| |
| |
Preface | |
| |
| |
| |
Introduction and Review | |
| |
| |
| |
Economic Questions and Data | |
| |
| |
| |
Economic Questions We Examine | |
| |
| |
| |
Does Reducing Class Size Improve Elementary School Education? | |
| |
| |
| |
Is There Racial Discrimination in the Market for Home Loans? | |
| |
| |
| |
How Much Do Cigarette Taxes Reduce Smoking? | |
| |
| |
| |
What Will the Rate of Inflation Be Next Year? | |
| |
| |
Quantitative Questions, Quantitative Answers | |
| |
| |
| |
Causal Effects and Idealized Experiments | |
| |
| |
Estimation of Causal Effects | |
| |
| |
Forecasting and Causality | |
| |
| |
| |
Data: Sources and Types | |
| |
| |
Experimental versus Observational Data | |
| |
| |
Cross-Sectional Data | |
| |
| |
Time Series Data | |
| |
| |
Panel Data | |
| |
| |
| |
Review of Probability | |
| |
| |
| |
Random Variables and Probability Distributions | |
| |
| |
Probabilities, the Sample Space, and Random Variables | |
| |
| |
Probability Distribution of a Discrete Random Variable | |
| |
| |
Probability Distribution of a Continuous Random Variable | |
| |
| |
| |
Expected Values, Mean, and Variance | |
| |
| |
The Expected Value of a Random Variable | |
| |
| |
The Standard Deviation and Variance | |
| |
| |
Mean and Variance of a Linear Function of a Random Variable | |
| |
| |
Other Measures of the Shape of a Distribution | |
| |
| |
| |
Two Random Variables | |
| |
| |
Joint and Marginal Distributions | |
| |
| |
Conditional Distributions | |
| |
| |
Independence | |
| |
| |
Covariance and Correlation | |
| |
| |
The Mean and Variance of Sums of Random Variables | |
| |
| |
| |
The Normal, Chi-Squared, Student t, and F Distributions | |
| |
| |
The Normal Distribution | |
| |
| |
The Chi-Squared Distribution | |
| |
| |
The Student t Distribution | |
| |
| |
The F Distribution | |
| |
| |
| |
Random Sampling and the Distribution of the Sample Average | |
| |
| |
Random Sampling | |
| |
| |
The Sampling Distribution of the Sample Average | |
| |
| |
| |
Large-Sample Approximations to Sampling Distributions | |
| |
| |
The Law of Large Numbers and Consistency | |
| |
| |
The Central Limit Theorem | |
| |
| |
| |
Derivation of Results in Key Concept 2.3 | |
| |
| |
| |
Review of Statistics | |
| |
| |
| |
Estimation of the Population Mean | |
| |
| |
Estimators and Their Properties | |
| |
| |
Properties of Y | |
| |
| |
The Importance of Random Sampling | |
| |
| |
| |
Hypothesis Tests Concerning the Population Mean | |
| |
| |
Null and Alternative Hypotheses | |
| |
| |
The p-Value | |
| |
| |
Calculating the p-Value When [sigma subscript Y] is Known | |
| |
| |
The Sample Variance, Sample Standard Deviation, and Standard Error | |
| |
| |
Calculating the p-Value When [sigma subscript Y] Is Unknown | |
| |
| |
The t-Statistic | |
| |
| |
Hypothesis Testing with a Prespecified Significance Level | |
| |
| |
One-Sided Alternatives | |
| |
| |
| |
Confidence Intervals for the Population Mean | |
| |
| |
| |
Comparing Means from Different Populations | |
| |
| |
Hypothesis Tests for the Difference Between Two Means | |
| |
| |
Confidence Intervals for the Difference Between Two Population Means | |
| |
| |
| |
Differences-of-Means Estimation of Causal Effects Using Experimental Data | |
| |
| |
The Causal Effect as a Difference of Conditional Expectations | |
| |
| |
Estimation of the Causal Effect Using Differences of Means | |
| |
| |
| |
Using the t-Statistic When the Sample Size Is Small | |
| |
| |
The t-Statistic and the Student t Distribution | |
| |
| |
Use of the Student t Distribution in Practice | |
| |
| |
| |
Scatterplot, the Sample Covariance, and the Sample Correlation | |
| |
| |
Scatterplots | |
| |
| |
Sample Covariance and Correlation | |
| |
| |
| |
The U.S. Current Population Survey | |
| |
| |
| |
Two Proofs That Y Is the Least Squares Estimator of [mu subscript Y] | |
| |
| |
| |
A Proof That the Sample Variance Is Consistent | |
| |
| |
| |
Fundamentals of Regression Analysis | |
| |
| |
| |
Linear Regression with One Regressor | |
| |
| |
| |
The Linear Regression Model | |
| |
| |
| |
Estimating the Coefficients of the Linear Regression Model | |
| |
| |
The Ordinary Least Squares Estimator | |
| |
| |
OLS Estimates of the Relationship Between Test Scores and the Student-Teacher Ratio | |
| |
| |
Why Use the OLS Estimator? | |
| |
| |
| |
Measures of Fit | |
| |
| |
The R[superscript 2] | |
| |
| |
The Standard Error of the Regression | |
| |
| |
Application to the Test Score Data | |
| |
| |
| |
The Least Squares Assumptions | |
| |
| |
| |
The Conditional Distribution of u[subscript i] Given X[subscript i] Has a Mean of Zero | |
| |
| |
| |
(X[subscript i], Y[subscript i]), = 1,..., n Are Independently and Identically Distributed | |
| |
| |
| |
Large Outliers Are Unlikely | |
| |
| |
Use of the Least Squares Assumptions | |
| |
| |
| |
The Sampling Distribution of the OLS Estimators | |
| |
| |
The Sampling Distribution of the OLS Estimators | |
| |
| |
| |
Conclusion | |
| |
| |
| |
The California Test Score Data Set | |
| |
| |
| |
Derivation of the OLS Estimators | |
| |
| |
| |
Sampling Distribution of the OLS Estimator | |
| |
| |
| |
Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals | |
| |
| |
| |
Testing Hypotheses About One of the Regression Coefficients | |
| |
| |
Two-Sided Hypotheses Concerning [Beta subscript 1] | |
| |
| |
One-Sided Hypotheses Concerning [Beta subscript 1] | |
| |
| |
Testing Hypotheses About the Intercept [Beta subscript 0] | |
| |
| |
| |
Confidence Intervals for a Regression Coefficient | |
| |
| |
| |
Regression When X Is a Binary Variable | |
| |
| |
Interpretation of the Regression Coefficients | |
| |
| |
| |
Heteroskedasticity and Homoskedasticity | |
| |
| |
What Are Heteroskedasticity and Homoskedasticity? | |
| |
| |
Mathematical Implications of Homoskedasticity | |
| |
| |
What Does This Mean in Practice? | |
| |
| |
| |
The Theoretical Foundations of Ordinary Least Squares | |
| |
| |
Linear Conditionally Unbiased Estimators and the Gauss-Markov Theorem | |
| |
| |
Regression Estimators Other Than OLS | |
| |
| |
| |
Using the t-Statistic in Regression When the Sample Size Is Small | |
| |
| |
The t-Statistic and the Student t Distribution | |
| |
| |
Use of the Student t Distribution in Practice | |
| |
| |
| |
Conclusion | |
| |
| |
| |
Formulas for OLS Standard Errors | |
| |
| |
| |
The Gauss-Markov Conditions and a Proof of the Gauss-Markov Theorem | |
| |
| |
| |
Linear Regression with Multiple Regressors | |
| |
| |
| |
Omitted Variable Bias | |
| |
| |
Definition of Omitted Variable Bias | |
| |
| |
A Formula for Omitted Variable Bias | |
| |
| |
Addressing Omitted Variable Bias by Dividing the Data into Groups | |
| |
| |
| |
The Multiple Regression Model | |
| |
| |
The Population Regression Line | |
| |
| |
The Population Multiple Regression Model | |
| |
| |
| |
The OLS Estimator in Multiple Regression | |
| |
| |
The OLS Estimator | |
| |
| |
Application to Test Scores and the Student-Teacher Ratio | |
| |
| |
| |
Measures of Fit in Multiple Regression | |
| |
| |
The Standard Error of the Regression (SER) | |
| |
| |
The R[superscript 2] | |
| |
| |
The "Adjusted R[superscript 2]" | |
| |
| |
Application to Test Scores | |
| |
| |
| |
The Least Squares Assumptions in Multiple Regression | |
| |
| |
| |
The Conditional Distribution of u[subscript i] Given X[subscript 1i], X[subscript 2i],..., X[subscript ki] Has a Mean of Zero | |
| |
| |
| |
(X[subscript 1i], X[subscript 2i],..., X[subscript ki], Y[subscript i]) i = 1,..., n Are i.i.d. | |
| |
| |
| |
Large Outliers Are Unlikely | |
| |
| |
| |
No Perfect Multicollinearity | |
| |
| |
| |
The Distribution of the OLS Estimators in Multiple Regression | |
| |
| |
| |
Multicollinearity | |
| |
| |
Examples of Perfect Multicollinearity | |
| |
| |
Imperfect Multicollinearity | |
| |
| |
| |
Conclusion | |
| |
| |
| |
Derivation of Equation (6.1) | |
| |
| |
| |
Distribution of the OLS Estimators When There Are Two Regressors and Homoskedastic Errors | |
| |
| |
| |
Hypothesis Tests and Confidence Intervals in Multiple Regression | |
| |
| |
| |
Hypothesis Tests and Confidence Intervals for a Single Coefficient | |
| |
| |
Standard Errors for the OLS Estimators | |
| |
| |
Hypothesis Tests for a Single Coefficient | |
| |
| |
Confidence Intervals for a Single Coefficient | |
| |
| |
Application to Test Scores and the Student-Teacher Ratio | |
| |
| |
| |
Tests of Joint Hypotheses | |
| |
| |
Testing Hypotheses on Two or More Coefficients | |
| |
| |
The F-Statistic | |
| |
| |
Application to Test Scores and the Student-Teacher Ratio | |
| |
| |
The Homoskedasticity-Only F-Statistic | |
| |
| |
| |
Testing Single Restrictions Involving Multiple Coefficients | |
| |
| |
| |
Confidence Sets for Multiple Coefficients | |
| |
| |
| |
Model Specification for Multiple Regression | |
| |
| |
Omitted Variable Bias in Multiple Regression | |
| |
| |
Model Specification in Theory and in Practice | |
| |
| |
Interpreting the R[superscript 2] and the Adjusted R[superscript 2] in Practice | |
| |
| |
| |
Analysis of the Test Score Data Set | |
| |
| |
| |
Conclusion | |
| |
| |
| |
The Bonferroni Test of a Joint Hypotheses | |
| |
| |
| |
Nonlinear Regression Functions | |
| |
| |
| |
A General Strategy for Modeling Nonlinear Regression Functions | |
| |
| |
Test Scores and District Income | |
| |
| |
The Effect on Y of a Change in X in Nonlinear Specifications | |
| |
| |
A General Approach to Modeling Nonlinearities Using Multiple Regression | |
| |
| |
| |
Nonlinear Functions of a Single Independent Variable | |
| |
| |
Polynomials | |
| |
| |
Logarithms | |
| |
| |
Polynomial and Logarithmic Models of Test Scores and District Income | |
| |
| |
| |
Interactions Between Independent Variables | |
| |
| |
Interactions Between Two Binary Variables | |
| |
| |
Interactions Between a Continuous and a Binary Variable | |
| |
| |
Interactions Between Two Continuous Variables | |
| |
| |
| |
Nonlinear Effects on Test Scores of the Student-Teacher Ratio | |
| |
| |
Discussion of Regression Results | |
| |
| |
Summary of Findings | |
| |
| |
Conclusion | |
| |
| |
| |
Regression Functions That Are Nonlinear in the Parameters | |
| |
| |
| |
Assessing Studies Based on Multiple Regression | |
| |
| |
| |
Internal and External Validity | |
| |
| |
Threats to Internal Validity | |
| |
| |
Threats to External Validity | |
| |
| |
| |
Threats to Internal Validity of Multiple Regression Analysis | |
| |
| |
Omitted Variable Bias | |
| |
| |
Misspecification of the Functional Form of the Regression Function | |
| |
| |
Errors-in-Variables | |
| |
| |
Sample Selection | |
| |
| |
Simultaneous Causality | |
| |
| |
Sources of Inconsistency of OLS Standard Errors | |
| |
| |
| |
Internal and External Validity When the Regression Is Used for Forecasting | |
| |
| |
Using Regression Models for Forecasting | |
| |
| |
Assessing the Validity of Regression Models for Forecasting | |
| |
| |
| |
Example: Test Scores and Class Size | |
| |
| |
External Validity | |
| |
| |
Internal Validity | |
| |
| |
Discussion and Implications | |
| |
| |
| |
Conclusion | |
| |
| |
| |
The Massachusetts Elementary School Testing Data | |
| |
| |
| |
Further Topics in Regression Analysis | |
| |
| |
| |
Regression with Panel Data | |
| |
| |
| |
Panel Data | |
| |
| |
Example: Traffic Deaths and Alcohol Taxes | |
| |
| |
| |
Panel Data with Two Time Periods: "Before and After" Comparisons | |
| |
| |
| |
Fixed Effects Regression | |
| |
| |
The Fixed Effects Regression Model | |
| |
| |
Estimation and Inference | |
| |
| |
Application to Traffic Deaths | |
| |
| |
| |
Regression with Time Fixed Effects | |
| |
| |
Time Effects Only | |
| |
| |
Both Entity and Time Fixed Effects | |
| |
| |
| |
The Fixed Effects Regression Assumptions and Standard Errors for Fixed Effects Regression | |
| |
| |
The Fixed Effects Regression Assumptions | |
| |
| |
Standard Errors for Fixed Effects Regression | |
| |
| |
| |
Drunk Driving Laws and Traffic Deaths | |
| |
| |
| |
Conclusion | |
| |
| |
| |
The State Traffic Fatality Data Set | |
| |
| |
| |
Standard Errors for Fixed Effects Regression with Serially Correlated Errors | |
| |
| |
| |
Regression with a Binary Dependent Variable | |
| |
| |
| |
Binary Dependent Variables and the Linear Probability Model | |
| |
| |
Binary Dependent Variables | |
| |
| |
The Linear Probability Model | |
| |
| |
| |
Probit and Logit Regression | |
| |
| |
Probit Regression | |
| |
| |
Logit Regression | |
| |
| |
Comparing the Linear Probability, Probit, and Logit Models | |
| |
| |
| |
Estimation and Inference in the Logit and Probit Models | |
| |
| |
Nonlinear Least Squares Estimation | |
| |
| |
Maximum Likelihood Estimation | |
| |
| |
Measures of Fit | |
| |
| |
| |
Application to the Boston HMDA Data | |
| |
| |
| |
Summary | |
| |
| |
| |
The Boston HMDA Data Set | |
| |
| |
| |
Maximum Likelihood Estimation | |
| |
| |
| |
Other Limited Dependent Variable Models | |
| |
| |
| |
Instrumental Variables Regression | |
| |
| |
| |
The IV Estimator with a Single Regressor and a Single Instrument | |
| |
| |
The IV Model and Assumptions | |
| |
| |
The Two Stage Least Squares Estimator | |
| |
| |
Why Does IV Regression Work? | |
| |
| |
The Sampling Distribution of the TSLS Estimator | |
| |
| |
Application to the Demand for Cigarettes | |
| |
| |
| |
The General IV Regression Model | |
| |
| |
TSLS in the General IV Model | |
| |
| |
Instrument Relevance and Exogeneity in the General IV Model | |
| |
| |
The IV Regression Assumptions and Sampling Distribution of the TSLS Estimator | |
| |
| |
Inference Using the TSLS Estimator | |
| |
| |
Application to the Demand for Cigarettes | |
| |
| |
| |
Checking Instrument Validity | |
| |
| |
| |
Instrument Relevance | |
| |
| |
| |
Instrument Exogeneity | |
| |
| |
| |
Application to the Demand for Cigarettes | |
| |
| |
| |
Where Do Valid Instruments Come From? | |
| |
| |
Three Examples | |
| |
| |
| |
Conclusion | |
| |
| |
| |
The Cigarette Consumption Panel Data Set | |
| |
| |
| |
Derivation of the Formula for the TSLS Estimator in Equation (12.4) | |
| |
| |
| |
Large-Sample Distribution of the TSLS Estimator | |
| |
| |
| |
Large-Sample Distribution of the TSLS Estimator When the Instrument Is Not Valid | |
| |
| |
| |
Instrumental Variables Analysis with Weak Instruments | |
| |
| |
| |
Experiments and Quasi-Experiments | |
| |
| |
| |
Idealized Experiments and Causal Effects | |
| |
| |
Ideal Randomized Controlled Experiments | |
| |
| |
The Differences Estimator | |
| |
| |
| |
Potential Problems with Experiments in Practice | |
| |
| |
Threats to Internal Validity | |
| |
| |
Threats to External Validity | |
| |
| |
| |
Regression Estimators of Causal Effects Using Experimental Data | |
| |
| |
The Differences Estimator with Additional Regressors | |
| |
| |
The Differences-in-Differences Estimator | |
| |
| |
Estimation of Causal Effects for Different Groups | |
| |
| |
Estimation When There Is Partial Compliance | |
| |
| |
Testing for Randomization | |
| |
| |
| |
Experimental Estimates of the Effect of Class Size Reductions | |
| |
| |
Experimental Design | |
| |
| |
Analysis of the STAR Data | |
| |
| |
Comparison of the Observational and Experimental Estimates of Class Size Effects | |
| |
| |
| |
Quasi-Experiments | |
| |
| |
Examples | |
| |
| |
Econometric Methods for Analyzing Quasi-Experiments | |
| |
| |
| |
Potential Problems with Quasi-Experiments | |
| |
| |
Threats to Internal Validity | |
| |
| |
Threats to External Validity | |
| |
| |
| |
Experimental and Quasi-Experimental Estimates in Heterogeneous Populations | |
| |
| |
Population Heterogeneity: Whose Causal Effect? | |
| |
| |
OLS with Heterogeneous Causal Effects | |
| |
| |
IV Regression with Heterogeneous Causal Effects | |
| |
| |
| |
Conclusion | |
| |
| |
| |
The Project STAR Data Set | |
| |
| |
| |
Extension of the Differences-in-Differences Estimator to Multiple Time Periods | |
| |
| |
| |
Conditional Mean Independence | |
| |
| |
| |
IV Estimation When the Causal Effect Varies Across Individuals | |
| |
| |
| |
Regression Analysis of Economic Time Series Data | |
| |
| |
| |
Introduction to Time Series Regression and Forecasting | |
| |
| |
| |
Using Regression Models for Forecasting | |
| |
| |
| |
Introduction to Time Series Data and Serial Correlation | |
| |
| |
The Rates of Inflation and Unemployment in the United States | |
| |
| |
Lags, First Differences, Logarithms, and Growth Rates | |
| |
| |
Autocorrelation | |
| |
| |
Other Examples of Economic Time Series | |
| |
| |
| |
Autoregressions | |
| |
| |
The First Order Autoregressive Model | |
| |
| |
The p[superscript th] Order Autoregressive Model | |
| |
| |
| |
Time Series Regression with Additional Predictors and the Autoregressive Distributed Lag Model | |
| |
| |
Forecasting Changes in the Inflation Rate Using Past Unemployment Rates | |
| |
| |
Stationarity | |
| |
| |
Time Series Regression with Multiple Predictors | |
| |
| |
Forecast Uncertainty and Forecast Intervals | |
| |
| |
| |
Lag Length Selection Using Information Criteria | |
| |
| |
Determining the Order of an Autoregression | |
| |
| |
Lag Length Selection in Time Series Regression with Multiple Predictors | |
| |
| |
| |
Nonstationarity I: Trends | |
| |
| |
What Is a Trend? | |
| |
| |
Problems Caused by Stochastic Trends | |
| |
| |
Detecting Stochastic Trends: Testing for a Unit AR Root | |
| |
| |
Avoiding the Problems Caused by Stochastic Trends | |
| |
| |
| |
Nonstationarity II: Breaks | |
| |
| |
What Is a Break? | |
| |
| |
Testing for Breaks | |
| |
| |
Pseudo Out-of-Sample Forecasting | |
| |
| |
Avoiding the Problems Caused by Breaks | |
| |
| |
| |
Conclusion | |
| |
| |
| |
Time Series Data Used in Chapter 14 | |
| |
| |
| |
Stationarity in the AR(1) Model | |
| |
| |
| |
Lag Operator Notation | |
| |
| |
| |
ARMA Models | |
| |
| |
| |
Consistency of the BIC Lag Length Estimator | |
| |
| |
| |
Estimation of Dynamic Causal Effects | |
| |
| |
| |
An Initial Taste of the Orange Juice Data | |
| |
| |
| |
Dynamic Causal Effects | |
| |
| |
Causal Effects and Time Series Data | |
| |
| |
Two Types of Exogeneity | |
| |
| |
| |
Estimation of Dynamic Causal Effects with Exogenous Regressors | |
| |
| |
The Distributed Lag Model Assumptions | |
| |
| |
Autocorrelated u[subscript t], Standard Errors, and Inference | |
| |
| |
Dynamic Multipliers and Cumulative Dynamic Multipliers | |
| |
| |
| |
Heteroskedasticity- and Autocorrelation-Consistent Standard Errors | |
| |
| |
Distribution of the OLS Estimator with Autocorrelated Errors | |
| |
| |
HAC Standard Errors | |
| |
| |
| |
Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors | |
| |
| |
The Distributed Lag Model with AR(1) Errors | |
| |
| |
OLS Estimation of the ADL Model | |
| |
| |
GLS Estimation | |
| |
| |
The Distributed Lag Model with Additional Lags and AR(p) Errors | |
| |
| |
| |
Orange Juice Prices and Cold Weather | |
| |
| |
| |
Is Exogeneity Plausible? Some Examples | |
| |
| |
U.S. Income and Australian Exports | |
| |
| |
Oil Prices and Inflation | |
| |
| |
Monetary Policy and Inflation | |
| |
| |
The Phillips Curve | |
| |
| |
| |
Conclusion | |
| |
| |
| |
The Orange Juice Data Set | |
| |
| |
| |
The ADL Model and Generalized Least Squares in Lag Operator Notation | |
| |
| |
| |
Additional Topics in Time Series Regression | |
| |
| |
| |
Vector Autoregressions | |
| |
| |
The VAR Model | |
| |
| |
A VAR Model of the Rates of Inflation and Unemployment | |
| |
| |
| |
Multiperiod Forecasts | |
| |
| |
Iterated Muliperiod Forecasts | |
| |
| |
Direct Multiperiod Forecasts | |
| |
| |
Which Method Should You Use? | |
| |
| |
| |
Orders of Integration and the DF-GLS Unit Root Test | |
| |
| |
Other Models of Trends and Orders of Integration | |
| |
| |
The DF-GLS Test for a Unit Root | |
| |
| |
Why Do Unit Root Tests Have Non-normal Distributions? | |
| |
| |
| |
Cointegration | |
| |
| |
Cointegration and Error Correction | |
| |
| |
How Can You Tell Whether Two Variables Are Cointegrated? | |
| |
| |
Estimation of Cointegrating Coefficients | |
| |
| |
Extension to Multiple Cointegrated Variables | |
| |
| |
Application to Interest Rates | |
| |
| |
| |
Volatility Clustering and Autoregressive Conditional Heteroskedasticity | |
| |
| |
Volatility Clustering | |
| |
| |
Autoregressive Conditional Heteroskedasticity | |
| |
| |
Application to Stock Price Volatility | |
| |
| |
| |
Conclusion | |
| |
| |
| |
U.S. Financial Data Used in Chapter 16 | |
| |
| |
| |
The Econometric Theory of Regression Analysis | |
| |
| |
| |
The Theory of Linear Regression with One Regressor | |
| |
| |
| |
The Extended Least Squares Assumptions and the OLS Estimator | |
| |
| |
The Extended Least Squares Assumptions | |
| |
| |
The OLS Estimator | |
| |
| |
| |
Fundamentals of Asymptotic Distribution Theory | |
| |
| |
Convergence in Probability and the Law of Large Numbers | |
| |
| |
The Central Limit Theorem and Convergence in Distribution | |
| |
| |
Slutsky's Theorem and the Continuous Mapping Theorem | |
| |
| |
Application to the t-Statistic Based on the Sample Mean | |
| |
| |
| |
Asymptotic Distribution of the OLS Estimator and t-Statistic | |
| |
| |
Consistency and Asymptotic Normality of the OLS Estimators | |
| |
| |
Consistency of Heteroskedasticity-Robust Standard Errors | |
| |
| |
Asymptotic Normality of the Heteroskedasticity-Robust t-Statistic | |
| |
| |
| |
Exact Sampling Distributions When the Errors Are Normally Distributed | |
| |
| |
Distribution of [Beta subscript 1] with Normal Errors | |
| |
| |
Distribution of the Homoskedasticity-only t-Statistic | |
| |
| |
| |
Weighted Least Squares | |
| |
| |
WLS with Known Heteroskedasticity | |
| |
| |
WLS with Heteroskedasticity of Known Functional Form | |
| |
| |
Heteroskedasticity-Robust Standard Errors or WLS? | |
| |
| |
| |
The Normal and Related Distributions and Moments of Continuous Random Variables | |
| |
| |
| |
Two Inequalities | |
| |
| |
| |
The Theory of Multiple Regression | |
| |
| |
| |
The Linear Multiple Regression Model and OLS Estimator in Matrix Form | |
| |
| |
The Multiple Regression Model in Matrix Notation | |
| |
| |
The Extended Least Squares Assumptions | |
| |
| |
The OLS Estimator | |
| |
| |
| |
Asymptotic Distribution of the OLS Estimator and t-Statistic | |
| |
| |
The Multivariate Central Limit Theorem | |
| |
| |
Asymptotic Normality of [Beta] | |
| |
| |
Heteroskedasticity-Robust Standard Errors | |
| |
| |
Confidence Intervals for Predicted Effects | |
| |
| |
Asymptotic Distribution of the t-Statistic | |
| |
| |
| |
Tests of Joint Hypotheses | |
| |
| |
Joint Hypotheses in Matrix Notation | |
| |
| |
Asymptotic Distribution of the F-Statistic | |
| |
| |
Confidence Sets for Multiple Coefficients | |
| |
| |
| |
Distribution of Regression Statistics with Normal Errors | |
| |
| |
Matrix Representations of OLS Regression Statistics | |
| |
| |
Distribution of [Beta] with Normal Errors | |
| |
| |
Distribution of [Characters not reproducible] | |
| |
| |
Homoskedasticity-Only Standard Errors | |
| |
| |
Distribution of the t-Statistic | |
| |
| |
Distribution of the F-Statistic | |
| |
| |
| |
Efficiency of the OLS Estimator with Homoskedastic Errors | |
| |
| |
The Gauss-Markov Conditions for Multiple Regression | |
| |
| |
Linear Conditionally Unbiased Estimators | |
| |
| |
The Gauss-Markov Theorem for Multiple Regression | |
| |
| |
| |
Generalized Least Squares | |
| |
| |
The GLS Assumptions | |
| |
| |
GLS When [Omega] Is Known | |
| |
| |
GLS When [Omega] Contains Unknown Parameters | |
| |
| |
The Zero Conditional Mean Assumption and GLS | |
| |
| |
| |
Instrumental Variables and Generalized Method of Moments Estimation | |
| |
| |
The IV Estimator in Matrix Form | |
| |
| |
Asymptotic Distribution of the TSLS Estimator | |
| |
| |
Properties of TSLS When the Errors Are Homoskedastic | |
| |
| |
Generalized Method of Moments Estimation in Linear Models | |
| |
| |
| |
Summary of Matrix Algebra | |
| |
| |
| |
Multivariate Distributions | |
| |
| |
| |
Derivation of the Asymptotic Distribution of [Beta] | |
| |
| |
| |
Derivations of Exact Distributions of OLS Test Statistics with Normal Errors | |
| |
| |
| |
Proof of the Gauss-Markov Theorem for Multiple Regression | |
| |
| |
| |
Proof of Selected Results for IV and GMM Estimation | |
| |
| |
Appendix | |
| |
| |
References | |
| |
| |
Answers to "Review the Concepts" Questions | |
| |
| |
Glossary | |
| |
| |
Index | |