51 lines
3.0 KiB
Matlab
51 lines
3.0 KiB
Matlab
function [P,H0inv,Hpinv] = fn_rlrpostr(xtx,xty,yty,Ptld,H0invtld,Hpinvtld,Ui,Vi)
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% [P,H0inv,Hpinv] = fn_rlrpostr(xtx,xty,yty,Ptld,H0tld,Hptld,Ui,Vi)
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%
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% Exporting random (i.e., random prior) Bayesian posterior matrices with linear restrictions
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% See Waggoner and Zha's Gibbs sampling paper
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%
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% xtx: X'X: k-by-k where k=ncoef
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% xty: X'Y: k-by-nvar
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% yty: Y'Y: nvar-by-nvar
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% Ptld: cell(nvar,1), transformation matrix that affects the (random walk) prior mean of A+ conditional on A0.
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% H0invtld: cell(nvar,1), transformed inv covaraince for free parameters in A0(:,i).
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% Hpinvtld: cell(nvar,1), transformed inv covaraince for free parameters in A+(:,i);
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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. Imported from dnrprior.m.
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% Vi: nvar-by-1 cell. In each cell, k-by-ri orthonormal basis for the null of the ith
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% equation lagged restriction matrix where k (ncoef) is a total number of RHS variables and
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% ri is the number of free parameters. With this transformation, we have fi = Vi*gi
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% or Vi'*fi = gi where fi is a vector of total original parameters and gi is a
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% vector of free parameters. There must be at least one free parameter left for
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% the ith equation. Imported from dnrprior.m.
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%-----------------
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% P: cell(nvar,1). In each cell, the transformation matrix that affects the posterior mean of A+ conditional on A0.
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% In other words, the posterior mean (of g_i) = P{i}*b_i where g_i is a column vector of free parameters
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% of A+(:,i)) given b_i (b_i is a column vector of free parameters of A0(:,i)).
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% H0inv: cell(nvar,1). Not divided by T yet. In each cell, inverse of posterior covariance matrix H0.
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% The exponential term is b_i'*inv(H0)*b_i for the ith equation where b_i = U_i*a0_i.
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% It resembles old SpH or Sbd in the exponent term in posterior of A0, but not divided by T yet.
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% Hpinv: cell(nvar,1). In each cell, posterior inverse of covariance matrix Hp (A+) for the free parameters
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% g_i = V_i*A+(:,i) in the ith equation.
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%
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% Tao Zha, February 2000
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nvar = size(yty,1);
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P = cell(nvar,1); % tld: tilda
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H0inv = cell(nvar,1); % posterior inv(H0), resemble old SpH, but not divided by T yet.
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Hpinv = cell(nvar,1); % posterior inv(Hp).
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for n=1:nvar % one for each equation
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Hpinv{n} = Vi{n}'*xtx*Vi{n} + Hpinvtld{n};
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P1 = Vi{n}'*xty*Ui{n} + Hpinvtld{n}*Ptld{n};
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P{n} = Hpinv{n}\P1;
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H0inv{n} = Ui{n}'*yty*Ui{n} + H0invtld{n} + Ptld{n}'*Hpinvtld{n}*Ptld{n} ...
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- P1'*P{n}; %P{n} = (Hpinv{n}\P1);
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end
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