Chapter I
LINEAR ALGEBRA AND MATRIX METHODS IN
ECONOMETRICS
HENRI THEIL*
University of Florida
Contents
1. Introduction
2. Why are matrix methods useful in econometrics?
2.1. Linear systems and quadratic forms
2.2. Vectors and matrices in statistical theory
2.3. Least squares in the standard linear model
2.4. Vectors and matrices in consumption theory
3. Partitioned matrices
3. I, The algebra of partitioned matrices
3.2. Block-recursive systems
3.3. Income and price derivatives revisited
4. Kronecker products and the vectorization of matrices
4. I. The algebra of Kronecker products
4.2. Joint generalized least-squares estimation of several equations
4.3. Vectorization of matrices
5. Differential demand and supply systems
5.1. A differential consumer demand system
5.2. A comparison with simultaneous equation systems
5.3. An extension to the inputs of a firm: A singularity problem
5.4. A differential input demand system
5.5. Allocation systems
5.6. Extensions
6. Definite and semidefinite square matrices
6. I. Covariance matrices and Gauss-Markov further considered
6.2. Maxima and minima
6.3. Block-diagonal definite matrices
7. Diagonalizations
7.1. ne standard diagonalization of a square matrix
1.2. Special cases
7.3. Aitken’s theorem
7.4. The Cholesky decomposition
7.5. Vectors written as diagonal matrices
7.6. A simultaneous diagonalization of two square matrices
7.7. Latent roots of an asymmetric matrix
8. Principal components and extensions
8.1. Principal components
8.2. Derivations
8.3. Further discussion of principal components
8.4. The independence transformation in microeconomic theory
8.5. An example
8.6. A principal component interpretation
9. The modeling of a disturbance covariance matrix
9.1. Rational random behavior
9.2. The asymptotics of rational random behavior
9.3. Applications to demand and supply
10. The Moore-Penrose inverse
10.1. Proof of the existence and uniqueness
10.2. Special cases
10.3. A generalization of Aitken’s theorem
10.4. Deleting an equation from an allocation model
Appendix A: Linear independence and related topics
Appendix B: The independence transformation
Appendix C: Rational random behavior
References

Chapter 2
STATISTICAL THEORY AND ECONOMETRICS
ARNOLD ZELLNER*
University of Chicago
Contents
1. Introduction and overview 68
2. Elements of probability theory 69
2.1. Probability models for observations 70
2.2. Definitions of probability 71
2.3. Axiom systems for probability theory 74
2.4. Random variables and probability models 82
2.5. Elements of asymptotic theory 110
3. Estimation theory 117
3. I. Point estimation 117
3.2. Criteria for point estimation 118
4. Interval estimation: Confidence bounds, intervals, and regions 152
4. I. Confidence bounds 152
4.2. Confidence intervals 154
4.3. Confidence regions 156
5. Prediction 158
5.1. Sampling theory prediction techniques 159
5.2. Bayesian prediction techniques 162
6. Statistical analysis of hypotheses 164
6.1. Types of hypotheses 164
6.2. Sampling theory testing procedures 165
6.3. Bayesian analysis of hypotheses 169
7. Summary and concluding remarks 172
References 174