Introduction to Econometrics, Brief Edition Book

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