43 lines
1.8 KiB
Matlab
43 lines
1.8 KiB
Matlab
function [g,badg] = fn_a0freegrad(b,Ui,nvar,n0,fss,H0inv)
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% [g,badg] = a0freegrad(b,Ui,nvar,n0,fss,H0inv)
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% Analytical gradient for a0freefun.m in use of csminwel.m. See Dhrymes's book.
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%
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% b: sum(n0)-by-1 vector of A0 free parameters
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% Ui: nvar-by-1 cell. In each cell, nvar-by-qi orthonormal basis for the null of the ith
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% equation contemporaneous restriction matrix where qi is the number of free parameters.
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% With this transformation, we have ai = Ui*bi or Ui'*ai = bi where ai is a vector
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% of total original parameters and bi is a vector of free parameters. When no
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% restrictions are imposed, we have Ui = I. There must be at least one free
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% parameter left for the ith equation.
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% nvar: number of endogeous variables
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% n0: nvar-by-1, ith element represents the number of free A0 parameters in ith equation
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% fss: nSample-lags (plus ndobs if dummies are included)
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% H0inv: cell(nvar,1). In each cell, posterior inverse of covariance inv(H0) for the ith equation,
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% resembling old SpH in the exponent term in posterior of A0, but not divided by T yet.
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%---------------
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% g: sum(n0)-by-1 analytical gradient for a0freefun.m
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% badg: 0, the value that is used in csminwel.m
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%
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% Tao Zha, February 2000. Revised, August 2000
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b=b(:); n0 = n0(:);
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A0 = zeros(nvar);
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n0cum = [0;cumsum(n0)];
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g = zeros(n0cum(end),1);
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badg = 0;
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%*** The derivative of the exponential term w.r.t. each free paramater
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for kj = 1:nvar
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bj = b(n0cum(kj)+1:n0cum(kj+1));
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g(n0cum(kj)+1:n0cum(kj+1)) = H0inv{kj}*bj;
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A0(:,kj) = Ui{kj}*bj;
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end
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B=inv(A0');
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%*** Add the derivative of -Tlog|A0| w.r.t. each free paramater
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for ki = 1:sum(n0)
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n = max(find( (ki-n0cum)>0 )); % note, 1<=n<=nvar
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g(ki) = g(ki) - fss*B(:,n)'*Ui{n}(:,ki-n0cum(n));
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end
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