function [yhat,Estr,rcon,Rcon,u,v,d] = fn_fcstcnd(valuecon,stepcon,varcon,nstepsm,... nconstr,eq_ms,nvar,lags,phil,Sband,yfore_h,imf3s_h,Bh_h,forep) % ******* There are some BUG problems when calling this fucntion. ******* 3/15/2004. % [yhat,Estr,rcon,Rcon,u,v,d] = fn_fcstcnd(valuecon,stepcon,varcon,nstepsm,... % nconstr,eq_ms,nvar,lags,phil,Sband,yfore_h,imf3s_h,Bh_h,forep) % % 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) levels (i.e., arithmetic averages of log(y) over % the period of stepcon{i}. 5/22/01. % Unconditional forecast when imf3s_h, etc is fixed and nconstr=0. % Note that 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). Meantime, consult or use the old code % fidencond.m. % % 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 % nstepsm: maximum number of steps in all DLS constraints % nconstr: number of DLS constraints % eq_ms: Equation location of MS shocks. If [], all shocks. % 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) % Sband: 1: draws from random shocks E; 0: no random shocks For now (4/27/01), no option % for Aband because I don't think it works best to do both Aband and Sband in one function. % yfore_h: uncondtional forecasts: forep-by-nvar. Never used when nconstr=0. % In this case, may 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, may set it to []; % 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) % eq_Cms: equation location of MS shocks % ------ % yhat: conditional forecasts: forep-by-nvar % Estr: backed-out structural shocks (from constrained Gaussians) % 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) % %% See Zha's note "Forecast (1)" p. 5, RATS manual (some errors in RATS), etc. %% 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 % % See the old code fidencond.m. I wond't use fn_fcstidcnd?.m, 5/22/01. % Copyright (c) March 1998 by Tao Zha. Revised November 1998, May 2001 (Delete A0_h as % input arg so that previous programs may not be compatible). DLSIdShock = ~isempty(eq_ms); % if not empty, the MS shock is identified as in DLS impsteps=size(imf3s_h,1); if (forep # 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 Econ = 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) %---------------------------------------------------- A0in = reshape(imf3s_h(1,:,:),nvar,nvar); % <<>>1 nvar shocks-by-nvar responses if nconstr % Conditional forecasts. for i=1:nconstr rcon(i)=length(stepcon{i})*valuecon(i) - sum(yfore_h(stepcon{i},varcon(i)),1); %<<>>2 Automatically taking care of average conditions. Rmat = zeros(nstepsm,nvar); r2mat = zeros(nstepsm,1); % simply one identified equation % 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),eq_ms) = Rmat(1:stepcon{i}(j),eq_ms) + ... imf3s_h(stepcon{i}(j):-1:1,eq_ms,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 % [u d v]=svd(Rcon,0); %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 % Unconditional forecasts. Econ = 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. In other words, no future shocks. %--------------------------------------- if (~Sband) %| (nconstr & (length(eq_ms)==1)) % 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 Ures = Estr*A0in; % <<>>1 nstepsm-by-nvar % Ures: reduced-form residuals. Row--steps; Column--n shocks % Note: We don't use /A0_h so as to eliminate small discrepancies to be % completely compatible with the computation of Rmat and Estr, which uses A0in. % ** 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. %--------------------------------------- else if nconstr % Conditional forecasts. %-------------- % Condition on variables with all shocks backed out. Straight DLS forecasts. No A random but S random. %-------------- if ~DLSIdShock 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 Estr1 = Econ + Stdcon*randn(kts,1); % Draws of constrained (conditioned) shocks. Estr2 = reshape(Estr1,nvar,nstepsm); Estr2 = Estr2'; % transpose so that % Estr2: structural shocks. Row--nstepsm, Column--n shocks Estr = [Estr2;randn(forep-nstepsm,nvar)]; % Second concatenated part: draws of free shocks. % Now, forep-by-nvar -- ready for forecasts %-------------- % Condition on variables with identified MS shocks backed out, no A random but S random. %-------------- else % other shocks are indepedent of the eq_ms shock % 3/20/99 The following may be problematic because Osk should depend % on u (A0_h and Bh_h) in general. I have NOT worked out any good version. %/* % Osk = randn(kts,1); % other shocks % for j=1:nstepsm % Osk(nvar*(j-1)+eq_ms)=0; % no shock to the MS or identified equation % end % Estr = Econ + Osk; % Econ is non zero only at position % % eq_ms*j where j=1:nstepsm % Estr = reshape(Estr,nvar,nstepsm); % Estr = Estr'; % transpose so that % % Estr: structural shocks. Row--steps, Column--n shocks % Estr = [Estr;randn(forep-nstepsm,nvar)]; % % Now, forep-by-nvar -- ready for forecasts % disp('DLS') 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 tmp1=zeros(nvar,nstepsm); tmp1(eq_ms,:)=randn(1,nstepsm); tmp2=tmp1(:); %Estr1 = Econ + Stdcon*randn(kts,1); %jnk = reshape(Stdcon*tmp2,nvar,nstepsm) Estr1 = Econ + Stdcon*tmp2; 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 else % Unconditional forecasts. Estr = randn(forep,nvar); % Unconditional draws. end Ures = Estr*A0in; % <<>>1 nstepsm-by-nvar % Ures: reduced-form residuals. Row--steps; Column--n shocks % Note: We don't use /A0_h so as to eliminate small discrepancies to be % completely compatible with the computation of Rmat and Estr, which uses A0in. % ** 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