Mathematical statistics for applied econometrics/ Charles B. Moss

1. Defining Mathematical Statistics --
1.1. Mathematical Statistics and Econometrics --
1.1.1. Econometrics and Scientific Discovery --
1.1.2. Econometrics and Planning --
1.2. Mathematical Statistics and Modeling Economic Decisions --
1.3. Chapter Summary --
1.4. Review Questions --
I. Defining Random Variables --
2. Introduction to Statistics, Probability, and Econometrics --
2.1. Two Definitions of Probability for Econometrics --
2.1.1. Counting Techniques --
2.1.2. Axiomatic Foundations --
2.2. What Is Statistics? --
2.3. Chapter Summary --
2.4. Review Questions --
2.5. Numerical Exercises --
3. Random Variables and Probability Distributions --
3.1. Uniform Probability Measure --
3.2. Random Variables and Distributions --
3.2.1. Discrete Random Variables --
3.2.2. Continuous Random Variables --
3.3. Conditional Probability and Independence --
3.3.1. Conditional Probability and Independence for Discrete Random Variables Note continued: 3.3.2. Conditional Probability and Independence for Continuous Random Variables --
3.4. Cumulative Distribution Function --
3.5. Some Useful Distributions --
3.6. Change of Variables --
3.7. Derivation of the Normal Distribution Function --
3.8. An Applied Sabbatical --
3.9. Chapter Summary --
3.10. Review Questions --
3.11. Numerical Exercises --
4. Moments and Moment-Generating Functions --
4.1. Expected Values --
4.2. Moments --
4.3. Covariance and Correlation --
4.4. Conditional Mean and Variance --
4.5. Moment-Generating Functions --
4.5.1. Moment-Generating Functions for Specific Distributions --
4.6. Chapter Summary --
4.7. Review Questions --
4.8. Numerical Exercises --
5. Binomial and Normal Random Variables --
5.1. Bernoulli and Binomial Random Variables --
5.2. Univariate Normal Distribution --
5.3. Linking the Normal Distribution to the Binomial --
5.4. Bivariate and Multivariate Normal Random Variables --
5.4.1. Bivariate Normal Random Variables Note continued: 5.4.2. Multivariate Normal Distribution --
5.5. Chapter Summary --
5.6. Review Questions --
5.7. Numerical Exercises --
II. Estimation --
6. Large Sample Theory --
6.1. Convergence of Statistics --
6.2. Modes of Convergence --
6.2.1. Almost Sure Convergence --
6.2.2. Convergence in Probability --
6.2.3. Convergence in the rth Mean --
6.3. Laws of Large Numbers --
6.4. Asymptotic Normality --
6.5. Wrapping Up Loose Ends --
6.5.1. Application of Holder's Inequality --
6.5.2. Application of Chebychev's Inequality --
6.5.3. Normal Approximation of the Binomial --
6.6. Chapter Summary --
6.7. Review Questions --
6.8. Numerical Exercises --
7. Point Estimation --
7.1. Sampling and Sample Image --
7.2. Familiar Estimators --
7.2.1. Estimators in General --
7.2.2. Nonparametric Estimation --
7.3. Properties of Estimators --
7.3.1. Measures of Closeness --
7.3.2. Mean Squared Error --
7.3.3. Strategies for Choosing an Estimator --
7.3.4. Best Linear Unbiased Estimator Note continued: 7.3.5. Asymptotic Properties --
7.3.6. Maximum Likelihood --
7.4. Sufficient Statistics --
7.4.1. Data Reduction --
7.4.2. Sufficiency Principle --
7.5. Concentrated Likelihood Functions --
7.6. Normal Equations --
7.7. Properties of Maximum Likelihood Estimators --
7.8. Chapter Summary --
7.9. Review Questions --
7.10. Numerical Exercises --
8. Interval Estimation --
8.1. Confidence Intervals --
8.2. Bayesian Estimation --
8.3. Bayesian Confidence Intervals --
8.4. Chapter Summary --
8.5. Review Questions --
8.6. Numerical Exercises --
9. Testing Hypotheses --
9.1. Type I and Type II Errors --
9.2. Neyman --
Pearson Lemma --
9.3. Simple Tests against a Composite --
9.4.Composite against a Composite --
9.5. Testing Hypotheses about Vectors --
9.6. Delta Method --
9.7. Chapter Summary --
9.8. Review Questions --
9.9. Numerical Exercises --
III. Econometric Applications --
10. Elements of Matrix Analysis --
10.1. Review of Elementary Matrix Algebra --
10.1.1. Basic Definitions Note continued: 10.1.2. Vector Spaces --
10.2. Projection Matrices --
10.3. Idempotent Matrices --
10.4. Eigenvalues and Eigenvectors --
10.5. Kronecker Products --
10.6. Chapter Summary --
10.7. Review Questions --
10.8. Numerical Exercises --
11. Regression Applications in Econometrics --
11.1. Simple Linear Regression --
11.1.1. Least Squares: A Mathematical Solution --
11.1.2. Best Linear Unbiased Estimator: A Statistical Solution --
11.1.3. Conditional Normal Model --
11.1.4. Variance of the Ordinary Least Squares Estimator --
11.2. Multivariate Regression --
11.2.1. Variance of Estimator --
11.2.2. Gauss --
Markov Theorem --
11.3. Linear Restrictions --
11.3.1. Variance of the Restricted Estimator --
11.3.2. Testing Linear Restrictions --
11.4. Exceptions to Ordinary Least Squares --
11.4.1. Heteroscedasticity --
11.4.2. Two Stage Least Squares and Instrumental Variables --
11.4.3. Generalized Method of Moments Estimator --
11.5. Chapter Summary --
11.6. Review Questions Note continued: 11.7. Numerical Exercises --
12. Survey of Nonlinear Econometric Applications --
12.1. Nonlinear Least Squares and Maximum Likelihood --
12.2. Bayesian Estimation --
12.2.1. Basic Model --
12.2.2. Conditioning and Updating --
12.2.3. Simple Estimation by Simulation --
12.3. Least Absolute Deviation and Related Estimators --
12.3.1. Least Absolute Deviation --
12.3.2. Quantile Regression --
12.4. Chapter Summary --
12.5. Review Questions --
12.6. Numerical Exercises --
13. Conclusions --
Appendix A Symbolic Computer Programs --
A.1. Maxima --
A.2. Mathematica["! --
Appendix B Change of Variables for Simultaneous Equations --
B.1. Linear Change in Variables --
B.2. Estimating a System of Equations --
Appendix C Fourier Transformations --
C.1. Continuing the Example --
C.2. Fourier Approximation --
Appendix D Farm Interest Rate Data --
Appendix E Nonlinear Optimization --
E.1. Hessian Matrix of Three-Parameter Cobb --
Douglas --
E.2. Bayesian Estimation