Probability and Statistics for Economists

Bruce Hansen

A comprehensive and up-to-date introduction to the mathematics that all economics students need to know

Hardcover

Sale Price:: $49.00/£40.60
Price:: ~~$70.00~~/~~£58.00~~
ISBN:
Published:: Jun 28, 2022
Copyright:: 2022
Illus:: 111 b/w illus. 6 tables.

ebook

Sale Price:: $49.00/£40.60
Price:: ~~$70.00~~/~~£58.00~~
ISBN:
Published:: Jun 28, 2022
Copyright:: 2022
Illus:: 111 b/w illus. 6 tables.

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Probability theory is the quantitative language used to handle uncertainty and is the foundation of modern statistics. Probability and Statistics for Economists provides graduate and PhD students with an essential introduction to mathematical probability and statistical theory, which are the basis of the methods used in econometrics. This incisive textbook teaches fundamental concepts, emphasizes modern, real-world applications, and gives students an intuitive understanding of the mathematics that every economist needs to know.

Covers probability and statistics with mathematical rigor while emphasizing intuitive explanations that are accessible to economics students of all backgrounds
Discusses random variables, parametric and multivariate distributions, sampling, the law of large numbers, central limit theory, maximum likelihood estimation, numerical optimization, hypothesis testing, and more
Features hundreds of exercises that enable students to learn by doing
Includes an in-depth appendix summarizing important mathematical results as well as a wealth of real-world examples
Can serve as a core textbook for a first-semester PhD course in econometrics and as a companion book to Bruce E. Hansen’s Econometrics
Also an invaluable reference for researchers and practitioners

Preface
Acknowledgments
Mathematical Preparation
Notation
1 Basic Probability Theory
- 1.1 Introduction
- 1.2 Outcomes and Events
- 1.3 Probability Function
- 1.4 Properties of the Probability Function
- 1.5 Equally Likely Outcomes
- 1.6 Joint Events
- 1.7 Conditional Probability
- 1.8 Independence
- 1.9 Law of Total Probability
- 1.10 Bayes Rule
- 1.11 Permutations and Combinations
- 1.12 Sampling with and without Replacement
- 1.13 Poker Hands
- 1.14 Sigma Fields*
- 1.15 Technical Proofs*
- 1.16 Exercises
2 Random Variables
- 2.1 Introduction
- 2.2 Random Variables
- 2.3 Discrete Random Variables
- 2.4 Transformations
- 2.5 Expectation
- 2.6 Finiteness of Expectations
- 2.7 Distribution Function
- 2.8 Continuous Random Variables
- 2.9 Quantiles
- 2.10 Density Functions
- 2.11 Transformations of Continuous Random Variables
- 2.12 Non-Monotonic Transformations
- 2.13 Expectation of Continuous Random Variables
- 2.14 Finiteness of Expectations
- 2.15 Unifying Notation
- 2.16 Mean and Variance
- 2.17 Moments
- 2.18 Jensen’s Inequality
- 2.19 Applications of Jensen’s Inequality*
- 2.20 Symmetric Distributions
- 2.21 Truncated Distributions
- 2.22 Censored Distributions
- 2.23 Moment Generating Function
- 2.24 Cumulants
- 2.25 Characteristic Function
- 2.26 Expectation: Mathematical Details*
- 2.27 Exercises
3 Parametric Distributions
- 3.1 Introduction
- 3.2 Bernoulli Distribution
- 3.3 Rademacher Distribution
- 3.4 Binomial Distribution
- 3.5 Multinomial Distribution
- 3.6 Poisson Distribution
- 3.7 Negative Binomial Distribution
- 3.8 Uniform Distribution
- 3.9 Exponential Distribution
- 3.10 Double Exponential Distribution
- 3.11 Generalized Exponential Distribution
- 3.12 Normal Distribution
- 3.13 Cauchy Distribution
- 3.14 Student t Distribution
- 3.15 Logistic Distribution
- 3.16 Chi-Square Distribution
- 3.17 Gamma Distribution
- 3.18 F Distribution
- 3.19 Non-Central Chi-Square
- 3.20 Beta Distribution
- 3.21 Pareto Distribution
- 3.22 Lognormal Distribution
- 3.23 Weibull Distribution
- 3.24 Extreme Value Distribution
- 3.25 Mixtures of Normals
- 3.26 Technical Proofs*
- 3.27 Exercises
4 Multivariate Distributions
- 4.1 Introduction
- 4.2 Bivariate Random Variables
- 4.3 Bivariate Distribution Functions
- 4.4 Probability Mass Function
- 4.5 Probability Density Function
- 4.6 Marginal Distribution
- 4.7 Bivariate Expectation
- 4.8 Conditional Distribution for Discrete X
- 4.9 Conditional Distribution for Continuous X
- 4.10 Visualizing Conditional Densities
- 4.11 Independence
- 4.12 Covariance and Correlation
- 4.13 Cauchy-Schwarz Inequality
- 4.14 Conditional Expectation
- 4.15 Law of Iterated Expectations
- 4.16 Conditional Variance
- 4.17 H ölder’s and Minkowski’s Inequalities*
- 4.18 Vector Notation
- 4.19 Triangle Inequalities*
- 4.20 Multivariate Random Vectors
- 4.21 Pairs of Multivariate Vectors
- 4.22 Multivariate Transformations
- 4.23 Convolutions
- 4.24 Hierarchical Distributions
- 4.25 Existence and Uniqueness of the Conditional Expectation*
- 4.26 Identiﬁcation
- 4.27 Exercises
5 Normal and Related Distributions
- 5.1 Introduction
- 5.2 Univariate Normal
- 5.3 Moments of the Normal Distribution
- 5.4 Normal Cumulants
- 5.5 Normal Quantiles
- 5.6 Truncated and Censored Normal Distributions
- 5.7 Multivariate Normal
- 5.8 Properties of the Multivariate Normal
- 5.9 Chi-Square, t,F , and Cauchy Distributions
- 5.10 Hermite Polynomials*
- 5.11 Technical Proofs*
- 5.12 Exercises
6 Sampling
- 6.1 Introduction
- 6.2 Samples
- 6.3 Empirical Illustration
- 6.4 Statistics, Parameters, and Estimators
- 6.5 Sample Mean
- 6.6 Expected Value of Transformations
- 6.7 Functions of Parameters
- 6.8 Sampling Distribution
- 6.9 Estimation Bias
- 6.10 Estimation Variance
- 6.11 Mean Squared Error
- 6.12 Best Unbiased Estimator
- 6.13 Estimation of Variance
- 6.14 Standard Error
- 6.15 Multivariate Means
- 6.16 Order Statistics∗
- 6.17 Higher Moments of Sample Mean*
- 6.18 Normal Sampling Model
- 6.19 Normal Residuals
- 6.20 Normal Variance Estimation
- 6.21 Studentized Ratio
- 6.22 Multivariate Normal Sampling
- 6.23 Exercises
7 Law of Large Numbers
- 7.1 Introduction
- 7.2 Asymptotic Limits
- 7.3 Convergence in Probability
- 7.4 Chebyshev’s Inequality
- 7.5 Weak Law of Large Numbers
- 7.6 Counterexamples
- 7.7 Examples
- 7.8 Illustrating Chebyshev’s Inequality
- 7.9 Vector-Valued Moments
- 7.10 Continuous Mapping Theorem
- 7.11 Examples
- 7.12 Uniformity Over Distributions*
- 7.13 Almost Sure Convergence and the Strong Law*
- 7.14 Technical Proofs*
- 7.15 Exercises
8 Central Limit Theory
- 8.1 Introduction
- 8.2 Convergence in Distribution
- 8.3 Sample Mean
- 8.4 A Moment Investigation
- 8.5 Convergence of the Moment Generating Function
- 8.6 Central Limit Theorem
- 8.7 Applying the Central Limit Theorem
- 8.8 Multivariate Central Limit Theorem
- 8.9 Delta Method
- 8.10 Examples
- 8.11 Asymptotic Distribution for Plug-In Estimator
- 8.12 Covariance Matrix Estimation
- 8.13 t -Ratios
- 8.14 Stochastic Order Symbols
- 8.15 Technical Proofs*
- 8.16 Exercises
9 Advanced Asymptotic Theory*
- 9.1 Introduction
- 9.2 Heterogeneous Central Limit Theory
- 9.3 Multivariate Heterogeneous Central Limit Theory
- 9.4 Uniform Central Limit Theory
- 9.5 Uniform Integrability
- 9.6 Uniform Stochastic Bounds
- 9.7 Convergence of Moments
- 9.8 Edgeworth Expansion for the Sample Mean
- 9.9 Edgeworth Expansion for Smooth Function Model
- 9.10 Cornish-Fisher Expansions
- 9.11 Technical Proofs*
10 Maximum Likelihood Estimation
- 10.1 Introduction
- 10.2 Parametric Model
- 10.3 Likelihood
- 10.4 Likelihood Analog Principle
- 10.5 Invariance Property
- 10.6 Examples
- 10.7 Score, Hessian, and Information
- 10.8 Examples
- 10.9 Cram ér-Rao Lower Bound
- 10.10 Examples
- 10.11 Cram ér-Rao Bound for Functions of Parameters
- 10.12 Consistent Estimation
- 10.13 Asymptotic Normality
- 10.14 Asymptotic Cram ér-Rao Eﬃciency
- 10.15 Variance Estimation
- 10.16 Kullback-Leibler Divergence
- 10.17 Approximating Models
- 10.18 Distribution of the MLE under Misspeciﬁcation
- 10.19 Variance Estimation under Misspeciﬁcation
- 10.20 Technical Proofs*
- 10.21 Exercises
11 Method of Moments
- 11.1 Introduction
- 11.2 Multivariate Means
- 11.3 Moments
- 11.4 Smooth Functions
- 11.5 Central Moments
- 11.6 Best Unbiased Estimation
- 11.7 Parametric Models
- 11.8 Examples of Parametric Models
- 11.9 Moment Equations
- 11.10 Asymptotic Distribution for Moment Equations
- 11.11 Example: Euler Equation
- 11.12 Empirical Distribution Function
- 11.13 Sample Quantiles
- 11.14 Robust Variance Estimation
- 11.15 Technical Proofs*
- 11.16 Exercises
12 Numerical Optimization
- 12.1 Introduction
- 12.2 Numerical Function Evaluation and Diﬀerentiation
- 12.3 Root Finding
- 12.4 Minimization in One Dimension
- 12.5 Failures of Minimization
- 12.6 Minimization in Multiple Dimensions
- 12.7 Constrained Optimization
- 12.8 Nested Minimization
- 12.9 Tips and Tricks
- 12.10 Exercises
13 Hypothesis Testing
- 13.1 Introduction
- 13.2 Hypotheses
- 13.3 Acceptance and Rejection
- 13.4 Type I and Type II Errors
- 13.5 One-Sided Tests
- 13.6 Two-Sided Tests
- 13.7 What Does “Accept ℍ0” Mean about ℍ0?
- 13.8 t Test with Normal Sampling
- 13.9 Asymptotic t Test
- 13.10 Likelihood Ratio Test for Simple Hypotheses
- 13.11 Neyman-Pearson Lemma
- 13.12 Likelihood Ratio Test against Composite Alternatives
- 13.13 Likelihood Ratio and t Tests
- 13.14 Statistical Signiﬁcance
- 13.15 p-Value
- 13.16 Composite Null Hypothesis
- 13.17 Asymptotic Uniformity
- 13.18 Summary
- 13.19 Exercises
14 Conﬁdence Intervals
- 14.1 Introduction
- 14.2 Deﬁnitions
- 14.3 Simple Conﬁdence Intervals
- 14.4 Conﬁdence Intervals for the Sample Mean under Normal Sampling
- 14.5 Conﬁdence Intervals for the Sample Mean under Non-Normal Sampling
- 14.6 Conﬁdence Intervals for Estimated Parameters
- 14.7 Conﬁdence Interval for the Variance
- 14.8 Conﬁdence Intervals by Test Inversion
- 14.9 Use of Conﬁdence Intervals
- 14.10 Uniform Conﬁdence Intervals
- 14.11 Exercises
15 Shrinkage Estimation
- 15.1 Introduction
- 15.2 Mean Squared Error
- 15.3 Shrinkage
- 15.4 James-Stein Shrinkage Estimator
- 15.5 Numerical Calculation
- 15.6 Interpretation of the Stein Eﬀect
- 15.7 Positive-Part Estimator
- 15.8 Summary
- 15.9 Technical Proofs*
- 15.10 Exercises
16 Bayesian Methods
- 16.1 Introduction
- 16.2 Bayesian Probability Model
- 16.3 Posterior Density
- 16.4 Bayesian Estimation
- 16.5 Parametric Priors
- 16.6 Normal-Gamma Distribution
- 16.7 Conjugate Prior
- 16.8 Bernoulli Sampling
- 16.9 Normal Sampling
- 16.10 Credible Sets
- 16.11 Bayesian Hypothesis Testing
- 16.12 Sampling Properties in the Normal Model
- 16.13 Asymptotic Distribution
- 16.14 Technical Proofs*
- 16.15 Exercises
17 Nonparametric Density Estimation
- 17.1 Introduction
- 17.2 Histogram Density Estimation
- 17.3 Kernel Density Estimator
- 17.4 Bias of Density Estimator
- 17.5 Variance of Density Estimator
- 17.6 Variance Estimation and Standard Errors
- 17.7 Integrated Mean Squared Error of Density Estimator
- 17.8 Optimal Kernel
- 17.9 Reference Bandwidth
- 17.10 Sheather-Jones Bandwidth*
- 17.11 Recommendations for Bandwidth Selection
- 17.12 Practical Issues in Density Estimation
- 17.13 Computation
- 17.14 Asymptotic Distribution
- 17.15 Undersmoothing
- 17.16 Technical Proofs*
- 17.17 Exercises
18 Empirical Process Theory
- 18.1 Introduction
- 18.2 Framework
- 18.3 Glivenko-Cantelli Theorem
- 18.4 Packing, Covering, and Bracketing Numbers
- 18.5 Uniform Law of Large Numbers
- 18.6 Functional Central Limit Theory
- 18.7 Conditions for Asymptotic Equicontinuity
- 18.8 Donsker’s Theorem
- 18.9 Technical Proofs*
- 18.10 Exercises
Appendix: Mathematics Reference
- A.1 Limits
- A.2 Series
- A.3 Factorials
- A.4 Exponentials
- A.5 Logarithms
- A.6 Diﬀerentiation
- A.7 Mean Value Theorem
- A.8 Integration
- A.9 Gaussian Integral
- A.10 Gamma Function
- A.11 Matrix Algebra
References
Index

Probability and Statistics for Economists

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Probability and Statistics for Economists

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