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Preface xix
Introduction and Review 1
Economic Questions and Data 3
Economic Questions We Examine 4
Does Reducing Class Size Improve Elementary School Education? 4
What Are the Economic Returns to Education? 5
Quantitative Questions, Quantitative Answers 6
Causal Effects and Idealized Experiments 6
Estimation of Causal Effects 6
Forecasting and Causality 8
Data: Sources and Types 8
Experimental versus Observational Data 8
Cross-Sectional Data 9
Time Series Data 10
Panel Data 11
Review of Probability 15
Random Variables and Probability Distributions 16
Probabilities, the Sample Space, and Random Variables 16
Probability Distribution of a Discrete Random Variable 17
Probability Distribution of a Continuous Random Variable 19
Expected Values, Mean, and Variance 21
The Expected Value of a Random Variable 21
The Standard Deviation and Variance 22
Mean and Variance of a Linear Function of a Random Variable 23
Other Measures of the Shape ofa Distribution 24
Two Random Variables 27
Joint and Marginal Distributions 27
Conditional Distributions 28
Independence 32
Covariance and Correlation 32
The Mean and Variance of Sums of Random Variables 33
The Normal, Chi-Squared, Student t, and F Distributions 37
The Normal Distribution 37
The Chi-Squared Distribution 41
The Student t Distribution 42
The F Distribution 42
Random Sampling and the Distribution of the Sample Average 43
Random Sampling 43
The Sampling Distribution of the Sample Average 44
Large-Sample Approximations to Sampling Distributions 46
The Law of Large Numbers and Consistency 47
The Central Limit Theorem 50
Derivation of Results in Key Concept 2.3 61
Review of Statistics 63
Estimation of the Population Mean 64
Estimators and Their Properties 65
Properties of Y 66
The Importance of Random Sampling 68
Hypothesis Tests Concerning the Population Mean 69
Null and Alternative Hypotheses 70
The p-Value 70
Calculating the p-Value When [sigma subscript Y] Is Known 72
The Sample Variance, Sample Standard Deviation, and Standard Error 73
Calculating the p-Value When [sigma subscript Y] Is Unknown 74
The t-Statistic 75
Hypothesis Testing with a Prespecified Significance Level 76
One-Sided Alternatives 78
Confidence Intervals for the Population Mean 79
Comparing Means from Different Populations 81
Hypothesis Tests for the Difference Between Two Means 81
Confidence Intervals for the Difference Between Two Population Means 82
Differences-of-Means Estimation of Causal Effects Using Experimental Data 83
The Causal Effect as a Difference of Conditional Expectations 83
Estimation of the Causal Effect Using Differences of Means 85
Using the t-Statistic When the Sample Size Is Small 86
The t-Statistic and the Student t Distribution 86
Use of the Student t Distribution in Practice 90
Scatterplot, the Sample Covariance, and the Sample Correlation 90
Scatterplots 91
Sample Covariance and Correlation 92
The U.S. Current Population Survey 103
Two Proofs That Y Is the Least Squares Estimator of [mu subscript Y] 104
A Proof That the Sample Variance Is Consistent 105
Fundamentals of Regression Analysis 107
Linear Regression with One Regressor 109
The Linear Regression Model 110
Estimating the Coefficients of the Linear Regression Model 114
The Ordinary Least Squares Estimator 116
OLS Estimates of the Relationship Between Test Scores and the Student-Teacher Ratio 118
Why Use the OLS Estimator? 119
Measures of Fit 121
The R[superscript 2] 121
The Standard Error of the Regression 122
Application to the Test Score Data 123
The Least Squares Assumptions 124
The Conditional Distribution of u[subscript i] Given X[subscript i] Has a Mean of Zero 124
(X[subscript i], Y[subscript i]), i = 1, ..., n Are Independently and Identically Distributed 126
Large Outliers Are Unlikely 127
Use of the Least Squares Assumptions 128
The Sampling Distribution of the OLS Estimators 129
The Sampling Distribution of the OLS Estimators 130
Conclusion 133
The California Test Score Data Set 141
Derivation of the OLS Estimators 141
Sampling Distribution of the OLS Estimator 142
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals 146
Testing Hypotheses About One of the Regression Coefficients 147
Two-Sided Hypotheses Concerning [Beta subscript 1] 147
One-Sided Hypotheses Concerning [Beta subscript 1] 151
Testing Hypotheses About the Intercept [Beta subscript 0] 153
Confidence Intervals for a Regression Coefficient 153
Regression When X Is a Binary Variable 156
Interpretation of the Regression Coefficients 156
Heteroskedasticity and Homoskedasticity 158
What Are Heteroskedasticity and Homoskedasticity? 158
Mathematical Implications of Homoskedasticity 161
What Does This Mean in Practice? 162
The Theoretical Foundations of Ordinary Least Squares 164
Linear Conditionally Unbiased Estimators and the Gauss-Markov Theorem 165
Regression Estimators Other Than OLS 166
Using the t-Statistic in Regression When the Sample Size is Small 167
The t-Statistic and the Student t Distribution 168
Use of the Student t Distribution in Practice 168
Conclusion 169
Formulas for OLS Standard Errors 178
The Gauss-Markov Conditions and a Proof of the Gauss-Markov Theorem 180
Linear Regression with Multiple Regressors 184
Omitted Variable Bias 184
Definition of Omitted Variable Bias 185
A Formula for Omitted Variable Bias 187
Addressing Omitted Variable Bias by Dividing the Data into Groups 189
The Multiple Regression Model 191
The Population Regression Line 191
The Population Multiple Regression Model 192
The OLS Estimator in Multiple Regression 194
The OLS Estimator 195
Application to Test Scores and the Student-Teacher Ratio 196
Measures of Fit in Multiple Regression 198
The Standard Error of the Regression (SER) 198
The R[superscript 2] 198
The "Adjusted R[superscript 2]" 199
Application to Test Scores 200
The Least Squares Assumptions in Multiple Regression 200
The Conditional Distribution of u[subscript i] Given X[subscript 1i], [subscript 2i], ..., X[subscript ki] Has a Mean of Zero 201
(X[subscript 1i], X[subscript 2i], ..., X[subscript ki], Y[subscript i]) i = 1, ..., n Are i.i.d. 201
Large Outliers Are Unlikely 201
No Perfect Multicollinearity 201
The Distribution of the OLS Estimators in Multiple Regression 203
Multicollinearity 204
Examples of Perfect Multicollinearity 204
Imperfect Multicollinearity 207
Conclusion 208
Derivation of Equation (6.1) 216
Distribution of the OLS Estimators When There Are Two Regressors and Homoskedastic Errors 216
The OLS Estimator With Two Regressors 217
Hypothesis Tests and Confidence Intervals in Multiple Regression 218
Hypothesis Tests and Confidence Intervals for a Single Coefficient 219
Standard Errors for the OLS Estimators 219
Hypothesis Tests for a Single Coefficient 219
Confidence Intervals for a Single Coefficient 221
Application to Test Scores and the Student-Teacher Ratio 221
Tests of Joint Hypotheses 223
Testing Hypotheses on Two or More Coefficients 223
The F-Statistic 225
Application to Test Scores and the Student-Teacher Ratio 227
The Homoskedasticity-Only F-Statistic 228
Testing Single Restrictions Involving Multiple Coefficients 230
Confidence Sets for Multiple Coefficients 232
Model Specification for Multiple Regression 233
Omitted Variable Bias in Multiple Regression 234
Model Specification in Theory and in Practice 234
Interpreting the R[superscript 2] and tine Adjusted R[superscript 2] in Practice 235
Analysis of the Test Score Data Set 237
Conclusion 242
The Bonferroni Test of a Joint Hypotheses 249
Nonlinear Regression Functions 252
A General Strategy for Modeling Nonlinear Regression Functions 254
Test Scores and District Income 254
The Effect on Y of a Change in X in Nonlinear Specifications 258
A General Approach to Modeling Nonlinearities Using Multiple Regression 262
Nonlinear Functions of a Single Independent Variable 262
Polynomials 263
Logarithms 265
Polynomial and Logarithmic Models of Test Scores and District Income 273
Interactions Between Independent Variables 275
Interactions Between Two Binary Variables 275
Interactions Between a Continuous and a Binary Variable 278
Interactions Between Two Continuous Variables 284
Nonlinear Effects on Test Scores of the Student-Teacher Ratio 288
Discussion of Regression Results 289
Summary of Findings 293
Conclusion 294
Regression Functions That Are Nonlinear in the Parameters 305
Assessing Studies Based on Multiple Regression 310
Internal and External Validity 311
Threats to Internal Validity 311
Threats to External Validity 312
Threats to Internal Validity of Multiple Regression Analysis 314
Omitted Variable Bias 314
Misspecification of the Functional Form of the Regression Function 317
Errors-in-Variables 317
Sample Selection 320
Simultaneous Causality 322
Sources of Inconsistency of OLS Standard Errors 323
Internal and External Validity When the Regression Is Used for Forecasting 325
Using Regression Models for Forecasting 325
Assessing the Validity of Regression Models for Forecasting 326
Example: Test Scores and Class Size 327
External Validity 327
Internal Validity 334
Discussion and Implications 335
Conclusion 336
The Massachusetts Elementary School Testing Data 342
Conducting a Regression Study Using Economic Data 343
Choosing a Topic 344
Collecting the Data 345
Finding a Data Set 345
Time Series Data and Panel Data 346
Preparing the Data for Regression Analysis 347
Conducting Your Regression Analysis 347
Writing Up Your Results 348
Appendix 351
References 359
Answers to "Review the Concepts" Questions 361
Glossary 365
Index 371
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Add Introduction to Econometrics, Brief Edition, In keeping with their successful introductory econometrics text, Stock and Watson motivate each methodological topic with a real-world policy application that uses data, so that readers apply the theory immediately. Introduction to Econometrics, Brief,, Introduction to Econometrics, Brief Edition to the inventory that you are selling on WonderClubX
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Add Introduction to Econometrics, Brief Edition, In keeping with their successful introductory econometrics text, Stock and Watson motivate each methodological topic with a real-world policy application that uses data, so that readers apply the theory immediately. Introduction to Econometrics, Brief,, Introduction to Econometrics, Brief Edition to your collection on WonderClub |