PROBABILITY AND STATISTICS FOR ECONOMISTS

1 PROBABILITYAND STATISTICSFOR ECONOMISTSBRUCE E. HANSENC ontentsPrefacexAcknowledgementsxiMathema tical PreparationxiiNotationxiii1 Basic PROBABILITY .. and Events .. Function.. of the PROBABILITY Function .. Outcomes .. Events .. PROBABILITY .. of Total PROBABILITY .. Rule.. and Combinations .. With and Without Replacement .. Hands .. Fields* .. Proofs* ..182 Random .. Variables .. Random Variables .. of Expectations .. Function .. Random Variables .. Functions .. of Continuous Random Variables.. Transformations.

2 Of Continuous Random Variables .. of Expectations .. Notation .. and Variance .. s Inequality .. of Jensen s Inequality* .. Distributions .. Distributions .. Distributions .. Generating Function .. Function .. : Mathematical Details*..523 Parametric .. Distribution .. Distribution.. Distribution .. Distribution .. Distribution.. Binomial Distribution .. Distribution .. Distribution .. Exponential Distribution .. Exponential Distribution .. Distribution .. Distribution .. t Distribution .. Distribution.

3 Distribution .. Distribution .. Distribution .. Chi-Square .. Distribution.. Distribution .. Distribution.. Distribution .. Value Distribution.. of Normals .. Proofs* ..714 Multivariate .. Random Variables .. Distribution Functions .. Mass Function .. Density Function.. Distribution .. Expectation .. Distribution for DiscreteX.. Distribution for ContinuousX.. Conditional Densities .. and Correlation .. Expectation .. of Iterated Expectations .. Variance .. lder s and Minkowski s Inequalities*.

4 Notation .. Inequalities* .. Random Vectors .. of Multivariate Vectors .. Transformations .. Distributions .. and Uniqueness of the Conditional Expectation* .. 1095 Normal and Related .. Normal .. of the Normal Distribution .. Cumulants .. Quantiles .. and Censored Normal Distributions .. Normal .. of the Multivariate Normal .. , t, F, and Cauchy Distributions .. Polynomials* .. Proofs* .. Illustration .. , Parameters, Estimators .. Mean.

5 Value of Transformations .. of Parameters .. Distribution.. Bias.. Variance .. Squared Error .. Unbiased Estimator .. of Variance .. Error .. Means .. STATISTICS .. Moments of Sample Mean* .. Sampling Model .. Residuals .. Variance Estimation .. Ratio .. Normal Sampling .. 1467 Law of Large .. Limits .. in PROBABILITY .. s Inequality .. Law of Large Numbers .. Chebyshev s.. Moments .. Mapping Theorem .. Over Distributions*.

6 Sure Convergence and the Strong Law* .. Proofs* .. 1628 Central Limit .. in Distribution .. Mean .. Moment Investigation .. of the Moment Generating Function .. Limit Theorem .. the Central Limit Theorem .. Central Limit Theorem .. Method .. Distribution for Plug-In Estimator .. Matrix Estimation .. Order Symbols.. Proofs* .. 1769 Advanced Asymptotic Theory* .. Central Limit Theory .. Heterogeneous CLTs .. CLT .. Integrability .. Stochastic Bounds.

7 Of Moments .. Expansion for the Sample Mean .. Expansion for Smooth Function Model.. Expansions .. Proofs .. 18810 Maximum Likelihood .. Model .. Analog Principle .. Property .. , Hessian, and Information.. r-Rao Lower Bound .. Examples .. Cram r-Rao Bound for Functions of Parameters .. Consistent Estimation.. Asymptotic Normality .. Asymptotic Cram r-Rao Efficiency .. Variance Estimation .. Kullback-Leibler Divergence .. Approximating Models.

8 Distribution of the MLE under Mis-Specification .. Variance Estimation under Mis-Specification .. Technical Proofs* .. Exercises .. 22111 Method of .. Means .. Functions .. Moments .. Unbiased Estimation .. Models .. of Parametric Models .. Equations .. Asymptotic Distribution for Moment Equations .. Example: Euler Equation .. Empirical Distribution Function .. Sample Quantiles .. Robust Variance Estimation .. Technical Proofs*.

9 Exercises .. 24612 Numerical .. Function Evaluation and Differentiation .. Finding .. in One Dimension .. of Minimization .. in Multiple Dimensions .. Optimization .. Minimization .. and Tricks .. Exercises .. 26913 Hypothesis .. and Rejection .. I and II Error .. Tests .. Tests .. Does AcceptH0 Mean AboutH0? .. Test with Normal Sampling .. t-test .. Likelihood Ratio Test for Simple Hypotheses .. Neyman-Pearson Lemma.

10 Likelihood Ratio Test Against Composite Alternatives .. Likelihood Ratio and t tests.. Statistical Significance .. P-Value .. Composite Null Hypothesis .. Asymptotic Uniformity .. Summary .. Exercises .. 29314 Confidence .. Confidence Intervals .. Intervals for the Sample Mean under Normal Sampling.. Intervals for the Sample Mean under non-Normal Sampling.. Intervals for Estimated Parameters .. Interval for the Variance .. Intervals by Test Inversion.