Introduction to econometrics

Cover
Title Page
Copyright Page
Brief Contents
Contents
Key Concepts
General Interest Boxes
Preface
Acknowledgments
Global Acknowledgments
Chapter 1: Economic Questions and Data
	1.1. Economic Questions We Examine
		Question #1: Does Reducing Class Size Improve Elementary School Education?
		Question #2: Is There Racial Discrimination in the Market for Home Loans?
		Question #3: Does Healthcare Spending Improve Health Outcomes?
		Question #4: By How Much Will U.S. GDP Grow Next Year?
		Quantitative Questions, Quantitative Answers
	1.2. Causal Effects and Idealized Experiments
		Estimation of Causal Effects
		Prediction, Forecasting, and Causality
	1.3. Data: Sources and Types
		Experimental versus Observational Data
		Cross-Sectional Data
		Time Series Data
		Panel Data
Chapter 2: Review of Probability
	2.1. Random Variables and Probability Distributions
		Probabilities, the Sample Space, and Random Variables
		Probability Distribution of a Discrete Random Variable
		Probability Distribution of a Continuous Random Variable
	2.2. Expected Values, Mean, and Variance
		The Expected Value of a Random Variable
		The Standard Deviation and Variance
		Mean and Variance of a Linear Function of a Random Variable
		Other Measures of the Shape of a Distribution
		Standardized Random Variables
	2.3. Two Random Variables
		Joint and Marginal Distributions
		Conditional Distributions
		Independence
		Covariance and Correlation
		The Mean and Variance of Sums of Random Variables
	2.4. The Normal, Chi-Squared, Student t, and F Distributions
		The Normal Distribution
		The Chi-Squared Distribution
		The Student t Distribution
		The F Distribution
	2.5. Random Sampling and the Distribution of the Sample Average
		Random Sampling
		The Sampling Distribution of the Sample Average
	2.6. Large-Sample Approximations to Sampling Distributions
		The Law of Large Numbers and Consistency
		The Central Limit Theorem
		Appendix 2.1: Derivation of Results in Key Concept 2.3
		Appendix 2.2: The Conditional Mean as the Minimum Mean Squared Error Predictor
Chapter 3: Review of Statistics
	3.1. Estimation of the Population Mean
		Estimators and Their Properties
		Properties of Y
		The Importance of Random Sampling
	3.2. Hypothesis Tests Concerning the Population Mean
		Null and Alternative Hypotheses
		The p-Value
		Calculating the p-Value When sY Is Known
		The Sample Variance, Sample Standard Deviation, and Standard Error
		Calculating the p-Value When sY Is Unknown
		The t-Statistic
		Hypothesis Testing with a Prespecified Significance Level
		One-Sided Alternatives
	3.3. Confidence Intervals for the Population Mean
	3.4. Comparing Means from Different Populations
		Hypothesis Tests for the Difference Between Two Means
		Confidence Intervals for the Difference Between Two Population Means
	3.5. Differences-of-Means Estimation of Causal Effects Using Experimental Data
		The Causal Effect as a Difference of Conditional Expectations
		Estimation of the Causal Effect Using Differences of Means
	3.6. Using the t-Statistic When the Sample Size Is Small
		The t-Statistic and the Student t Distribution
		Use of the Student t Distribution in Practice
	3.7. Scatterplots, the Sample Covariance, and the Sample Correlation
		Scatterplots
		Sample Covariance and Correlation
	Appendix 3.1: The U.S. Current Population Survey
	Appendix 3.2: Two Proofs That Y Is the Least Squares Estimator of µY
	Appendix 3.3: A Proof That the Sample Variance Is Consistent
Chapter 4: Linear Regression with One Regressor
	4.1. The Linear Regression Model
	4.2. Estimating the Coefficients of the Linear Regression Model
		The Ordinary Least Squares Estimator
		OLS Estimates of the Relationship Between Test Scores and the Student–Teacher Ratio
		Why Use the OLS Estimator?
	4.3. Measures of Fit and Prediction Accuracy
		The R2
		The Standard Error of the Regression
		Prediction Using OLS
		Application to the Test Score Data
	4.4. The Least Squares Assumptions for Causal Inference
		Assumption 1: The Conditional Distribution of ui Given Xi Has a Mean of Zero
		Assumption 2: (Xi, Yi), i = 1,
		Assumption 3: Large Outliers Are Unlikely
		Use of the Least Squares Assumptions
	4.5. The Sampling Distribution of the OLS Estimators
	4.6. Conclusion
	Appendix 4.1: The California Test Score Data Set
	Appendix 4.2: Derivation of the OLS Estimators
	Appendix 4.3: Sampling Distribution of the OLS Estimator
	Appendix 4.4: The Least Squares Assumptions for Prediction
Chapter 5: Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals
	5.1. Testing Hypotheses About One of the Regression Coefficients
		Two-Sided Hypotheses Concerning ß1
		One-Sided Hypotheses Concerning ß1
		Testing Hypotheses About the Intercept ß0
	5.2. Confidence Intervals for a Regression Coefficient
	5.3. Regression When X Is a Binary Variable
		Interpretation of the Regression Coefficients
	5.4. Heteroskedasticity and Homoskedasticity
		What Are Heteroskedasticity and Homoskedasticity?
		Mathematical Implications of Homoskedasticity
		What Does This Mean in Practice?
	5.5. The Theoretical Foundations of Ordinary Least Squares
		Linear Conditionally Unbiased Estimators and the Gauss–Markov Theorem
		Regression Estimators Other Than OLS
	5.6. Using the t-Statistic in Regression When the Sample Size Is Small
		The t-Statistic and the Student t Distribution
		Use of the Student t Distribution in Practice
	5.7. Conclusion
	Appendix 5.1: Formulas for OLS Standard Errors
	Appendix 5.2: The Gauss–Markov Conditions and a Proof of the Gauss–Markov Theorem
Chapter 6: Linear Regression with Multiple Regressors
	6.1. Omitted Variable Bias
		Definition of Omitted Variable Bias
		A Formula for Omitted Variable Bias
		Addressing Omitted Variable Bias by Dividing the Data into Groups
	6.2. The Multiple Regression Model
		The Population Regression Line
		The Population Multiple Regression Model
	6.3. The OLS Estimator in Multiple Regression
		The OLS Estimator
		Application to Test Scores and the Student–Teacher Ratio
	6.4. Measures of Fit in Multiple Regression
		The Standard Error of the Regression (SER)
		The R2
		The Adjusted R2
		Application to Test Scores
	6.5. The Least Squares Assumptions for Causal Inference in Multiple Regression
		Assumption 1: The Conditional Distribution of ui Given X1i, X2i,
		Assumption 2: (X1i, X2i,
		Assumption 3: Large Outliers Are Unlikely
		Assumption 4: No Perfect Multicollinearity
	6.6. The Distribution of the OLS Estimators in Multiple Regression
	6.7. Multicollinearity
		Examples of Perfect Multicollinearity
		Imperfect Multicollinearity
	6.8. Control Variables and Conditional Mean Independence
		Control Variables and Conditional Mean Independence
	6.9. Conclusion
	Appendix 6.1: Derivation of Equation (6.1)
	Appendix 6.2: Distribution of the OLS Estimators When There Are Two Regressors and Homoskedastic Errors
	Appendix 6.3: The Frisch–Waugh Theorem
	Appendix 6.4: The Least Squares Assumptions for Prediction with Multiple Regressors
	Appendix 6.5: Distribution of OLS Estimators in Multiple Regression with Control Variables
Chapter 7: Hypothesis Tests and Confidence Intervals in Multiple Regression
	7.1. Hypothesis Tests and Confidence Intervals for a Single Coefficient
		Standard Errors for the OLS Estimators
		Hypothesis Tests for a Single Coefficient
		Confidence Intervals for a Single Coefficient
		Application to Test Scores and the Student–Teacher Ratio
	7.2. Tests of Joint Hypotheses
		Testing Hypotheses on Two or More Coefficients
		The F-Statistic
		Application to Test Scores and the Student–Teacher Ratio
		The Homoskedasticity-Only F-Statistic
	7.3. Testing Single Restrictions Involving Multiple Coefficients
	7.4. Confidence Sets for Multiple Coefficients
	7.5. Model Specification for Multiple Regression
		Model Specification and Choosing Control Variables
		Interpreting the R2 and the Adjusted R2 in Practice
	7.6. Analysis of the Test Score Data Set
	7.7. Conclusion
	Appendix 7.1: The Bonferroni Test of a Joint Hypothesis
Chapter 8: Nonlinear Regression Functions
	8.1. A General Strategy for Modeling Nonlinear Regression Functions
		Test Scores and District Income
		The Effect on Y of a Change in X in Nonlinear Specifications
		A General Approach to Modeling Nonlinearities Using Multiple Regression
	8.2. Nonlinear Functions of a Single Independent Variable
		Polynomials
		Logarithms
		Polynomial and Logarithmic Models of Test Scores and District Income
	8.3. Interactions Between Independent Variables
		Interactions Between Two Binary Variables
		Interactions Between a Continuous and a Binary Variable
		Interactions Between Two Continuous Variables
	8.4. Nonlinear Effects on Test Scores of the Student–Teacher Ratio
		Discussion of Regression Results
		Summary of Findings
	8.5. Conclusion
	Appendix 8.1: Regression Functions That Are Nonlinear in the Parameters
	Appendix 8.2: Slopes and Elasticities for Nonlinear Regression Functions
Chapter 9: Assessing Studies Based on Multiple Regression
	9.1. Internal and External Validity
		Threats to Internal Validity
		Threats to External Validity
	9.2. Threats to Internal Validity of Multiple Regression Analysis
		Omitted Variable Bias
		Misspecification of the Functional Form of the Regression Function
		Measurement Error and Errors-in-Variables Bias
		Missing Data and Sample Selection
		Simultaneous Causality
		Sources of Inconsistency of OLS Standard Errors
	9.3. Internal and External Validity When the Regression Is Used for Prediction
	9.4. Example: Test Scores and Class Size
		External Validity
		Internal Validity
		Discussion and Implications
	9.5. Conclusion
	Appendix 9.1: The Massachusetts Elementary School Testing Data
Chapter 10: Regression with Panel Data
	10.1. Panel Data
		Example: Traffic Deaths and Alcohol Taxes
	10.2. Panel Data with Two Time Periods: “Before and After” Comparisons
	10.3. Fixed Effects Regression
		The Fixed Effects Regression Model
		Estimation and Inference
		Application to Traffic Deaths
	10.4. Regression with Time Fixed Effects
		Time Effects Only
		Both Entity and Time Fixed Effects
	10.5. The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression
		The Fixed Effects Regression Assumptions
		Standard Errors for Fixed Effects Regression
	10.6. Drunk Driving Laws and Traffic Deaths
	10.7. Conclusion
	Appendix 10.1: The State Traffic Fatality Data Set
	Appendix 10.2: Standard Errors for Fixed Effects Regression
Chapter 11: Regression with a Binary Dependent Variable
	11.1. Binary Dependent Variables and the Linear Probability Model
		Binary Dependent Variables
		The Linear Probability Model
	11.2. Probit and Logit Regression
		Probit Regression
		Logit Regression
		Comparing the Linear Probability, Probit, and Logit Models
	11.3. Estimation and Inference in the Logit and Probit Models
		Nonlinear Least Squares Estimation
		Maximum Likelihood Estimation
		Measures of Fit
	11.4. Application to the Boston HMDA Data
	11.5. Conclusion
	Appendix 11.1: The Boston HMDA Data Set
	Appendix 11.2: Maximum Likelihood Estimation
	Appendix 11.3: Other Limited Dependent Variable Models
Chapter 12: Instrumental Variables Regression
	12.1. The IV Estimator with a Single Regressor and a Single Instrument
		The IV Model and Assumptions
		The Two Stage Least Squares Estimator
		Why Does IV Regression Work?
		The Sampling Distribution of the TSLS Estimator
		Application to the Demand for Cigarettes
	12.2. The General IV Regression Model
		TSLS in the General IV Model
		Instrument Relevance and Exogeneity in the General IV Model
		The IV Regression Assumptions and Sampling Distribution of the TSLS Estimator
		Inference Using the TSLS Estimator
		Application to the Demand for Cigarettes
	12.3. Checking Instrument Validity
		Assumption 1: Instrument Relevance
		Assumption 2: Instrument Exogeneity
	12.4. Application to the Demand for Cigarettes
	12.5. Where Do Valid Instruments Come From?
		Three Examples
	12.6. Conclusion
	Appendix 12.1: The Cigarette Consumption Panel Data Set
	Appendix 12.2: Derivation of the Formula for the TSLS Estimator in Equation (12.4)
	Appendix 12.3: Large-Sample Distribution of the TSLS Estimator
	Appendix 12.4: Large-Sample Distribution of the TSLS Estimator When the Instrument Is Not Valid
	Appendix 12.5: Instrumental Variables Analysis with Weak Instruments
	Appendix 12.6: TSLS with Control Variables
Chapter 13: Experiments and Quasi-Experiments
	13.1. Potential Outcomes, Causal Effects, and Idealized Experiments
		Potential Outcomes and the Average Causal Effect
		Econometric Methods for Analyzing Experimental Data
	13.2. Threats to Validity of Experiments
		Threats to Internal Validity
		Threats to External Validity
	13.3. Experimental Estimates of the Effect of Class Size Reductions
		Experimental Design
		Analysis of the STAR Data
		Comparison of the Observational and Experimental Estimates of Class Size Effects
	13.4. Quasi-Experiments
		Examples
		The Differences-in-Differences Estimator
		Instrumental Variables Estimators
		Regression Discontinuity Estimators
	13.5. Potential Problems with Quasi-Experiments
		Threats to Internal Validity
		Threats to External Validity
	13.6. Experimental and Quasi-Experimental Estimates in Heterogeneous Populations
		OLS with Heterogeneous Causal Effects
		IV Regression with Heterogeneous Causal Effects
	13.7. Conclusion
	Appendix 13.1: The Project STAR Data Set
	Appendix 13.2: IV Estimation When the Causal Effect Varies Across Individuals
	Appendix 13.3: The Potential Outcomes Framework for Analyzing Data from Experiments
Chapter 14: Prediction with Many Regressors and Big Data
	14.1. What Is “Big Data”?
	14.2. The Many-Predictor Problem and OLS
		The Mean Squared Prediction Error
		The First Least Squares Assumption for Prediction
		The Predictive Regression Model with Standardized Regressors
		The MSPE of OLS and the Principle of Shrinkage
		Estimation of the MSPE
	14.3. Ridge Regression
		Shrinkage via Penalization and Ridge Regression
		Estimation of the Ridge Shrinkage Parameter by Cross Validation
		Application to School Test Scores
	14.4. The Lasso
		Shrinkage Using the Lasso
		Application to School Test Scores
	14.5. Principal Components
		Principals Components with Two Variables
		Principal Components with k Variables
		Application to School Test Scores
	14.6. Predicting School Test Scores with Many Predictors
	14.7. Conclusion
	Appendix 14.1: The California School Test Score Data Set
	Appendix 14.2: Derivation of Equation (14.4) for k = 1
	Appendix 14.3: The Ridge Regression Estimator When k = 1
	Appendix 14.4: The Lasso Estimator When k = 1
	Appendix 14.5: Computing Out-of-Sample Predictions in the Standardized Regression Model
Chapter 15: Introduction to Time Series Regression and Forecasting
	15.1. Introduction to Time Series Data and Serial Correlation
		Real GDP in the United States
		Lags, First Differences, Logarithms, and Growth Rates
		Autocorrelation
		Other Examples of Economic Time Series
	15.2. Stationarity and the Mean Squared Forecast Error
		Stationarity
		Forecasts and Forecast Errors
		The Mean Squared Forecast Error
	15.3. Autoregressions
		The First-Order Autoregressive Model
		The pth-Order Autoregressive Model
	15.4. Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model
		Forecasting GDP Growth Using the Term Spread
		The Autoregressive Distributed Lag Model
		The Least Squares Assumptions for Forecasting with Multiple Predictors
	15.5. Estimation of the MSFE and Forecast Intervals
		Estimation of the MSFE
		Forecast Uncertainty and Forecast Intervals
	15.6. Estimating the Lag Length Using Information Criteria
		Determining the Order of an Autoregression
		Lag Length Selection in Time Series Regression with Multiple Predictors
	15.7. Nonstationarity I: Trends
		What Is a Trend?
		Problems Caused by Stochastic Trends
		Detecting Stochastic Trends: Testing for a Unit AR Root
		Avoiding the Problems Caused by Stochastic Trends
	15.8. Nonstationarity II: Breaks
		What Is a Break?
		Testing for Breaks
		Detecting Breaks Using Pseudo Out-of-Sample Forecasts
		Avoiding the Problems Caused by Breaks
	15.9. Conclusion
	Appendix 15.1: Time Series Data Used in Chapter 15
	Appendix 15.2: Stationarity in the AR(1) Model
	Appendix 15.3: Lag Operator Notation
	Appendix 15.4: ARMA Models
	Appendix 15.5: Consistency of the BIC Lag Length Estimator
Chapter 16: Estimation of Dynamic Causal Effects
	16.1. An Initial Taste of the Orange Juice Data
	16.2. Dynamic Causal Effects
		Causal Effects and Time Series Data
		Two Types of Exogeneity
	16.3. Estimation of Dynamic Causal Effects with Exogenous Regressors
		The Distributed Lag Model Assumptions
		Autocorrelated ut, Standard Errors, and Inference
		Dynamic Multipliers and Cumulative Dynamic Multipliers
	16.4. Heteroskedasticity- and Autocorrelation-Consistent Standard Errors
		Distribution of the OLS Estimator with Autocorrelated Errors
		HAC Standard Errors
	16.5. Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors
		The Distributed Lag Model with AR(1) Errors
		OLS Estimation of the ADL Model
		GLS Estimation
	16.6. Orange Juice Prices and Cold Weather
	16.7. Is Exogeneity Plausible? Some Examples
		U.S. Income and Australian Exports
		Oil Prices and Inflation
		Monetary Policy and Inflation
		The Growth Rate of GDP and the Term Spread
	16.8. Conclusion
	Appendix 16.1: The Orange Juice Data Set
	Appendix 16.2: The ADL Model and Generalized Least Squares in Lag Operator Notation
Chapter 17: Additional Topics in Time Series Regression
	17.1. Vector Autoregressions
		The VAR Model
		A VAR Model of the Growth Rate of GDP and the Term Spread
	17.2. Multi-period Forecasts
		Iterated Multi-period Forecasts
		Direct Multi-period Forecasts
		Which Method Should You Use?
	17.3. Orders of Integration and the Nonnormality of Unit Root Test Statistics
		Other Models of Trends and Orders of Integration
		Why Do Unit Root Tests Have Nonnormal Distributions?
	17.4. Cointegration
		Cointegration and Error Correction
		How Can You Tell Whether Two Variables Are Cointegrated?
		Estimation of Cointegrating Coefficients
		Extension to Multiple Cointegrated Variables
	17.5. Volatility Clustering and Autoregressive Conditional Heteroskedasticity
		Volatility Clustering
		Realized Volatility
		Autoregressive Conditional Heteroskedasticity
		Application to Stock Price Volatility
	17.6. Forecasting with Many Predictors Using Dynamic Factor Models and Principal Components
		The Dynamic Factor Model
		The DFM: Estimation and Forecasting
		Application to U.S. Macroeconomic Data
	17.7. Conclusion
	Appendix 17.1: The Quarterly U.S. Macro Data Set
Chapter 18: The Theory of Linear Regression with One Regressor
	18.1. The Extended Least Squares Assumptions and the OLS Estimator
		The Extended Least Squares Assumptions
		The OLS Estimator
	18.2. Fundamentals of Asymptotic Distribution Theory
		Convergence in Probability and the Law of Large Numbers
		The Central Limit Theorem and Convergence in Distribution
		Slutsky’s Theorem and the Continuous Mapping Theorem
		Application to the t-Statistic Based on the Sample Mean
	18.3. Asymptotic Distribution of the OLS Estimator and t-Statistic
		Consistency and Asymptotic Normality of the OLS Estimators
		Consistency of Heteroskedasticity-Robust Standard Errors
		Asymptotic Normality of the Heteroskedasticity-Robust t-Statistic
	18.4. Exact Sampling Distributions When the Errors Are Normally Distributed
		Distribution of b n 1 with Normal Errors
		Distribution of the Homoskedasticity-Only t-Statistic
	18.5. Weighted Least Squares
		WLS with Known Heteroskedasticity
		WLS with Heteroskedasticity of Known Functional Form
		Heteroskedasticity-Robust Standard Errors or WLS?
	Appendix 18.1: The Normal and Related Distributions and Moments of Continuous Random Variables
	Appendix 18.2: Two Inequalities
Chapter 19: The Theory of Multiple Regression
	19.1. The Linear Multiple Regression Model and OLS Estimator in Matrix Form
		The Multiple Regression Model in Matrix Notation
		The Extended Least Squares Assumptions
		The OLS Estimator
	19.2. Asymptotic Distribution of the OLS Estimator and t-Statistic
		The Multivariate Central Limit Theorem
		Asymptotic Normality of b n
		Heteroskedasticity-Robust Standard Errors
		Confidence Intervals for Predicted Effects
		Asymptotic Distribution of the t-Statistic
	19.3. Tests of Joint Hypotheses
		Joint Hypotheses in Matrix Notation
		Asymptotic Distribution of the F-Statistic
		Confidence Sets for Multiple Coefficients
	19.4. Distribution of Regression Statistics with Normal Errors
		Matrix Representations of OLS Regression Statistics
		Distribution of b n with Independent Normal Errors
		Distribution of su 2 N
		Homoskedasticity-Only Standard Errors
		Distribution of the t-Statistic
		Distribution of the F-Statistic
	19.5. Efficiency of the OLS Estimator with Homoskedastic Errors
		The Gauss–Markov Conditions for Multiple Regression
		Linear Conditionally Unbiased Estimators
		The Gauss–Markov Theorem for Multiple Regression
	19.6. Generalized Least Squares
		The GLS Assumptions
		GLS When O Is Known
		GLS When O Contains Unknown Parameters
		The Conditional Mean Zero Assumption and GLS
	19.7. Instrumental Variables and Generalized Method of Moments Estimation
		The IV Estimator in Matrix Form
		Asymptotic Distribution of the TSLS Estimator
		Properties of TSLS When the Errors Are Homoskedastic
		Generalized Method of Moments Estimation in Linear Models
	Appendix 19.1: Summary of Matrix Algebra
	Appendix 19.2: Multivariate Distributions
	Appendix 19.3: Derivation of the Asymptotic Distribution of b n
	Appendix 19.4: Derivations of Exact Distributions of OLS Test Statistics with Normal Errors
	Appendix 19.5: Proof of the Gauss–Markov Theorem for Multiple Regression
	Appendix 19.6: Proof of Selected Results for IV and GMM Estimation
	Appendix 19.7: Regression with Many Predictors: MSPE, Ridge Regression, and Principal Components Analysis
Appendix
References
Glossary
Index