Page 137
5.4. Independence of Linear and Quadratic Transformations of Multivariate Normal Random Variables......Page 139
5.5. Distributions of Quadratic Forms of Multivariate Normal Random Variables......Page 140
5.6.1. Estimation......Page 141
5.6.2. Confidence Intervals......Page 144
5.6.3. Testing Parameter Hypotheses......Page 147
5.7.2. Least-Squares Estimation......Page 149
5.7.3. Hypotheses Testing......Page 153
5.8. Exercises......Page 155
5.A. Proof of Theorem 5.8......Page 156
6.1. Introduction......Page 159
6.2. Convergence in Probability and the Weak Law of Large Numbers......Page 162
6.3. Almost-Sure Convergence and the Strong Law of Large Numbers......Page 165
6.4.2.1. Consistency of M-Estimators......Page 167
6.4.2.2. Generalized Slutsky’s Theorem......Page 170
6.5. Convergence in Distribution......Page 171
6.6. Convergence of Characteristic Functions......Page 176
6.7. The Central Limit Theorem......Page 177
6.8. Stochastic Boundedness, Tightness, and the… Notations......Page 179
6.9. Asymptotic Normality of M-Estimators......Page 181
6.10. Hypotheses Testing......Page 184
6.11. Exercises......Page 185
6.A. Proof of the Uniform Weak Law of Large Numbers......Page 186
6.B.1. Preliminary Results......Page 189
6.B.3. Kolmogorov’s Strong Law of Large Numbers......Page 191
6.B.5. The Uniform Strong Law of Large Numbers and Its Applications......Page 194
6.C. Convergence of Characteristic Functions and Distributions......Page 196
7.1. Stationarity and the Wold Decomposition......Page 201
7.2. Weak Laws of Large Numbers for Stationary Processes......Page 205
7.3. Mixing Conditions......Page 208
7.4.2. Random Functions Depending on Infinite-Dimensional Random Vectors......Page 209
7.5.1. Introduction......Page 212
7.5.2. A Generic Central Limit Theorem......Page 213
7.5.3. Martingale Difference Central Limit Theorems......Page 218
7.6. Exercises......Page 220
7.A.1. Introduction......Page 221
7.A.2. A Hilbert Space of Random Variables......Page 222
7.A.3. Projections......Page 223
7.A.5. Proof of the Wold Decomposition......Page 225
8.1. Introduction......Page 227
8.2. Likelihood Functions......Page 229
8.3.2. Linear Regression with Normal Errors......Page 231
8.3.3. Probit and Logit Models......Page 233
8.3.4. The Tobit Model......Page 234
8.4.2. First-and Second-Order Conditions......Page 236
8.4.3. Generic Conditions for Consistency and Asymptotic Normality......Page 238
8.4.4. Asymptotic Normality in the Time Series Case......Page 241
8.4.5. Asymptotic Efifciency of the ML Estimator......Page 242
8.5.1. The Pseudo t-Test and the Wald Test......Page 244
8.5.2. The Likelihood Ratio Test......Page 245
8.5.3. The Lagrange Multiplier Test......Page 247
8.6. Exercises......Page 248
I.1. Vectors in Euclidean Space......Page 251
I.2. Vector Spaces......Page 254
I.3. Matrices......Page 257
I.4. The Inverse and Transpose of Matrix......Page 260
I.5. Elementary Matrices and Permutation Matrices......Page 263
I.6.1. Gaussian Elimination of a Square a Matrix......Page 266
I.6.2. The Gauss–Jordan Iteration for Inverting a Matrix......Page 270
I.7. Gaussian Elimination of Nonsquare Matrix......Page 274
I.8. Subspaces Spanned by the Columns and Rows of a Matrix......Page 275
I.9. Projections, Projection Matrices, and Idempotent Matrices......Page 278
I.10. Inner Product, Orthogonal Bases, and Orthogonal Matrices......Page 279
I.11. Determinants: Geometric Interpretation and Basic Properties......Page 282
I.12. Determinants of Block-Triangular Matrices......Page 290
I.13. Determinants and Cofactors......Page 291
I.14. Inverse of Matrix in Terms of Cofactors......Page 294
I.15.1. Eigenvalues......Page 295
I.15.2. Eigenvectors......Page 296
I.15.3. Eigenvalues and Eigenvectors of Symmetric Matrices......Page 297
I.16. Positive Definite and Semidefinite Matrices......Page 299
I.17. Generalized Eigenvalues and Eigenvectors......Page 300
I.18. Exercises......Page 302
II.1.1. General Set Operations......Page 305
II.1.2. Sets in Euclidean Spaces......Page 306
II.2. Supremum and Infimum......Page 307
II.3. Limsup and Liminf......Page 308
II.4. Continuity of Concave and Convex Functions......Page 309
II.5. Compactness......Page 310
II.6. Uniform Continuity......Page 312
II.7. Derivatives of Vector and Matrix Functions......Page 313
II.9. Taylor’s Theorem......Page 316
II.10. Optimization......Page 318
III.1. The Complex Number System......Page 320
III.2. The Complex Exponential Function......Page 323
III.4. Series Expansion of the Complex Logarithm......Page 325
III.5. Complex Integration......Page 327
Appendix IV – Tables of Critical Values......Page 328
References......Page 337
Index......Page 339