Preface
Acknowledgments
Notation
1 Introduction
1.1 What Is Econometrics?
1.2 The Probability Approach to Econometrics
1.3 Econometric Terms
1.4 Observational Data
1.5 Standard Data Structures
1.6 Econometric Software
1.7 Replication
1.8 Data Files for Textbook
1.9 Reading the Book
Part I Regression
2 Conditional Expectation and Projection
2.1 Introduction
2.2 The Distribution of Wages
2.3 Conditional Expectation
2.4 Logs and Percentages
2.5 Conditional Expectation Function
2.6 Continuous Variables
2.7 Law of Iterated Expectations
2.8 CEF Error
2.9 Intercept-Only Model
2.10 Regression Variance
2.11 Best Predictor
2.12 Conditional Variance
2.13 Homoskedasticity and Heteroskedasticity
2.14 Regression Derivative
2.15 Linear CEF
2.16 Linear CEF with Nonlinear Effects
2.17 Linear CEF with Dummy Variables
2.18 Best Linear Predictor
2.19 Illustrations of Best Linear Predictor
2.20 Linear Predictor Error Variance
2.21 Regression Coefficients
2.22 Regression Subvectors
2.23 Coefficient Decomposition
2.24 Omitted Variable Bias
2.25 Best Linear Approximation
2.26 Regression to the Mean
2.27 Reverse Regression
2.28 Limitations of the Best Linear Projection
2.29 Random Coefficient Model
2.30 Causal Effects
2.31 Existence and Uniqueness of the Conditional Expectation*
2.32 Identification*
2.33 Technical Proofs*
2.34 Exercises
3 The Algebra of Least Squares
3.1 Introduction
3.2 Samples
3.3 Moment Estimators
3.4 Least Squares Estimator
3.5 Solving for Least Squares with One Regressor
3.6 Solving for Least Squares with Multiple Regressors
3.7 Illustration
3.8 Least Squares Residuals
3.9 Demeaned Regressors
3.10 Model in Matrix Notation
3.11 Projection Matrix
3.12 Annihilator Matrix
3.13 Estimation of Error Variance
3.14 Analysis of Variance
3.15 Projections
3.16 Regression Components
3.17 Regression Components (Alternative Derivation)*
3.18 Residual Regression
3.19 Leverage Values
3.20 Leave-One-Out Regression
3.21 Influential Observations
3.22 CPS Dataset
3.23 Numerical Computation
3.24 Collinearity Errors
3.25 Programming
3.26 Exercises
4 Least Squares Regression
4.1 Introduction
4.2 Random Sampling
4.3 Sample Mean
4.4 Linear Regression Model
4.5 Expectation of Least Squares Estimator
4.6 Variance of Least Squares Estimator
4.7 Unconditional Moments
4.8 Gauss-Markov Theorem
4.9 Generalized Least Squares
4.10 Residuals
4.11 Estimation of Error Variance
4.12 Mean-Squared Forecast Error
4.13 Covariance Matrix Estimation under Homoskedasticity
4.14 Covariance Matrix Estimation under Heteroskedasticity
4.15 Standard Errors
4.16 Estimation with Sparse Dummy Variables
4.17 Computation
4.18 Measures of Fit
4.19 Empirical Example
4.20 Multicollinearity
4.21 Clustered Sampling
4.22 Inference with Clustered Samples
4.23 At What Level to Cluster?
4.24 Technical Proofs*
4.25 Exercises
5 Normal Regression
5.1 Introduction
5.2 The Normal Distribution
5.3 Multivariate Normal Distribution
5.4 Joint Normality and Linear Regression
5.5 Normal Regression Model
5.6 Distribution of OLS Coefficient Vector
5.7 Distribution of OLS Residual Vector
5.8 Distribution of Variance Estimator
5.9 t-Statistic
5.10 Confidence Intervals for Regression Coefficients
5.11 Confidence Intervals for Error Variance
5.12 t-Test
5.13 Likelihood Ratio Test
5.14 Information Bound for Normal Regression
5.15 Exercises
Part II Large Sample Methods
6 A Review of Large Sample Asymptotics
6.1 Introduction
6.2 Modes of Convergence
6.3 Weak Law of Large Numbers
6.4 Central Limit Theorem
6.5 Continuous Mapping Theorem and Delta Method
6.6 Smooth Function Model
6.7 Stochastic Order Symbols
6.8 Convergence of Moments
7 Asymptotic Theory for Least Squares
7.1 Introduction
7.2 Consistency of Least Squares Estimator
7.3 Asymptotic Normality
7.4 Joint Distribution
7.5 Consistency of Error Variance Estimators
7.6 Homoskedastic Covariance Matrix Estimation
7.7 Heteroskedastic Covariance Matrix Estimation
7.8 Summary of Covariance Matrix Notation
7.9 Alternative Covariance Matrix Estimators*
7.10 Functions of Parameters
7.11 Asymptotic Standard Errors
7.12 t-Statistic
7.13 Confidence Intervals
7.14 Regression Intervals
7.15 Forecast Intervals
7.16 Wald Statistic
7.17 Homoskedastic Wald Statistic
7.18 Confidence Regions
7.19 Edgeworth Expansion*
7.20 Uniformly Consistent Residuals*
7.21 Asymptotic Leverage*
7.22 Exercises
8 Restricted Estimation
8.1 Introduction
8.2 Constrained Least Squares
8.3 Exclusion Restriction
8.4 Finite Sample Properties
8.5 Minimum Distance
8.6 Asymptotic Distribution
8.7 Variance Estimation and Standard Errors
8.8 Efficient Minimum Distance Estimator
8.9 Exclusion Restriction Revisited
8.10 Variance and Standard Error Estimation
8.11 Hausman Equality
8.12 Example: Mankiw, Romer, and Weil (1992)
8.13 Misspecification
8.14 Nonlinear Constraints
8.15 Inequality Restrictions
8.16 Technical Proofs*
8.17 Exercises
9 Hypothesis Testing
9.1 Introduction
9.2 Hypotheses
9.3 Acceptance and Rejection
9.4 Type I Error
9.5 t-Tests
9.6 Type II Error and Power
9.7 Statistical Significance
9.8 p-Values
9.9 t-Ratios and the Abuse of Testing
9.10 Wald Tests
9.11 Homoskedastic Wald Tests
9.12 Criterion-Based Tests
9.13 Minimum Distance Tests
9.14 Minimum Distance Tests under Homoskedasticity
9.15 F Tests
9.16 Hausman Tests
9.17 Score Tests
9.18 Problems with Tests of Nonlinear Hypotheses
9.19 Monte Carlo Simulation
9.20 Confidence Intervals by Test Inversion
9.21 Multiple Tests and Bonferroni Corrections
9.22 Power and Test Consistency
9.23 Asymptotic Local Power
9.24 Asymptotic Local Power, Vector Case
9.25 Exercises
10 Resampling Methods
10.1 Introduction
10.2 Example
10.3 Jackknife Estimation of Variance
10.4 Example
10.5 Jackknife for Clustered Observations
10.6 The Bootstrap Algorithm
10.7 Bootstrap Variance and Standard Errors
10.8 Percentile Interval
10.9 The Bootstrap Distribution
10.10 The Distribution of the Bootstrap Observations
10.11 The Distribution of the Bootstrap Sample Mean
10.12 Bootstrap Asymptotics
10.13 Consistency of the Bootstrap Estimate of Variance
10.14 Trimmed Estimator of Bootstrap Variance
10.15 Unreliability of Untrimmed Bootstrap Standard Errors
10.16 Consistency of the Percentile Interval
10.17 Bias-Corrected Percentile Interval
10.18 BCa Percentile Interval
10.19 Percentile-t Interval
10.20 Percentile-t Asymptotic Refinement
10.21 Bootstrap Hypothesis Tests
10.22 Wald-Type Bootstrap Tests
10.23 Criterion-Based Bootstrap Tests
10.24 Parametric Bootstrap
10.25 How Many Bootstrap Replications?
10.26 Setting the Bootstrap Seed
10.27 Bootstrap Regression
10.28 Bootstrap Regression Asymptotic Theory
10.29 Wild Bootstrap
10.30 Bootstrap for Clustered Observations
10.31 Technical Proofs*
10.32 Exercises
Part III Multiple Equation Models
11 Multivariate Regression
11.1 Introduction
11.2 Regression Systems
11.3 Least Squares Estimator
11.4 Expectation and Variance of Systems Least Squares
11.5 Asymptotic Distribution
11.6 Covariance Matrix Estimation
11.7 Seemingly Unrelated Regression
11.8 Equivalence of SUR and Least Squares
11.9 Maximum Likelihood Estimator
11.10 Restricted Estimation
11.11 Reduced Rank Regression
11.12 Principal Component Analysis
11.13 Factor Models
11.14 Approximate Factor Models
11.15 Factor Models with Additional Regressors
11.16 Factor-Augmented Regression
11.17 Multivariate Normal*
11.18 Exercises
12 Instrumental Variables
12.1 Introduction
12.2 Overview
12.3 Examples
12.4 Endogenous Regressors
12.5 Instruments
12.6 Example: College Proximity
12.7 Reduced Form
12.8 Identification
12.9 Instrumental Variables Estimator
12.10 Demeaned Representation
12.11 Wald Estimator
12.12 Two-Stage Least Squares
12.13 Limited Information Maximum Likelihood
12.14 Split-Sample IV and JIVE
12.15 Consistency of 2SLS
12.16 Asymptotic Distribution of 2SLS
12.17 Determinants of 2SLS Variance
12.18 Covariance Matrix Estimation
12.19 LIML Asymptotic Distribution
12.20 Functions of Parameters
12.21 Hypothesis Tests
12.22 Finite Sample Theory
12.23 Bootstrap for 2SLS
12.24 The Peril of Bootstrap 2SLS Standard Errors
12.25 Clustered Dependence
12.26 Generated Regressors
12.27 Regression with Expectation Errors
12.28 Control Function Regression
12.29 Endogeneity Tests
12.30 Subset Endogeneity Tests
12.31 Overidentification Tests
12.32 Subset Overidentification Tests
12.33 Bootstrap Overidentification Tests
12.34 Local Average Treatment Effects
12.35 Identification Failure
12.36 Weak Instruments
12.37 Many Instruments
12.38 Testing for Weak Instruments
12.39 Weak Instruments with k2 > 1
12.40 Example: Acemoglu, Johnson, and Robinson (2001)
12.41 Example: Angrist and Krueger (1991)
12.42 Programming
12.43 Exercises
13 Generalized Method of Moments
13.1 Introduction
13.2 Moment Equation Models
13.3 Method of Moments Estimators
13.4 Overidentified Moment Equations
13.5 Linear Moment Models
13.6 GMM Estimator
13.7 Distribution of GMM Estimator
13.8 Efficient GMM
13.9 Efficient GMM versus 2SLS
13.10 Estimation of the Efficient Weight Matrix
13.11 Iterated GMM
13.12 Covariance Matrix Estimation
13.13 Clustered Dependence
13.14 Wald Test
13.15 Restricted GMM
13.16 Nonlinear Restricted GMM
13.17 Constrained Regression
13.18 Multivariate Regression
13.19 Distance Test
13.20 Continuously Updated GMM
13.21 Overidentification Test
13.22 Subset Overidentification Tests
13.23 Endogeneity Test
13.24 Subset Endogeneity Test
13.25 Nonlinear GMM
13.26 Bootstrap for GMM
13.27 Conditional Moment Equation Models
13.28 Technical Proofs*
13.29 Exercises
Part IV Dependent and Panel Data
14 Time Series
14.1 Introduction
14.2 Examples
14.3 Differences and Growth Rates
14.4 Stationarity
14.5 Transformations of Stationary Processes
14.6 Convergent Series
14.7 Ergodicity
14.8 Ergodic Theorem
14.9 Conditioning on Information Sets
14.10 Martingale Difference Sequences
14.11 CLT for Martingale Differences
14.12 Mixing
14.13 CLT for Correlated Observations
14.14 Linear Projection
14.15 White Noise
14.16 The Wold Decomposition
14.17 Lag Operator
14.18 Autoregressive Wold Representation
14.19 Linear Models
14.20 Moving Average Process
14.21 Infinite-Order Moving Average Process
14.22 First-Order Autoregressive Process
14.23 Unit Root and Explosive AR(1) Processes
14.24 Second-Order Autoregressive Process
14.25 AR(p) Process
14.26 Impulse Response Function
14.27 ARMA and ARIMA Processes
14.28 Mixing Properties of Linear Processes
14.29 Identification
14.30 Estimation of Autoregressive Models
14.31 Asymptotic Distribution of Least Squares Estimator
14.32 Distribution under Homoskedasticity
14.33 Asymptotic Distribution under General Dependence
14.34 Covariance Matrix Estimation
14.35 Covariance Matrix Estimation under General Dependence
14.36 Testing the Hypothesis of No Serial Correlation
14.37 Testing for Omitted Serial Correlation
14.38 Model Selection
14.39 Illustrations
14.40 Time Series Regression Models
14.41 Static, Distributed Lag, and Autoregressive Distributed Lag Models
14.42 Time Trends
14.43 Illustration
14.44 Granger Causality
14.45 Testing for Serial Correlation in Regression Models
14.46 Bootstrap for Time Series
14.47 Technical Proofs*
14.48 Exercises
15 Multivariate Time Series
15.1 Introduction
15.2 Multiple Equation Time Series Models
15.3 Linear Projection
15.4 Multivariate Wold Decomposition
15.5 Impulse Response
15.6 VAR(1) Model
15.7 VAR(p) Model
15.8 Regression Notation
15.9 Estimation
15.10 Asymptotic Distribution
15.11 Covariance Matrix Estimation
15.12 Selection of Lag Length in a VAR
15.13 Illustration
15.14 Predictive Regressions
15.15 Impulse Response Estimation
15.16 Local Projection Estimator
15.17 Regression on Residuals
15.18 Orthogonalized Shocks
15.19 Orthogonalized Impulse Response Function
15.20 Orthogonalized Impulse Response Estimation
15.21 Illustration
15.22 Forecast Error Decomposition
15.23 Identification of Recursive VARs
15.24 Oil Price Shocks
15.25 Structural VARs
15.26 Identification of Structural VARs
15.27 Long-Run Restrictions
15.28 Blanchard and Quah (1989) Illustration
15.29 External Instruments
15.30 Dynamic Factor Models
15.31 Technical Proofs*
15.32 Exercises
16 Nonstationary Time Series
16.1 Introduction
16.2 Partial Sum Process and Functional Convergence
16.3 Beveridge-Nelson Decomposition
16.4 Functional CLT
16.5 Orders of Integration
16.6 Means, Local Means, and Trends
16.7 Demeaning and Detrending
16.8 Stochastic Integrals
16.9 Estimation of an AR(1)
16.10 AR(1) Estimation with an Intercept
16.11 Sample Covariances of Integrated and Stationary Processes
16.12 AR(p) Models with a Unit Root
16.13 Testing for a Unit Root
16.14 KPSS Stationarity Test
16.15 Spurious Regression
16.16 NonStationary VARs
16.17 Cointegration
16.18 Role of Intercept and Trend
16.19 Cointegrating Regression
16.20 VECM Estimation
16.21 Testing for Cointegration in a VECM
16.22 Technical Proofs*
16.23 Exercises
17 Panel Data
17.1 Introduction
17.2 Time Indexing and Unbalanced Panels
17.3 Notation
17.4 Pooled Regression
17.5 One-Way Error Component Model
17.6 Random Effects
17.7 Fixed Effects Model
17.8 Within Transformation
17.9 Fixed Effects Estimator
17.10 Differenced Estimator
17.11 Dummy Variables Regression
17.12 Fixed Effects Covariance Matrix Estimation
17.13 Fixed Effects Estimation in Stata
17.14 Between Estimator
17.15 Feasible GLS
17.16 Intercept in Fixed Effects Regression
17.17 Estimation of Fixed Effects
17.18 GMM Interpretation of Fixed Effects
17.19 Identification in the Fixed Effects Model
17.20 Asymptotic Distribution of Fixed Effects Estimator
17.21 Asymptotic Distribution for Unbalanced Panels
17.22 Heteroskedasticity-Robust Covariance Matrix Estimation
17.23 Heteroskedasticity-Robust Estimation—Unbalanced Case
17.24 Hausman Test for Random vs. Fixed Effects
17.25 Random Effects or Fixed Effects?
17.26 Time Trends
17.27 Two-Way Error Components
17.28 Instrumental Variables
17.29 Identification with Instrumental Variables
17.30 Asymptotic Distribution of Fixed Effects 2SLS Estimator
17.31 Linear GMM
17.32 Estimation with Time-Invariant Regressors
17.33 Hausman-Taylor Model
17.34 Jackknife Covariance Matrix Estimation
17.35 Panel Bootstrap
17.36 Dynamic Panel Models
17.37 The Bias of Fixed Effects Estimation
17.38 Anderson-Hsiao Estimator
17.39 Arellano-Bond Estimator
17.40 Weak Instruments
17.41 Dynamic Panels with Predetermined Regressors
17.42 Blundell-Bond Estimator
17.43 Forward Orthogonal Transformation
17.44 Empirical Illustration
17.45 Exercises
18 Difference in Differences
18.1 Introduction
18.2 Minimum Wage in New Jersey
18.3 Identification
18.4 Multiple Units
18.5 Do Police Reduce Crime?
18.6 Trend Specification
18.7 Do Blue Laws Affect Liquor Sales?
18.8 Check Your Code: Does Abortion Impact Crime?
18.9 Inference
18.10 Exercises
Part V Nonparametric Methods
19 Nonparametric Regression
19.1 Introduction
19.2 Binned Means Estimator
19.3 Kernel Regression
19.4 Local Linear Estimator
19.5 Local Polynomial Estimator
19.6 Asymptotic Bias
19.7 Asymptotic Variance
19.8 AIMSE
19.9 Reference Bandwidth
19.10 Estimation at a Boundary
19.11 Nonparametric Residuals and Prediction Errors
19.12 Cross-Validation Bandwidth Selection
19.13 Asymptotic Distribution
19.14 Undersmoothing
19.15 Conditional Variance Estimation
19.16 Variance Estimation and Standard Errors
19.17 Confidence Bands
19.18 The Local Nature of Kernel Regression
19.19 Application to Wage Regression
19.20 Clustered Observations
19.21 Application to Test Scores
19.22 Multiple Regressors
19.23 Curse of Dimensionality
19.24 Partially Linear Regression
19.25 Computation
19.26 Technical Proofs*
19.27 Exercises
20 Series Regression
20.1 Introduction
20.2 Polynomial Regression
20.3 Illustrating Polynomial Regression
20.4 Orthogonal Polynomials
20.5 Splines
20.6 Illustrating Spline Regression
20.7 The Global/Local Nature of Series Regression
20.8 Stone-Weierstrass and Jackson Approximation Theory
20.9 Regressor Bounds
20.10 Matrix Convergence
20.11 Consistent Estimation
20.12 Convergence Rate
20.13 Asymptotic Normality
20.14 Regression Estimation
20.15 Undersmoothing
20.16 Residuals and Regression Fit
20.17 Cross-Validation Model Selection
20.18 Variance and Standard Error Estimation
20.19 Clustered Observations
20.20 Confidence Bands
20.21 Uniform Approximations
20.22 Partially Linear Model
20.23 Panel Fixed Effects
20.24 Multiple Regressors
20.25 Additively Separable Models
20.26 Nonparametric Instrumental Variables Regression
20.27 NPIV Identification
20.28 NPIV Convergence Rate
20.29 Nonparametric vs. Parametric Identification
20.30 Example: Angrist and Lavy (1999)
20.31 Technical Proofs*
20.32 Exercises
21 Regression Discontinuity
21.1 Introduction
21.2 Sharp Regression Discontinuity
21.3 Identification
21.4 Estimation
21.5 Inference
21.6 Bandwidth Selection
21.7 RDD with Covariates
21.8 A Simple RDD Estimator
21.9 Density Discontinuity Test
21.10 Fuzzy Regression Discontinuity
21.11 Estimation of FRD
21.12 Exercises
Part VI Nonlinear Methods
22 M-Estimators
22.1 Introduction
22.2 Examples
22.3 Identification and Estimation
22.4 Consistency
22.5 Uniform Law of Large Numbers
22.6 Asymptotic Distribution
22.7 Asymptotic Distribution under Broader Conditions*
22.8 Covariance Matrix Estimation
22.9 Technical Proofs*
22.10 Exercises
23 Nonlinear Least Squares
23.1 Introduction
23.2 Identification
23.3 Estimation
23.4 Asymptotic Distribution
23.5 Covariance Matrix Estimation
23.6 Panel Data
23.7 Threshold Models
23.8 Testing for Nonlinear Components
23.9 Computation
23.10 Technical Proofs*
23.11 Exercises
24 Quantile Regression
24.1 Introduction
24.2 Median Regression
24.3 Least Absolute Deviations
24.4 Quantile Regression
24.5 Example Quantile Shapes
24.6 Estimation
24.7 Asymptotic Distribution
24.8 Covariance Matrix Estimation
24.9 Clustered Dependence
24.10 Quantile Crossings
24.11 Quantile Causal Effects
24.12 Random Coefficient Representation
24.13 Nonparametric Quantile Regression
24.14 Panel Data
24.15 IV Quantile Regression
24.16 Technical Proofs*
24.17 Exercises
25 Binary Choice
25.1 Introduction
25.2 Binary Choice Models
25.3 Models for the Response Probability
25.4 Latent Variable Interpretation
25.5 Likelihood
25.6 Pseudo-True Values
25.7 Asymptotic Distribution
25.8 Covariance Matrix Estimation
25.9 Marginal Effects
25.10 Application
25.11 Semiparametric Binary Choice
25.12 IV Probit
25.13 Binary Panel Data
25.14 Technical Proofs*
25.15 Exercises
26 Multiple Choice
26.1 Introduction
26.2 Multinomial Response
26.3 Multinomial Logit
26.4 Conditional Logit
26.5 Independence of Irrelevant Alternatives
26.6 Nested Logit
26.7 Mixed Logit
26.8 Simple Multinomial Probit
26.9 General Multinomial Probit
26.10 Ordered Response
26.11 Count Data
26.12 BLP Demand Model
26.13 Technical Proofs*
26.14 Exercises
27 Censoring and Selection
27.1 Introduction
27.2 Censoring
27.3 Censored Regression Functions
27.4 The Bias of Least Squares Estimation
27.5 Tobit Estimator
27.6 Identification in Tobit Regression
27.7 CLAD and CQR Estimators
27.8 Illustrating Censored Regression
27.9 Sample Selection Bias
27.10 Heckman’s Model
27.11 Nonparametric Selection
27.12 Panel Data
27.13 Exercises
28 Model Selection, Stein Shrinkage, and Model Averaging
28.1 Introduction
28.2 Model Selection
28.3 Bayesian Information Criterion
28.4 Akaike Information Criterion for Regression
28.5 Akaike Information Criterion for Likelihood
28.6 Mallows Criterion
28.7 Hold-Out Criterion
28.8 Cross-Validation Criterion
28.9 K-Fold Cross-Validation
28.10 Many Selection Criteria Are Similar
28.11 Relation with Likelihood Ratio Testing
28.12 Consistent Selection
28.13 Asymptotic Selection Optimality
28.14 Focused Information Criterion
28.15 Best Subset and Stepwise Regression
28.16 The MSE of Model Selection Estimators
28.17 Inference after Model Selection
28.18 Empirical Illustration
28.19 Shrinkage Methods
28.20 James-Stein Shrinkage Estimator
28.21 Interpretation of the Stein Effect
28.22 Positive Part Estimator
28.23 Shrinkage Toward Restrictions
28.24 Group James-Stein
28.25 Empirical Illustrations
28.26 Model Averaging
28.27 Smoothed BIC and AIC
28.28 Mallows Model Averaging
28.29 Jackknife (CV) Model Averaging
28.30 Granger-Ramanathan Averaging
28.31 Empirical Illustration
28.32 Technical Proofs*
28.33 Exercises
29 Machine Learning
29.1 Introduction
29.2 Big Data, High Dimensionality, and Machine Learning
29.3 High-Dimensional Regression
29.4 p-norms
29.5 Ridge Regression
29.6 Statistical Properties of Ridge Regression
29.7 Illustrating Ridge Regression
29.8 Lasso
29.9 Lasso Penalty Selection
29.10 Lasso Computation
29.11 Asymptotic Theory for the Lasso
29.12 Approximate Sparsity
29.13 Elastic Net
29.14 Post-Lasso
29.15 Regression Trees
29.16 Bagging
29.17 Random Forests
29.18 Ensembling
29.19 Lasso IV
29.20 Double Selection Lasso
29.21 Post-Regularization Lasso
29.22 Double/Debiased Machine Learning
29.23 Technical Proofs*
29.24 Exercises
Appendixes
A Matrix Algebra
A.1 Notation
A.2 Complex Matrices
A.3 Matrix Addition
A.4 Matrix Multiplication
A.5 Trace
A.6 Rank and Inverse
A.7 Orthogonal and Orthonormal Matrices
A.8 Determinant
A.9 Eigenvalues
A.10 Positive Definite Matrices
A.11 Idempotent Matrices
A.12 Singular Values
A.13 Matrix Decompositions
A.14 Generalized Eigenvalues
A.15 Extrema of Quadratic Forms
A.16 Cholesky Decomposition
A.17 QR Decomposition
A.18 Solving Linear Systems
A.19 Algorithmic Matrix Inversion
A.20 Matrix Calculus
A.21 Kronecker Products and the Vec Operator
A.22 Vector Norms
A.23 Matrix Norms
B Useful-Inequalities
B.1 Inequalities for Real Numbers
B.2 Inequalities for Vectors
B.3 Inequalities for Matrices
B.4 Probability Inequalities
B.5 Proofs*
References
Index
