Contents
List of Figures xvii
Preface xix
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 s2
Estimate of Var(b 1 X)
1.4 Hypothesis Testing under Normality
Normally Distributed Error Terms
Testing Hypotheses about Individual Regression Coefficients
Decision Rule for the t-Test
Confidence Interval
vi Contents
p-Value 3 8
Linear Hypotheses 39
The F-Test 40
A More Convenient Expression for F 42
t versus F 43
An Example of a Test Statistic Whose Distribution Depends on X 45
1.5 Relation to Maximum Likelihood 47
The Maximum Likelihood Principle 47
Conditional versus Unconditional Likelihood 47
The Log Likelihood for the Regression Model 48
ML via Concentrated Likelihood 48
Cramer-Rao Bound for the Classical Regression Model 49
The F-Test as a Likelihood Ratio Test 52
Quasi-Maximum Likelihood 53
1.6 Generalized Least Squares (GLS) 54
Consequence of Relaxing Assumption 1.4 55
Efficient Estimation with Known V 55
A Special Case: Weighted Least Squares (WLS) 58
Limiting Nature of GLS 58
1.7 Application: Returns to Scale in Electricity Supply 60
The Electricity Supply Industry 60
The Data 60
Why Do We Need Econometrics? 61
The Cobb-Douglas Technology 62
How Do We Know Things Are Cobh-Douglas? 63
Are the OLS Assumptions Satisfied? 64
Restricted Least Squares 65
Testing the Homogeneity of the Cost Function 65
Detour: A Cautionary Note on R~ 67
Testing Constant Returns to Scale 67
Importance of Plotting Residuals 68
Subsequent Developments 68
Problem Set 7 1
Answers to Selected Questions 84
Large-Sample Theory 88
2.1 Review of Limit Theorems for Sequences of Random Variables 88
Various Modes of Convergence 89
Three Useful Results 92
Contents vii
Viewing Estimators as Sequences of Random Variables
Laws of Large Numbers and Central Limit Theorems
2.2 Fundamental Concepts in Time-Series Analysis
Need for Ergodic Stationarity
Various Classes of Stochastic Processes
Different Formulation of Lack of Serial Dependence
The CLT for Ergodic Stationary Martingale Differences Sequences
2.3 Large-Sample Distribution of the OLS Estimator
The Model
Asymptotic Distribution of the OLS Estimator
s2 IS Consistent
2.4 Hypothesis Testing
Testing Linear Hypotheses
The Test Is Consistent
Asymptotic Power
Testing Nonlinear Hypotheses
2.5 Estimating E(E?x~x;) Consistently
Using Residuals for the Errors
Data Matrix Representation of S
Finite-Sample Considerations
2.6 Implications of Conditional Homoskedasticity
Conditional versus Unconditional Homoskedasticity
Reduction to Finite-Sample Formulas
Large-Sample Distribution of t and F Statistics
Variations of Asymptotic Tests under Conditional
Homoskedasticity
2.7 Testing Conditional Homoskedasticity
2.8 Estimation with Parameterized Conditional Heteroskedasticity
(optional)
The Functional Form
WLS with Known a
Regression of e? on zi Provides a Consistent Estimate of a
WLS with Estimated a
OLS versus WLS
2.9 Least Squares Projection
Optimally Predicting the Value of the Dependent Variable
Best Linear Predictor
OLS Consistently Estimates the Projection Coefficients
Contents
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
2.1 1 Application: Rational Expectations Econometrics
The Efficient Market Hypotheses
Testable Implications
Testing for Serial Correlation
Is the Nominal Interest Rate the Optimal Predictor?
R, 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
3 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
Errors-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 Defined
Method of Moments
Generalized Method of Moments
Sampling Error
Contents
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
J 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
4 Multiple-Equation GMM
4.1 The Multiple-Equation Model
Linearity
Stationarity and Ergodicity
Orthogonality Conditions
Identification
Contents
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
5 Panel Data
5.1 The Error-Components Model
Error Components
Group Means
A Reparameterization
5.2 The Fixed-Effects Estimator
Contents
The Formula
Large-Sample Properties
Digression: When rli 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 cri
Consistent Estimation of Speed of Convergence
Appendix 5.A: Distribution of Hausman Statistic
Problem Set
Answers to Selected Questions
6 Serial Correlation
6.1 Modeling Serial Correlation: Linear Processes
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(p7 q)
ARMA(p, q) with Common Roots
Invertibility
Autocovariance-Generating Function and the Spectrum
6.3 Vector Processes
6.4 Estimating Autoregressions
Estimation of AR(1)
Estimation of AR(p)
Choice of Lag Length
Estimation of VARs
Estimation of ARMA(p, q)
xii Contents
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
7 Extremum Estimators
7.1 Extremum Estimators
"Measurability" of (?
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
Contents
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-Raphson
Gauss-Newton
Writing Newton-Raphson and Gauss-Newton in a Common
Format
Equations Nonlinear in Parameters Only
Problem Set
Answers to Selected Questions
8 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
X ~ V Contents
Consistency and Asymptotic Normality 525
8.5 FIML 526
The Multiple-Equation Model with Common Instruments Restated 526
The Complete System of Simultaneous Equations 529
Relationship between (ro, Bo) and J0 530
The FIML Likelihood Function 53 1
The FIML Concentrated Likelihood Function 532
Testing Overidentifying Restrictions 533
Properties of the FIML Estimator 533
ML Estimation of the SUR Model 535
8.6 LIML 538
LIML Defined 538
Computation of LIML 540
LIML versus 2SLS 542
8.7 Serially Correlated Observations 543
Two Questions 543
Unconditional ML for Dependent Observations 545
ML Estimation of AR(1) Processes 546
Conditional ML Estimation of AR(1) Processes 547
Conditional ML Estimation of AR(p) and VAR(p) Processes 549
Problem Set 55 1
9 Unit-Root Econometrics
9.1 Modeling Trends
Integrated Processes
Why Is It Important to Know if the Process Is I(1)?
Which Should Be Taken as the Null, I(0) or 1(1)?
Other Approaches to Modeling Trends
9.2 Tools for Unit-Root Econometrics
Linear I(0) Processes
Approximating I(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 I(1) Null
Incorporating the Intercept
Incorporating Time Trend
Contents
9.4 Augmented Dickey-Fuller Tests
The Augmented Autoregression
Limiting Distribution of the OLS Estimator
Deriving Test Statistics
Testing Hypotheses about 5
What to Do When p Is Unknown?
A Suggestion for the Choice of pm(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 I(0) and 1(1) Processes
The Beveridge-Nelson Decomposition
Cointegration Defined
10.2 Alternative Representations of Cointegrated Systems
Phillips'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
Contents
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