(Econometrics I) Probability Theory and Statistics for Economists Yongmiao Hong Economics 6190 is the first course of the first year doctoral econometrics sequence at Cornell University.It provides an introduction to probability theory and statistics. Why do we need to teach probability and statistics to doctoral students in economics?Put it simply,it provides necessary probability and statistics background for first-year graduate stu- dents for their courses in econometrics,microeconomics,and macroeconomics.Statistics and mathematics are basic analytic tools in economics.Statistics is an essential tool to study situa- tions involving uncertainties,in the same way as calculus is essential to characterize optimizing behaviors in economics.For example,probability theory is needed in study of game theory.In macroeconomics,as Robert Lucas points out,the introduction of stochastic factors can provide much new insights into dynamic economic systems.In certain sense,a course on probability and statistics might not be called Econometrics I,because they are necessary analytic tools in every field of economics.Of course,the demand for probability and statistics varies from field to field in economics,with econometrics most heavily using it. Specifically,those who are attracted to theoretical econometrics would be expected to study probability and statistics in more depth by taking further graduate courses in mathematics and statistics.Those who are not attracted to should find this course in adequate preparation in the theory of probability and statistics for both their applied courses in econometrics and their courses in microeconomics and macroeconomics. The purpose of this course is to develop a deep understanding of probability and statistics and a solid intuition for statistical concepts.One year of calculus is a prerequisite for understanding the materials in this book,and an additional year of advanced calculus and some basic background in probability and statistics will be very helpful.The analysis is conducted in a relatively rigorous manner.Formal proofs will be given for some important theorems,because the proofs themselves can aid understanding and in some cases,the proof techniques or methods have practical value. Many students taking this course are experiencing the ideas of statistics for the first time.It will be helpful for them to spend some time learning how the mathematical ideas of statistics carry over into the world of applications in economics and finance.Thus in addition to developing a
(Econometrics I) Probability Theory and Statistics for Economists Yongmiao Hong Economics 6190 is the first course of the first year doctoral econometrics sequence at Cornell University. It provides an introduction to probability theory and statistics. Why do we need to teach probability and statistics to doctoral students in economics? Put it simply, it provides necessary probability and statistics background for first-year graduate students for their courses in econometrics, microeconomics, and macroeconomics. Statistics and mathematics are basic analytic tools in economics. Statistics is an essential tool to study situations involving uncertainties, in the same way as calculus is essential to characterize optimizing behaviors in economics. For example, probability theory is needed in study of game theory. In macroeconomics, as Robert Lucas points out, the introduction of stochastic factors can provide much new insights into dynamic economic systems. In certain sense, a course on probability and statistics might not be called Econometrics I, because they are necessary analytic tools in every field of economics. Of course, the demand for probability and statistics varies from field to field in economics, with econometrics most heavily using it. Specifically, those who are attracted to theoretical econometrics would be expected to study probability and statistics in more depth by taking further graduate courses in mathematics and statistics. Those who are not attracted to should find this course in adequate preparation in the theory of probability and statistics for both their applied courses in econometrics and their courses in microeconomics and macroeconomics. The purpose of this course is to develop a deep understanding of probability and statistics and a solid intuition for statistical concepts. One year of calculus is a prerequisite for understanding the materials in this book, and an additional year of advanced calculus and some basic background in probability and statistics will be very helpful. The analysis is conducted in a relatively rigorous manner. Formal proofs will be given for some important theorems, because the proofs themselves can aid understanding and in some cases, the proof techniques or methods have practical value. Many students taking this course are experiencing the ideas of statistics for the first time. It will be helpful for them to spend some time learning how the mathematical ideas of statistics carry over into the world of applications in economics and finance. Thus in addition to developing a 1
fundamental understanding of probability and mathematical statistics that are most relevant to modern econometrics,this course also tries to develop a sound intuition for statistical concepts from economic perspective.For example,why are statistical concepts (e.g.,conditional mean, conditional variance)useful in economics?What are economic intuition and interpretation for the probability and statistical relations?Emphasis on economic motivation and intuition is a most important feature that distinguishes this course from other courses on probability and statistics. Prerequisites:Advanced Calculus including Multivariate Calculus,Probability and Statistics equivalent to Math 471 and Math 472,as well as intermediate macroeconomics and microeco- nomics. Textbook:Probability Theory and Statistics for Economists by Yongmiao Hong,2017,World Scientific Publishing Company. Requirements:The course grade will be based on homework (10%),Midterm (35%),and Final (55%). Teaching Schedule:Tuesday and Thursday,4:55pm-6:10pm,online. Instructor Information:Yongmiao Hong,424 Uris Hall,Email addresses:yh20@cornell.edu yhong.cornell@gmail.com. Office Hours in Fall 2019:3:00-4:30pm,Thursday,or by appointment,online.Zoom meeting ID:95755574484,Password:1q476k TA Information:Siguang Li,Email:s12932@cornell.edu. 2
fundamental understanding of probability and mathematical statistics that are most relevant to modern econometrics, this course also tries to develop a sound intuition for statistical concepts from economic perspective. For example, why are statistical concepts (e.g., conditional mean, conditional variance) useful in economics? What are economic intuition and interpretation for the probability and statistical relations? Emphasis on economic motivation and intuition is a most important feature that distinguishes this course from other courses on probability and statistics. Prerequisites: Advanced Calculus including Multivariate Calculus, Probability and Statistics equivalent to Math 471 and Math 472, as well as intermediate macroeconomics and microeconomics. Textbook: Probability Theory and Statistics for Economists by Yongmiao Hong, 2017, World Scientific Publishing Company. Requirements: The course grade will be based on homework (10%), Midterm (35%), and Final (55%). Teaching Schedule: Tuesday and Thursday, 4:55pm-6:10pm, online. Instructor Information: Yongmiao Hong, 424 Uris Hall, Email addresses: yh20@cornell.edu & yhong.cornell@gmail.com. Office Hours in Fall 2019: 3:00-4:30pm, Thursday, or by appointment, online. Zoom meeting ID: 957 5557 4484, Password: 1q476k TA Information: Siguang Li, Email: sl2932@cornell.edu. 2
Outline of Contents Chapter 1 Introduction to Statistics and Econometrics Chapter 2 Foundation of Probability Theory 2.1 Random Experiments 2.2 Basic Concepts of Probability 2.3 Review of Set Theory 2.4 Fundamental Probability Laws 2.5 Methods of Counting 2.6 Conditional Probability and Independence Chapter 3 Random Variables and Univariate Probability Distributions 3.1 Random Variables and Distribution Functions 3.2 Discrete Random Variable 3.3 Continuous Random Variables 3.4 Functions of a Random Variable 3.5 Mathematical Expectations 3.6 Moment Generating Function 3.7 Characteristic Function Chapter 4 Important Parametric Distributions 4.1 Introduction 4.2 Discrete Distributions 4.3 Continuous Probability Distributions Chapter 5 Random Vectors and Multivariate Probability Distribution 5.1 Random vectors and Joint Probability Distributions 5.2 Marginal Distributions 5.3 Conditional Distributions 5.4 Independence 5.5 Empirical Applications 5.6 Bivariate Transformation 5.7 Expectations Under Multivariate Distributions 5.8 Implications of Independence 5.9 Conditional Expectations 3
Outline of Contents Chapter 1 Introduction to Statistics and Econometrics Chapter 2 Foundation of Probability Theory 2.1 Random Experiments 2.2 Basic Concepts of Probability 2.3 Review of Set Theory 2.4 Fundamental Probability Laws 2.5 Methods of Counting 2.6 Conditional Probability and Independence Chapter 3 Random Variables and Univariate Probability Distributions 3.1 Random Variables and Distribution Functions 3.2 Discrete Random Variable 3.3 Continuous Random Variables 3.4 Functions of a Random Variable 3.5 Mathematical Expectations 3.6 Moment Generating Function 3.7 Characteristic Function Chapter 4 Important Parametric Distributions 4.1 Introduction 4.2 Discrete Distributions 4.3 Continuous Probability Distributions Chapter 5 Random Vectors and Multivariate Probability Distribution 5.1 Random vectors and Joint Probability Distributions 5.2 Marginal Distributions 5.3 Conditional Distributions 5.4 Independence 5.5 Empirical Applications 5.6 Bivariate Transformation 5.7 Expectations Under Multivariate Distributions 5.8 Implications of Independence 5.9 Conditional Expectations 3
Chapter 6 Introduction to Sampling Theory and Statistics 6.1 Population and Random Sample 6.2 The Sampling Distribution of the Sample Mean 6.3 The Sampling Distribution of the Sample Variance 6.4 Student's t Distribution 6.5 Snedecor's F Distribution 6.6 Sufficient Statistics Chapter 7 Convergence Concepts and Limit Theories 7.1 Limits and Orders of Magnitude:A Review 7.2 Motivation for Convergence Concepts 7.3 Convergence in Quadratic Mean and Lp-convergence 7.4 Convergence in Probability 7.5 Almost Sure Convergence 7.6 Convergence in Distribution Chapter 8 Parameter Estimation and Evaluation 8.1 Population and Distribution Model 8.2 Maximum Likelihood Estimation 8.3 Method of Moments and Generalized Method of Moments 8.4 Mean Squared Error Criterion 8.5 Best Unbiased Estimators Chapter 9 Hypothesis Testing 9.1 Introduction to Hypothesis Testing 9.2 The Wald Test 9.3 The Lagrangian Multiplier Test 9.4 The Likelihood Ratio Test 9.5 A Simple Example Chapter 10 Big data,Machine Learning and Statistics 10.1 Features of Big Data 10.2 Machine Learning and Its Nature 10.3 Big Data and Sampling Theory 10.4 Statistical Significance and Economic Importance 10.5 Big Data,Machine Learning and Model Uncertainty 4
Chapter 6 Introduction to Sampling Theory and Statistics 6.1 Population and Random Sample 6.2 The Sampling Distribution of the Sample Mean 6.3 The Sampling Distribution of the Sample Variance 6.4 Student’s t Distribution 6.5 Snedecor’s F Distribution 6.6 Sufficient Statistics Chapter 7 Convergence Concepts and Limit Theories 7.1 Limits and Orders of Magnitude: A Review 7.2 Motivation for Convergence Concepts 7.3 Convergence in Quadratic Mean and Lp-convergence 7.4 Convergence in Probability 7.5 Almost Sure Convergence 7.6 Convergence in Distribution Chapter 8 Parameter Estimation and Evaluation 8.1 Population and Distribution Model 8.2 Maximum Likelihood Estimation 8.3 Method of Moments and Generalized Method of Moments 8.4 Mean Squared Error Criterion 8.5 Best Unbiased Estimators Chapter 9 Hypothesis Testing 9.1 Introduction to Hypothesis Testing 9.2 The Wald Test 9.3 The Lagrangian Multiplier Test 9.4 The Likelihood Ratio Test 9.5 A Simple Example Chapter 10 Big data, Machine Learning and Statistics 10.1 Features of Big Data 10.2 Machine Learning and Its Nature 10.3 Big Data and Sampling Theory 10.4 Statistical Significance and Economic Importance 10.5 Big Data, Machine Learning and Model Uncertainty 4
10.6 Big Data,Machine Learning and Sufficiency 10.7 Machine Learning and Nonparametric Analysis 10.8 Machine Learning and Causality Chapter 11 Conclusion 11.1 Summary 11.2 What next? 5
10.6 Big Data, Machine Learning and Sufficiency 10.7 Machine Learning and Nonparametric Analysis 10.8 Machine Learning and Causality Chapter 11 Conclusion 11.1 Summary 11.2 What next? 5
Student Personal Information Please provide your following personal information and submit it to TA:(i)name and email address,(ii)department and school,and (iii)status and background (e.g.,MA or PHD student, relevant courses taken).A course email list will be created to make announcements of course information (e.g.,homework assignments).Thank you! 6
Student Personal Information Please provide your following personal information and submit it to TA: (i) name and email address, (ii) department and school, and (iii) status and background (e.g., MA or PHD student, relevant courses taken). A course email list will be created to make announcements of course information (e.g., homework assignments). Thank you! 6