2.1 An Economic Model
2.2 An Econometric Model
2.2.1 Data Generating Process
2.2.2 The Random Error and Strict Exogeneity
2.2.3 The Regression Function
2.2.4 Random Error Variation
2.2.5 Variation in x
2.2.6 Error Normality
2.2.7 Generalizing the Exogeneity Assumption
2.2.8 Error Correlation
2.2.9 Summarizing the Assumptions
2.3 Estimating the Regression Parameters
2.3.1 The Least Squares Principle
2.3.2 Other Economic Models
2.4 Assessing the Least Squares Estimators
2.4.1 The Estimator b2
2.4.2 The Expected Values of b1 and b2
2.4.3 Sampling Variation
2.4.4 The Variances and Covariance of b1 and b2
2.5 The Gauss–Markov Theorem
2.6 The Probability Distributions of the Least Squares Estimators
2.7 Estimating the Variance of the Error Term
2.7.1 Estimating the Variances and Covariance of the Least Squares Estimators
2.7.2 Interpreting the Standard Errors
2.8 Estimating Nonlinear Relationships
2.8.1 Quadratic Functions
2.8.2 Using a Quadratic Model
2.8.3 A Log-Linear Function
2.8.4 Using a Log-Linear Model
2.8.5 Choosing a Functional Form
2.9 Regression with Indicator Variables
2.10 The Independent Variable
2.10.1 Random and Independent x
2.10.2 Random and Strictly Exogenous x
2.10.3 Random Sampling
2.11 Exercises
2.11.1 Problems
2.11.2 Computer Exercises
Appendix 2A Derivation of the Least Squares Estimates
Appendix 2B Deviation from the Mean Form of
b2
Appendix 2C
b2 Is a Linear Estimator
Appendix 2D Derivation of Theoretical Expression for
b2
Appendix 2E Deriving the Conditional Variance of
b2
Appendix 2F Proof of the Gauss–Markov Theorem
Appendix 2G Proofs of Results Introduced in Section 2.10
2G.1 The Implications of Strict Exogeneity
2G.2 The Random and Independent x Case
2G.3 The Random and Strictly Exogenous x Case
2G.4 Random Sampling
Appendix 2H Monte Carlo Simulation
2H.1 The Regression Function
2H.2 The Random Error
2H.3 Theoretically True Values
2H.4 Creating a Sample of Data
2H.5 Monte Carlo Objectives
2H.6 Monte Carlo Results
2H.7 Random-x Monte Carlo Results
5.1 Introduction
5.1.1 The Economic Model
5.1.2 The Econometric Model
5.1.3 The General Model
5.1.4 Assumptions of the Multiple Regression Model
5.2 Estimating the Parameters of the Multiple Regression Model
5.2.1 Least Squares Estimation Procedure
5.2.2 Estimating the Error Variance σ2
5.2.3 Measuring Goodness-of-Fit
5.2.4 Frisch–Waugh–Lovell (FWL) Theorem
5.3 Finite Sample Properties of the Least Squares Estimator
5.3.1 The Variances and Covariances of the Least Squares Estimators
5.3.2 The Distribution of the Least Squares Estimators
5.4 Interval Estimation
5.4.1 Interval Estimation for a Single Coefficient
5.4.2 Interval Estimation for a Linear Combination of Coefficients
5.5 Hypothesis Testing
5.5.1 Testing the Significance of a Single Coefficient
5.5.2 One-Tail Hypothesis Testing for a Single Coefficient
5.5.3 Hypothesis Testing for a Linear Combination of Coefficients
5.6 Nonlinear Relationships
5.7 Large Sample Properties of the Least Squares Estimator
5.7.1 Consistency
5.7.2 Asymptotic Normality
5.7.3 Relaxing Assumptions
5.7.4 Inference for a Nonlinear Function of Coefficients
5.8 Exercises
5.8.1 Problems
5.8.2 Computer Exercises
Appendix 5A Derivation of Least Squares Estimators
Appendix 5B The Delta Method
5B.1 Nonlinear Function of a Single Parameter
5B.2 Nonlinear Function of Two Parameters
Appendix 5C Monte Carlo Simulation
5C.1 Least Squares Estimation with Chi-Square Errors
5C.2 Monte Carlo Simulation of the Delta Method
Appendix 5D Bootstrapping
5D.1 Resampling
5D.2 Bootstrap Bias Estimate
5D.3 Bootstrap Standard Error
5D.4 Bootstrap Percentile Interval Estimate
5D.5 Asymptotic Refinement
7.1 Indicator Variables
7.1.1 Intercept Indicator Variables
7.1.2 Slope-Indicator Variables
7.2 Applying Indicator Variables
7.2.1 Interactions Between Qualitative Factors
7.2.2 Qualitative Factors with Several Categories
7.2.3 Testing the Equivalence of Two Regressions
7.2.4 Controlling for Time
7.3 Log-Linear Models
7.3.1 A Rough Calculation
7.3.2 An Exact Calculation
7.4 The Linear Probability Model
7.5 Treatment Effects
7.5.1 The Difference Estimator
7.5.2 Analysis of the Difference Estimator
7.5.3 The Differences-in-Differences Estimator
7.6 Treatment Effects and Causal Modeling
7.6.1 The Nature of Causal Effects
7.6.2 Treatment Effect Models
7.6.3 Decomposing the Treatment Effect
7.6.4 Introducing Control Variables
7.6.5 The Overlap Assumption
7.6.6 Regression Discontinuity Designs
7.7 Exercises
7.7.1 Problems
7.7.2 Computer Exercises
Appendix 7A Details of Log-Linear Model Interpretation
Appendix 7B Derivation of the Differences-in-Differences Estimator
Appendix 7C The Overlap Assumption: Details
8 Heteroskedasticity
8.1 The Nature of Heteroskedasticity
8.2 Heteroskedasticity in the Multiple Regression Model
8.2.1 The Heteroskedastic Regression Model
8.2.2 Heteroskedasticity Consequences for the OLS Estimator
8.3 Heteroskedasticity Robust Variance Estimator
8.4 Generalized Least Squares: Known Form of Variance
8.4.1 Transforming the Model: Proportional Heteroskedasticity
8.4.2 Weighted Least Squares: Proportional Heteroskedasticity
8.5 Generalized Least Squares: Unknown Form of Variance
8.5.1 Estimating the Multiplicative Model
8.6 Detecting Heteroskedasticity
8.6.1 Residual Plots
8.6.2 The Goldfeld–Quandt Test
8.6.3 A General Test for Conditional Heteroskedasticity
8.6.4 The White Test
8.6.5 Model Specification and Heteroskedasticity
8.7 Heteroskedasticity in the Linear Probability Model
8.8 Exercises
8.8.1 Problems
8.8.2 Computer Exercises
Appendix 8A Properties of the Least Squares Estimator
Appendix 8B Lagrange Multiplier Tests for Heteroskedasticity
Appendix 8C Properties of the Least Squares Residuals
8C.1 Details of Multiplicative Heteroskedasticity Model
Appendix 8D Alternative Robust Sandwich Estimators
Appendix 8E Monte Carlo Evidence: OLS, GLS, and FGLS
9 Regression with Time-Series Data: Stationary Variables
9.1 Introduction
9.1.1 Modeling Dynamic Relationships
9.1.2 Autocorrelations
9.2 Stationarity and Weak Dependence
9.3 Forecasting
9.3.1 Forecast Intervals and Standard Errors
9.3.2 Assumptions for Forecasting
9.3.3 Selecting Lag Lengths
9.3.4 Testing for Granger Causality
9.4 Testing for Serially Correlated Errors
9.4.1 Checking the Correlogram of the Least Squares Residuals
9.4.2 Lagrange Multiplier Test
9.4.3 Durbin–Watson Test
9.5 Time-Series Regressions for Policy Analysis
9.5.1 Finite Distributed Lags
9.5.2 HAC Standard Errors
9.5.3 Estimation with AR(1) Errors
9.5.4 Infinite Distributed Lags
9.6 Exercises
9.6.1 Problems
9.6.2 Computer Exercises
Appendix 9A The Durbin–Watson Test
9A.1 The Durbin–Watson Bounds Test
Appendix 9B Properties of an AR(1) Error
10 Endogenous Regressors and Moment-Based Estimation
10.1 Least Squares Estimation with Endogenous Regressors
10.1.1 Large Sample Properties of the OLS Estimator
10.1.2 Why Least Squares Estimation Fails
10.1.3 Proving the Inconsistency of OLS
10.2 Cases in Which
x and
e are Contemporaneously Correlated
10.2.1 Measurement Error
10.2.2 Simultaneous Equations Bias
10.2.3 Lagged-Dependent Variable Models with Serial Correlation
10.2.4 Omitted Variables
10.3 Estimators Based on the Method of Moments
10.3.1 Method of Moments Estimation of a Population Mean and Variance
10.3.2 Method of Moments Estimation in the Simple Regression Model
10.3.3 Instrumental Variables Estimation in the Simple Regression Model
10.3.4 The Importance of Using Strong Instruments
10.3.5 Proving the Consistency of the IV Estimator
10.3.6 IV Estimation Using Two-Stage Least Squares (2SLS)
10.3.7 Using Surplus Moment Conditions
10.3.8 Instrumental Variables Estimation in the Multiple Regression Model
10.3.9 Assessing Instrument Strength Using the First-Stage Model
10.3.10 Instrumental Variables Estimation in a General Model
10.3.11 Additional Issues When Using IV Estimation
10.4 Specification Tests
10.4.1 The Hausman Test for Endogeneity
10.4.2 The Logic of the Hausman Test
10.4.3 Testing Instrument Validity
10.5 Exercises
10.5.1 Problems
10.5.2 Computer Exercises
Appendix 10A Testing for Weak Instruments
10A.1 A Test for Weak Identification
10A.2 Testing for Weak Identification: Conclusions
Appendix 10B Monte Carlo Simulation
10B.1 Illustrations Using Simulated Data
10B.2 The Sampling Properties of IV/2SLS
11 Simultaneous Equations Models
11.1 A Supply and Demand Model
11.2 The Reduced-Form Equations
11.3 The Failure of Least Squares Estimation
11.3.1 Proving the Failure of OLS
11.4 The Identification Problem
11.5 Two-Stage Least Squares Estimation
11.5.1 The General Two-Stage Least Squares Estimation Procedure
11.5.2 The Properties of the Two-Stage Least Squares Estimator
11.6 Exercises
11.6.1 Problems
11.6.2 Computer Exercises
Appendix 11A 2SLS Alternatives
11A.1 The k-Class of Estimators
11A.2 The LIML Estimator
11A.3 Monte Carlo Simulation Results
12 Regression with Time-Series Data: Nonstationary Variables
12.1 Stationary and Nonstationary Variables
12.1.1 Trend Stationary Variables
12.1.2 The First-Order Autoregressive Model
12.1.3 Random Walk Models
12.2 Consequences of Stochastic Trends
12.3 Unit Root Tests for Stationarity
12.3.1 Unit Roots
12.3.2 Dickey–Fuller Tests
12.3.3 Dickey–Fuller Test with Intercept and No Trend
12.3.4 Dickey–Fuller Test with Intercept and Trend
12.3.5 Dickey–Fuller Test with No Intercept and No Trend
12.3.6 Order of Integration
12.3.7 Other Unit Root Tests
12.4 Cointegration
12.4.1 The Error Correction Model
12.5 Regression When There Is No Cointegration
12.6 Summary
12.7 Exercises
12.7.1 Problems
12.7.2 Computer Exercises
13 Vector Error Correction and Vector Autoregressive Models
13.1 VEC and VAR Models
13.2 Estimating a Vector Error Correction Model
13.3 Estimating a VAR Model
13.4 Impulse Responses and Variance Decompositions
13.4.1 Impulse Response Functions
13.4.2 Forecast Error Variance Decompositions
13.5 Exercises
13.5.1 Problems
13.5.2 Computer Exercises
Appendix 13A The Identification Problem
14 Time-Varying Volatility and ARCH Models
14.1 The ARCH Model
14.2 Time-Varying Volatility
14.3 Testing, Estimating, and Forecasting
14.4 Extensions
14.4.1 The GARCH Model—Generalized ARCH
14.4.2 Allowing for an Asymmetric Effect
14.4.3 GARCH-in-Mean and Time-Varying Risk Premium
14.4.4 Other Developments
14.5 Exercises
14.5.1 Problems
14.5.2 Computer Exercises
15 Panel Data Models
15.1 The Panel Data Regression Function
15.1.1 Further Discussion of Unobserved Heterogeneity
15.1.2 The Panel Data Regression Exogeneity Assumption
15.1.3 Using OLS to Estimate the Panel Data Regression
15.2 The Fixed Effects Estimator
15.2.1 The Difference Estimator: T = 2
15.2.2 The Within Estimator: T = 2
15.2.3 The Within Estimator: T > 2
15.2.4 The Least Squares Dummy Variable Model
15.3 Panel Data Regression Error Assumptions
15.3.1 OLS Estimation with Cluster-Robust Standard Errors
15.3.2 Fixed Effects Estimation with Cluster-Robust Standard Errors
15.4 The Random Effects Estimator
15.4.1 Testing for Random Effects
15.4.2 A Hausman Test for Endogeneity in the Random Effects Model
15.4.3 A Regression-Based Hausman Test
15.4.4 The Hausman–Taylor Estimator
15.4.5 Summarizing Panel Data Assumptions
15.4.6 Summarizing and Extending Panel Data Model Estimation
15.5 Exercises
15.5.1 Problems
15.5.2 Computer Exercises
Appendix 15A Cluster-Robust Standard Errors: Some Details
Appendix 15B Estimation of Error Components
16 Qualitative and Limited Dependent Variable Models
16.1 Introducing Models with Binary Dependent Variables
16.1.1 The Linear Probability Model
16.2 Modeling Binary Choices
16.2.1 The Probit Model for Binary Choice
16.2.2 Interpreting the Probit Model
16.2.3 Maximum Likelihood Estimation of the Probit Model
16.2.4 The Logit Model for Binary Choices
16.2.5 Wald Hypothesis Tests
16.2.6 Likelihood Ratio Hypothesis Tests
16.2.7 Robust Inference in Probit and Logit Models
16.2.8 Binary Choice Models with a Continuous Endogenous Variable
16.2.9 Binary Choice Models with a Binary Endogenous Variable
16.2.10 Binary Endogenous Explanatory Variables
16.2.11 Binary Choice Models and Panel Data
16.3 Multinomial Logit
16.3.1 Multinomial Logit Choice Probabilities
16.3.2 Maximum Likelihood Estimation
16.3.3 Multinomial Logit Postestimation Analysis
16.4 Conditional Logit
16.4.1 Conditional Logit Choice Probabilities
16.4.2 Conditional Logit Postestimation Analysis
16.5 Ordered Choice Models
16.5.1 Ordinal Probit Choice Probabilities
16.5.2 Ordered Probit Estimation and Interpretation
16.6 Models for Count Data
16.6.1 Maximum Likelihood Estimation of the Poisson Regression Model
16.6.2 Interpreting the Poisson Regression Model
16.7 Limited Dependent Variables
16.7.1 Maximum Likelihood Estimation of the Simple Linear Regression Model
16.7.2 Truncated Regression
16.7.3 Censored Samples and Regression
16.7.4 Tobit Model Interpretation
16.7.5 Sample Selection
16.8 Exercises
16.8.1 Problems
16.8.2 Computer Exercises
Appendix 16A Probit Marginal Effects: Details
16A.1 Standard Error of Marginal Effect at a Given Point
16A.2 Standard Error of Average Marginal Effect
Appendix 16B Random Utility Models
16B.1 Binary Choice Model
16B.2 Probit or Logit?
Appendix 16C Using Latent Variables
16C.1 Tobit (Tobit Type I)
16C.2 Heckit (Tobit Type II)
Appendix 16D A Tobit Monte Carlo Experiment
Appendix A Mathematical Tools
A.1 Some Basics
A.1.1 Numbers
A.1.2 Exponents
A.1.3 Scientific Notation
A.1.4 Logarithms and the Number e
A.1.5 Decimals and Percentages
A.1.6 Logarithms and Percentages
A.2 Linear Relationships
A.2.1 Slopes and Derivatives
A.2.2 Elasticity
A.3 Nonlinear Relationships
A.3.1 Rules for Derivatives
A.3.2 Elasticity of a Nonlinear Relationship
A.3.3 Second Derivatives
A.3.4 Maxima and Minima
A.3.5 Partial Derivatives
A.3.6 Maxima and Minima of Bivariate Functions
A.4 Integrals
A.4.1 Computing the Area Under a Curve
A.5 Exercises
Appendix B Probability Concepts
B.1 Discrete Random Variables
B.1.1 Expected Value of a Discrete Random Variable
B.1.2 Variance of a Discrete Random Variable
B.1.3 Joint, Marginal, and Conditional Distributions
B.1.4 Expectations Involving Several Random Variables
B.1.5 Covariance and Correlation
B.1.6 Conditional Expectations
B.1.7 Iterated Expectations
B.1.8 Variance Decomposition
B.1.9 Covariance Decomposition
B.2 Working with Continuous Random Variables
B.2.1 Probability Calculations
B.2.2 Properties of Continuous Random Variables
B.2.3 Joint, Marginal, and Conditional Probability Distributions
B.2.4 Using Iterated Expectations with Continuous Random Variables
B.2.5 Distributions of Functions of Random Variables
B.2.6 Truncated Random Variables
B.3 Some Important Probability Distributions
B.3.1 The Bernoulli Distribution
B.3.2 The Binomial Distribution
B.3.3 The Poisson Distribution
B.3.4 The Uniform Distribution
B.3.5 The Normal Distribution
B.3.6 The Chi-Square Distribution
B.3.7 The t-distribution
B.3.8 The F-distribution
B.3.9 The Log-Normal Distribution
B.4 Random Numbers
B.4.1 Uniform Random Numbers
B.5 Exercises
Appendix C Review of Statistical Inference
C.1 A Sample of Data
C.2 An Econometric Model
C.3 Estimating the Mean of a Population
C.3.1 The Expected Value of Ȳ
C.3.2 The Variance of Ȳ
C.3.3 The Sampling Distribution of Ȳ
C.3.4 The Central Limit Theorem
C.3.5 Best Linear Unbiased Estimation
C.4 Estimating the Population Variance and Other Moments
C.4.1 Estimating the Population Variance
C.4.2 Estimating Higher Moments
C.5 Interval Estimation
C.5.1 Interval Estimation: σ2 Known
C.5.2 Interval Estimation: σ2 Unknown
C.6 Hypothesis Tests About a Population Mean
C.6.1 Components of Hypothesis Tests
C.6.2 One-Tail Tests with Alternative “Greater Than” (>)
C.6.3 One-Tail Tests with Alternative “Less Than” (<)
C.6.4 Two-Tail Tests with Alternative “Not Equal To” (≠)
C.6.5 The p-Value
C.6.6 A Comment on Stating Null and Alternative Hypotheses
C.6.7 Type I and Type II Errors
C.6.8 A Relationship Between Hypothesis Testing and Confidence Intervals
C.7 Some Other Useful Tests
C.7.1 Testing the Population Variance
C.7.2 Testing the Equality of Two Population Means
C.7.3 Testing the Ratio of Two Population Variances
C.7.4 Testing the Normality of a Population
C.8 Introduction to Maximum Likelihood Estimation
C.8.1 Inference with Maximum Likelihood Estimators
C.8.2 The Variance of the Maximum Likelihood Estimator
C.8.3 The Distribution of the Sample Proportion
C.8.4 Asymptotic Test Procedures
C.9 Algebraic Supplements
C.9.1 Derivation of Least Squares Estimator
C.9.2 Best Linear Unbiased Estimation
C.10 Kernel Density Estimator
C.11 Exercises
C.11.1 Problems
C.11.2 Computer Exercises
Appendix D Statistical Tables
Table D.1 Cumulative Probabilities for the Standard Normal Distribution Φ(z) = P(Z ≤ z)
Table D.2 Percentiles of the t-distribution
Table D.3 Percentiles of the Chi-square Distribution
Table D.4 95th Percentile for the F-distribution
Table D.5 99th Percentile for the F-distribution
Table D.6 Standard Normal pdf Values Φ(z)
Index
