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function [phi,y,fss,xtx,xty,yty,Bols] = xydata(xdgel,lags,mu)
% [phi,y,xtx,xty,yty,Bols,fss] = xydata(xdgel,lags,mu)
% Construct matrices X, Y, X'X, etc. so that y = X*B + U, where y is T-by-1, X T-by-ncoef,
% and B is ncoef-by-1.
%
% xdgel: the general matrix of the original data (no manipulation involved)
% with sample size including lags
% lags: the lag length in the AR(p) process
% mu: 2-by-1 vector of hyperparameters for dummy observations (FRA's parameter settings)
% mu(1): weight on nvar sums of coeffs dummy observations (unit roots); (5)
% mu(2): weight on single dummy initial observation including constant
% (cointegration, unit roots, and stationarity); (5)
%---------------
% phi: X: T-by-ncoef where ncoef=nvar*lags + constant and T=fss
% y: Y: T-by-nvar where T=fss
% fss: effective sample size (including dummies if mu is provided) exclusing all lags
% xtx: X'X (Data)
% xty: X'Y (Data)
% yty: Y'Y (Data)
% Bols: ncoef-by-1: OLS estimate of B
%
%function [Gb,Sbd,Bh,SpH,fss,ndobs,phi,y,nvar,ncoef,xxhpc,a0indx,na0p,...
% idmat0,idmatpp] = szasbvar(idfile,q_m,lags,xdgel,mu)
% Now version: [Gb,Sbd,Bh,SpH,fss,ndobs,phi,y,nvar,ncoef,xxhpc,a0indx,na0p,...
% idmat0,idmatpp] = szasbvar(idfile,q_m,lags,xdgel,mu)
%
% Estimating the Bayesian VAR of Sims and Zha with asymmetric prior (as)
%
% idfile: Identification filename with rows corresponding to equations.
% But all the output will be transposed to columns
% corresponding to equations.
% q_m: quarter or month
% lags: the maximum length of lag
% nhp: number of haperparameters
% xdgel: the general matrix of the original data (no manipulation involved)
% with sample size including lags
% mu: 6-by-1 vector of hyperparameters (the following numbers for Atlanta Fed's forecast)
% mu(1): overall tightness and also for A0; (0.57)
% mu(2): relative tightness for A+; (0.13)
% mu(3): relative tightness for the constant term; (0.1)
% mu(4): tightness on lag decay; (1)
% mu(5): weight on nvar sums of coeffs dummy observations (unit roots); (5)
% mu(6): weight on single dummy initial observation including constant
% (cointegration, unit roots, and stationarity); (5)
% --------------------
% --------------------
% Gb: cell(nvar,1). Each cell, postmultiplied by A0(:,i), is used to compute A+(:,i)
% where A+ is k-by-m, where k--ncnoef, m--nvar, if asymmetric prior
% Sbd: cell(nvar,1). Sbd=diag(S(1), ..., S(m)). Already divided by 'fss' for the
% posterior of a0 or A0(:) in Waggoner and Zha when one has asymmetric prior.
% Note,"bd" stands for block diagonal.
% Bh: reduced form B, k(mp+1)-by-m, column--equation. Bh=NaN if asymmetric prior.
% In the form of y=X*B+U where B=[B1|B2| ...|Bp|C]
% where Bi: m-by-m (i=1,..,p--lag length), C: 1-by-ml, constant.
% SpH: divided by T, the final S in the Waggoner-Zha exponential part of p(A0|y)
% SpH=NaN if asymmetric prior
% fss: in-sample-size (for forecasting). Including dummy observations
% ndobs: number of dummy observations, ndobs=nvar+1
% phi: X in the form of y = X*B+U. Row: nSmaple-lags+ndobs. Column: ncoef
% y: y in the form y = X*B+U. Including the dummy observations too,
% T-lags+ndobs-by-nvar.
% nvar: number of variables
% ncoef: number of coefficients in *each* equation. RHS coefficients only, nvar*lags+1
% xxhpc: chol(X'X+inv(H_p_tilde)): upper triangular but its transpose
% is lower triagular Choleski
% a0indx: the index number for free parameters in A0. Column meaning equation
% na0p: number of free parameters in A0
% idmat0: identification matrix for A0 with asymmetric prior; column -- equation.
% idmatpp: asymmetric prior variance for A+ without constant; column -- equation.
%
% Revisions by CAS 9/27/96: If we transmit idmat, rather than a0indx, we can scale
% the elements of idmat0 or idmatp so that it carries information about relative
% prior variances.
% Revsions by TZ, 10/13/96: Efficiency improvement by streamlining the previous
% code according to the general setup in Sims and Zha's IER and according
% to Zha's notes Forecast (0) (p.3) and Forecast (1) (p.9).
%
%
% NOTE: "nSample" is something I deleted as an input recently, so it may not
% be compatible with old programs, 10/15/98 by TAZ.
%
% NOTE2: I put "mu" as an input arg and delete "nhp" as an input arg, so it may
% not be compatible with old programs, 03/06/99 by TAZ
%
%
% Copyright (C) 1997-2012 Lutz Kilian and Tao Zha
%
% This free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% It is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% If you did not received a copy of the GNU General Public License
% with this software, see <http://www.gnu.org/licenses/>.
%
[nSample,nvar]=size(xdgel); % sample size including lags and number of variables
ncoef=nvar*lags+1; % number of coefficients in *each* equation, RHS coefficients only,
% with only constant
ndobs=nvar+1; % number of dummy observations
sb = lags+1; % the beginning of original sample without dummies
if nargin==3 % activate the option of dummy observations
fss = nSample+ndobs-lags;
% Effective sample size including dummies but exclusing all lags
%
% **** nvar prior dummy observations with the sum of coefficients
% ** construct X for Y = X*B + U where phi = X: (T-lags)*k, Y: (T-lags)*nvar
% ** columns: k = # of [nvar for 1st lag, ..., nvar for last lag, const]
% ** Now, T=T+ndobs -- added with "ndobs" dummy observations
%
phi = zeros(fss,ncoef);
const = ones(fss,1);
const(1:nvar) = zeros(nvar,1);
% the first nvar periods: no or zero constant (designed for dummies)!
phi(:,ncoef) = const;
%
xdgelint = mean(xdgel(1:lags,:),1); % mean of the first lags initial conditions
%**** Dummies for phi
for k=1:nvar
for m=1:lags
phi(ndobs,nvar*(m-1)+k) = xdgelint(k);
phi(k,nvar*(m-1)+k) = xdgelint(k);
% <<>> multiply hyperparameter later
end
end
%* True data for phi
for k=1:lags
phi(ndobs+1:fss,nvar*(k-1)+1:nvar*k) = xdgel(sb-k:nSample-k,:);
% row: T-lags; column: [nvar for 1st lag, ..., nvar for last lag, const]
% Thus, # of columns is nvar*lags+1 = ncoef.
end
%
% ** Y with "ndobs" dummies added
y = zeros(fss,nvar);
%* Dummies for y
for k=1:nvar
y(ndobs,k) = xdgelint(k);
y(k,k) = xdgelint(k);
% multiply hyperparameter later
end
%* True data for y
y(ndobs+1:fss,:) = xdgel(sb:nSample,:);
%
%*** Dummies once again (finfal version)
phi(1:nvar,:) = 1*mu(1)*phi(1:nvar,:); % standard Sims and Zha prior
y(1:nvar,:) = mu(1)*y(1:nvar,:); % standard Sims and Zha prior
%
phi(nvar+1,:) = mu(2)*phi(nvar+1,:);
y(nvar+1,:) = mu(2)*y(nvar+1,:);
else % dummy observations
fss = nSample-lags;
% Effective sample size with no dummies but exclusing all lags
%
% **** nvar prior dummy observations with the sum of coefficients
% ** construct X for Y = X*B + U where phi = X: (T-lags)*k, Y: (T-lags)*nvar
% ** columns: k = # of [nvar for 1st lag, ..., nvar for last lag, const]
% ** Now, T=T+ndobs -- added with "ndobs" dummy observations
%
phi = zeros(fss,ncoef);
const = ones(fss,1);
phi(:,ncoef) = const;
%* True data for phi
for k=1:lags
phi(1:fss,nvar*(k-1)+1:nvar*k) = xdgel(sb-k:nSample-k,:);
% row: T-lags; column: [nvar for 1st lag, ..., nvar for last lag, const]
% Thus, # of columns is nvar*lags+1 = ncoef.
end
%
%* True data for y
y(1:fss,:) = xdgel(sb:nSample,:);
end
%*** data input for the posterior density
[xq,xr]=qr(phi,0);
xtx=xr'*xr; % X'X
xty=phi'*y; % X'Y
yty = y'*y; % Y'Y
Bols = xr\(xr'\xty); % inv(X'X)*X'Y
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