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Handbook of Econometrics Vols1-5 _ Chapter 1

Chapter I LINEAR ALGEBRA AND MATRIX METHODS IN ECONOMETRICS 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 | Chapter 1 LINEAR ALGEBRA AND MATRIX METHODS IN ECONOMETRICS HENRI THEIL University of Florida Contents 1. Introduction 5 2. Why are matrix methods useful in econometrics 5 2.1. Linear systems and quadratic forms 5 2.2. Vectors and matrices in statistical theory 7 2.3. Least squares in the standard linear model 8 2.4. Vectors and matrices in consumption theory 10 3. Partitioned matrices 12 3.1. The algebra of partitioned matrices 12 3.2. Block-recursive systems 14 3.3. Income and price derivatives revisited 15 4. Kronecker products and the vectorization of matrices 16 4.1. The algebra of Kronecker products 16 4.2. Joint generalized least-squares estimation of several equations 17 4.3. Vectorization of matrices 19 5. Differential demand and supply systems 20 5.1. A differential consumer demand system 20 5.2. A comparison with simultaneous equation systems 22 5.3. An extension to the inputs of a firm A singularity problem 23 5.4. A differential input demand system 23 5.5. Allocation systems 25 5.6. Extensions 25 6. Definite and semidefinite square matrices 27 6.1. Covariance matrices and Gauss-Markov further considered ll 6.2. Maxima and minima 29 6.3. Block-diagonal definite matrices 30 7. Diagonalizations 30 7.1. The standard diagonalization of a square matrix 30 Research supported in part by NSF Grant SOC76-82718. The author is indebted to Kenneth Clements Reserve Bank of Australia Sydney and Michael Intriligator University of California Los Angeles for comments on an earlier draft of this chapter. Handbook of Econometrics Volume I Edited by Z. Griliches and M.D. Intriligator North-Holland Publishing Company 1983 4 H. Theil 7.2. Spécial cases 32 7.3. Aitken s theorem 33 7.4. The Cholesky decomposition 34 7.5. Vectors written as diagonal matrices 34 7.6. A simultaneous diagonalization of two square matrices 35 7.7. Latent roots of an asymmetric matrix 36 8. Principal components and extensions 37 8.1. Principal components 37 8.2. Derivations 38 8.3. Further .