Preface

About the Author

**Chapter 1 The Nature of Econometrics and Economic Data**

1.1 What Is Econometrics?

1.2 Steps in Empirical Economic Analysis

1.3 The Structure of Economic Data

Cross-Sectional Data

Time Series Data

Pooled Cross Sections

Panel or Longitudinal Data

A Comment on Data Structures

1.4 Causality and the Notion of Ceteris Paribus in Econometric Analysis

Summary

Key Terms

Problems

Computer Exercises

**PART 1 Regression Analysis with Cross-Sectional Data**

**Chapter 2 The Simple Regression Model**

2.1 Definition of the Simple Regression Model

2.2 Deriving the Ordinary Least Squares Estimates

A Note on Terminology

2.3 Properties of OLS on Any Sample of Data

Fitted Values and Residuals

Algebraic Properties of OLS Statistics

Goodness-of-Fit

2.4 Units of Measurement and Functional Form

The Effects of Changing Units of Measurement on OLS Statistics

Incorporating Nonlinearities in Simple Regression

The Meaning of "Linear" Regression

2.5 Expected Values and Variances of the OLS Estimators

Unbiasedness of OLS

Variances of the OLS Estimators

Estimating the Error Variance

2.6 Regression through the Origin and Regression on a Constant

Summary

Key Terms

Problems

Computer Exercises

Appendix 2A

**Chapter 3 Multiple Regression Analysis: Estimation**

3.1 Motivation for Multiple Regression

The Model with Two Independent Variables

The Model with *k* Independent Variables

3.2 Mechanics and Interpretation of Ordinary Least Squares

Obtaining the OLS Estimates

Interpreting the OLS Regression Equation

On the Meaning of "Holding Other Factors Fixed" in Multiple Regression

Changing More Than One Independent Variable Simultaneously

OLS Fitted Values and Residuals

A "Partialling Out" Interpretation of Multiple Regression

Comparison of Simple and Multiple Regression Estimates

Goodness-of-Fit

Regression through the Origin

3.3 The Expected Value of the OLS Estimators

Including Irrelevant Variables in a Regression Model

Omitted Variable Bias: The Simple Case

Omitted Variable Bias: More General Cases

3.4 The Variance of the OLS Estimators

The Components of the OLS Variances

Multicollinearity

Variances in Misspecified Models

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

Summary

Key Terms

Problems

Computer Exercises

Appendix 3A

**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

Testing against One-Sided Alternatives

Two-Sided Alternatives

Testing Other Hypotheses about β_{j}

Computing *p*-Values for *t* Tests

A Reminder on the Language of Classical Hypothesis Testing

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

Testing Exclusion Restrictions

Relationship between *F* and *t* Statistics

The *R*-Squared Form of the *F* Statistic

Computing *p*-Values for *F* Tests

The *F* Statistic for Overall Significance of a Regression

Testing General Linear Restrictions

4.6 Reporting Regression Results

Summary

Key Terms

Problems

Computer Exercises

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

5.1 Consistency

Deriving the Inconsistency in OLS

5.2 Asymptotic Normality and Large Sample Inference

Other Large Sample Tests: The Lagrange Multiplier Statistic

5.3 Asymptotic Efficiency of OLS

Summary

Key Terms

Problems

Computer Exercises

Appendix 5A

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

6.1 Effects of Data Scaling on OLS Statistics

Beta Coefficients

6.2 More on Functional Form

More on Using Logarithmic Functional Forms

Models with Quadratics

Models with Interaction Terms

Computing Average Partial Effects

6.3 More on Goodness-of-Fit and Selection of Regressors

Adjusted *R*-Squared

Using Adjusted *R*-Squared to Choose between Nonnested Models

Controlling for Too Many Factors in Regression Analysis

Adding Regressors to Reduce the Error Variance

6.4 Prediction and Residual Analysis

Confidence Intervals for Predictions

Residual Analysis

Predicting *y* When *log*(*y*) Is the Dependent Variable

Predicting *y* When the Dependent Variable Is *log*(*y*):

Summary

Key Terms

Problems

Computer Exercises

Appendix 6A

**Chapter 7 Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables**

7.1 Describing Qualitative Information

7.2 A Single Dummy Independent Variable

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

7.3 Using Dummy Variables for Multiple Categories

Incorporating Ordinal Information by Using Dummy Variables

7.4 Interactions Involving Dummy Variables

Interactions among Dummy Variables

Allowing for Different Slopes

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

7.7 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

Computing Heteroskedasticity-Robust LM Tests

8.3 Testing for Heteroskedasticity

The White Test for Heteroskedasticity

8.4 Weighted Least Squares Estimation

The Heteroskedasticity Is Known up to a Multiplicative Constant

The Heteroskedasticity Function Must Be Estimated: Feasible GLS

What If the Assumed Heteroskedasticity Function Is Wrong?

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

RESET as a General Test for Functional Form Misspecification

Tests against Nonnested Alternatives

9.2 Using Proxy Variables for Unobserved Explanatory Variables

Using Lagged Dependent Variables as Proxy Variables

A Different Slant on Multiple Regression

9.3 Models with Random Slopes

9.4 Properties of OLS under Measurement Error

Measurement Error in the Dependent Variable

Measurement Error in an Explanatory Variable

9.5 Missing Data, Nonrandom Samples, and Outlying Observations

Missing Data

Nonrandom Samples

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

Static Models

Finite Distributed Lag Models

A Convention about the Time Index

10.3 Finite Sample Properties of OLS under Classical Assumptions

Unbiasedness of OLS

The Variances of the OLS Estimators and the Gauss-Markov Theorem

Inference under the Classical Linear Model Assumptions

10.4 Functional Form, Dummy Variables, and Index Numbers

10.5 Trends and Seasonality

Characterizing Trending Time Series

Using Trending Variables in Regression Analysis

A Detrending Interpretation of Regressions with a Time Trend

Computing *R*-Squared When the Dependent Variable Is Trending

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

Stationary and Nonstationary Time Series

Weakly Dependent Time Series

11.2 Asymptotic Properties of OLS

11.3 Using Highly Persistent Time Series in Regression Analysis

Highly Persistent Time Series

Transformations on Highly Persistent Time Series

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

Unbiasedness and Consistency

Efficiency and Inference

Goodness-of-Fit

Serial Correlation in the Presence of Lagged Dependent Variables

12.2 Testing for Serial Correlation

A

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

The Durbin-Watson Test under Classical Assumptions

Testing for AR(1) Serial Correlation without Strictly Exogenous Regressors

Testing for Higher Order Serial Correlation

12.3 Correcting for Serial Correlation with Strictly Exogenous Regressors

Obtaining the Best Linear Unbiased Estimator in the AR(1) Model

Feasible GLS Estimation with AR(1) Errors

Comparing OLS and FGLS

Correcting for Higher Order Serial Correlation

12.4 Differencing and Serial Correlation

12.5 Serial Correlation-Robust Inference after OLS

12.6 Heteroskedasticity in Time Series Regressions

Heteroskedasticity-Robust Statistics

Testing for Heteroskedasticity

Autoregressive Conditional Heteroskedasticity

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

The Chow Test for Structural Change across Time

13.2 Policy Analysis with Pooled Cross Sections

13.3 Two-Period Panel Data Analysis

Organizing Panel Data

13.4 Policy Analysis with Two-Period Panel Data

13.5 Differencing with More Than Two Time Periods

Potential Pitfalls in First Differencing Panel Data

Summary

Key Terms

Problems

Computer Exercises

Appendix 13A

**Chapter 14 Advanced Panel Data Methods**

14.1 Fixed Effects Estimation

The Dummy Variable Regression

Fixed Effects or First Differencing?

Fixed Effects with Unbalanced Panels

14.2 Random Effects Models

Random Effects or Fixed Effects?

14.3 The Correlated Random Effects Approach

Unbalanced Panels

14.4 Applying Panel Data Methods to Other Data Structures

Summary

Key Terms

Problems

Computer Exercises

Appendix 14A

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

15.1 Motivation: Omitted Variables in a Simple Regression Model

Statistical Inference with the IV Estimator

Properties of IV with a Poor Instrumental Variable

Computing *R*-Squared after IV Estimation

15.2 IV Estimation of the Multiple Regression Model

15.3 Two Stage Least Squares

A Single Endogenous Explanatory Variable

Multicollinearity and 2SLS

Detecting Weak Instruments

Multiple Endogenous Explanatory Variables

Testing Multiple Hypotheses after 2SLS Estimation

15.4 IV Solutions to Errors-in-Variables Problems

15.5 Testing for Endogeneity and Testing Overidentifying Restrictions

Testing for Endogeneity

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

Appendix 15A

**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

Identification in a Two-Equation System

Estimation by 2SLS

16.4 Systems with More Than Two Equations

Identification in Systems with Three or More Equations

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

Specifying Logit and Probit Models

Maximum Likelihood Estimation of Logit and Probit Models

Testing Multiple Hypotheses

Interpreting the Logit and Probit Estimates

17.2 The Tobit Model for Corner Solution Responses

Interpreting the Tobit Estimates

Specification Issues in Tobit Models

17.3 The Poisson Regression Model

17.4 Censored and Truncated Regression Models

Censored Regression Models

Truncated Regression Models

17.5 Sample Selection Corrections

When Is OLS on the Selected Sample Consistent?

Incidental Truncation

Summary

Key Terms

Problems

Computer Exercises

Appendix 17A

Appendix 17B

**Chapter 18 Advanced Time Series Topics**

18.1 Infinite Distributed Lag Models

The Geometric (or Koyck) Distributed Lag

Rational Distributed Lag Models

18.2 Testing for Unit Roots

18.3 Spurious Regression

18.4 Cointegration and Error Correction Models

Cointegration

Error Correction Models

18.5 Forecasting

Types of Regression Models Used for Forecasting

One-Step-Ahead Forecasting

Comparing One-Step-Ahead Forecasts

Multiple-Step-Ahead Forecasts

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

Deciding on the Appropriate Data Set

Entering and Storing Your Data

Inspecting, Cleaning, and Summarizing Your Data

19.4 Econometric Analysis

19.5 Writing an Empirical Paper

Introduction

Conceptual (or Theoretical) Framework

Econometric Models and Estimation Methods

The Data

Results

Conclusions

Style Hints

Summary

Key Terms

Sample Empirical Projects

List of Journals

Data Sources

**APPENDIX 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

Quadratic Functions

The Natural Logarithm

The Exponential Function

A.5 Differential Calculus

Summary

Key Terms

Problems

**APPENDIX B Fundamentals of Probability**

B.1 Random Variables and Their Probability Distributions

Discrete Random Variables

Continuous Random Variables

B.2 Joint Distributions, Conditional Distributions, and Independence

Joint Distributions and Independence

Conditional Distributions

B.3 Features of Probability Distributions

A Measure of Central Tendency: The Expected Value

Properties of Expected Values

Another Measure of Central Tendency: The Median

Measures of Variability: Variance and Standard Deviation

Variance

Standard Deviation

Standardizing a Random Variable

Skewness and Kurtosis

B.4 Features of Joint and Conditional Distributions

Measures of Association: Covariance and Correlation

Covariance

Correlation Coefficient

Variance of Sums of Random Variables

Conditional Expectation

Properties of Conditional Expectation

Conditional Variance

B.5 The Normal and Related Distributions

The Normal Distribution

The Standard Normal Distribution

Additional Properties of the Normal Distribution

The Chi-Square Distribution

The *t* Distribution

The *F* Distribution

Summary

Key Terms

Problems

**APPENDIX C Fundamentals of Mathematical Statistics**

C.1 Populations, Parameters, and Random Sampling

Sampling

C.2 Finite Sample Properties of Estimators

Estimators and Estimates

Unbiasedness

The Sampling Variance of Estimators

Efficiency

C.3 Asymptotic or Large Sample Properties of Estimators

Consistency

Asymptotic Normality

C.4 General Approaches to Parameter Estimation

Method of Moments

Maximum Likelihood

Least Squares

C.5 Interval Estimation and Confidence Intervals

The Nature of Interval Estimation

Confidence Intervals for the Mean from a Normally Distributed Population

A Simple Rule of Thumb for a 95% Confidence Interval

Asymptotic Confidence Intervals for Nonnormal Populations

C.6 Hypothesis Testing

Fundamentals of Hypothesis Testing

Testing Hypotheses about the Mean in a Normal Population

Asymptotic Tests for Nonnormal Populations

Computing and Using *p*-Values

The Relationship between Confidence Intervals and Hypothesis Testing

Practical versus Statistical Significance

C.7 Remarks on Notation

Summary

Key Terms

Problems

**APPENDIX D Summary of Matrix Algebra**

D.1 Basic Definitions

D.2 Matrix Operations

Matrix Addition

Scalar Multiplication

Matrix Multiplication

Transpose

Partitioned Matrix Multiplication

Trace

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

Expected Value

Variance-Covariance Matrix

Multivariate Normal Distribution

Chi-Square Distribution

*t* Distribution

*F* Distribution

Summary

Key Terms

Problems

**APPENDIX E The Linear Regression Model in Matrix Form**

E.1 The Model and Ordinary Least Squares Estimation

The Frisch-Waugh Theorem

E.2 Finite Sample Properties of OLS

E.3 Statistical Inference

E.4 Some Asymptotic Analysis

Wald Statistics for Testing Multiple Hypotheses

Summary

Key Terms

Problems

**APPENDIX F Answers to Chapter Questions**

**APPENDIX G Statistical Tables**

References

Glossary

Index