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function [yhat,Estr,rcon,Rcon,u,v,d] = fcstidcnd2(valuecon,stepcon,varcon,nstepsm,...
nconstr,eq_ms,nvar,lags,phil,Aband,Sband,yfore_h,imf3s_h,A0_h,Bh_h,...
forep,TLindx,TLnumber,nCms,eq_Cms,nconadd,eq_oth,radd,Radd)
% [yhat,Estr,rcon,Rcon,u,v,d] = fcstidcnd2(valuecon,stepcon,varcon,nstepsm,...
% nconstr,eq_ms,nvar,lags,phil,Aband,Sband,yfore_h,imf3s_h,A0_h,Bh_h,...
% forep,TLindx,TLnumber,nCms,eq_Cms,nconadd,eq_oth,radd,Radd)
%
% 5/2/99: This one is much, much slower than fidcnderr.m when eq_ms=[] and
% nconstr>0 and length(eq_ms)*nstepsm>nconstr. Seems to me that fcstidcnd2.m
% is designed for the situation which eq_ms=2. The slow speed is due to
% "Radd" added. When I have time, I need to check to see if I can speed up
% this M file.
%
% Conditional forecasting in the identified model with or without error bands
% It handles conditions on average values as well, so "valuecon" must be
% expressed at average (NOT sum) level. Including unconditional forecasts
% when nconstr=nCms=0. Maybe slower than "fcstidcnd.m" when length(eq_ms)*nstepsm=nconstr
% because of Radd added. But "fsctidcnd.m" is incorrect when length(eq_ms)
% *nstepsm>nconstr (situations like the average annual FFR constraint). 3/22/99
% Ref.: Zha Forecast (I), pp. 17-17c.
%
% valuecon: vector of values conditioned
% stepcon: sequence (cell) of steps conditioned; if length(stepcon{i}) > 1, the condition
% is then an arithmetic average of log(y) over the stepcon{i} period.
% varcon: vector of variables conditioned
% nconstr: number of DLS constraints
% nstepsm: maximum number of steps in all DLS constraints
% nvar: number of variables in the BVAR model
% lags: number of lags in the BVAR model
% phil: the 1-by-(nvar*lags+1) data matrix where k=nvar*lags+1
% (last period plus lags before the beginning of forecast)
% eq_ms: identified equation or shock (in terms of number). If equ_ms=[], then "fidencond" or
% 'fcstidcond' is, similar to RATS, to compute forecasts with *all* shocks.
% Aband: 1: draws from A0 and Bh; 0: no draws
% Sband: 1: draws from random shocks E; 0: no random shocks
% yfore_h: uncondtional forecasts: forep-by-nvar. Never used when nconstr=0.
% In this case, set it to [];
% imf3s_h: 3-dimensional impulse responses matrix: impsteps-by-nvar shocks-by-nvar responses
% Never used when nconstr=0. In this case, set it to [];
% A0_h: A0 contemporaneous parameter matrix
% Bh_h: reduced-form parameter matrix: k-by-nvar, y(t) = X(t)*Bh+e(t)
% where X(t) is k-by-nvar and y(t) is 1-by-nvar
% forep: # of forecast periods (e.g., monthly for a monthly model)
% TLindx: 1-by-nCms vector of 1's and 0's, indicating tight or loose; 1: tighter, 0: looser
% Used only when /* (MS draws) is activated. Right now, MS shocks are deterministic.
% TLnumber: 1-by-nCms vector, lower bound for tight and upper bound for loose
% nCms: # of LZ conditions
% eq_Cms: equation location of MS shocks
% nconadd: number of DLS added constraints DIRECTLY on structural shocks
% for identified version where eq_ms is not empty. Note
% nconadd=0 when eq_ms is empty.
% eq_oth: index for other structural shocks than those in eq_ms.
% If eq_ms=[], eq_oth=[]. Note length(eq_oth)+length(eq_ms)=nvar
% radd: sparse nconadd-by-1; nconadd values added to rcon in the text later
% Radd: sparce nvar*nstepsm-by-nconadd; added to Rcon in the text later
% ------
% yhat: conditional forecasts: forep-by-nvar
% Estr: backed-out structural shocks (from N(0,1))
% rcon: vector - the difference between valuecon and log(yfore) (unconditional forecasts)
% Rcon: k-by-q (q constranits and k=nvar*max(nsteps)) so that
% Rcon'*e = rcon where e is k-by-1
% [u,d,v]: svd(Rcon,0)
%
% Written by Tao Zha March 1998. Revised November 1998;
% 3/20/99 Disenabled draws of MS shcoks. To enable it, activate /* part
% 3/20/99 Added A0_h and forep and deleted Cms as input argument.
% 3/21/99 Added nconadd and eq_oth.
% Previous programs may not be compatible.
%
% Copyright (C) 1997-2012 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/>.
%
%
%% See Zha's note "Forecast (1)" p. 5, RATS manual (some errors in RATS), etc.
%% See also Zha Forecast (I), pp. 17-17c.
%
%% Some notations: y(t+1) = y(t)B1 + e(t+1)inv(A0). e(t+1) is 1-by-n.
%% Let r(t+1)=e(t+1)inv(A0) + e(t+2)C + .... where inv(A0) is impulse
%% response at t=1, C at t=2, etc. The row of inv(A0) or C is
%% all responses to one shock.
%% Let r be q-by-1 (such as r(1) = r(t+1)
%% = y(t+1) (constrained) - y(t+1) (forecast)).
%% Use impulse responses to find out R (k-by-q) where k=nvar*nsteps
%% where nsteps the largest constrained step. The key of the program
%% is to creat R using impulse responses
%% Optimal solution for shock e where R'*e=r and e is k-by-1 is
%% e = R*inv(R'*R)*r and k>=q
%
DLSIdShock = ~isempty(eq_ms); % if not empty, the MS shock is identified as in DLS
impsteps=size(imf3s_h,1);
if (forep<nstepsm) | (impsteps<nstepsm)
disp('Increase # of forecast or impulse steps!!')
disp('Or decrease # of constraints (nconstr) or constrained steps (stepcon(i))!!')
error('Maximum of conditional steps > # of forecast or impulse steps!!')
end
kts = nvar*nstepsm; % k -- ts: total shocks some of which are restricted and others
% are free.
%*** initializing
Rcon = zeros(kts,nconstr); % R: k-by-q where q include possible additional
% constraints directly on structural shocks for identified models.
% These addtional constraints will be added later.
Econ = sparse(zeros(kts,1)); % E: k-by-1
rcon = zeros(nconstr,1); % r: q-by-1
%rcon=valuecon-diag(yfore(stepcon,varcon)); % another way is to use "loop" below.
tcwc = nvar*lags; % total coefficients without constant
phi=phil;
%----------------------------------------------------
% Form rcon, Rcon, and Econ (the mean of structural shocks)
%----------------------------------------------------
if nconstr
A0in = reshape(imf3s_h(1,:,:),nvar,nvar); % nvar shocks-by-nvar responses
for i=1:nconstr
rcon(i)=length(stepcon{i})*valuecon(i) - ...
sum(yfore_h(stepcon{i},varcon(i)),1); %<<>>
Rmat = zeros(nstepsm,nvar);
% Must be here inside the loop because it's matrix of one column of Rcon
for j=1:length(stepcon{i})
if DLSIdShock % Assuming the Fed can't see all other shocks within a month
Rmat(1:stepcon{i}(j),:) = Rmat(1:stepcon{i}(j),:) + ...
imf3s_h(stepcon{i}(j):-1:1,:,varcon(i));
% Rmat: row--nstepsm, column--nvar shocks (here all shocks except
% the identified one are set to zero) for a particular
% endogenous variable 'varcon(i)'. See Zha Forcast (1), pp.6-7
else % Rcon random with (A0,A+)
Rmat(1:stepcon{i}(j),:) = Rmat(1:stepcon{i}(j),:) + ...
imf3s_h(stepcon{i}(j):-1:1,:,varcon(i));
% Rmat: row--nstepsm, column--nvar shocks (here all shocks are
% *not* set to zero) for a particular endogenous
% variable 'varcon(i)'. See Zha Forcast (1), pp.6-7
end
end
Rmatt = Rmat'; % Now, nvar-by-nstepsm. I think here is where RATS has an error
% i.e. "OVERR" is not transposed when overlaid to "CAPR"
Rcon(:,i)=Rmatt(:); % Rcon: k-by-q where q=nconstr
end
rcon = [rcon;radd]; % added nconadd constrained values for identified model
% sparse because radd is sparse
Rcon = [Rcon Radd]; % added constraints on shocks for identified version
% sparse because Radd is sparse
Rcont=Rcon';
rinvrtr=Rcon/(Rcont*Rcon);
Econ = rinvrtr*rcon;
%/* Too slow, I believe, when q is large. 3/21/99
% [u d v]=svd(Rcon,0); %trial
% %???? Can we reduce the time by computing inv(R'*R) directly?
% % rtr = Rcon'*Rcon; %trial
% % rtrinv = inv(Rcon'*Rcon); %trial
% vd=v.*(ones(size(v,2),1)*diag(d)'); %trial
% dinv = 1./diag(d); % inv(diag(d))
% vdinv=v.*(ones(size(v,2),1)*dinv'); %trial
% rtr=vd*vd'; % R'*R
% rtrinv = vdinv*vdinv'; % inv(R'*R)
%
% Econ = Rcon*rtrinv*rcon; % E = R*inv(R'R)*r; the mean of structural shocks
else
Econ = sparse(zeros(kts,1)); % the mean of shocks is zero under no variable condition
Rcon = NaN;
rcon = NaN;
u = NaN;
d = NaN;
v = NaN;
end
%---------------------------------------
% No uncertainty at all or only random (A0,A+)
% In other words, no future shocks
%---------------------------------------
if (~Sband) | (nconstr & length(eq_ms)*nstepsm==nconstr)
% length(eq_ms)==1 implies one-one mapping between MS shocks and, say, FFR
% if nstepsm==nconstr. If this condition does not hold, this procedure
% is incorrect. I don't have time to fix it now (3/20/99). So I use
% this as a proximation
Estr = reshape(Econ,nvar,nstepsm);
Estr = Estr'; % transpose so that
% Estr: structural shocks. Row--steps, Column--n shocks
Estr = [Estr;zeros(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
Estr(1:nCms,eq_Cms) = TLnumber(:);
Ures = Estr/A0_h; % nstepsm-by-nvar
% Ures: reduced-form residuals. Row--steps; Column--n shocks
% ** reconstruct x(t) for y(t+h) = x(t+h-1)*B
% ** where phi = x(t+h-1) with last column being constant
%
yhat = zeros(forep,nvar);
for k=1:forep
yhat(k,:) = phi*Bh_h + Ures(k,:);
phi(1,nvar+1:tcwc) = phi(1,1:tcwc-nvar);
phi(1,1:nvar) = yhat(k,:);
%
end
%---------------------------------------
% With random future shocks and possibly (A0,A+) depending
% on if imf3s_h is random
%---------------------------------------
else
%--------------
% Condition on variables and A random
%--------------
if nconstr & Aband
warning(' ')
disp('This situation (both E and A random) is still under construction')
disp('It is closely related to Waggoner and Zha ReStat Gibbs sampling method')
disp('and related to "if DLSIdShock" in the following')
disp('Please press ctrl-c to abort')
pause
elseif nconstr
%--------------
% Condition on variables and DLS MS shock, no A random but S random
%--------------
if DLSIdShock % other shocks are indepedent of the eq_ms shock
Ome=speye(kts)-rinvrtr*Rcont;
Ome(find(abs(Ome)<1e-10))=0; % tighter the tolerance is, the longer it
% takes for svds to compute. E.g., size(Ome,1)*eps is perhaps very
% tight. After all, when the number in Ome is very small relative to
% 1 (which is the variance of structural shocks Estr), we can treat it
% as zero.
[u1 d1 v1] = svds(Ome,kts-nconstr-nconadd);
% We have exactly nconstr+nconadd zero singular values
Stdcon = u1*diag(sqrt(diag(abs(d1)))); % find a square root of Ome
% see Zha's forecast (1), p.17
Stdcon = [Stdcon sparse(zeros(kts,kts-size(d1,1)))];
Estr1 = Econ + Stdcon*randn(kts,1); % We must have rand(kts,1). Assigning
% some of randn(kts,1) to be zero according to Radd is WRONG.
% For this argument, see Zha Forecast (I) p.17a
Estr2 = reshape(Estr1,nvar,nstepsm);
% nvar-by-nstepsm; Needed to be transposed later
Estr = [Estr2';randn(forep-nstepsm,nvar)]; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
% Now, forep-by-nvar -- ready for forecasts
else
Ome=speye(kts)-rinvrtr*Rcont;
Ome(find(abs(Ome)<1e-10))=0; % tighter the tolerance is, the longer it
% takes for svds to compute. E.g., size(Ome,1)*eps is perhaps very
% tight. After all, when the number in Ome is very small relative to
% 1 (which is the variance of structural shocks Estr), we can treat it
% as zero.
[u1 d1 v1] = svds(Ome,kts-nconstr-nconadd);
% We have exactly nconstr+nconadd zero singular values
Stdcon = u1*diag(sqrt(diag(abs(d1)))); % find a square root of Ome
% see Zha's forecast (1), p.17
Stdcon = [Stdcon sparse(zeros(kts,nconstr+nconadd))];
%*** The following very inefficient
% Ome = eye(kts) - u*u'; % note, I-u*u' = I - R*inv(R'*R)*R'
% %[u1 d1 v1] = svd(Ome); % too slow
% [u1 d1] = eig(Ome);
% Stdcon = u1*diag(sqrt(diag(abs(d1)))); % lower triagular chol of conditional variance
% % see Zha's forecast (1), p.17
%--------------
% Condition on variables and LZ MS shock, no A random but S random
% This section has not be tested yet, 10/14/98
%--------------
if nCms
Estr1 = Econ + Stdcon*randn(kts,1);
Estr2 = reshape(Estr1,nvar,nstepsm);
Estr2 = Estr2'; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
Estr = [Estr2;randn(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
Estr(1:nCms,eq_Cms) = TLnumber(:);
%/* draw MS shocks
% for k=1:nCms
% if TLindx(k) % tighter
% while (Estr(k,eq_Cms)<TLnumber(k))
% Estr(k,eq_Cms) = randn(1,1);
% end
% else % looser
% while (Estr(k,eq_Cms)>TLnumber(k))
% Estr(k,eq_Cms) = randn(1,1);
% end
% end
% end
%--------------
% Condition on variables only, no A random but S random
%--------------
else
Estr1 = Econ + Stdcon*randn(kts,1);
Estr2 = reshape(Estr1,nvar,nstepsm);
Estr2 = Estr2'; % transpose so that
% Estr2: structural shocks. Row--nstepsm, Column--n shocks
Estr = [Estr2;randn(forep-nstepsm,nvar)];
% Now, forep-by-nvar -- ready for forecasts
end
end
%--------------
% Condition on LZ MS shocks only, S random and possibly A random depending on
% if A0_h and Bh_h are random
%--------------
else
if nCms
Estr = randn(forep,nvar);
% Now, forep-by-nvar -- ready for forecasts
Estr(:,eq_Cms)=0;
Estr(1:nCms,eq_Cms) = TLnumber(:);
%/* draw MS shocks
% for k=1:nCms
% if TLindx(k) % tighter
% while (Estr(k,eq_Cms)<TLnumber(k))
% Estr(k,eq_Cms) = randn(1,1);
% end
% else % looser
% while (Estr(k,eq_Cms)>TLnumber(k))
% Estr(k,eq_Cms) = randn(1,1);
% end
% end
% end
else
Estr = randn(forep,nvar); % Unconditional forecast
% Now, forep-by-nvar -- ready for forecasts
end
end
%
Ures = Estr/A0_h; % nstepsm-by-nvar
% Ures: reduced-form residuals. Row--steps; Column--n shocks
% ** reconstruct x(t) for y(t+h) = x(t+h-1)*B
% ** where phi = x(t+h-1) with last column being constant
%
yhat = zeros(forep,nvar);
for k=1:forep
yhat(k,:) = phi*Bh_h + Ures(k,:);
phi(1,nvar+1:tcwc) = phi(1,1:tcwc-nvar);
phi(1,1:nvar) = yhat(k,:);
end
end
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