Textbook: The Econometrics of Multi-dimensional Panels: Theory and Applications
Advanced Studies in Theoretical and Applied Econometrics; 54
Author(s): Laszlo Matyas (editor)


This textbook presents the econometric foundations and applications of multi-dimensional panels, including modern methods of big data analysis. In light of the big data revolution and the emergence of higher dimensional panel data sets, it provides new results to synthesize existing knowledge on the field. The first, theoretical part of the volume is providing the econometric foundations to deal with these new high-dimensional panel data sets. It not only synthesizes our current knowledge, but mostly, presents new research results. The second empirical part of the book provides insight into the most relevant applications in this area. These chapters are a mixture of surveys and new results, always focusing on the econometric problems and feasible solutions.
This second extended and revised edition provides an update of all existent chapters to reflect on new developments in the area as well as several new chapters on topics such as machine learning, nonparametric models, networks, and multi-dimensional panels in health economics. The book serves as a standard reference work, a textbook for graduate students in economics, and a source of background material for professionals conducting empirical studies.


Contents

Chapter 1 Fixed Effects Models
1.1 Introduction
1.2 Models with Different Types of Heterogeneity
1.3 Least Squares Estimation of the Models
1.4 Model Selection
1.5 Incomplete Panels
1.6 The Within Estimator
1.6.1 The Equivalence of the LSDV and theWithin Estimator
1.6.2 Incomplete Panels and the Within Estimator
1.6.2.1 No Self-flow Data
1.6.2.2 General Incompleteness
1.7 Heteroscedasticity and Cross-correlation
1.7.1 The New Covariance Matrices and the GLS Estimator
1.7.2 Estimation of the Variance Components and the Cross-correlations
1.8 Extensions to Higher Dimensions
1.8.1 Different Forms of Heterogeneity
1.8.2 Least Squares and the Within Estimators
1.8.3 Incomplete Panels
1.9 Varying Coefficients Models
References
Chapter 2 When and How Much Do Fixed Effects Matter?
2.1 Introduction
2.2 Effect of the Fixed Effects on Parameters of Interest
2.2.1 Misspecification in 2D
2.2.2 Misspecification in 3D and Higher Dimensions
2.3 Monte Carlo Evidence
2.4 Machine Learning Tools to Learn the Fixed Effects Structure
2.4.1 The Impossibility of Learning the Fully Interacting Fixed Effects
2.4.2 Learning High-dimensional Fixed Effects Structures
2.5 Conclusion
Appendix
Properties of the Residual Maker
2D case
Proof of Corollary 2.1
Example for the Autoregressive Process in 3D
References
Chapter 3 Random Effects Models
3.1 Introduction
3.2 Different Model Specifications
3.2.1 Various Heterogeneity Formulations
3.2.2 Spectral Decomposition of the Covariance Matrices
3.3 FGLS Estimation
3.4 Unbalanced Data
3.4.1 Structure of the Covariance Matrices
3.4.2 The Inverse of the Covariance Matrices
3.4.3 Estimation of the Variance Components
3.5 Extensions
3.5.1 4D and Beyond
3.5.2 Mixed FE-RE Models
3.6 Testing
3.7 Conclusion
Appendix 1
Example for normalizing with 1: Model (3.14), T → ∞
Example for normalizing with √N1N2/A: Model (3.2), N1,N2 → ∞
Appendix 2: Proof of formula (3.19)
Appendix 3: Inverse of (3.34), and the estimation of the variance components
References
Chapter 4 Estimation of Sparse Variance-Covariance Matrix
4.1 Introduction
4.2 Basic Methods
4.3 Multi-dimensional Panel Data and Sparse Variance-Covariance Matrix
4.4 Estimating Variance-Covariance Matrix
4.4.1 Estimating Variance-Covariance Matrices in Higher Dimensions
4.4.1.1 Case 1:
4.4.1.2 Case 2:
4.4.1.3 Case 3:
4.4.1.4 Case 4:
4.4.2 Shrinkage Approximation
4.4.3 Regularistion
4.5 Testing for Misspecification in the Unobserved Heterogeneity
4.5.1 Finite Sample Performance of the Misspecification Test
4.6 Further Considerations
4.6.1 Unbalanced Panel
4.6.2 Higher Dimension
4.6.3 Computation
Appendix
References
Chapter 5 Models with Endogenous Regressors
5.1 Introduction
5.2 The Hausman-Taylor-like Instrument Variable Estimator
5.2.1 A Simple Approach
5.2.2 Sources of Endogeneity
5.2.3 The Hausman-Taylor Estimator
5.2.3.1 Extending the Hausman-Taylor Two-Stage Least Squares Estimator
5.2.3.2 The More Efficient Hausman-Taylor Estimator
5.2.4 Time Varying Individual Specific Effects
5.2.5 Properties
5.2.6 Common Correlated Effects Pooled with Hausman-Taylor
5.2.7 Using External Instruments
5.3 The Non-linear Generalized Method of Moments Estimator
5.4 Mixed Effects Models
5.5 Exogeneity Tests
5.5.1 Testing for Endogeneity
5.5.2 Testing for Instrument Validity
5.5.3 Testing in the Case of Fixed Effects
5.5.3.1 Improper Model Specifications
5.5.3.2 Conventional Endogeneity
5.6 Further Considerations
5.6.1 Incomplete Data
5.6.2 Notes on Higher-dimensional Panels
Appendix: Proofs
Proof of Proposition 5.1
Proof of Corollary 5.1
References
Chapter 6 Dynamic Models and Reciprocity
6.1 Introduction
6.2 Dynamics
6.2.1 Estimation
6.2.2 Monte Carlo Experiments
6.3 Reciprocity
6.3.1 Within Estimator
6.3.2 Estimation
6.3.3 No Self-flow
6.4 Combining Dynamics and Reciprocity
6.4.1 Monte Carlo Experiments
6.5 Extensions
6.5.1 Generalized Reciprocity
6.5.2 Higher Dimensions
Appendix
References
Chapter 7 Random Coefficients Models
7.1 Introduction
7.2 The Linear Model for Three Dimensions
7.2.1 The Model
7.3 Identification and Estimation
7.3.1 Feasible Generalized Least Squares (FGLS)
7.3.1.1 Variance-Covariance Estimation
7.3.1.2 Method 1: Using Within Dimensions Variation
7.3.1.3 Method 2: Using the Overall Variation
7.3.1.4 Minimum Norm Quadratic Unbiased Estimation (MINQUE)
7.3.1.5 Statistical properties of the Estimators
7.3.2 Maximum Likelihood Estimation
7.3.2.1 The Unrestricted Maximum Likelihood
7.3.2.2 Restricted Maximum Likelihood Estimation
7.4 Inference: Varying (Random) or Constant Coefficients?
7.4.1 Testing for Methods 1 and 2
7.4.1.1 Test in Steps
7.4.1.2 Joint Test for the Presence of Random Elements
7.4.2 Testing in the Case of MLE
7.5 Prediction of the Coefficients
7.6 Bayesian Approach
7.7 Extensions within the Linear Model
7.7.1 Alternative Model Formulations
7.7.2 Incomplete Panels
7.7.3 Cross-Section Dependence
7.7.4 Random Coefficients Correlated with the Explanatory Variables
7.7.5 Some Random and Some ‘Fixed’ Coefficients?
7.7.6 Higher Dimensions
7.8 Misspecification of the Variance-Covariance Matrix
7.9 Non Linear Extension: RC Probit Model
7.10 A Simulation Experiment
7.11 Conclusions
References
Chapter 8 Nonparametric Models with Random Effects
8.1 Introduction
8.2 The Pooled Local Linear Estimator
8.3 Two-step Local Linear Estimator
8.3.1 Weighted Local Linear Estimator
8.3.2 Two-step Estimator
8.4 Pairwise Random Effects
8.4.1 Cases (i)–(iii): The Sample Size Increases in One Index Only
8.4.2 Cases (iv)-(vi): The Sample Size Increases in Two out of the Three Indices
8.4.3 Case (vii): The Sample Size Increases in All Three Indices
8.5 Some Extensions
8.5.1 Mixed Fixed and Random Effects Models
8.5.2 Four and Higher-dimensional Cases
8.5.3 Fixed Effects Models
8.6 Conclusion
Acknowledgements
Appendix
References
Chapter 9 Nonparametric Models with Fixed Effects
9.1 Introduction
9.2 Random Effects Models
9.3 Fixed Effects Models
9.3.1 Parametric Estimation
9.3.2 Nonparametric Estimation
9.3.2.1 Within Estimator
9.3.2.2 Pairwise Estimator
9.3.3 Semiparametric Estimation
9.4 Inference
9.4.1 A More General Central Limit Theorem for U-Statistics
9.4.2 Parametric vs Nonparametric
9.4.3 Parametric vs Varying-Coefficient
9.4.4 Testing for Relevance
9.4.5 Hausman Test
9.5 Potential Extensions
9.6 Empirical Illustration: Environmental Kuznets Curve
9.6.1 Data
9.6.2 Specification Tests
9.6.3 Gradient Estimates
9.7 Conclusion
Acknowledgements
Appendix
Assumptions
Asymptotic Results for the Within Estimator
Lemmas with Corresponding Proofs
References
Chapter 10 Multi-dimensional Panels in Quantile Regression Models
10.1 Introduction
10.2 Fixed Effects Models
10.2.1 Estimation and Implementation
10.2.2 Inference Procedures
10.2.2.1 A random-weighted bootstrap method
Practical implementation
Percentile confidence interval
Variance-covariance matrix estimation
10.2.3 Smoothed Quantile Regression Panel Data
Bias Correction Analytical Method
Jackknife
10.3 Random Effects Models
10.3.1 Model
10.3.2 Estimation and Implementation
10.3.3 Inference Procedures
10.3.4 Correlated Random Effects Models
10.4 Quantile Models for Network Data
10.5 Specific Guidelines for Practitioners
References
Chapter 11 Multi-Dimensional Models for Spatial Panels
11.1 Introduction
11.2 Spatial Models
11.2.1 The Baseline Model
11.2.2 Unobserved Heterogeneity
11.3 Spatial Estimation Methods
11.3.1 Maximum Likelihood Estimation
11.3.2 GM, FGLS and Instrumental Variables Approaches
The GM Spatial FGLS Estimator
11.4 Testing for Spatial Dependence
11.5 Prediction with Spatial Models
11.6 Some Further Topics
11.6.1 Heterogenous Coefficients Spatial Models
11.6.2 Time-Space Models
11.7 Conclusion
References
Chapter 12 The Econometrics of Gravity Models in International Trade
12.1 Introduction
12.2 Generic Theoretical Background
12.3 Specific Problems with Estimating Gravity Models
12.3.1 Heteroskedasticity
12.3.2 Modelling the Mass Point at Zero Bilateral Trade Flows
12.3.3 Dynamics
12.3.4 Spatial Data: Interdependence of Bilateral Trade Flows Conditional on Exogenous Determinants
12.3.5 Endogenous Regressors
12.3.6 Ratio Estimators
12.3.7 Binary Regressors and Event-study Designs
12.3.8 High-dimensional Data Designs
12.3.9 Nonparametric Analysis and Machine Learning
12.3.10 Inference
12.4 Conclusion
References
Chapter 13 Modelling Housing Using Multi-dimensional Panel Data
13.1 Introduction
13.2 Discrete Choice Models and Hedonic Price Functions: A Quick Overview
13.3 Multi-dimensional Models of Housing Hedonic Price Functions: Some Examples
13.4 Multi-dimensional Models of Residential Mobility and Location Choice: Some Examples
13.5 Multi-dimensional Dynamic Models of Housing Models
13.6 From Variational Inference to Multi-dimensional Housing Models
13.7 Conclusion
Appendix: Inverse G-Wishart distributions
References
Chapter 14 Modelling Migration
14.1 Introduction and Objectives
14.2 Micro-foundations of the Gravity Model of Migration
14.3 Data Limitations and Estimation Issues
14.3.1 Data and Measurement Issues
14.3.2 Missing and Incomplete Data
14.3.3 Logs and Zeros
14.3.4 Multilateral Resistance to Migration
14.3.5 Endogeneity
14.3.6 Spatial Models
14.4 Concluding Remarks
Acknowledgements
References
Chapter 15 Multi-dimensional Panels in Health Economics with an Application on Antibiotic Consumption
15.1 Introduction
15.2 Literature
15.2.1 Explaining Variation in Healthcare Use with Multi-dimensional Fixed Effects Models
15.2.1.1 Patient and Place Effects
15.2.1.2 Patient and Provider Effects
15.2.2 Data Limitations and Empirical Issues
15.2.2.1 Data Availability
15.2.2.2 Healthcare Use Distributions: Zeros and Heavy Right Tails
15.2.2.3 Limited Mobility Bias
15.3 Application on Antibiotic Consumption
15.3.1 Literature
15.3.2 Data
15.3.2.1 Movers
15.3.3 Decomposing Antibiotic Consumption into Placeand Patient-specific Components
15.3.4 Effect of Primary Care Availability on Antibiotic Use
15.4 Conclusions
Acknowledgements
References
Chapter 16 Can Machine Learning Beat Gravity in Flow Prediction?
16.1 Introduction
16.2 Gravity Models for Flow Prediction
16.3 Machine Learning Models for Flow Prediction
16.3.1 Random Forest Model
16.3.2 Neural Networks
16.3.3 Graph Neural Network
16.4 Empirical Evaluation of the Models
16.4.1 Data Description and Characteristics
16.4.2 Evaluation Metrics
16.4.3 Experimental Results
16.4.4 Discussion
16.5 Concluding Remarks
References
Index