Preface

PART ONE Introduction and Review

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: How Much Do Cigarette Taxes Reduce Smoking?

Question #4: What Will the Rate of Inflation Be Next Year?

Quantitative Questions, Quantitative Answers

1.2 Causal Effects and Idealized Experiments

Estimation of Causal Effects

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

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 Distributions

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 the Sampling Distributions

The Law of Large Numbers and Consistency

The Central Limit Theorem

APPENDIX 2.1 Derivation of Results in Key Concept 2.3

CHAPTER 3 Review of Statistics

3.1 Estimation of the Population Mean

Estimators and Their Properties

Properties of *Y*-bar

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 σ_{ϒ} Is Known

The Sample Variance, Sample Standard Deviation, and Standard Error

Calculating the *p*-Value When σ_{ϒ} 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*-bar Is the Least Squares Estimator of *μ*_{ϒ}

APPENDIX 3.3 A Proof That the Sample Variance is Consistent

PART TWO Fundamentals of Regression Analysis

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

The *R*^{2}

The Standard Error of the Regression

Application to the Test Score Data

4.4 The Least Squares Assumptions

Assumption #1: The Conditional Distribution of *u*_{i}
Given *X*_{i} Has a Mean of Zero

Assumption #2: (*X*_{i}, *X*_{i}) *i* = 1,…, *n*, Are Independently and Identically Distributed

Assumption #3: Large Outliers Are Unlikely

Use of the Least Squares Assumptions

4.5 Sampling Distribution of the OLS Estimators

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

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 *Β*_{1}

One-Sided Hypotheses Concerning *Β*_{1}

Testing Hypotheses About the Intercept *Β*_{0}

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 *R*^{2}

The “Adjusted *R*^{2}”

Application to Test Scores

6.5 The Least Squares Assumptions in Multiple Regression

Assumption #1: The Conditional Distribution of *u*_{i}
Given *X*_{1i ′},
*X*_{2i ′},…,
*X*_{ki} Has a Mean of Zero

Assumption #2: (*X*_{1i ′},
*X*_{2i ′},…,
*X*_{ki},*Y*_{i}), *i* = 1,…,*n*, Are i.i.d.

Assumption #3: Large Outliers Are Unlikely

Assumption #4: No Perfect Multicollinearity

6.6 The Distribution of OLS Estimators in Multiple Regression

6.7 Multicollinearity

Examples of Perfect Multicollinearity

Imperfect Multicollinearity

6.8 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

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

Omitted Variable Bias in Multiple Regression

The Role of Control Variables in Multiple Regression

Model Specification in Theory and Practice

Interpreting the *R*^{2} and the Adjusted
*R*^{2} in Practice

7.6 Analysis of the Test Score Data Set

7.7 Conclusion

APPENDIX 7.1 The Bonferroni Test of a Joint Hypothesis

APPENDIX 7.2 Conditional Mean Independence

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 Forecasting

Using Regression Models for Forecasting

Assessing the Validity of Regression Models for Forecasting

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

PART THREE Further Topics in Regression Analysis

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

PART FOUR Regression Analysis of Economic Time Series Data

CHAPTER 14 Introduction to Time Series Regression and Forecasting

14.1 Using Regression Models for Forecasting

14.2 Introduction to Time Series Data and Serial Correlation

The Rates of Inflation and Unemployment in the United States

Lags, First Differences, Logarithms, and Growth Rates

Autocorrelation

Other Examples of Economic Time Series

14.3 Autoregressions

The First Order Autoregressive Model

The *p*^{th} Order Autoregressive Model

14.4 Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model

Forecasting Changes in the Inflation Rate Using Past Unemployment Rates

Stationarity

Time Series Regression with Multiple Predictors

Forecast Uncertainty and Forecast Intervals

14.5 Lag Length Selection Using Information Criteria

Determining the Order of an Autoregression

Lag Length Selection in Time Series Regression with Multiple Predictors

14.6 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

14.7 Nonstationarity II: Breaks

What Is a Break?

Testing for Breaks

Pseudo Out-of-Sample Forecasting

Avoiding the Problems Caused by Breaks

14.8 Conclusion

APPENDIX 14.1 Time Series Data Used in Chapter 14

APPENDIX 14.2 Stationarity in the AR(1) Model

APPENDIX 14.3 Lag Operator Notation

APPENDIX 14.4 ARMA Models

APPENDIX 14.5 Consistency of the BIC Lag Length Estimator

CHAPTER 15 Estimation of Dynamic Causal Effects

15.1 An Initial Taste of the Orange Juice Data

15.2 Dynamic Causal Effects

Causal Effects and Time Series Data

Two Types of Exogeneity

15.3 Estimation of Dynamic Causal Effects with Exogenous Regressors

The Distributed Lag Model Assumptions

Autocorrelated *u*_{t}, Standard Errors, and Inference

Dynamic Multipliers and Cumulative Dynamic Multipliers

15.4 Heteroskedasticity- and Autocorrelation-Consistent Standard Errors

Distribution of the OLS Estimator with Autocorrelated Errors

HAC Standard Errors

15.5 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

The Distributed Lag Model with Additional Lags and AR(*p*) Errors

15.6 Orange Juice Prices and Cold Weather

15.7 Is Exogeneity Plausible? Some Examples

U.S. Income and Australian Exports

Oil Prices and Inflation

Monetary Policy and Inflation

The Phillips Curve

15.8 Conclusion

APPENDIX 15.1 The Orange Juice Data Set

APPENDIX 15.2 The ADL Model and Generalized Least Squares in Lag Operator Notation

CHAPTER 16 Additional Topics in Time Series Regression

16.1 Vector Autoregressions

The VAR Model

A VAR Model of the Rates of Inflation and Unemployment

16.2 Multiperiod Forecasts

Iterated Multiperiod Forecasts

Direct Multiperiod Forecasts

Which Method Should You Use?

16.3 Orders of Integration and the DF-GLS Unit Root Test

Other Models of Trends and Orders of Integration

The DF-GLS Test for a Unit Root

Why Do Unit Root Tests Have Nonnormal Distributions?

16.4 Cointegration

Cointegration and Error Correction

How Can You Tell Whether Two Variables are Cointegrated?

Estimation of Cointegrating Coefficients

Extension to Multiple Cointegrated Variables

Application to Interest Rates

16.5 Volatility Clustering and Autoregressive Conditional Heteroskedasticity

Volatility Clustering

Autoregressive Conditional Heteroskedasticity

Application to Stock Price Volatility

16.6 Conclusion

APPENDIX 16.1 U.S. Financial Data Used in Chapter 16

PART FIVE The Econometric Theory of Regression Analysis

CHAPTER 17 The Theory of Linear Regression with One Regressor

17.1 The Extended Least Squares Assumptions and the OLS Estimator

The Extended Least Squares Assumptions

The OLS Estimator

17.2 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

17.3 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

17.4 Exact Sampling Distributions When the Errors Are Normally Distributed

Distribution of *Β*-hat_{1} with Normal Errors

Distribution of the Homoskedasticity-Only *t*-Statistic

17.5 Weighted Least Squares

WLS with Known Heteroskedasticity

WLS with Heteroskedasticity of Known Functional Form

Heteroskedasticity-Robust Standard Errors or WLS?

APPENDIX 17.1 The Normal and Related Distributions and Moments of Continuous Random Variables

APPENDIX 17.2 Two Inequalities

CHAPTER 18 The Theory of Multiple Regression

18.1 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

18.2 Asymptotic Distribution of the OLS Estimator and

*t*-Statistic

The Multivariate Central Limit Theorem

Asymptotic Normality of *Β*-hat

Heteroskedasticity-Robust Standard Errors

Confidence Intervals for Predicted Effects

Asymptotic Distribution of the *t*-Statistic

18.3 Tests of Joint Hypotheses

Joint Hypotheses in Matrix Notation

Asymptotic Distribution of the *F*-Statistic

Confidence Sets for Multiple Coefficients

18.4 Distribution of Regression Statistics with Normal Errors

Matrix Representations of OLS Regression Statistics

Distribution of *Β*-hat for Normal Errors

Distribution of *s*^{2}_{û}

Homoskedasticity-Only Standard Errors

Distribution of the *t*-Statistic

Distribution of the *F*-Statistic

18.5 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

18.6 Generalized Least Squares

The GLS Assumptions

GLS When Ω Is Known

GLS When Ω Contains Unknown Parameters

The Zero Conditional Mean Assumption and GLS

18.7 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 18.1 Summary of Matrix Algebra

APPENDIX 18.2 Multivariate Distributions

APPENDIX 18.3 Derivation of the Asymptotic Distribution of *Β*-hat

APPENDIX 18.4 Derivations of Exact Distributions of OLS Test Statistics with Normal Errors

APPENDIX 18.5 Proof of the Gauss–Markov Theorem for Multiple Regression

APPENDIX 18.6 Proof of Selected Results for IV and GMM Estimation

Appendix

References

Glossary

Index