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| Preface | |
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| Introduction and Review | |
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| Economic Questions and Data | |
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| Economic Questions We Examine | |
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| Does Reducing Class Size Improve Elementary School Education? | |
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| What Are the Economic Returns to Education? | |
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| Quantitative Questions, Quantitative Answers | |
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| Causal Effects and Idealized Experiments | |
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| Estimation of Causal Effects | |
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| Forecasting and Causality | |
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| Data: Sources and Types | |
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| Experimental versus Observational Data | |
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| Cross-Sectional Data | |
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| Time Series Data | |
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| Panel Data | |
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| Review of Probability | |
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| Random Variables and Probability Distributions | |
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| Probabilities, the Sample Space, and Random Variables | |
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| Probability Distribution of a Discrete Random Variable | |
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| Probability Distribution of a Continuous Random Variable | |
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| Expected Values, Mean, and Variance | |
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| The Expected Value of a Random Variable | |
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| The Standard Deviation and Variance | |
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| Mean and Variance of a Linear Function of a Random Variable | |
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| Other Measures of the Shape of a Distribution | |
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| Two Random Variables | |
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| Joint and Marginal Distributions | |
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| Conditional Distributions | |
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| Independence | |
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| Covariance and Correlation | |
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| The Mean and Variance of Sums of Random Variables | |
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| The Normal, Chi-Squared, Student t, and F Distributions | |
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| The Normal Distribution | |
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| The Chi-Squared Distribution | |
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| The Student t Distribution | |
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| The F Distribution | |
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| Random Sampling and the Distribution of the Sample Average | |
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| Random Sampling | |
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| The Sampling Distribution of the Sample Average | |
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| Large-Sample Approximations to Sampling Distributions | |
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| The Law of Large Numbers and Consistency | |
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| The Central Limit Theorem | |
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| Derivation of Results in Key Concept 2.3 | |
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| Review of Statistics | |
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| Estimation of the Population Mean | |
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| Estimators and Their Properties | |
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| Properties of Y | |
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| The Importance of Random Sampling | |
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| Hypothesis Tests Concerning the Population Mean | |
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| Null and Alternative Hypotheses | |
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| The p-Value | |
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| Calculating the p-Value When [sigma subscript Y] Is Known | |
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| The Sample Variance, Sample Standard Deviation, and Standard Error | |
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| Calculating the p-Value When [sigma subscript Y] Is Unknown | |
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| The t-Statistic | |
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| Hypothesis Testing with a Prespecified Significance Level | |
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| One-Sided Alternatives | |
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| Confidence Intervals for the Population Mean | |
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| Comparing Means from Different Populations | |
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| Hypothesis Tests for the Difference Between Two Means | |
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| Confidence Intervals for the Difference Between Two Population Means | |
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| Differences-of-Means Estimation of Causal Effects Using Experimental Data | |
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| The Causal Effect as a Difference of Conditional Expectations | |
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| Estimation of the Causal Effect Using Differences of Means | |
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| Using the t-Statistic When the Sample Size Is Small | |
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| The t-Statistic and the Student t Distribution | |
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| Use of the Student t Distribution in Practice | |
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| Scatterplot, the Sample Covariance, and the Sample Correlation | |
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| Scatterplots | |
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| Sample Covariance and Correlation | |
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| The U.S. Current Population Survey | |
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| Two Proofs That Y Is the Least Squares Estimator of [mu subscript Y] | |
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| A Proof That the Sample Variance Is Consistent | |
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| Fundamentals of Regression Analysis | |
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| Linear Regression with One Regressor | |
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| The Linear Regression Model | |
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| Estimating the Coefficients of the Linear Regression Model | |
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| The Ordinary Least Squares Estimator | |
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| OLS Estimates of the Relationship Between Test Scores and the Student-Teacher Ratio | |
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| Why Use the OLS Estimator? | |
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| Measures of Fit | |
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| The R[superscript 2] | |
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| The Standard Error of the Regression | |
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| Application to the Test Score Data | |
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| The Least Squares Assumptions | |
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| The Conditional Distribution of u[subscript i] Given X[subscript i] Has a Mean of Zero | |
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| (X[subscript i], Y[subscript i]), i = 1, ..., n Are Independently and Identically Distributed | |
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| Large Outliers Are Unlikely | |
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| Use of the Least Squares Assumptions | |
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| The Sampling Distribution of the OLS Estimators | |
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| The Sampling Distribution of the OLS Estimators | |
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| Conclusion | |
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| The California Test Score Data Set | |
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| Derivation of the OLS Estimators | |
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| Sampling Distribution of the OLS Estimator | |
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| Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals | |
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| Testing Hypotheses About One of the Regression Coefficients | |
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| Two-Sided Hypotheses Concerning [Beta subscript 1] | |
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| One-Sided Hypotheses Concerning [Beta subscript 1] | |
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| Testing Hypotheses About the Intercept [Beta subscript 0] | |
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| Confidence Intervals for a Regression Coefficient | |
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| Regression When X Is a Binary Variable | |
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| Interpretation of the Regression Coefficients | |
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| Heteroskedasticity and Homoskedasticity | |
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| What Are Heteroskedasticity and Homoskedasticity? | |
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| Mathematical Implications of Homoskedasticity | |
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| What Does This Mean in Practice? | |
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| The Theoretical Foundations of Ordinary Least Squares | |
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| Linear Conditionally Unbiased Estimators and the Gauss-Markov Theorem | |
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| Regression Estimators Other Than OLS | |
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| Using the t-Statistic in Regression When the Sample Size is Small | |
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| The t-Statistic and the Student t Distribution | |
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| Use of the Student t Distribution in Practice | |
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| Conclusion | |
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| Formulas for OLS Standard Errors | |
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| The Gauss-Markov Conditions and a Proof of the Gauss-Markov Theorem | |
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| Linear Regression with Multiple Regressors | |
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| Omitted Variable Bias | |
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| Definition of Omitted Variable Bias | |
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| A Formula for Omitted Variable Bias | |
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| Addressing Omitted Variable Bias by Dividing the Data into Groups | |
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| The Multiple Regression Model | |
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| The Population Regression Line | |
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| The Population Multiple Regression Model | |
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| The OLS Estimator in Multiple Regression | |
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| The OLS Estimator | |
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| Application to Test Scores and the Student-Teacher Ratio | |
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| Measures of Fit in Multiple Regression | |
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| The Standard Error of the Regression (SER) | |
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| The R[superscript 2] | |
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| The "Adjusted R[superscript 2]" | |
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| Application to Test Scores | |
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| The Least Squares Assumptions in Multiple Regression | |
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| The Conditional Distribution of u[subscript i] Given X[subscript 1i], [subscript 2i], ..., X[subscript ki] Has a Mean of Zero | |
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| (X[subscript 1i], X[subscript 2i], ..., X[subscript ki], Y[subscript i]) i = 1, ..., n Are i.i.d. | |
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| Large Outliers Are Unlikely | |
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| No Perfect Multicollinearity | |
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| The Distribution of the OLS Estimators in Multiple Regression | |
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| Multicollinearity | |
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| Examples of Perfect Multicollinearity | |
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| Imperfect Multicollinearity | |
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| Conclusion | |
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| Derivation of Equation (6.1) | |
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| Distribution of the OLS Estimators When There Are Two Regressors and Homoskedastic Errors | |
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| The OLS Estimator With Two Regressors | |
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| Hypothesis Tests and Confidence Intervals in Multiple Regression | |
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| Hypothesis Tests and Confidence Intervals for a Single Coefficient | |
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| Standard Errors for the OLS Estimators | |
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| Hypothesis Tests for a Single Coefficient | |
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| Confidence Intervals for a Single Coefficient | |
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| Application to Test Scores and the Student-Teacher Ratio | |
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| Tests of Joint Hypotheses | |
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| Testing Hypotheses on Two or More Coefficients | |
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| The F-Statistic | |
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| Application to Test Scores and the Student-Teacher Ratio | |
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| The Homoskedasticity-Only F-Statistic | |
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| Testing Single Restrictions Involving Multiple Coefficients | |
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| Confidence Sets for Multiple Coefficients | |
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| Model Specification for Multiple Regression | |
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| Omitted Variable Bias in Multiple Regression | |
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| Model Specification in Theory and in Practice | |
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| Interpreting the R[superscript 2] and tine Adjusted R[superscript 2] in Practice | |
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| Analysis of the Test Score Data Set | |
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| Conclusion | |
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| The Bonferroni Test of a Joint Hypotheses | |
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| Nonlinear Regression Functions | |
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| A General Strategy for Modeling Nonlinear Regression Functions | |
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| Test Scores and District Income | |
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| The Effect on Y of a Change in X in Nonlinear Specifications | |
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| A General Approach to Modeling Nonlinearities Using Multiple Regression | |
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| Nonlinear Functions of a Single Independent Variable | |
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| Polynomials | |
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| Logarithms | |
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| Polynomial and Logarithmic Models of Test Scores and District Income | |
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| Interactions Between Independent Variables | |
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| Interactions Between Two Binary Variables | |
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| Interactions Between a Continuous and a Binary Variable | |
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| Interactions Between Two Continuous Variables | |
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| Nonlinear Effects on Test Scores of the Student-Teacher Ratio | |
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| Discussion of Regression Results | |
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| Summary of Findings | |
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| Conclusion | |
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| Regression Functions That Are Nonlinear in the Parameters | |
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| Assessing Studies Based on Multiple Regression | |
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| Internal and External Validity | |
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| Threats to Internal Validity | |
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| Threats to External Validity | |
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| Threats to Internal Validity of Multiple Regression Analysis | |
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| Omitted Variable Bias | |
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| Misspecification of the Functional Form of the Regression Function | |
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| Errors-in-Variables | |
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| Sample Selection | |
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| Simultaneous Causality | |
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| Sources of Inconsistency of OLS Standard Errors | |
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| Internal and External Validity When the Regression Is Used for Forecasting | |
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| Using Regression Models for Forecasting | |
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| Assessing the Validity of Regression Models for Forecasting | |
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| Example: Test Scores and Class Size | |
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| External Validity | |
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| Internal Validity | |
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| Discussion and Implications | |
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| Conclusion | |
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| The Massachusetts Elementary School Testing Data | |
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| Conducting a Regression Study Using Economic Data | |
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| Choosing a Topic | |
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| Collecting the Data | |
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| Finding a Data Set | |
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| Time Series Data and Panel Data | |
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| Preparing the Data for Regression Analysis | |
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| Conducting Your Regression Analysis | |
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| Writing Up Your Results | |
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| Appendix | |
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| References | |
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| Answers to "Review the Concepts" Questions | |
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| Glossary | |
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| Index | |