<P><STRONG>INTRODUCTION TO THE MATHEMATICAL AND STATISTICAL FOUNDATIONS OF ECONOMETRICS<br></STRONG>HERMAN J. BIERENS<br> <br><STRONG>Cambridge</STRONG>, New York, Melbourne, Madrid, Cape Town, Singapore, S&atilde;o Paulo<br>Cambridge University Press<br>The Edinburgh Building, Cambridge , UK<br>First published in print format<br>- ----<br>- ----<br>- ----<br><STRONG>&copy; Herman J. Bierens 2005<br></STRONG>Information on this title: <a href="http://www.cambridge.org/9780521834315" target="_blank" ><a href="http://www.cambridge.org/9780521834315" target="_blank" ><FONT color=#000000>www.cambridge.org/9780521834315</FONT></A></A><br>This book is in copyright. Subject to statutory exception and to the provision of<br>relevant collective licensing agreements, no reproduction of any part may take place<br>without the written permission of Cambridge University Press.<br>- ---<br>- ---<br>- ---<br>Cambridge University Press has no responsibility for the persistence or accuracy of<br>s for external or third-party internet websites referred to in this book, and does not<br>guarantee that any content on such websites is, or will remain, accurate or appropriate.<br>Published in the United States of America by Cambridge University Press, New York<br><a href="http://www.cambridge.org/" target="_blank" ><a href="http://www.cambridge.org/" target="_blank" ><FONT color=#000000>www.cambridge.org</FONT></A></A><br>hardback<br>paperback<br>paperback<br>eBook (NetLibrary)<br>eBook (NetLibrary)<br>hardback</P>
<P>Contents<br>Preface page xv<br>1 Probability and Measure 1<br>1.1 The Texas Lotto 1<br>1.1.1 Introduction 1<br>1.1.2 Binomial Numbers 2<br>1.1.3 Sample Space 3<br>1.1.4Algebras and Sigma-Algebras of Events 3<br>1.1.5 Probability Measure 4<br>1.2 Quality Control 6<br>1.2.1 Sampling without Replacement 6<br>1.2.2 Quality Control in Practice 7<br>1.2.3 Sampling with Replacement 8<br>1.2.4Limits of the Hypergeometric and Binomial<br>Probabilities 8<br>1.3 Why Do We Need Sigma-Algebras of Events ? 10<br>1.4Proper ties of Algebras and Sigma-Algebras 11<br>1.4.1 General Properties 11<br>1.4.2 Borel Sets 14<br>1.5 Properties of Probability Measures 15<br>1.6 The Uniform Probability Measure 16<br>1.6.1 Introduction 16<br>1.6.2 Outer Measure 17<br>1.7 Lebesgue Measure and Lebesgue Integral 19<br>1.7.1 Lebesgue Measure 19<br>1.7.2 Lebesgue Integral 19<br>1.8 Random Variables and Their Distributions 20<br>1.8.1 Random Variables and Vectors 20<br>1.8.2 Distribution Functions 23<br>1.9 Density Functions 25</P>
<P>1.10 Conditional Probability, Bayes’ Rule,<br>and Independence 27<br>1.10.1 Conditional Probability 27<br>1.10.2 Bayes’ Rule 27<br>1.10.3 Independence 28<br>1.11 Exercises 30<br>Appendix 1.A – Common Structure of the Proofs of Theorems<br>1.6 and 1.10 32<br>Appendix 1.B – Extension of an Outer Measure to a<br>Probability Measure 32<br>2 Borel Measurability, Integration, and Mathematical<br>Expectations 37<br>2.1 Introduction 37<br>2.2 Borel Measurability 38<br>2.3 Integrals of Borel-Measurable Functions with Respect<br>to a Probability Measure 42<br>2.4General Measurability and Integrals of Random<br>Variables with Respect to Probability Measures 46<br>2.5 Mathematical Expectation 49<br>2.6 Some Useful Inequalities Involving Mathematical<br>Expectations 50<br>2.6.1 Chebishev’s Inequality 51<br>2.6.2 Holder’s Inequality 51<br>2.6.3 Liapounov’s Inequality 52<br>2.6.4Mink owski’s Inequality 52<br>2.6.5 Jensen’s Inequality 52<br>2.7 Expectations of Products of Independent Random<br>Variables 53<br>2.8 Moment-Generating Functions and Characteristic<br>Functions 55<br>2.8.1 Moment-Generating Functions 55<br>2.8.2 Characteristic Functions 58<br>2.9 Exercises 59<br>Appendix 2.A – Uniqueness of Characteristic Functions 61<br>3 Conditional Expectations 66<br>3.1 Introduction 66<br>3.2 Properties of Conditional Expectations 72<br>3.3 Conditional Probability Measures and Conditional<br>Independence 79<br>3.4Conditioning on Increasing Sigma-Algebras 80</P>
<P>3.5 Conditional Expectations as the Best Forecast Schemes 80<br>3.6 Exercises 82<br>Appendix 3.A – Proof of Theorem 3.12 83<br>4 Distributions and Transformations 86<br>4.1 Discrete Distributions 86<br>4.1.1 The Hypergeometric Distribution 86<br>4.1.2 The Binomial Distribution 87<br>4.1.3 The Poisson Distribution 88<br>4.1.4 The Negative Binomial Distribution 88<br>4.2 Transformations of Discrete Random Variables and<br>Vectors 89<br>4.3 Transformations of Absolutely Continuous Random<br>Variables 90<br>4.4 Transformations of Absolutely Continuous Random<br>Vectors 91<br>4.4.1 The Linear Case 91<br>4.4.2 The Nonlinear Case 94<br>4.5 The Normal Distribution 96<br>4.5.1 The Standard Normal Distribution 96<br>4.5.2 The General Normal Distribution 97<br>4.6 Distributions Related to the Standard Normal<br>Distribution 97<br>4.6.1 The Chi-Square Distribution 97<br>4.6.2 The Student’s t Distribution 99<br>4.6.3 The Standard Cauchy Distribution 100<br>4.6.4 The F Distribution 100<br>4.7 The Uniform Distribution and Its Relation to the<br>Standard Normal Distribution 101<br>4.8 The Gamma Distribution 102<br>4.9 Exercises 102<br>Appendix 4.A – Tedious Derivations 104<br>Appendix 4.B – Proof of Theorem 4.4 106<br>5 The Multivariate Normal Distribution and Its Application<br>to Statistical Inference 110<br>5.1 Expectation and Variance of Random Vectors 110<br>5.2 The Multivariate Normal Distribution 111<br>5.3 Conditional Distributions of Multivariate Normal<br>Random Variables 115<br>5.4Independence of Linear and Quadratic Transformations<br>of Multivariate Normal Random Variables 117</P>
<P>5.5 Distributions of Quadratic Forms of Multivariate<br>Normal Random Variables 118<br>5.6 Applications to Statistical Inference under Normality 119<br>5.6.1 Estimation 119<br>5.6.2 Confidence Intervals 122<br>5.6.3 Testing Parameter Hypotheses 125<br>5.7 Applications to Regression Analysis 127<br>5.7.1 The Linear Regression Model 127<br>5.7.2 Least-Squares Estimation 127<br>5.7.3 Hypotheses Testing 131<br>5.8 Exercises 133<br>Appendix 5.A – Proof of Theorem 5.8 134<br>6 Modes of Convergence 137<br>6.1 Introduction 137<br>6.2 Convergence in Probability and the Weak Law of Large<br>Numbers 140<br>6.3 Almost-Sure Convergence and the Strong Law of Large<br>Numbers 143<br>6.4The Uniform Law of Large Numbers and Its<br>Applications 145<br>6.4.1 The Uniform Weak Law of Large Numbers 145<br>6.4.2 Applications of the Uniform Weak Law of<br>Large Numbers 145<br>6.4.2.1 Consistency of M-Estimators 145<br>6.4.2.2 Generalized Slutsky’s Theorem 148<br>6.4.3 The Uniform Strong Law of Large Numbers<br>and Its Applications 149<br>6.5 Convergence in Distribution 149<br>6.6 Convergence of Characteristic Functions 154<br>6.7 The Central Limit Theorem 155<br>6.8 Stochastic Boundedness, Tightness, and the Op and op<br>Notations 157<br>6.9 Asymptotic Normality of M-Estimators 159<br>6.10 Hypotheses Testing 162<br>6.11 Exercises 163<br>Appendix 6.A – Proof of the Uniform Weak Law of<br>Large Numbers 164<br>Appendix 6.B – Almost-Sure Convergence and Strong Laws of<br>Large Numbers 167<br>Appendix 6.C – Convergence of Characteristic Functions and<br>Distributions 174</P>
<P>7 Dependent Laws of Large Numbers and Central Limit<br>Theorems 179<br>7.1 Stationarity and the Wold Decomposition 179<br>7.2 Weak Laws of Large Numbers for Stationary Processes 183<br>7.3 Mixing Conditions 186<br>7.4Unifor m Weak Laws of Large Numbers 187<br>7.4.1 Random Functions Depending on<br>Finite-Dimensional Random Vectors 187<br>7.4.2 Random Functions Depending on<br>Infinite-Dimensional Random Vectors 187<br>7.4.3 Consistency of M-Estimators 190<br>7.5 Dependent Central Limit Theorems 190<br>7.5.1 Introduction 190<br>7.5.2 A Generic Central Limit Theorem 191<br>7.5.3 Martingale Difference Central Limit Theorems 196<br>7.6 Exercises 198<br>Appendix 7.A – Hilbert Spaces 199<br>8 Maximum Likelihood Theory 205<br>8.1 Introduction 205<br>8.2 Likelihood Functions 207<br>8.3 Examples 209<br>8.3.1 The Uniform Distribution 209<br>8.3.2 Linear Regression with Normal Errors 209<br>8.3.3 Probit and Logit Models 211<br>8.3.4The Tobit Model 212<br>8.4Asymptotic Properties of ML Estimators 214<br>8.4.1 Introduction 214<br>8.4.2 First- and Second-Order Conditions 214<br>8.4.3 Generic Conditions for Consistency and<br>Asymptotic Normality 216<br>8.4.4 Asymptotic Normality in the Time Series Case 219<br>8.4.5 Asymptotic Efficiency of the ML Estimator 220<br>8.5 Testing Parameter Restrictions 222<br>8.5.1 The Pseudo t-Test and the Wald Test 222<br>8.5.2 The Likelihood Ratio Test 223<br>8.5.3 The Lagrange Multiplier Test 225<br>8.5.4Selecting a Test 226<br>8.6 Exercises 226<br>I Review of Linear Algebra 229<br>I.1 Vectors in a Euclidean Space 229<br>I.2 Vector Spaces 232</P>
<P>I.3 Matrices 235<br>I.4The Inverse and Transpose of a Matrix 238<br>I.5 Elementary Matrices and Permutation Matrices 241<br>I.6 Gaussian Elimination of a Square Matrix and the<br>Gauss–Jordan Iteration for Inverting a Matrix 244<br>I.6.1 Gaussian Elimination of a Square Matrix 244<br>I.6.2 The Gauss–Jordan Iteration for Inverting a<br>Matrix 248<br>I.7 Gaussian Elimination of a Nonsquare Matrix 252<br>I.8 Subspaces Spanned by the Columns and Rows<br>of a Matrix 253<br>I.9 Projections, Projection Matrices, and Idempotent<br>Matrices 256<br>I.10 Inner Product, Orthogonal Bases, and Orthogonal<br>Matrices 257<br>I.11 Determinants: Geometric Interpretation and<br>Basic Properties 260<br>I.12 Determinants of Block-Triangular Matrices 268<br>I.13 Determinants and Cofactors 269<br>I.14In verse of a Matrix in Terms of Cofactors 272<br>I.15 Eigenvalues and Eigenvectors 273<br>I.15.1 Eigenvalues 273<br>I.15.2 Eigenvectors 274<br>I.15.3 Eigenvalues and Eigenvectors of Symmetric<br>Matrices 275<br>I.16 Positive Definite and Semidefinite Matrices 277<br>I.17 Generalized Eigenvalues and Eigenvectors 278<br>I.18 Exercises 280<br>II Miscellaneous Mathematics 283<br>II.1 Sets and Set Operations 283<br>II.1.1 General Set Operations 283<br>II.1.2 Sets in Euclidean Spaces 284<br>II.2 Supremum and Infimum 285<br>II.3 Limsup and Liminf 286<br>II.4Continuity of Concave and Convex Functions 287<br>II.5 Compactness 288<br>II.6 Uniform Continuity 290<br>II.7 Derivatives of Vector and Matrix Functions 291<br>II.8 The Mean Value Theorem 294<br>II.9 Taylor’s Theorem 294<br>II.10 Optimization 296</P>
<P>III A Brief Review of Complex Analysis 298<br>III.1 The Complex Number System 298<br>III.2 The Complex Exponential Function 301<br>III.3 The Complex Logarithm 303<br>III.4Series Expansion of the Complex Logarithm 303<br>III.5 Complex Integration 305<br>IV Tables of Critical Values 306<br>References 315<br>Index 317</P>
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