A ist of ast pages Papers on No. 70, 1982 OptiJnization under non1inear COnstraints by Leif Jansson anderik Meander October, 1982
- 2 - OP"'nUZA'I'ION UNDER NONLIHEAR CONS'I'RAIN'I'S Leif Jansson The Ministry of Economic 103 33 Stockhom Sweden rs Erik Meander The Industria Institute for Economic and Socia Research 114 53 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.
- 3-2. 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 - The derivative of L with respect to a free parame~ ter i can be written as L 08. 1 = 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. 1 3. 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 - 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 - 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) = 28. - R2. + 08. 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 - 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) 160-169.
WORKING PAPERS (Missing numbers indicate pubication esewhere) 1976. 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 1979 20. A Putty-Cay Mode of Oemand Uncertainty and Investment by James W. Abrecht and Abert G. Hart 1980 25. 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 1913-1977 by Märtha Josefsson and Johan Örtengren
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