1 The Geometry of Least Square

1.1 Introduction

1.2 The Geometry of Least Squares

1.3 Restrictions and Reparametrizations

1.4 The Frisch-Waugh-Lovell Theorem

1.5 Computing OLS Estimates

1.6 Influential Observations and Leverage

1.7 Further Reading and Conclusion

2 Nonlinear Regression Models and Nonlinear Least Squares

2.1 Introduction

2.2 The Geometry of Nonlinear Least Squares

2.3 Identification in Nonlinear Regression Models

2.4 Models and Data-Generating Processes

2.5 Linear and Nonlinear Regression Functions

2.6 Error Terms

2.7 Conclusion

3 Inference in Nonlinear Regression Models

3.1 Introduction

3.2 Covariance Matrix Estimation

3.3 Confidence Intervals and Confidence Regions

3.4 Hypothesis Testing: Introduction

3.5 Hypothesis Testing in Linear Regression Models

3.6 Hypothesis Testing in Nonlinear Regression Models

3.7 Restrictions and Pretest Estimators

Conclusion

4 Introduction to Asymptotic Theory and Methods

4.1 Introduction

4.2 Sequences, Limits, and Convergence

4.3 Rates of Convergence

4.4 Data-Generating Processes and Asymptotic Theory

4.5 Consistency and Laws of Large Numbers

4.6 Asymptotic Normality and Central Limit Theorems

4.7 Some Useful Results

4.8 Conclusion

5 Asymptotic Methods and Nonlinear Least Squares

5.1 Introduction

5.2 Asymptotic Identifiability

5.3 Consistency of the NLS Estimator

5.4 Asymptotic Normality of the NLS Estimator

5.5 Asymptotic Efficiency of Nonlinear Least Squares

5.6 Properties of Nonlinear Least Squares Residuals

5.7 Test Statistics Based on NLS Estimates

5.8 Further Reading and Conclusion

6 The Gauss–Newton Regression

6.1 Introduction

6.2 Computing Covariance Matrices

6.3 Collinearity in Nonlinear Regression Models

6.4 Testing Restrictions

6.5 Diagnostic Tests for Linear Regression Models

6.6 One-Step Efficient Estimation

6.7 Hypothesis Tests Using Any Consistent Estimates

6.8 Nonlinear Estimation Using the GNR

6.9 Further Reading

7 Instrumental Variables

7.1 Introduction

7.2 Errors in Variables

7.3 Simultaneous Equations

7.4 Instrumental Variables: The Linear Case

7.5 Two-Stage Least Squares

7.6 Instrumental Variables: The Nonlinear Case

7.7 Hypothesis Tests Based on the GNR

7.8 Identification and Overidentifying Restrictions

7.9 Durbin-Wu-Hausman Tests

7.10 Conclusion

8 The Method of Maximum Likelihood

8.1 Introduction

8.2 Fundamental Concepts and Notation

8.3 Transformations and Reparametrizations

8.4 Consistency

8.5 The Asymptotic Distribution of the ML Estimator

8.6 The Information Matrix Equality

8.7 Concentrating the Loglikelihood Function

8.8 Asymptotic Efficiency of the ML Estimator

8.9 The Three Classical Test Statistics

8.10 Nonlinear Regression Models

8.11 Conclusion

9 Maximum Likelihood and Generalized Least Squares

9.1 Introduction

9.2 Generalized Least Squares

9.3 The Geometry of GLS

9.4 The Gauss-Newton Regression

9.5 Feasible Generalized Least Squares

9.6 Maximum Likelihood and GNLS

9.7 Introduction to Multivariate Regression Models

9.8 GLS Estimation of Multivariate Regression Models

9.9 ML Estimation of Multivariate Regression Models

9.10 Modeling Time-Series/Cross-Section Data

9.11 Conclusion

10 Serial Correlation

10.1 Introduction

10.2 Serial Correlation and Least Squares Estimation

10.3 Estimating Regression Models with AR(1) Errors

10.4 Standard Errors and Covariance Matrices

10.5 Higher-Order AR Processes

10.6 Initial Observations in Models with AR Errors

10.7 Moving Average and ARMA Processes

10.8 Testing for Serial Correlation

10.9 Common Factor Restrictions

10.10 Instrumental Variables and Serial Correlation

10.11 Serial Correlation and Multivariate Models

10.12 Conclusion

11 Tests Based on the Gauss-Newton Regression

11.1 Introduction

11.2 Tests for Equality of Two Parameter Vectors

11.3 Testing Nonnested Regression Models

11.4 Tests Based on Comparing Two Sets of Estimates

11.5 Testing for Heteroskedasticity

11.6 A Heteroskedasticity-Robust Version of the GNR

11.7 Conclusion

12 Interpreting Tests in Regression Directions

12.1 Introduction

12.2 Size and Power

12.3 Drifting DGPs

12.4 The Asymptotic Distribution of Test Statistics

12.5 The Geometry of Test Power

12.6 Asymptotic Relative Efficiency

12.7 Interpreting Test Statistics that Reject the Null

12.8 Test Statistics that Do Not Reject the Null

12.9 Conclusion

13 The Classical Hypothesis Tests

13.1 Introduction

13.2 The Geometry of the Classical Test Statistics

13.3 Asymptotic Equivalence of the Classical Tests

13.4 Classical Tests and Linear Regression Models

13.5 Alternative Covariance Matrix Estimators

13.6 Classical Test Statistics

13.7 The Outer-Product-of-the-Gradient Regression

13.8 Further Reading and Conclusion

14 Transforming the Dependent Variable

14.1 Introduction

14.2 The Box-Cox Transformation

14.3 The Role of Jacobian Terms in ML Estimation

14.4 Double-Length Artificial Regressions

14.5 The DLR and Models Involving Transformations

14.6 Testing Linear and Loglinear Regression Models

14.7 Other Transformations

14.8 Conclusion

15 Qualitative and Limited Dependent Variables

15.1 Introduction

15.2 Binary Response Models

15.3 Estimation of Binary Response Models

15.4 An Artificial Regression

15.5 Models for More than Two Discrete Responses

15.6 Models for Truncated Data

15.7 Models for Censored Data

15.8 Sample Selectivity

15.9 Conclusion

16 Heteroskedasticity and Related Topics

16.1 Introduction

16.2 Least Squares and Heteroskedasticity

16.3 Covariance Matrix Estimation

16.4 Autoregressive Conditional Heteroskedasticity

16.5 Testing for Heteroskedasticity

16.6 Skedastic Directions and Regression Directions

16.7 Tests for Skewness and Excess Kurtosis

16.8 Conditional Moment Tests

16.9 Information Matrix Tests

16.10 Conclusion

17 The Generalized Method of Moments

17.1 Introduction and Definitions

17.2 Criterion Functions and M-Estimators

17.3 Efficient GMM Estimators

17.4 Estimation with Conditional Moments

17.5 Covariance Matrix Estimation

17.6 Inference with GMM Models

17.7 Conclusion

18 Simultaneous Equations Models

18.1 Introduction

18.2 Exogeneity and Causality

18.3 Identification in Simultaneous Equations Models

18.4 Full-Information Maximum Likelihood

18.5 Limited-Information Maximum Likelihood

18.6 Three-Stage Least Squares

18.7 Nonlinear Simultaneous Equations Models

18.8 Conclusion

19 Regression Models for Time-Series Data

19.1 Introduction

19.2 Spurious Regressions

19.3 Distributed Lags

19.4 Dynamic Regression Models

19.5 Vector Autoregressions

19.6 Seasonal Adjustment

19.7 Modeling Seasonality

19.8 Conclusion

20 Unit Roots and Cointegration

20.1 Introduction

20.2 Testing for Unit Roots

20.3 Asymptotic Theory for Unit Root Tests

20.4 Serial Correlation and Other Problems

20.5 Cointegration

20.6 Testing for Cointegration

20.7 Model-Building with Cointegrated Variables

20.8 Vector Autoregressions and Cointegration

20.9 Conclusion

21 Monte Carlo Experiments

21.1 Introduction

21.2 Generating Pseudo-Random Numbers

21.3 Generating Pseudo-Random Variates

21.4 Designing Monte Carlo Experiments

21.5 Variance Reduction: Antithetic Variates

21.6 Variance Reduction: Control Variates

21.7 Response Surfaces

21.8 The Bootstrap and Related Methods

21.9 Conclusion

Appendices

A Matrix Algebra

A.1 Introduction

A.2 Elementary Facts about Matrices

A.3 The Geometry of Vectors

A.4 Matrices as Mappings of Linear Spaces

A.5 Partitioned Matrices

A.6 Determinants

A.7 Positive Definite Matrices

A.8 Eigenvalues and Eigenvectors

B Results from Probability Theory

B.1 Introduction

B.2 Random Variables and Probability Distributions

B.3 Moments of Random Variables

B.4 Some Standard Probability Distributions

References

Author Index

Subject Index