List of Figures Preface

1 Finite-Sample Properties of OLS

1.1 The Classical Linear Regression Model

The Linearity Assumption

Matrix Notation

The Strict Exogeneity Assumption

Implications of Strict Exogeneity

Strict Exogeneity in Time-Series Models

Other Assumptions of the Model

The Classical Regression Model for Random Samples

"Fixed" Regressors

1.2 The Algebra of Least Squares

OLS Minimizes the Sum of Squared Residuals

Normal Equations

Two Expressions for the OLS Estimator

More Concepts and Algebra

Influential Analysis (optional)

A Note on the Computation of OLS Estimates

1.3 Finite-Sample Properties of OLS

Finite-Sample Distribution of b

Finite-Sample Properties of *s*^{2}

Estimate of Var(b | X)

1.4 Hypothesis Testing under Normality

Normally Distributed Error Terms

Testing Hypotheses about Individual Regression Coefficients Decision Rule for the ?-Test Confidence Interval

2.10 Testing for Serial Correlation

Box-Pierce and Ljung-Box

Sample Autocorrelations Calculated from Residuals Testing with Predetermined, but Not Strictly Exogenous,

Regressors An Auxiliary Regression-Based Test

2.11 Application: Rational Expectations Econometrics

The Efficient Market Hypotheses

Testable Implications

Testing for Serial Correlation

Is the Nominal Interest Rate the Optimal Predictor?

*R*_{t} Is Not Strictly Exogenous

Subsequent Developments

2.12 Time Regressions

The Asymptotic Distribution of the OLS Estimator

Hypothesis Testing for Time Regressions

Appendix 2. A: Asymptotics with Fixed Regressors Appendix 2.B: Proof of Proposition

2.10 Problem Set Answers to Selected Questions

Single-Equation GMM

3.1 Endogeneity Bias: Working's Example

A Simultaneous Equations Model of Market Equilibrium Endogeneity Bias Observable Supply Shifters

3.2 More Examples

A Simple Macroeconometric Model Error s-in -Variables Production Function

3.3 The General Formulation

Regressors and Instruments

Identification

Order Condition for Identification

The Assumption for Asymptotic Normality

3.4 Generalized Method of Moments Denned

Method of Moments

Generalized Method of Moments Sampling Error

3.5 Large-Sample Properties of GMM

Asymptotic Distribution of the GMM Estimator

Estimation of Error Variance

Hypothesis Testing Estimation of S Efficient GMM Estimator Asymptotic Power Small-Sample Properties

3.6 Testing Overidentifying Restrictions

Testing Subsets of Orthogonality Conditions

3.7 Hypothesis Testing by the Likelihood-Ratio Principle

The *LR *Statistic for the Regression Model

Variable Addition Test (optional)

3.8 Implications of Conditional Homoskedasticity

Efficient GMM Becomes 2SLS / Becomes Sargan's Statistic Small-Sample Properties of 2SLS Alternative Derivations of 2SLS When Regressors Are Predetermined Testing a Subset of Orthogonality Conditions Testing Conditional Homoskedasticity Testing for Serial Correlation

3.9 Application: Returns from Schooling

The NLS-Y Data

The Semi-Log Wage Equation

Omitted Variable Bias

IQ as the Measure of Ability

Errors-in-Variables

2SLS to Correct for the Bias

Subsequent Developments Problem Set Answers to Selected Questions

Multiple-Equation GMM

4.1 The Multiple-Equation Model Linearity

Stationarity and Ergodicity Orthogonality Conditions Identification

The Assumption for Asymptotic Normality

Connection to the "Complete" System of Simultaneous Equations

4.2 Multiple-Equation GMM Defined

4.3 Large-Sample Theory

4.4 Single-Equation versus Multiple-Equation Estimation

When Are They "Equivalent"?

Joint Estimation Can Be Hazardous

4.5 Special Cases of Multiple-Equation GMM: FIVE, 3SLS, and SUR

Conditional Homoskedasticity

Full-Information Instrumental Variables Efficient (FIVE) Three-Stage Least Squares (3SLS) Seemingly Unrelated Regressions (SUR) SUR versus OLS

4.6 Common Coefficients

The Model with Common Coefficients

The GMM Estimator

Imposing Conditional Homoskedasticity

Pooled OLS

Beautifying the Formulas

The Restriction That Isn't

4.7 Application: Interrelated Factor Demands

The Translog Cost Function

Factor Shares

Substitution Elasticities

Properties of Cost Functions

Stochastic Specifications

The Nature of Restrictions

Multivariate Regression Subject to Cross-Equation Restrictions

Which Equation to Delete?

Results Problem Set Answers to Selected Questions

Panel Data

5.1 The Error-Components Model

Error Components

Group Means

A Reparameterization

5.2 The Fixed-Effects Estimator

The Formula Large-Sample Properties Digression: When n Is Spherical Random Effects versus Fixed Effects Relaxing Conditional Homoskedasticity

5.3 Unbalanced Panels (optional)

"Zeroing Out" Missing Observations

Zeroing Out versus Compression

No Selectivity Bias

5.4 Application: International Differences in Growth Rates

Derivation of the Estimation Equation

Appending the Error Term

Treatment of a

Consistent Estimation of Speed of Convergence Appendix 5.A: Distribution of Hausman Statistic Problem Set Answers to Selected Questions

Serial Correlation

6.1 Modeling Serial Correlation: Linear Processes *MA(q)*

MA(oo) as a Mean Square Limit

Filters

Inverting Lag Polynomials

6.2 ARMA Processes

AR(1) and Its MA(oo) Representation

Autocovariances of AR(1)

*AR(p) *and Its MA(oo) Representation

ARMA(/7, *q)*

ARMA(p, *q) *with Common Roots

Invertibility

Autocovanance-Generating Function and the Spectrum

6.3 Vector Processes

6.4 Estimating Autoregressions Estimation of AR(1) Estimation of AR(/>) Choice of Lag Length Estimation of VARs Estimation of ARMAQ?, *q)*

6.5 Asymptotics for Sample Means of Serially Correlated Processes

LLN for Covariance-Stationary Processes

Two Central Limit Theorems Multivariate Extension

6.6 Incorporating Serial Correlation in GMM

The Model and Asymptotic Results

Estimating S When Autocovariances Vanish after Finite Lags

Using Kernels to Estimate S

VARHAC

6.7 Estimation under Conditional Homoskedasticity (Optional)

Kernel-Based Estimation of S under Conditional Homoskedasticity

Data Matrix Representation of Estimated Long-Run Variance

Relation to GLS

6.8 Application: Forward Exchange Rates as Optimal Predictors

The Market Efficiency Hypothesis

Testing Whether the Unconditional Mean Is Zero

Regression Tests Problem Set Answers to Selected Questions

Extremum Estimators

7.1 Extremum Estimators

"Measurability" of *9*

Two Classes of Extremum Estimators Maximum Likelihood (ML) Conditional Maximum Likelihood Invariance of ML Nonlinear Least Squares (NLS) Linear and Nonlinear GMM

7.2 Consistency

Two Consistency Theorems for Extremum Estimators Consistency of M-Estimators Concavity after Reparameterization Identification in NLS and ML Consistency of GMM

7.3 Asymptotic Normality

Asymptotic Normality of M-Estimators

Consistent Asymptotic Variance Estimation

Asymptotic Normality of Conditional ML

Two Examples

Asymptotic Normality of GMM

GMM versus ML

Expressing the Sampling Error in a Common Format

7.4 Hypothesis Testing

The Null Hypothesis

The Working Assumptions

The Wald Statistic

The Lagrange Multiplier (LM) Statistic

The Likelihood Ratio (LR) Statistic

Summary of the Trinity

7.5 Numerical Optimization

Newton-Raphs on Gauss-Newton

Writing Newton-Raphson and Gauss-Newton in a Common Format

Equations Nonlinear in Parameters Only Problem Set Answers to Selected Questions

Examples of Maximum Likelihood

8.1 Qualitative Response (QR) Models Score and Hessian for Observation *t *Consistency

Asymptotic Normality

8.2 Truncated Regression Models The Model

Truncated Distributions The Likelihood Function Reparameterizing the Likelihood Function Verifying Consistency and Asymptotic Normality Recovering Original Parameters

8.3 Censored Regression (Tobit) Models Tobit Likelihood Function Reparameterization

8.4 Multivariate Regressions

The Multivariate Regression Model Restated The Likelihood Function Maximizing the Likelihood Function

Consistency and Asymptotic Normality

8.5 FIML

The Multiple-Equation Model with Common Instruments Restated

The Complete System of Simultaneous Equations

Relationship between (Го, Bo) and *S*_{o}

The FIML Likelihood Function

The FIML Concentrated Likelihood Function

Testing Overidentifying Restrictions

Properties of the FIML Estimator

ML Estimation of the SUR Model

8.6 LIML

LIML Defined

Computation of LIML

LTML versus 2SLS

8.7 Serially Correlated Observations

Two Questions

Unconditional ML for Dependent Observations ML Estimation of AR(1) Processes Conditional ML Estimation of AR(1) Processes Conditional ML Estimation of AR(/?) and VAR(p) Processes Problem Set

Unit-Root Econometrics

9.1 Modeling Trends

Integrated Processes

Why Is It Important to Know if the Process Is 1(1)? Which Should Be Taken as the Null, 1(0) or 1(1)? Other Approaches to Modeling Trends

9.2 Tools for Unit-Root Econometrics

Linear 1(0) Processes

Approximating 1(1) by a Random Walk

Relation to ARMA Models

The Wiener Process A Useful Lemma

9.3 Dickey-Fuller Tests

The AR(1) Model

Deriving the Limiting Distribution under the 1(1) Null Incorporating the Intercept Incorporating Time Trend

9.4 Augmented Dickey-Fuller Tests

The Augmented Autoregression

Limiting Distribution of the OLS Estimator

Deriving Test Statistics

Testing Hypotheses about £

What to Do When *p *Is Unknown?

A Suggestion for the Choice of *Pmex(T)*

Including the Intercept in the Regression

Incorporating Time Trend

Summary of the DF and ADF Tests and Other Unit-Root Tests

9.5 Which Unit-Root Test to Use?

Local-to-Unity Asymptotics

Small-Sample Properties

9.6 Application: Purchasing Power Parity

The Embarrassing Resiliency of the Random Walk Model? Problem Set Answers to Selected Questions

10 Cointegration

10.1 Cointegrated Systems

Linear Vector 1(0) and 1(1) Processes The Beveridge-Nelson Decomposition Cointegration Defined

10.2 Alternative Representations of Cointegrated Systems

Ph
illips's Triangular Representation

VAR and Cointegration

The Vector Error-Correction Model (VECM)

Johansen's ML Procedure

10.3 Testing the Null of No Cointegration

Spurious Regressions

The Residual-Based Test for Cointegration Testing the Null of Cointegration

10.4 Inference on Cointegrating Vectors

The SOLS Estimator

The Bivariate Example

Continuing with the Bivariate Example

Allowing for Serial Correlation

General Case

Other Estimators and Finite-Sample Properties

10.5 Application: The Demand for Money in the United States

The Data (m — *p, y, R) *as a Cointegrated System

DOLS

Unstable Money Demand? Problem Set

Appendix A: Partitioned Matrices and Kronecker Products

Addition and Multiplication of Partitioned Matrices Inverting Partitioned Matrices

Index