Page 12 -

Contents 11

3.5 Differences-of-Means Estimation of Causal Effects Using
Experimental Data 130

The Causal Effect as a Difference of Conditional Expectations 131
Estimation of the Causal Effect Using Differences of Means 131

3.6 Using the t-Statistic When the Sample Size Is Small 133

The t-Statistic and the Student t Distribution 133
Use of the Student t Distribution in Practice 135

3.7 Scatterplots, the Sample Covariance, and the Sample
Correlation 137

Scatterplots 137
Sample Covariance and Correlation 138
Appendix 3.1 The U.S. Current Population Survey 152
Appendix 3.2 Two Proofs That Y Is the Least Squares Estimator of μY 153
Appendix 3.3 A Proof That the Sample Variance Is Consistent 154

Part Two Fundamentals of Regression Analysis
Chapter 4 Linear Regression with One Regressor 155

4.1 The Linear Regression Model 155

4.2 Estimating the Coefficients of the Linear Regression
Model 160

The Ordinary Least Squares Estimator 162
OLS Estimates of the Relationship Between Test Scores and the Student–

Teacher Ratio 164
Why Use the OLS Estimator? 165

4.3 Measures of Fit 167

The R2 167
The Standard Error of the Regression 168
Application to the Test Score Data 169

4.4 The Least Squares Assumptions 170

Assumption #1: The Conditional Distribution of ui Given Xi Has a Mean of Zero 170
Assumption #2: (Xi, Yi), i = 1,…, n, Are Independently and Identically

Distributed 172
Assumption #3: Large Outliers Are Unlikely 173
Use of the Least Squares Assumptions 174