59 lines
3.9 KiB
Matlab
Executable File
59 lines
3.9 KiB
Matlab
Executable File
function [Tinv,UT,VHphalf,PU,VPU] = fn_gibbsrvar_setup(H0inv, Ui, Hpinv, Pmat, Vi, nvar, fss)
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% [Tinv,UT,VHphalf,PU,VPU] = fn_gibbsrvar_setup.m(H0inv, Ui, Hpinv, Pmat, Vi, fss, nvar)
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% Global setup outside the Gibbs loop to be used by fn_gibbsvar().
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% Reference: "A Gibbs sampler for structural VARs" by D.F. Waggoner and T. Zha, ``
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% Journal of Economic Dynamics & Control (JEDC) 28 (2003) 349-366.
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% See Note Forecast (2) pp. 44-51, 70-71, and Theorem 1 and Section 3.1 in the WZ JEDC paper.
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%
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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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% 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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% 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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% Pmat: 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) = Pmat{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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% 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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% nvar: number of endogenous variables or rank of A0.
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% fss: effective sample size (in the exponential term) = nSample - lags + ndobs (ndobs = # of dummy observations
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% is set to 0 when fn_rnrprior_covres_dobs() is used where dummy observations are included as part of the explicit prior.
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%-------------
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% Tinv: cell(nvar,1). In each cell, inv(T_i) for T_iT_i'=S_i where S_i is defined on p.355 of the WZ JEDC paper.
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% UT: cell(nvar,1). In each cell, U_i*T_i.
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% VHphalf: cell(nvar,1). In each cell, V_i*sqrt(Hp_i).
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% PU: cell(nvar,1). In each cell, Pmat{i}*U_i where Pmat{i} = P_i defined in (13) on p.353 of the WZ JEDC paper.
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% VPU: cell(nvar,1). In each cell, V_i*P_i*U_i
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%
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% Written by Tao Zha, September 2004.
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% Copyright (c) 2004 by Waggoner and Zha
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%--- For A0.
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Tinv = cell(nvar,1); % in each cell, inv(T_i) for T_iT_i'=S_i where S_i is defined on p.355 of the WZ JEDC paper.
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UT = cell(nvar,1); % in each cell, U_i*T_i.
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%--- For A+.
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VHphalf = cell(nvar,1); % in each cell, V_i*sqrt(Hp_i).
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PU = cell(nvar,1); % in each cell, Pmat{i}*U_i where Pmat{i} = P_i defined in (13) on p.353 of the WZ JEDC paper.
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VPU = cell(nvar,1); % in each cell, V_i*P_i*U_i
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%
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for ki=1:nvar
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%--- For A0.
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Tinv{ki} = chol(H0inv{ki}/fss); % Tinv_i'*Tinv_i = inv(S_i) ==> T_i*T_i' = S_i where S_i = H0inv{i}/fss is defined on p.355 of the WZ JEDC paper.
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UT{ki} = Ui{ki}/Tinv{ki}; % n-by-qi: U_i*T_i in (14) on p. 255 of the WZ JEDC paper.
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%--- For A+.
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VHphalf{ki} = Vi{ki}/chol(Hpinv{ki}); % where chol(Hpinv_i)*chol(Hpinv_i)'=Hpinv_i.
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PU{ki} = Pmat{ki}*Ui{ki}';
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VPU{ki} = Vi{ki}*PU{ki};
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end
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