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12 Contents

4.5 Sampling Distribution of the OLS Estimators 175

The Sampling Distribution of the OLS Estimators 176

4.6 Conclusion 179
Appendix 4.1 The California Test Score Data Set 187
Appendix 4.2 Derivation of the OLS Estimators 187
Appendix 4.3 Sampling Distribution of the OLS Estimator 188

Chapter 5 Regression with a Single Regressor: Hypothesis Tests and
Confidence Intervals 192

5.1 Testing Hypotheses About One of the Regression
Coefficients 192

Two-Sided Hypotheses Concerning β1 193
One-Sided Hypotheses Concerning β1 196
Testing Hypotheses About the Intercept β0 198

5.2 Confidence Intervals for a Regression Coefficient 199

5.3 Regression When X Is a Binary Variable 201

Interpretation of the Regression Coefficients 201

5.4 Heteroskedasticity and Homoskedasticity 203

What Are Heteroskedasticity and Homoskedasticity? 204
Mathematical Implications of Homoskedasticity 206
What Does This Mean in Practice? 207

5.5 The Theoretical Foundations of Ordinary Least Squares 209

Linear Conditionally Unbiased Estimators and the Gauss–Markov
Theorem 210

Regression Estimators Other Than OLS 211

5.6 Using the t-Statistic in Regression When the Sample Size
Is Small 212

The t-Statistic and the Student t Distribution 212
Use of the Student t Distribution in Practice 213

5.7 Conclusion 214
Appendix 5.1 Formulas for OLS Standard Errors 223
Appendix 5.2 The Gauss–Markov Conditions and a Proof of the
Gauss–Markov Theorem 224