3-1 Motivation for Multiple Regression

3-1a The Model with Two Independent Variables

3-1b The Model with *k* Independent Variables

3-2 Mechanics and Interpretation of Ordinary Least Squares

3-2a Obtaining the OLS Estimates

3-2b Interpreting the OLS Regression Equation

3-2c On the Meaning of "Holding Other Factors Fixed" in Multiple Regression

3-2d Changing More Than One Independent Variable Simultaneously

3-2e OLS Fitted Values and Residuals

3-2f A "Partialling Out" Interpretation of Multiple Regression

3-2g Comparison of Simple and Multiple Regression Estimates

3-2h Goodness-of-Fit

3-2i Regression through the Origin

3-3 The Expected Value of the OLS Estimators

3-3a Including Irrelevant Variables in a Regression Model

3-3b Omitted Variable Bias: The Simple Case

3-3c Omitted Variable Bias: More General Cases

3-4 The Variance of the OLS Estimators

3-4a The Components of the OLS Variances: Multicollinearity

3-4b Variances in Misspecified Models

3-4c Estimating σ^{2}: Standard Errors of the OLS Estimators

3-5 Efficiency of OLS: The Gauss-Markov Theorem

3-6 Some Comments on the Language of Multiple Regression Analysis

3-7 Several Scenarios for Applying Multiple Regression

3-7a Prediction

3-7b Efficient Markets

3-7c Measuring the Tradeoff between Two Variables

3-7d Testing for Ceteris Paribus Group Differences

3-7e Potential Outcomes, Treatment Effects, and Policy Analysis

Summary

Key Terms

Problems

Computer Exercises

**Chapter 4 Multiple Regression Analysis: Inference**

4-1 Sampling Distributions of the OLS Estimators

4-2 Testing Hypotheses about a Single Population Parameter: The

*t* Test

4-2a Testing against One-Sided Alternatives

4-2b Two-Sided Alternatives

4-2c Testing Other Hypotheses about β_{j}

4-2d Computing *p*-Values for *t* Tests

4-2e A Reminder on the Language of Classical Hypothesis Testing

4-2f Economic, or Practical, versus Statistical Significance

4-3 Confidence Intervals

4-4 Testing Hypotheses about a Single Linear Combination of the Parameters

4-5 Testing Multiple Linear Restrictions: The

*F* Test

4-5a Testing Exclusion Restrictions

4-5b Relationship between *F* and *t* Statistics

4-5c The *R*-Squared Form of the *F* Statistic

4-5d Computing *p*-Values for *F* Tests

4-5e The *F* Statistic for Overall Significance of a Regression

4-5f Testing General Linear Restrictions

4-6 Reporting Regression Results

4-7 Revisiting Causal Effects and Policy Analysis

Summary

Key Terms

Problems

Computer Exercises

**Chapter 5 Multiple Regression Analysis: OLS Asymptotics**

5-1 Consistency

5-1a Deriving the Inconsistency in OLS

5-2 Asymptotic Normality and Large Sample Inference

5-2a Other Large Sample Tests: The Lagrange Multiplier Statistic

5-3 Asymptotic Efficiency of OLS

Summary

Key Terms

Problems

Computer Exercises

**Chapter 6 Multiple Regression Analysis: Further Issues**

6-1 Effects of Data Scaling on OLS Statistics

6-1a Beta Coefficients

6-2 More on Functional Form

6-2a More on Using Logarithmic Functional Forms

6-2b Models with Quadratics

6-2c Models with Interaction Terms

6-3d Computing Average Partial Effects

6-3 More on Goodness-of-Fit and Selection of Regressors

6-3a Adjusted *R*-Squared

6-3b Using Adjusted *R*-Squared to Choose between Nonnested Models

6-3c Controlling for Too Many Factors in Regression Analysis

6-3d Adding Regressors to Reduce the Error Variance

6-4 Prediction and Residual Analysis

6-4a Confidence Intervals for Predictions

6-4b Residual Analysis

6-4c Predicting *y* When *log*(*y*) Is the Dependent Variable

6-4d Predicting *y* When the Dependent Variable Is *log*(*y*):

Summary

Key Terms

Problems

Computer Exercises

**Chapter 7 Multiple Regression Analysis with Qualitative Information**

7-1 Describing Qualitative Information

7-2 A Single Dummy Independent Variable

7-2a Interpreting Coefficients on Dummy Explanatory Variables When the Dependent Variable Is log(*y*)

7-3 Using Dummy Variables for Multiple Categories

7-3a Incorporating Ordinal Information by Using Dummy Variables

7-4 Interactions Involving Dummy Variables

7-4a Interactions among Dummy Variables

7-4b Allowing for Different Slopes

7-4c Testing for Differences in Regression Functions across Groups

7-5 A Binary Dependent Variable: The Linear Probability Model

7-6 More on Policy Analysis and Program Evaluation

7-6a Program Evaluation and Unrestricted Regression Adjustment

7-7 Interpreting Regression Results with Discrete Dependent Variables

Summary

Key Terms

Problems

Computer Exercises

**Chapter 8 Heteroskedasticity**

8-1 Consequences of Heteroskedasticity for OLS

8-2 Heteroskedasticity-Robust Inference after OLS Estimation

8-2a Computing Heteroskedasticity-Robust LM Tests

8-3 Testing for Heteroskedasticity

8-3a The White Test for Heteroskedasticity

8-4 Weighted Least Squares Estimation

8-4a The Heteroskedasticity Is Known up to a Multiplicative Constant

8-4b The Heteroskedasticity Function Must Be Estimated: Feasible GLS

8-4c What If the Assumed Heteroskedasticity Function Is Wrong?

8-4d Prediction and Prediction Intervals with Heteroskedasticity

8-5 The Linear Probability Model Revisited

Summary

Key Terms

Problems

Computer Exercises

**Chapter 9 More on Specification and Data Issues**

9-1 Functional Form Misspecification

9-1a RESET as a General Test for Functional Form Misspecification

9-1b Tests against Nonnested Alternatives

9-2 Using Proxy Variables for Unobserved Explanatory Variables

9-2a Using Lagged Dependent Variables as Proxy Variables

9-2b A Different Slant on Multiple Regression

9-2c Potential Outcomes and Proxy Variables

9-3 Models with Random Slopes

9-4 Properties of OLS under Measurement Error

9-4a Measurement Error in the Dependent Variable

9-4b Measurement Error in an Explanatory Variable

9-5 Missing Data, Nonrandom Samples, and Outlying Observations

9-5a Missing Data

9-5b Nonrandom Samples

9-5c Outliers and Influential Observations

9-6 Least Absolute Deviations Estimation

Summary

Key Terms

Problems

Computer Exercises

**Part 2 Regression Analysis with Time Series Data**

**Chapter 10 Basic Regression Analysis with Time Series Data**

10-1 The Nature of Time Series Data

10-2 Examples of Time Series Regression Models

10-2a Static Models

10-2b Finite Distributed Lag Models

10-2c A Convention about the Time Index

10-3 Finite Sample Properties of OLS under Classical Assumptions

10-3a Unbiasedness of OLS

10-3b The Variances of the OLS Estimators and the Gauss-Markov Theorem

10-3c Inference under the Classical Linear Model Assumptions

10-4 Functional Form, Dummy Variables, and Index Numbers

10-5 Trends and Seasonality

10-5a Characterizing Trending Time Series

10-5b Using Trending Variables in Regression Analysis

10-5c A Detrending Interpretation of Regressions with a Time Trend

10-5d Computing *R*-Squared When the Dependent Variable Is Trending

10-5e Seasonality

Summary

Key Terms

Problems

Computer Exercises

**Chapter 11 Further Issues in Using OLS with Time Series Data**

11-1 Stationary and Weakly Dependent Time Series

11-1a Stationary and Nonstationary Time Series

11-1b Weakly Dependent Time Series

11-2 Asymptotic Properties of OLS

11-3 Using Highly Persistent Time Series in Regression Analysis

11-3a Highly Persistent Time Series

11-3b Transformations on Highly Persistent Time Series

11-3c Deciding Whether a Time Series Is I(1)

11-4 Dynamically Complete Models and the Absence of Serial Correlation

11-5 The Homoskedasticity Assumption for Time Series Models

Summary

Key Terms

Problems

Computer Exercises

**Chapter 12 Serial Correlation and Heteroskedasticity in Time Series Regressions**

12-1 Properties of OLS with Serially Correlated Errors

12-1a Unbiasedness and Consistency

12-1b Efficiency and Inference

12-1c Goodness-of-Fit

12-1d Serial Correlation in the Presence of Lagged Dependent Variables

12-2 Serial Correlation-Robust Inference after OLS

12-3 Testing for Serial Correlation

12-3a A

*t* Test for AR(1) Serial Correlation with Strictly Exogenous Regressors

12-3b The Durbin-Watson Test under Classical Assumptions

12-3c Testing for AR(1) Serial Correlation without Strictly Exogenous Regressors

12-3d Testing for Higher-Order Serial Correlation

12-4 Correcting for Serial Correlation with Strictly Exogenous Regressors

12-4a Obtaining the Best Linear Unbiased Estimator in the AR(1) Model

12-4b Feasible GLS Estimation with AR(1) Errors

12-4c Comparing OLS and FGLS

12-4d Correcting for Higher-Order Serial Correlation

12-4e What if the Serial Correlation Models Is Wrong?

12-5 Differencing and Serial Correlation

12-6 Heteroskedasticity in Time Series Regressions

12-6a Heteroskedasticity-Robust Statistics

12-6b Testing for Heteroskedasticity

12-6c Autoregressive Conditional Heteroskedasticity

12-6d Heteroskedasticity and Serial Correlation in Regression Models

Summary

Key Terms

Problems

Computer Exercises

**Part 3 Advanced Topics**

**Chapter 13 Pooling Cross Sections across Time: Simple Panel Data Methods**

13-1 Pooling Independent Cross Sections across Time

13-1a The Chow Test for Structural Change across Time

13-2 Policy Analysis with Pooled Cross Sections

13-2a Adding an Additional Control Group

13-2b A General Framework for Policy Analysis with Pooled Cross Sections

13-3 Two-Period Panel Data Analysis

13-3a Organizing Panel Data

13-4 Policy Analysis with Two-Period Panel Data

13-5 Differencing with More Than Two Time Periods

13-5a Potential Pitfalls in First Differencing Panel Data

Summary

Key Terms

Problems

Computer Exercises

**Chapter 14 Advanced Panel Data Methods**

14-1 Fixed Effects Estimation

14-1a The Dummy Variable Regression

14-1b Fixed Effects or First Differencing?

14-1c Fixed Effects with Unbalanced Panels

14-2 Random Effects Models

14-2a Random Effects of Pooled OLS?

14-2b Random Effects or Fixed Effects?

14-3 The Correlated Random Effects Approach

14-3a Unbalanced Panels

14-4 General Policy Analysis with Panel Data

14-4a Advanced Considerations with Policy Analysis

14-5 Applying Panel Data Methods to Other Data Structures

Summary

Key Terms

Problems

Computer Exercises

**Chapter 15 Instrumental Variables Estimation and Two-Stage Least Squares**

15-1 Motivation: Omitted Variables in a Simple Regression Model

15-1a Statistical Inference with the IV Estimator

15-1b Properties of IV with a Poor Instrumental Variable

15-1c Computing *R*-Squared after IV Estimation

15-2 IV Estimation of the Multiple Regression Model

15-3 Two-Stage Least Squares

15-3a A Single Endogenous Explanatory Variable

15-3b Multicollinearity and 2SLS

15-3c Detecting Weak Instruments

15-3d Multiple Endogenous Explanatory Variables

15-3e Testing Multiple Hypotheses after 2SLS Estimation

15-4 IV Solutions to Errors-in-Variables Problems

15-5 Testing for Endogeneity and Testing Overidentifying Restrictions

15-5a Testing for Endogeneity

15-5b Testing Overidentification Restrictions

15-6 2SLS with Heteroskedasticity

15-7 Applying 2SLS to Time Series Equations

15-8 Applying 2SLS to Pooled Cross Sections and Panel Data

Summary

Key Terms

Problems

Computer Exercises

**Chapter 16 Simultaneous Equations Models**

16-1 The Nature of Simultaneous Equations Models

16-2 Simultaneity Bias in OLS

16-3 Identifying and Estimating a Structural Equation

16-3a Identification in a Two-Equation System

16-3b Estimation by 2SLS

16-4 Systems with More Than Two Equations

16-4a Identification in Systems with Three or More Equations

16-4b Estimation

16-5 Simultaneous Equations Models with Time Series

16-6 Simultaneous Equations Models with Panel Data

Summary

Key Terms

Problems

Computer Exercises

**Chapter 17 Limited Dependent Variable Models and Sample Selection Corrections**

17-1 Logit and Probit Models for Binary Response

17-1a Specifying Logit and Probit Models

17-1b Maximum Likelihood Estimation of Logit and Probit Models

17-1c Testing Multiple Hypotheses

17-1d Interpreting the Logit and Probit Estimates

17-2 The Tobit Model for Corner Solution Responses

17-2a Interpreting the Tobit Estimates

17-2b Specification Issues in Tobit Models

17-3 The Poisson Regression Model

17-4 Censored and Truncated Regression Models

17-4a Censored Regression Models

17-4b Truncated Regression Models

17-5 Sample Selection Corrections

17-5a When Is OLS on the Selected Sample Consistent?

17-5b Incidental Truncation

Summary

Key Terms

Problems

Computer Exercises

**Chapter 18 Advanced Time Series Topics**

18-1 Infinite Distributed Lag Models

18-1a The Geometric (or Koyck) Distributed Lag Model

18-1b Rational Distributed Lag Models

18-2 Testing for Unit Roots

18-3 Spurious Regression

18-4 Cointegration and Error Correction Models

18-4a Cointegration

18-4b Error Correction Models

18-5 Forecasting

18-5a Types of Regression Models Used for Forecasting

18-5b One-Step-Ahead Forecasting

18-5c Comparing One-Step-Ahead Forecasts

18-5d Multiple-Step-Ahead Forecasts

18-5e Forecasting Trending, Seasonal, and Integrated Processes

Summary

Key Terms

Problems

Computer Exercises

**Chapter 19 Carrying Out an Empirical Project**

19-1 Posing a Question

19-2 Literature Review

19-3 Data Collection

19-3a Deciding on the Appropriate Data Set

19-3b Entering and Storing Your Data

19-3c Inspecting, Cleaning, and Summarizing Your Data

19-4 Econometric Analysis

19-5 Writing an Empirical Paper

19-5a Introduction

19-5b Conceptual (or Theoretical) Framework

19-5c Econometric Models and Estimation Methods

19-5d The Data

19-5e Results

19-5f Conclusions

19-5g Style Hints

Summary

Key Terms

Sample Empirical Projects

List of Journals

Data Sources

**MATH REFRESHER A Basic Mathematical Tools**

A-1 The Summation Operator and Descriptive Statistics

A-2 Properties of Linear Functions

A-3 Proportions and Percentages

A-4 Some Special Functions and Their Properties

A-4a Quadratic Functions

A-4b The Natural Logarithm

A-4c The Exponential Function

A-5 Differential Calculus

Summary

Key Terms

Problems

**MATH REFRESHER B Fundamentals of Probability**

B-1 Random Variables and Their Probability Distributions

B-1a Discrete Random Variables

B-1b Continuous Random Variables

B-2 Joint Distributions, Conditional Distributions, and Independence

B-2a Joint Distributions and Independence

B-2b Conditional Distributions

B-3 Features of Probability Distributions

B-3a A Measure of Central Tendency: The Expected Value

B-3b Properties of Expected Values

B-3c Another Measure of Central Tendency: The Median

B-3d Measures of Variability: Variance and Standard Deviation

B-3e Variance

B-3f Standard Deviation

B-3g Standardizing a Random Variable

B-3h Skewness and Kurtosis

B-4 Features of Joint and Conditional Distributions

B-4a Measures of Association: Covariance and Correlation

B-4b Covariance

B-4c Correlation Coefficient

B-4d Variance of Sums of Random Variables

B-4e Conditional Expectation

B-4f Properties of Conditional Expectation

B-4g Conditional Variance

B-5 The Normal and Related Distributions

B-5a The Normal Distribution

B-5b The Standard Normal Distribution

B-5c Additional Properties of the Normal Distribution

B-5d The Chi-Square Distribution

B-5e The *t* Distribution

B-5f The *F* Distribution

Summary

Key Terms

Problems

**MATH REFRESHER C Fundamentals of Mathematical Statistics**

C-1 Populations, Parameters, and Random Sampling

C-1a Sampling

C-2 Finite Sample Properties of Estimators

C-2a Estimators and Estimates

C-2b Unbiasedness

C-2c The Sampling Variance of Estimators

C-2d Efficiency

C-3 Asymptotic or Large Sample Properties of Estimators

C-3a Consistency

C-3b Asymptotic Normality

C-4 General Approaches to Parameter Estimation

C-4a Method of Moments

C-4b Maximum Likelihood

C-4c Least Squares

C-5 Interval Estimation and Confidence Intervals

C-5a The Nature of Interval Estimation

C-5b Confidence Intervals for the Mean from a Normally Distributed Population

C-5c A Simple Rule of Thumb for a 95% Confidence Interval

C-5d Asymptotic Confidence Intervals for Nonnormal Populations

C-6 Hypothesis Testing

C-6a Fundamentals of Hypothesis Testing

C-6b Testing Hypotheses about the Mean in a Normal Population

C-6c Asymptotic Tests for Nonnormal Populations

C-6d Computing and Using *p*-Values

C-6e The Relationship between Confidence Intervals and Hypothesis Testing

C-6f Practical versus Statistical Significance

C-7 Remarks on Notation

Summary

Key Terms

Problems

**ADVANCED TREATMENT D Summary of Matrix Algebra**

D-1 Basic Definitions

D-2 Matrix Operations

D-2a Matrix Addition

D-2b Scalar Multiplication

D-2c Matrix Multiplication

D-2d Transpose

D-2e Partitioned Matrix Multiplication

D-2f Trace

D-2g Inverse

D-3 Linear Independence and Rank of a Matrix

D-4 Quadratic Forms and Positive Definite Matrices

D-5 Idempotent Matrices

D-6 Differentiation of Linear and Quadratic Forms

D-7 Moments and Distributions of Random Vectors

D-7a Expected Value

D-7b Variance-Covariance Matrix

D-7c Multivariate Normal Distribution

D-7d Chi-Square Distribution

D-7e *t* Distribution

D-7f *F* Distribution

Summary

Key Terms

Problems

**ADVANCED TREATMENT E The Linear Regression Model in Matrix Form**

E-1 The Model and Ordinary Least Squares Estimation

E-1a The Frisch-Waugh Theorem

E-2 Finite Sample Properties of OLS

E-3 Statistical Inference

E-4 Some Asymptotic Analysis

E-4a Wald Statistics for Testing Multiple Hypotheses

Summary

Key Terms

Problems

Answers to Going Further Questions

Statistical Tables

References

Glossary

Index