list of last pages Leif Jansson anderik Mellander

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1 A ist of ast pages Papers on No. 70, 1982 OptiJnization under non1inear COnstraints by Leif Jansson anderik Meander October, 1982

2 - 2 - OP"'nUZA'I'ION UNDER NONLIHEAR CONS'I'RAIN'I'S Leif Jansson The Ministry of Economic Stockhom Sweden rs Erik Meander The Industria Institute for Economic and Socia Research Stockhom Sweden In this paper atimesaving method is proposed for maximizing ikeihood functions when the parameter space is subject to noninear constraints, expressabe as second order poynomias. The suggested approach is especiay attractive when deaing with systems with many parameters. 1. Introduction In this note the impement.ation of second order poynomia constraints on a parameter matrix is given a simpe agebraic form. First order partiai derivatives of the constrained matrix are aso given simpe vector forms. The formuas presented are easiytransformed into computer agorithms. One area of specia interest for the appication of the above resuts is the maximization of the ikeihood function of a system of simutaneous equations.

3 A max.immbj. 1ike1ihood function Consider a standard simutaneous equation system compacty written as BY' + CZ' = AX' = E' ( ) where A = (B IC) is the n, (n+m) parameter matrix, X' = (Yz)' is the (n+m),t matrix of observaendogenous and predetermined tions on the varia bes and the coumns of vectors of unobservabe errors are assumed to E', et (t=1,2,..,t) are random disturbances. The have a time independent mutivariate norma distribution, et NID ( O, ). I f the variance-covariance matrix is unconstrained the og-ikeihood function can be concentrated with respect to giving L = k + T og B - 1/2 T og (AX'XA') Koopmans and Hood (1953) determinant of matrix M. where denotes the A common way of maximizing the og-ikeihood function L is to use an iterative gradient search method where the derivatives are numericay cacuated. This approach aows for a fexibe specification of the equations and makes it easy to specify noninear restrictions, but the number of evauations of the og-ikeihood function is in each iteration proportiona to the number of parameters to be estirnated. Thus, with many parameters to estimate there can be considerabe gains in computing time if the first order derivatives instead can be given a simpe anaytica form.

4 - 4 - The derivative of L with respect to a free parame~ ter i can be written as L = vec A ' ~e~ ) 1 -' e vec (( B I O) - QJ) ( 2 ) where QJ = [AX'XA,]- A(X'X) where vec M denotes the coumn vector whose eements are the coumns of M put on top of each other. Much of the cacuations to achieve the compicated matrix QJ is done to get t.he ikeihood vaue L. So when the derivative vector oa/08. can be 1 easiy cacuated aso 01./08. is given with itte extra computation Ca1cu1ations of derivatives The restricted parameters a.. of the matrix A can 1) be expressed as functions of k unrestricted parameters ej. Second order poynomia restrictions on the parameters of the matrix A as can then be written vec[a(8)] = ReF + R2eF2 + d where R = the [n(n+m),k] inear restriction matrix F = the coumn vector containing the k free parame'ters R2 = the [n(n+m),k(k+)/2] non-inear restriction matrix

5 - 5 - F2 ::::: the coumn vector of the k(k+)/2 possibe different products of the k free parameters 2 2 Le. F2' ::::: (G ' G I G 2, GIGk,..,G k ) d ::::: a coumn vector with n(n+m) constants The first order derivatives of the inear part of the restricted paramet.ers of the A' matrix with respect to G. become o 08. R F = R L11. where L11 is a coumn vector with eements j =,..., k and o., J if j =/; i if j ::::: i rrhus, the first order derivatives of R F with respect to G. are equa to the i:th coumn of the inear restriction matrix R. The non-inear part the derivatives becomes o og. R2 F2 = R2 L12. where L12 is a coumn vector with eement 'j' j :::::,., k (k+ ) /2 and O if G j and G j2 7:: O '. ::::: J G j if G j2 ::::: O. j2 if G j ::::: G. 2G. if G j ::::: G j2 ::::: G. 1

6 - 6 - Thus, the first order derivative of the non-inear part become inear combinations of the coumns of R2. The derivative of R2-P2 with respect to for instance, can aternativey be written as 8., o(r2 -P2) = R L: 8. - R2. '.J.' J J Jr In practice the R, R2 and d matrices wi contain mosty zeros and it woud be practicay impossibe to set them up as in the formuas. T~hey can, however, easiy be packed into a dense form by storing ony the nonzero eements of the R and R2 matrix together with their destination in the A matrix and the position of the free variabe parameters they are associated with.

7 - 7 - Reference Koopmans, T. C. and Hood, vv. C., 1953, The estimation of s imu1taneous economic reationships, in: W.C. Hood and T.C. Koopmans, eds., Studies in Econometric \1ethod (John Wi1ey, New York)

8 WORKING PAPERS (Missing numbers indicate pubication esewhere) Corporate and Persona Taxation and the Growing Firm by Uf Jakobsson 7. A Micro Macro Interactive Simuation Mode of the Swedish Economy. Preiminary mode specification by Gunnar Eiasson in coaboration with Gösta Oavi 8. Estimation and Anaysis with a WOI Production Function by Göran Eriksson, Uf Jakobsson and Leif Jansson 1977 I. A Comparative Study of Compete Systems of Oemand Functions by N Anders Kevmarken 1,2. The Linear Expenditure System and Oemand for Housing under Rent Controi by Per Högberg and N Anders Kevmarken t 4. Rates of Oepreciation of Human Capita Oue to Nonuse by Siv Gustafsson 15. Pay Differentias between Government and Private Sector Empoyees in Sweden by Siv Gustafsson A Putty-Cay Mode of Oemand Uncertainty and Investment by James W. Abrecht and Abert G. Hart On Unexpained Price Oifferences by Bo Axe 26. The West European Stee Industry - Structure and Competitiveness in Historica Perspective by Bo Carsson 27. Crises, Infation and Reative Prices in Sweden by Märtha Josefsson and Johan Örtengren

9 The Demand for Energy in Swedish Manufacturing by Joyce M. Dare:ay 34. Imperfect Information Equiibrium, Existence, Configuration and Stabiity by Bo Axe Va1ue Added Tax: Experience in Sweden by Göran Normann 3h. Energi} stabiitet och tiväxt i svensk ekonomi (Energy. Stabiity and Growth in the Swedish Economy) by Bengt-Christer Ysander 37. Pkking Wnners or Baiing out Losers? A study of the Swedish state hoding companv and its ro1e in the new Swedish industria poicy by Gunnar Eiasson and Bengt-Christer Ysander 38. Utiity in Loca Government Budgeting by Bengt-Christer Ysander 40. Wage Earners Funds and Rationai Expectations by Bo Axe 41. AVintage Mode for the Swedish Iron and Stee Industry by Leif Jansson 42. The Structure of the Isac Mode by Leif Jansson, Tomas Nordström and Bengt-Christer Ysander 43. An Econometric Mode of Loca Government and Budgeting by Bengt-Christer Ysander 44. Loca Authorities, Economic Stabiity and the Efficiency of Fisca Poicy by Tomas Nordström and Bengt-Christer Ysander 45. Growthy Exit and Entry of Firms by Göran Eriksson '-17. OH Prices and Economic Stabiity. The Macroeconomic Impact of Oi Price Shocks on the Swedish Economy by Bengt-Christer Ysander 48. An Examination of the Impact of Changes in the Prices of Fue1s and Primary Meta1s on Nordic Countries Using a Word Econometric Mode by K. S. Sarma

10 Fexibiity in Budget Poicy. Changing Probems and Requirements of Pubic Budgeting by A. Robinson and.b.-c. Ysander 51. On Price Easticities in Foreign Trade by Eva Christina Horwitz 52. Swedish Export Performance A Constant Market Shares Anaysis by Eva Christina Horwitz 53. Overshooting and Asymmetries in the Transmission of Foreign Price Shocks to the Swedish Economy by Hans Genberg 54. Pubic Budgets in Sweden. A Brief Account of Budget Structure and Budgeting Procedure by Bengt-Christer Ysander 55. Arbetsmarknad och strukturomvanding de nordiska änderna av Berti Homiund 56. Centra Controi of the Loca Government Sector in Sweden by Richard Murray 58. Industria Subsidies in Sweden: Macro-economic Effects and an Internationa Comparison by Bo Carsson 59. Longitudina Lessons from the Pane Study of Income Dynamics by Greg J. Duncan and James N. Morgan Stabiization and Growth Poicy with Uncertain OH Prices: Some Rues of Thumb by Mark Sharefkin 61. Var står den nationaekonomiska centrateorin idag? av Bo Axe 62. Missing Variabes and Two-stage Least-squares Estimation from More than One Data Set by N. Anders Kevmarken

11 Genera Search Market Equiibrium by James W. Abrecht and Bo Axe 64. The Structure and Working of the Isac Mode by Leif Jansson, Thomas Nordström and Bengt-Christer Ysander 65. Comparative Advantage and Deveopment Poicy Twenty Years Later by Anne O. Krueger 66. Eectronics, Econonomic Growth and Empoyment Revoution or Evoution by Gunnar Eiasson 67. Computabe Muti-Country Modes of Production and Trade by James M. Henderson 68. Payro Taxes and Wage Infation: The Swedish Experiences by Berti Homund 69. Reative Competitiveness of Foreign Subsidiary Operations of a Mutinationa Company by Anders Grufman 70. Optimization under noninear constraints by Leif Jansson and Erik Meander