Introduction to the mathematical and statistical foundations of econometrics:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2004
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Themes in modern econometrics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverzeichnis Seite 315 - 316 Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XVII, 323 S. graph. Darst. |
| ISBN: | 0521542243 0521834317 9780521834315 9780521542241 |
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| 100 | 1 | |a Bierens, Herman J. |d 1943- |e Verfasser |0 (DE-588)109761766 |4 aut | |
| 245 | 1 | 0 | |a Introduction to the mathematical and statistical foundations of econometrics |c Herman J. Bierens |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2004 | |
| 300 | |a XVII, 323 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Themes in modern econometrics | |
| 500 | |a Literaturverzeichnis Seite 315 - 316 | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 7 | |a Économétrie |2 rasuqam | |
| 650 | 4 | |a aEconometrics | |
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Datensatz im Suchindex
| _version_ | 1817704592190734336 |
|---|---|
| adam_text |
Contents
Preface
page XV
1
Probability and Measure
1
1.1
The Texas Lotto
1
1.1.1
Introduction
1
1.1.2
Binomial Numbers
2
1.1.3
Sample Space
3
1.1.4
Algebras and Sigma-Algebras of Events
3
1.1.5
Probability Measure
4
1.2
Quality Control
6
1.2.1
Sampling without Replacement
6
1.2.2
Quality Control in Practice
7
1.2.3
Sampling with Replacement
8
1.2.4
Limits of the Hypergeometric and Binomial
Probabilities
8
1.3
Why Do We Need Sigma-Algebras of Events
?
10
1.4
Properties of Algebras and Sigma-Algebras
11
1.4.1
General Properties
11
1.4.2
Borei
Sets
14
1.5
Properties of Probability Measures
15
1.6
The Uniform Probability Measure
16
1.6.1
Introduction
16
1.6.2
Outer Measure
17
1.7
Lebesgue Measure and Lebesgue Integral
19
1.7.1
Lebesgue Measure
19
1.7.2
Lebesgue Integral
19
1.8
Random Variables and Their Distributions
20
1.8.1
Random Variables and Vectors
20
1.8.2
Distribution Functions
23
1.9
Density Functions
25
vu
viii Contents
1.10
Conditional Probability,
Bayes'
Rule,
and Independence
27
1.10.1
Conditional Probability
27
1.10.2
Bayes'
Rule
27
1.10.3
Independence
28
1.11
Exercises
30
Appendix LA
-
Common Structure of the Proofs of Theorems
1.6
and
1.10 32
Appendix l.B
-
Extension of an Outer Measure to a
Probability Measure
32
2
Borei Measurability,
Integration, and Mathematical
Expectations
37
2.1
Introduction
37
2.2
Borei
Measurability
38
2.3
Integrals of Borel-Measurable Functions with Respect
to a Probability Measure
42
2.4
General Measurability and Integrals of Random
Variables with Respect to Probability Measures
46
2.5
Mathematical Expectation
49
2.6
Some Useful Inequalities Involving Mathematical
Expectations
50
2.6.1
Chebishev's Inequality
51
2.6.2
Holder's Inequality
51
2.6.3
Liapounov's Inequality
52
2.6.4
Minkowski's Inequality
52
2.6.5
Jensen's Inequality
52
2.7
Expectations of Products of Independent Random
Variables
53
2.8
Moment-Generating Functions and Characteristic
Functions
55
2.8.1
Moment-Generating Functions
55
2.8.2
Characteristic Functions
58
2.9
Exercises
59
Appendix 2.A
-
Uniqueness of Characteristic Functions
61
3
Conditional Expectations
66
3.1
Introduction
66
3.2
Properties of Conditional Expectations
72
3.3
Conditional Probability Measures and Conditional
Independence
79
3.4
Conditioning on Increasing Sigma-Algebras
80
Contents ix
3.5
Conditional Expectations as the Best Forecast Schemes
80
3.6
Exercises
82
Appendix
3
.A
-
Proof of Theorem
3.12 83
4
Distributions and Transformations
86
4.1
Discrete Distributions
86
4.1.1
The Hypergeometric Distribution
86
4.1.2
The Binomial Distribution
87
4.1.3
The
Poisson
Distribution
88
4.1.4
The Negative Binomial Distribution
88
4.2
Transformations of Discrete Random Variables and
Vectors
89
4.3
Transformations of Absolutely Continuous Random
Variables
90
4.4
Transformations of Absolutely Continuous Random
Vectors
91
4.4.1
The Linear Case
91
4.4.2
The Nonlinear Case
94
4.5
The Normal Distribution
96
4.5.1
The Standard Normal Distribution
96
4.5.2
The General Normal Distribution
97
4.6
Distributions Related to the Standard Normal
Distribution
97
4.6.1
The Chi-Square Distribution
97
4.6.2
The Student's
t
Distribution
99
4.6.3
The Standard Cauchy Distribution
100
4.6.4
The
F
Distribution
100
4.7
The Uniform Distribution and Its Relation to the
Standard Normal Distribution
101
4.8
The Gamma Distribution
102
4.9
Exercises
102
Appendix
4.
A
-
Tedious Derivations
104
Appendix 4.B
-
Proof of Theorem
4.4 106
5
The Multivariate Normal Distribution and Its Application
to Statistical Inference
110
5.1
Expectation and Variance of Random Vectors
110
5.2
The Multivariate Normal Distribution
111
5.3
Conditional Distributions of Multivariate Normal
Random Variables
115
5.4
Independence of Linear and Quadratic Transformations
of Multivariate Normal Random Variables
117
Contents
5.5
Distributions of Quadratic Forms of Multivariate
Normal Random Variables
118
5.6
Applications to Statistical Inference under Normality
119
5.6.1
Estimation
119
5.6.2
Confidence Intervals
122
5.6.3
Testing Parameter Hypotheses
125
5.7
Applications to Regression Analysis
127
5.7.1
The Linear Regression Model
127
5.7.2
Least-Squares Estimation
127
5.7.3
Hypotheses Testing
131
5.8
Exercises
133
Appendix 5.A
-
Proof of Theorem
5.8 134
Modes of Convergence
137
6.1
Introduction
137
6.2
Convergence in Probability and the Weak Law of Large
Numbers
140
6.3
Almost-Sure Convergence and the Strong Law of Large
Numbers
143
6.4
The Uniform Law of Large Numbers and Its
Applications
145
6.4.1
The Uniform Weak Law of Large Numbers
145
6.4.2
Applications of the Uniform Weak Law of
Large Numbers
145
6.4.2.1
Consistency of M-Estimators
145
6.4.2.2
Generalized Slutsky's Theorem
148
6.4.3
The Uniform Strong Law of Large Numbers
and Its Applications
149
6.5
Convergence in Distribution
149
6.6
Convergence of Characteristic Functions
154
6.7
The Central Limit Theorem
155
6.8
Stochastic Boundedness, Tightness, and the Op and op
Notations
157
6.9
Asymptotic Normality of M-Estimators
159
6.10
Hypotheses Testing
162
6.11
Exercises
163
Appendix
6.
A
-
Proof of the Uniform Weak Law of
Large Numbers
164
Appendix 6.B
-
Almost-Sure Convergence and Strong Laws of
Large Numbers
167
Appendix 6.C
-
Convergence of Characteristic Functions and
Distributions
174
Contents xi
7
Dependent
Laws of Large Numbers and Central Limit
Theorems
179
7.1
Stationarity and the Wold Decomposition
179
7.2
Weak Laws of Large Numbers for Stationary Processes
183
7.3
Mixing Conditions
186
7.4
Uniform Weak Laws of Large Numbers
187
7.4.1
Random Functions Depending on
Finite-Dimensional Random Vectors
187
7.4.2
Random Functions Depending on
Infinite-Dimensional Random Vectors
187
7.4.3
Consistency of M-Estimators
190
7.5
Dependent Central Limit Theorems
190
7.5.1
Introduction
190
7.5.2
A Generic Central Limit Theorem
191
7.5.3
Martingale Difference Central Limit Theorems
196
7.6
Exercises
198
Appendix 7.A
-
Hubert Spaces
199
8
Maximum Likelihood Theory
205
8.1
Introduction
205
8.2
Likelihood Functions
207
8.3
Examples
209
8.3.1
The Uniform Distribution
209
8.3.2
Linear Regression with Normal Errors
209
8.3.3
Probit
and Logit Models
211
8.3.4
The Tobit Model
212
8.4
Asymptotic Properties of ML Estimators
214
8.4.1
Introduction
214
8.4.2
First- and Second-Order Conditions
214
8.4.3
Generic Conditions for Consistency and
Asymptotic Normality
216
8.4.4
Asymptotic Normality in the Time Series Case
219
8.4.5
Asymptotic Efficiency of the ML Estimator
220
8.5
Testing Parameter Restrictions
222
8.5.1
The
Pseudo
/-Test and the
Wald Test 222
8.5.2
The Likelihood Ratio Test
223
8.5.3
The
Lagrange
Multiplier Test
225
8.5.4
Selecting a Test
226
8.6
Exercises
226
I Review of Linear Algebra
229
1.
1
Vectors in a Euclidean Space
229
1.2
Vector Spaces
232
xii Contents
1.3
Matrices
235
1.4
The Inverse and Transpose of a Matrix
23 8
1.5
Elementary Matrices and Permutation Matrices
241
1.6
Gaussian Elimination of a Square Matrix and the
Gauss-Jordan Iteration for Inverting a Matrix
244
1.6.1
Gaussian Elimination of a Square Matrix
244
1.6.2
The Gauss-Jordan Iteration for Inverting a
Matrix
248
1.7
Gaussian Elimination of
a
Nonsquare
Matrix
252
1.8
Subspaces Spanned by the Columns and Rows
of a Matrix
253
1.9
Projections, Projection Matrices, and Idempotent
Matrices
256
1.
10
Inner Product, Orthogonal Bases, and Orthogonal
Matrices
257
I.I
1
Determinants: Geometric Interpretation and
Basic Properties
260
1.12
Determinants of Block-Triangular Matrices
268
1.13
Determinants and Cofactors
269
1.
14
Inverse of a Matrix in Terms of Cofactors
272
1.15
Eigenvalues and Eigenvectors
273
1.15.1
Eigenvalues
273
1.15.2
Eigenvectors
274
1.15.3
Eigenvalues and Eigenvectors of Symmetric
Matrices
275
1.16
Positive Definite and Semidefinite Matrices
277
1.17
Generalized Eigenvalues and Eigenvectors
278
1.18
Exercises
280
II Miscellaneous Mathematics
283
II.
1
Sets and Set Operations
283
II.
1.1
General Set Operations
283
II.
1.2
Sets in Euclidean Spaces
284
11.2 Supremum and Infimum
285
11.3 Limsup and Liminf
286
11.4 Continuity of Concave and Convex Functions
287
11.5 Compactness
288
11.6 Uniform Continuity
290
11.7 Derivatives of Vector and Matrix Functions
291
11.8 The Mean Value Theorem
294
11.9
Taylor's Theorem
294
11.10
Optimization
296
Contents xiii
HI A
Brief
Review of Complex Analysis
298
111.1
The Complex Number System
298
Ш.2
The Complex Exponential Function
301
111.3 The Complex Logarithm
303
111.4 Series Expansion of the Complex Logarithm
303
111.5 Complex Integration
305
IV Tables of Critical Values
306
References
315
Index
317 |
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| author | Bierens, Herman J. 1943- |
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| spelling | Bierens, Herman J. 1943- Verfasser (DE-588)109761766 aut Introduction to the mathematical and statistical foundations of econometrics Herman J. Bierens 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2004 XVII, 323 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Themes in modern econometrics Literaturverzeichnis Seite 315 - 316 Hier auch später erschienene, unveränderte Nachdrucke Économétrie rasuqam aEconometrics Wirtschaftsmathematik (DE-588)4066472-7 gnd rswk-swf Ökonometrie (DE-588)4132280-0 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Ökonometrie (DE-588)4132280-0 s DE-188 Wirtschaftsmathematik (DE-588)4066472-7 s Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012955626&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bierens, Herman J. 1943- Introduction to the mathematical and statistical foundations of econometrics Économétrie rasuqam aEconometrics Wirtschaftsmathematik (DE-588)4066472-7 gnd Ökonometrie (DE-588)4132280-0 gnd |
| subject_GND | (DE-588)4066472-7 (DE-588)4132280-0 (DE-588)4123623-3 |
| title | Introduction to the mathematical and statistical foundations of econometrics |
| title_auth | Introduction to the mathematical and statistical foundations of econometrics |
| title_exact_search | Introduction to the mathematical and statistical foundations of econometrics |
| title_full | Introduction to the mathematical and statistical foundations of econometrics Herman J. Bierens |
| title_fullStr | Introduction to the mathematical and statistical foundations of econometrics Herman J. Bierens |
| title_full_unstemmed | Introduction to the mathematical and statistical foundations of econometrics Herman J. Bierens |
| title_short | Introduction to the mathematical and statistical foundations of econometrics |
| title_sort | introduction to the mathematical and statistical foundations of econometrics |
| topic | Économétrie rasuqam aEconometrics Wirtschaftsmathematik (DE-588)4066472-7 gnd Ökonometrie (DE-588)4132280-0 gnd |
| topic_facet | Économétrie aEconometrics Wirtschaftsmathematik Ökonometrie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012955626&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bierenshermanj introductiontothemathematicalandstatisticalfoundationsofeconometrics |