| |

| |

Preface | |

| |

| |

| |

Introduction and Review | |

| |

| |

| |

Economic Questions and Data | |

| |

| |

| |

Economic Questions We Examine | |

| |

| |

| |

Does Reducing Class Size Improve Elementary School Education? | |

| |

| |

| |

What Are the Economic Returns to Education? | |

| |

| |

Quantitative Questions, Quantitative Answers | |

| |

| |

| |

Causal Effects and Idealized Experiments | |

| |

| |

Estimation of Causal Effects | |

| |

| |

Forecasting and Causality | |

| |

| |

| |

Data: Sources and Types | |

| |

| |

Experimental versus Observational Data | |

| |

| |

Cross-Sectional Data | |

| |

| |

Time Series Data | |

| |

| |

Panel Data | |

| |

| |

| |

Review of Probability | |

| |

| |

| |

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 | |

| |

| |

| |

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 | |

| |

| |

| |

Two Random Variables | |

| |

| |

Joint and Marginal Distributions | |

| |

| |

Conditional Distributions | |

| |

| |

Independence | |

| |

| |

Covariance and Correlation | |

| |

| |

The Mean and Variance of Sums of Random Variables | |

| |

| |

| |

The Normal, Chi-Squared, Student t, and F Distributions | |

| |

| |

The Normal Distribution | |

| |

| |

The Chi-Squared Distribution | |

| |

| |

The Student t Distribution | |

| |

| |

The F Distribution | |

| |

| |

| |

Random Sampling and the Distribution of the Sample Average | |

| |

| |

Random Sampling | |

| |

| |

The Sampling Distribution of the Sample Average | |

| |

| |

| |

Large-Sample Approximations to Sampling Distributions | |

| |

| |

The Law of Large Numbers and Consistency | |

| |

| |

The Central Limit Theorem | |

| |

| |

| |

Derivation of Results in Key Concept 2.3 | |

| |

| |

| |

Review of Statistics | |

| |

| |

| |

Estimation of the Population Mean | |

| |

| |

Estimators and Their Properties | |

| |

| |

Properties of Y | |

| |

| |

The Importance of Random Sampling | |

| |

| |

| |

Hypothesis Tests Concerning the Population Mean | |

| |

| |

Null and Alternative Hypotheses | |

| |

| |

The p-Value | |

| |

| |

Calculating the p-Value When [sigma subscript Y] Is Known | |

| |

| |

The Sample Variance, Sample Standard Deviation, and Standard Error | |

| |

| |

Calculating the p-Value When [sigma subscript Y] Is Unknown | |

| |

| |

The t-Statistic | |

| |

| |

Hypothesis Testing with a Prespecified Significance Level | |

| |

| |

One-Sided Alternatives | |

| |

| |

| |

Confidence Intervals for the Population Mean | |

| |

| |

| |

Comparing Means from Different Populations | |

| |

| |

Hypothesis Tests for the Difference Between Two Means | |

| |

| |

Confidence Intervals for the Difference Between Two Population Means | |

| |

| |

| |

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 | |

| |

| |

| |

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 | |

| |

| |

| |

Scatterplot, the Sample Covariance, and the Sample Correlation | |

| |

| |

Scatterplots | |

| |

| |

Sample Covariance and Correlation | |

| |

| |

| |

The U.S. Current Population Survey | |

| |

| |

| |

Two Proofs That Y Is the Least Squares Estimator of [mu subscript Y] | |

| |

| |

| |

A Proof That the Sample Variance Is Consistent | |

| |

| |

| |

Fundamentals of Regression Analysis | |

| |

| |

| |

Linear Regression with One Regressor | |

| |

| |

| |

The Linear Regression Model | |

| |

| |

| |

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? | |

| |

| |

| |

Measures of Fit | |

| |

| |

The R[superscript 2] | |

| |

| |

The Standard Error of the Regression | |

| |

| |

Application to the Test Score Data | |

| |

| |

| |

The Least Squares Assumptions | |

| |

| |

| |

The Conditional Distribution of u[subscript i] Given X[subscript i] Has a Mean of Zero | |

| |

| |

| |

(X[subscript i], Y[subscript i]), i = 1, ..., n Are Independently and Identically Distributed | |

| |

| |

| |

Large Outliers Are Unlikely | |

| |

| |

Use of the Least Squares Assumptions | |

| |

| |

| |

The Sampling Distribution of the OLS Estimators | |

| |

| |

The Sampling Distribution of the OLS Estimators | |

| |

| |

| |

Conclusion | |

| |

| |

| |

The California Test Score Data Set | |

| |

| |

| |

Derivation of the OLS Estimators | |

| |

| |

| |

Sampling Distribution of the OLS Estimator | |

| |

| |

| |

Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals | |

| |

| |

| |

Testing Hypotheses About One of the Regression Coefficients | |

| |

| |

Two-Sided Hypotheses Concerning [Beta subscript 1] | |

| |

| |

One-Sided Hypotheses Concerning [Beta subscript 1] | |

| |

| |

Testing Hypotheses About the Intercept [Beta subscript 0] | |

| |

| |

| |

Confidence Intervals for a Regression Coefficient | |

| |

| |

| |

Regression When X Is a Binary Variable | |

| |

| |

Interpretation of the Regression Coefficients | |

| |

| |

| |

Heteroskedasticity and Homoskedasticity | |

| |

| |

What Are Heteroskedasticity and Homoskedasticity? | |

| |

| |

Mathematical Implications of Homoskedasticity | |

| |

| |

What Does This Mean in Practice? | |

| |

| |

| |

The Theoretical Foundations of Ordinary Least Squares | |

| |

| |

Linear Conditionally Unbiased Estimators and the Gauss-Markov Theorem | |

| |

| |

Regression Estimators Other Than OLS | |

| |

| |

| |

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 | |

| |

| |

| |

Conclusion | |

| |

| |

| |

Formulas for OLS Standard Errors | |

| |

| |

| |

The Gauss-Markov Conditions and a Proof of the Gauss-Markov Theorem | |

| |

| |

| |

Linear Regression with Multiple Regressors | |

| |

| |

| |

Omitted Variable Bias | |

| |

| |

Definition of Omitted Variable Bias | |

| |

| |

A Formula for Omitted Variable Bias | |

| |

| |

Addressing Omitted Variable Bias by Dividing the Data into Groups | |

| |

| |

| |

The Multiple Regression Model | |

| |

| |

The Population Regression Line | |

| |

| |

The Population Multiple Regression Model | |

| |

| |

| |

The OLS Estimator in Multiple Regression | |

| |

| |

The OLS Estimator | |

| |

| |

Application to Test Scores and the Student-Teacher Ratio | |

| |

| |

| |

Measures of Fit in Multiple Regression | |

| |

| |

The Standard Error of the Regression (SER) | |

| |

| |

The R[superscript 2] | |

| |

| |

The "Adjusted R[superscript 2]" | |

| |

| |

Application to Test Scores | |

| |

| |

| |

The Least Squares Assumptions in Multiple Regression | |

| |

| |

| |

The Conditional Distribution of u[subscript i] Given X[subscript 1i], [subscript 2i], ..., X[subscript ki] Has a Mean of Zero | |

| |

| |

| |

(X[subscript 1i], X[subscript 2i], ..., X[subscript ki], Y[subscript i]) i = 1, ..., n Are i.i.d. | |

| |

| |

| |

Large Outliers Are Unlikely | |

| |

| |

| |

No Perfect Multicollinearity | |

| |

| |

| |

The Distribution of the OLS Estimators in Multiple Regression | |

| |

| |

| |

Multicollinearity | |

| |

| |

Examples of Perfect Multicollinearity | |

| |

| |

Imperfect Multicollinearity | |

| |

| |

| |

Conclusion | |

| |

| |

| |

Derivation of Equation (6.1) | |

| |

| |

| |

Distribution of the OLS Estimators When There Are Two Regressors and Homoskedastic Errors | |

| |

| |

| |

The OLS Estimator With Two Regressors | |

| |

| |

| |

Hypothesis Tests and Confidence Intervals in Multiple Regression | |

| |

| |

| |

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 | |

| |

| |

| |

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 | |

| |

| |

| |

Testing Single Restrictions Involving Multiple Coefficients | |

| |

| |

| |

Confidence Sets for Multiple Coefficients | |

| |

| |

| |

Model Specification for Multiple Regression | |

| |

| |

Omitted Variable Bias in Multiple Regression | |

| |

| |

Model Specification in Theory and in Practice | |

| |

| |

Interpreting the R[superscript 2] and tine Adjusted R[superscript 2] in Practice | |

| |

| |

| |

Analysis of the Test Score Data Set | |

| |

| |

| |

Conclusion | |

| |

| |

| |

The Bonferroni Test of a Joint Hypotheses | |

| |

| |

| |

Nonlinear Regression Functions | |

| |

| |

| |

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 | |

| |

| |

| |

Nonlinear Functions of a Single Independent Variable | |

| |

| |

Polynomials | |

| |

| |

Logarithms | |

| |

| |

Polynomial and Logarithmic Models of Test Scores and District Income | |

| |

| |

| |

Interactions Between Independent Variables | |

| |

| |

Interactions Between Two Binary Variables | |

| |

| |

Interactions Between a Continuous and a Binary Variable | |

| |

| |

Interactions Between Two Continuous Variables | |

| |

| |

| |

Nonlinear Effects on Test Scores of the Student-Teacher Ratio | |

| |

| |

Discussion of Regression Results | |

| |

| |

Summary of Findings | |

| |

| |

| |

Conclusion | |

| |

| |

| |

Regression Functions That Are Nonlinear in the Parameters | |

| |

| |

| |

Assessing Studies Based on Multiple Regression | |

| |

| |

| |

Internal and External Validity | |

| |

| |

Threats to Internal Validity | |

| |

| |

Threats to External Validity | |

| |

| |

| |

Threats to Internal Validity of Multiple Regression Analysis | |

| |

| |

Omitted Variable Bias | |

| |

| |

Misspecification of the Functional Form of the Regression Function | |

| |

| |

Errors-in-Variables | |

| |

| |

Sample Selection | |

| |

| |

Simultaneous Causality | |

| |

| |

Sources of Inconsistency of OLS Standard Errors | |

| |

| |

| |

Internal and External Validity When the Regression Is Used for Forecasting | |

| |

| |

Using Regression Models for Forecasting | |

| |

| |

Assessing the Validity of Regression Models for Forecasting | |

| |

| |

| |

Example: Test Scores and Class Size | |

| |

| |

External Validity | |

| |

| |

Internal Validity | |

| |

| |

Discussion and Implications | |

| |

| |

| |

Conclusion | |

| |

| |

| |

The Massachusetts Elementary School Testing Data | |

| |

| |

| |

Conducting a Regression Study Using Economic Data | |

| |

| |

| |

Choosing a Topic | |

| |

| |

| |

Collecting the Data | |

| |

| |

Finding a Data Set | |

| |

| |

Time Series Data and Panel Data | |

| |

| |

Preparing the Data for Regression Analysis | |

| |

| |

| |

Conducting Your Regression Analysis | |

| |

| |

| |

Writing Up Your Results | |

| |

| |

Appendix | |

| |

| |

References | |

| |

| |

Answers to "Review the Concepts" Questions | |

| |

| |

Glossary | |

| |

| |

Index | |