Introduction to the Mathematical and Statistical Foundations of Econometrics

Description:

1. Probability and measure: 1.1 The Texas lotto; 1.2 Quality control; 1.3 Why do we need sigma-algebras of events?; 1.4 Properties of algebras and sigma-algebras; 1.5 Properties of probability measures; 1.6 The uniform probability measures; 1.7 Lebesque measure and Lebesque integral; 1.8 Random variables and their distributions; 1.9 Density functions; 1.10 Conditional probability, Bayes’s rule, and independence; 1.11 Exercises; 1.A Common structure of the proofs of Theorems 6 and 10; 1.B Extension of an outer measure to a probability measure; 2. Borel measurability, integration, and mathematical expectations: 2.1 Introduction; 2.2 Borel measurability; 2.3 Integral of Borel measurable functions with respect to a probability measure; 2.4 General measurability and integrals of random variables with respect to probability measures; 2.5 Mathematical expectation; 2.6 Some useful inequalities involving mathematical expectations; 2.7 Expectations of products of independent random variables; 2.8 Moment generating functions and characteristic functions; 2.9 Exercises; 2.A Uniqueness of characteristic functions; 3. Conditional expectations: 3.1 Introduction; 3.2 Properties of conditional expectations; 3.3 Conditional probability measures and conditional independence; 3.4 Conditioning on increasing sigma-algebras; 3.5 Conditional expectations as the best forecast schemes; 3.6 Exercises; 3.A Proof of theorem 3.12; 4. Distributions and transformations: 4.1 Discrete distributions; 4.2 Transformations of discrete random vectors; 4.3 Transformations of absolutely continuous random variables; 4.4 Transformations of absolutely continuous random vectors; 4.5 The normal distribution; 4.6 Distributions related to the normal distribution; 4.7 The uniform distribution and its relation to the standard normal distribution; 4.8 The gamma distribution; 4.9 Exercises; 4.A Tedious derivations; 4.B Proof of theorem 4.4; 5. The multivariate normal distribution and its application to statistical inference; 5.1 Expectation and variance of random vectors; 5.2 The multivariate normal distribution; 5.3 Conditional distributions of multivariate normal random variables; 5.4 Independence of linear and quadratic transformations of multivariate normal random variables; 5.5 Distribution of quadratic forms of multivariate normal random variables; 5.6 Applications to statistical inference under normality; 5.7 Applications to regression analysis; 5.8 Exercises; 5.A Proof of theorem 5.8; 6. Modes of convergence: 6.1 Introduction; 6.2 Convergence in probability and the weak law of large numbers; 6.3 Almost sure convergence, and the strong law of large numbers; 6.4 The uniform law of large numbers and its applications; 6.5 Convergence in distribution; 6.6 Convergence of characteristic functions; 6.7 The central limit theorem; 6.8 Stochastic boundedness, tightness, and the Op and op-notations; 6.9 Asymptotic normality of M-estimators; 6.10 Hypotheses testing; 6.11 Exercises; 6.A Proof of the uniform weak law of large numbers; 6.B Almost sure convergence and strong laws of large numbers; 6.C Convergence of characteristic functions and distributions; 7. Dependent laws of large numbers and central limit theorems: 7.1 Stationary and the world decomposition; 7.2 Weak laws of large numbers for stationary processes; 7.3 Mixing conditions; 7.4 Uniform weak laws of large numbers; 7.5 Dependent central limit theorems; 7.6 Exercises; 7.A Hilbert spaces; 8. Maximum likelihood theory; 8.1 Introduction; 8.2 Likelihood functions; 8.3 Examples; 8.4 Asymptotic properties if ML estimators; 8.5 Testing parameter restrictions; 8.6 Exercises.