Multivariate Reduced-Rank Regression: Theory, Methods and Applications

Preface to the Second Edition
Preface to the First Edition
Contents
About the Authors
1 Multivariate Linear Regression
1.1 Introduction
1.2 Multivariate Linear Regression Model and Least Squares Estimator
1.3 Further Inference Properties in the Multivariate Regression Model
1.4 Prediction in the Multivariate Linear Regression Model
1.5 Numerical Examples
1.5.1 Biochemical Data
1.5.2 Sales Performance Data
2 Reduced-Rank Regression Model
2.1 The Basic Reduced-Rank Model and Background
2.2 Some Examples of Application of the Reduced-Rank Model
2.3 Estimation of Parameters in the Reduced-Rank Model
2.4 Relation to Principal Components and Canonical Correlation Analysis
2.4.1 Principal Components Analysis
2.4.2 Application to Functional and Structural Relationships Models
2.4.3 Canonical Correlation Analysis
2.5 Asymptotic Distribution of Estimators in Reduced-Rank Model
2.6 Identification of Rank of the Regression Coefficient Matrix
2.7 Reduced-Rank Inverse Regression for Estimating Structural Dimension
2.8 Numerical Examples
2.9 Alternate Procedures for Analysis of Multivariate Regression Models
3 Reduced-Rank Regression Models with Two Sets of Regressors
3.1 Reduced-Rank Model of Anderson
3.2 Application to One-Way ANOVA and Linear Discriminant Analysis
3.3 Numerical Example Using Chemometrics Data
3.4 Both Regression Matrices of Lower Ranks: Model and Its Applications
3.5 Estimation and Inference for the Model
3.5.1 Efficient Estimator
3.5.2 An Alternative Estimator
3.5.3 Asymptotic Inference
3.6 Identification of Ranks of Coefficient Matrices
3.7 An Example on Ozone Data
3.8 Conclusion
4 Reduced-Rank Regression Model With Autoregressive Errors
4.1 Introduction and the Model
4.2 Example on the U.K. Economy: Basic Data and Their Descriptions
4.3 Maximum Likelihood Estimators for the Model
4.4 Computational Algorithms for Efficient Estimators
4.5 Alternative Estimators and Their Properties
4.5.1 A Comparison Between Efficient and Other Estimators
4.6 Identification of Rank of the Regression Coefficient Matrix
4.7 Inference for the Numerical Example
4.8 An Alternate Estimator with Kronecker Approximation
4.8.1 Computational Results
5 Multiple Time Series Modeling With Reduced Ranks
5.1 Introduction and Time Series Models
5.2 Reduced-Rank Autoregressive Models
5.2.1 Estimation and Inference
5.2.2 Relationship to Canonical Analysis of Box and Tiao
5.3 An Extended Reduced-Rank Autoregressive Model
5.4 Nested Reduced-Rank Autoregressive Models
5.4.1 Specification of Ranks
5.4.2 A Canonical Form
5.4.3 Maximum Likelihood Estimation
5.5 Numerical Example: U.S. Hog Data
5.6 Relationship Between Nonstationarity and Canonical Correlations
5.7 Cointegration for Nonstationary Series—Reduced Rank in Long Term
5.7.1 LS and ML Estimation and Inference
5.7.2 Likelihood Ratio Test for the Number of Cointegrating Relations
5.8 Unit Root and Cointegration Aspects for the U.S. Hog Data Example
6 The Growth Curve Model and Reduced-Rank Regression Methods
6.1 Introduction and the Growth Curve Model
6.2 Estimation of Parameters in the Growth Curve Model
6.3 Likelihood Ratio Testing of Linear Hypotheses in Growth Curve Model
6.4 An Extended Model for Growth Curve Data
6.5 Modification of Basic Growth Curve Model to Reduced-Rank Model
6.6 Reduced-Rank Growth Curve Models
6.6.1 Extensions of the Reduced-Rank Growth Curve Model
6.7 Application to One-way ANOVA and Linear Discriminant Analysis
6.8 A Numerical Example
6.9 Some Recent Developments
7 Seemingly Unrelated Regressions Models With Reduced Ranks
7.1 Introduction and the Seemingly Unrelated Regressions Model
7.2 Relation of Growth Curve Model to the Seemingly Unrelated Regressions Model
7.3 Reduced-Rank Coefficient in Seemingly Unrelated Regressions Model
7.4 Maximum Likelihood Estimators for Reduced-Rank Model
7.5 An Alternate Estimator and Its Properties
7.6 Identification of Rank of the Regression Coefficient Matrix
7.7 A Numerical Illustration with Scanner Data
7.8 Some Recent Developments
8 Applications of Reduced-Rank Regression in Financial Economics
8.1 Introduction to Asset Pricing Models
8.2 Estimation and Testing in the Asset Pricing Model
8.3 Additional Applications of Reduced-Rank Regression in Finance
8.4 Empirical Studies and Results on Asset Pricing Models
8.5 Related Topics
8.6 An Application
8.7 Cointegration and Pairs Trading
9 Partially Reduced-Rank Regression with Grouped Responses
9.1 Introduction: Partially Reduced-Rank Regression Model
9.2 Estimation of Parameters
9.3 Test for Rank and Inference Results
9.4 Procedures for Identification of Subset Reduced-Rank Structure
9.5 Illustrative Examples
9.6 Discussion and Extensions
10 High-Dimensional Reduced-Rank Regression
10.1 Introduction
10.2 Overview of High-Dimensional Regularized Regression
10.3 Framework of Singular Value Regularization
10.4 Reduced-Rank Regression via Adaptive Nuclear-Norm Penalization
10.4.1 Adaptive Nuclear Norm
10.4.2 Adaptive Nuclear-Norm Penalized Regression
10.4.3 Theoretical Analysis
10.4.3.1 Setup and Assumptions
10.4.3.2 Rank Consistency and Prediction Error Bound
10.5 Integrative Reduced-Rank Regression: Bridging Sparse and Low-Rank Models
10.5.1 Composite Nuclear-Norm Penalization
10.5.2 Theoretical Analysis
10.6 Applications
10.6.1 Breast Cancer Data
10.6.2 Longitudinal Studies of Aging
11 Unbiased Risk Estimation in Reduced-Rank Regression
11.1 Introduction
11.2 Degrees of Freedom
11.3 Degrees of Freedom of Reduced-Rank Estimation
11.4 Comparing Empirical and Exact Estimators
11.4.1 Simulation Setup
11.4.2 Comparing Estimators of the Degrees of Freedom
11.4.3 Performance on Estimating the Prediction Error
11.4.4 Performance on Model Selection
11.5 Applications
11.5.1 Norwegian Paper Quality Data
11.5.2 Arabidopsis Thaliana Data
12 Generalized Reduced-Rank Regression
12.1 Introduction
12.2 Robust Reduced-Rank Regression
12.2.1 Non-Robustness of Reduced-Rank Regression
12.2.2 Robustification with Sparse Mean Shift
12.2.3 Theoretical Analysis
12.3 Reduced-Rank Estimation with Incomplete Data
12.3.1 Noiseless Matrix Completion
12.3.2 Stable Matrix Completion
12.3.3 Computation
12.4 Generalized Reduced-Rank Regression with Mixed Outcomes
12.5 Applications
12.5.1 Arabidopsis Thaliana data
12.5.2 Longitudinal Studies of Aging
13 Sparse Reduced-Rank Regression
13.1 Introduction
13.2 Sparse Reduced-Rank Regression
13.2.1 Sparse Reduced-Rank Regression for Predictor Selection
13.2.2 Computation
13.2.3 Theoretical Analysis
13.3 Co-sparse Factor Regression
13.3.1 Model Formulation and Deflation Procedures
13.3.2 Co-sparse Unit-Rank Estimation
13.3.3 Theoretical Analysis
13.4 Applications
13.4.1 Yeast eQTL Mapping Analysis
13.4.2 Forecasting Macroeconomic and Financial Indices
Appendix
References
Subject Index
Reference Index