tailieunhanh - Handbook of Econometrics Vols1-5 _ Chapter 12

Chapter 12 COMPUTATIONAL PROBLEMS AND METHODS Although economic theory generally provides only loose restrictions on the distribution of observable quantities, much econometric work is based on tightly specified parametric models and likelihood based methods of inference. | Chapter 12 COMPUTATIONAL PROBLEMS AND METHODS RICHARD E. QUANDT Princeton University Contents 1. Introduction 701 2. Matrix methods 702 . Methods for solving A 3 c 703 . Singular value decomposition 706 . Sparse matrix methods 707 3. Common functions requiring optimization 707 . Likelihood functions 707 . Generalized distance functions 709 . Functions in optimal control 710 4. Algorithms for optimizing functions of many variables 710 . Introduction 710 . Methods employing no derivatives 712 . Methods employing first and second derivatives 717 . Methods employing first derivatives 721 5. Special purpose algorithms and simplifications 724 . Jacobi and Gauss-Seidel methods 724 . Parke s Algorithm A 727 . The EM algorithm 728 . Simplified Jacobian computation 729 6. Further aspects of algorithms 730 . Computation of derivatives 731 . Linear searches 735 . Stopping criteria 737 . Multiple optima 738 7. Particular problems in optimization 740 . Smoothing of non-differentiable functions 740 1 am indebted to David A. Belsley Angus Deaton Ray C. Fair Stephen M. Goldfeld Jerry A. Hausman and Mark Plant for constructive comments. Handbook of Econometrics Volume I Edited by Z. Griliches and . Intriligator North-Holland Publishing Company 1983 700 R. E. Quandt 12. Unbounded likelihood functions and other false optima 742 . Constraints on the parameters 744 8. Numerical integration 747 . Monte Carlo integration 749 . Polynomial approximations 750 . Evaluation of multivariate normal integrals 751 . Special cases of the multivariate normal integral 753 9. The generation of random numbers 755 . The generation of uniformly distributed variables 756 . The generation of normally distributed variables 757 References 760 Ch. 12 Computational Problems 701 1. Introduction The very substantial growth in econometric and statistical theory in the last 30 years has been at least matched by the explosive growth of