ITE / code / shared / embedded / E4 / imod.m

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function [ dts, corrm, varm, Im ] = imod(theta, din, z, aprox)
% imod - Computes the exact information matrix under normality.
% [dts, corrm, varm, Im] = imod(theta, din, z, aprox)
% theta > parameter vector.
% din > matrix which stores a description of the model dynamics.
% z > matrix of observable variables.
% aprox > If aprox = 0, computes the approximation of Watson and Engle (1983).
% dts < vector of standard deviations.
% corrm < correlation matrix.
% varm < covariance matrix.
% Im < information matrix
% If Im is not positive-definite, it is perturbed before computing
% the rest of the matrices.
%
% 11/3/97

% Copyright (C) 1997 Jaime Terceiro
% 
% This program is 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 2, or (at your option)
% any later version.
% 
% This program 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. 
% 
% You should have received a copy of the GNU General Public License
% along with this file.  If not, write to the Free Software Foundation,
% 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

global E4OPTION

if nargin < 3, e4error(3); end

if nargin < 4, aprox = 0; elseif aprox, e4warn(7); end

oldfil = E4OPTION(1); E4OPTION(1) = 1;
scaleb = E4OPTION(2);
zeps   = E4OPTION(15);
econd  = E4OPTION(4);

[Phi, Gam, E, H, D, C, Q, S, R] = thd2ss(theta, din);
n = size(z,1);
m = size(H,1);
r = size(Gam,2);

if econd == 5
   if r, econd = 2; else, econd = 3; end;
end
if econd == 2 & ~aprox, MV = 1; else, MV = 0; end

if size(z,2) ~= m+r, e4error(11); end

[x0, P0] = lfmodini(Phi,Gam,E,H,D,C,Q,S,R,z, MV);
CRCt=C*R*C';  ESCt=E*S*C';  EQEt=E*Q*E';

if size(theta,2) > 1  % look for fixed parameters in theta
   index = find(theta(:,2) == 0);
else
   index = (1:size(theta,1))';
end

np = size(index,1);
p  = size(P0,1);
Im = zeros(np, np);

dPhii = zeros(p*np, p);
sGam  = size(Gam,1);
dGami = zeros(sGam*np, size(Gam,2));
sE    = size(E,1);
dEi   = zeros(sE*np, size(E,2));
dHi   = zeros(m*np, p);
sD    = size(D,1);
dDi   = zeros(sD*np, size(D,2));
sC    = size(C,1);
dCi   = zeros(sC*np, size(C,2));
sQ    = size(Q,1);
dQi   = zeros(sQ*np, size(Q,2));
sS    = size(S,1);
dSi   = zeros(sS*np, size(S,2));
sR    = size(R,1);
dRi   = zeros(sR*np, size(R,2));

dx0i = zeros(p*np,1);
dP0i  = zeros(p*np,p);
dB1i  = zeros(m*np,m);

if MV
   x00 = x0;
   P00 = P0;
   Phibb0 = eye(p,p);
   dPhibb0 = zeros(p*np,p);
   WW  = zeros(p,p); WZ = zeros(p,1);
   dWW = zeros(p*np,p); dWZ = zeros(p*np,1);
end

if aprox | MV
   dz1i = zeros(m*np,1);
end

if ~aprox
%
   dK1i = zeros(p*np,m);
   Pc11 = zeros(p, p);
   Pc21 = zeros(p*np,p);
   Pc23 = zeros(p*(np+1)*np/2, p);

   if r | MV
      xc  = x0;
      dxc = zeros(p*np,1);
      dzc = zeros(m*np,1);
   else
      Pc11 = Pc11 + x0*x0';
   end
%
end

for i = 1:np
%
    j = i-1;

    [dPhii(j*p+1:i*p,:),dGam,dEi(j*sE+1:i*sE,:),...
     dHi(j*m+1:i*m,:),dD,dCi(j*sC+1:i*sC,:),...
     dQi(j*sQ+1:i*sQ,:),dSi(j*sS+1:i*sS,:),dRi(j*sR+1:i*sR,:)] =...
         ss_dv(theta,din,index(i));

    if r
       dGami(j*sGam+1:i*sGam,:) = dGam;
       dDi(j*sD+1:i*sD,:) = dD;
    end

    [dx0i(j*p+1:i*p,:), dP0i(j*p+1:i*p,:)] = ...
       gmodini(Phi,dPhii(j*p+1:i*p,:),...
               Gam, dGami(j*sGam+1:i*sGam,:),E,dEi(j*sE+1:i*sE,:),...
               H,dHi(j*m+1:i*m,:),D,dDi(j*sD+1:i*sD,:),C,dCi(j*sC+1:i*sC,:),...
               Q,dQi(j*sQ+1:i*sQ,:),S,dSi(j*sS+1:i*sS,:),...
               R,dRi(j*sR+1:i*sR,:),z,MV);

    if ~aprox
    %
       if ~r & ~MV
       %
          Pc21(j*p+1:i*p,:) = dx0i(j*p+1:i*p,:)*x0';

          for k=1:i
              l = k-1;
              ptri = (j*i/2 + l)*p + 1;
              ptrf = ptri + p - 1;

              Pc23(ptri:ptrf,:) = dx0i(j*p+1:i*p,:)*dx0i(l*p+1:k*p,:)';
          end
       %
       elseif ~MV
          dxc(j*p+1:i*p,:) = dx0i(j*p+1:i*p,:);
       end
    %
    end
%
end

for t = 1:n  % main loop
%
    B1   = H*P0*H' + CRCt;
    if scaleb, U = cholp(B1, abs(B1));
    else       U = cholp(B1); end
    iB1 = (eye(size(U))/U)/U';

    K1  = (Phi*P0*H' + ESCt)*iB1;
    Phib  = Phi - K1*H;
    
    if r, Gamb   = Gam  - K1*D; end

    if MV
       HPhi= H*Phibb0;
       if r
          z1  = z(t,1:m)' - H*x0 - D*z(t,m+1:m+r)';
       else
          z1  = z(t,:)' - H*x0;
       end
    end

    for i = 1:np
    %
        j = i-1;
        dB1i(j*m+1:i*m,:) = dHi(j*m+1:i*m,:)*P0*H'+ H*dP0i(j*p+1:i*p,:)*H'+ H*P0*dHi(j*m+1:i*m,:)' + ...
                          dCi(j*sC+1:i*sC,:)*R*C' + C*dRi(j*sR+1:i*sR,:)*C' + C*R*dCi(j*sC+1:i*sC,:)';

        if aprox
        %
           dK1i= (dPhii(j*p+1:i*p,:)*P0*H' + Phi*dP0i(j*p+1:i*p,:)*H' + ...
                 Phi*P0*dHi(j*m+1:i*m,:)' + ...
                 dEi(j*sE+1:i*sE,:)*S*C' + E*dSi(j*sS+1:i*sS,:)*C' + ...
                 E*S*dCi(j*sC+1:i*sC,:)' - K1*dB1i(j*m+1:i*m,:))*iB1;

           dPhibi= dPhii(j*p+1:i*p,:) - dK1i*H - K1*dHi(j*m+1:i*m,:);

           if r
              dz1i(j*m+1:i*m,:) = -dHi(j*m+1:i*m,:)*x0 -H*dx0i(j*p+1:i*p,:) - ...
                                  dDi(j*sD+1:i*sD,:)*z(t,m+1:m+r)';

              dGambi  = dGami(j*sGam+1:i*sGam,:) - dK1i*D - K1*dDi(j*sD+1:i*sD,:);

              dx0i(j*p+1:i*p,:) = dPhibi*x0 + Phib*dx0i(j*p+1:i*p,:) + ...
                                  dGambi*z(t,m+1:m+r)' + dK1i*z(t,1:m)';
           else
              dz1i(j*m+1:i*m,:) = -dHi(j*m+1:i*m,:)*x0 -H*dx0i(j*p+1:i*p,:);

              dx0i(j*p+1:i*p,:) = dPhibi*x0 + Phib*dx0i(j*p+1:i*p,:) + ...
                                  dK1i*z(t,1:m)';
           end

           dP0i(j*p+1:i*p,:) = dPhibi*P0*Phib' + Phib*dP0i(j*p+1:i*p,:)*Phib' + Phib*P0*dPhibi' + ...
                    dEi(j*sE+1:i*sE,:)*Q*E' + E*dQi(j*sQ+1:i*sQ,:)*E' + ...
                    E*Q*dEi(j*sE+1:i*sE,:)' + dK1i*CRCt*K1' + ...
                    K1*dCi(j*sC+1:i*sC,:)*R*C'*K1' + K1*C*dRi(j*sR+1:i*sR,:)*C'*K1' + ...
                    K1*C*R*dCi(j*sC+1:i*sC,:)'*K1' + K1*CRCt*dK1i' - ...
                    dK1i*ESCt' - K1*dCi(j*sC+1:i*sC,:)*S'*E' - ...
                    K1*C*dSi(j*sS+1:i*sS,:)'*E' - K1*C*S'*dEi(j*sE+1:i*sE,:)' - ...
                    dEi(j*sE+1:i*sE,:)*S*C'*K1' - E*dSi(j*sS+1:i*sS,:)*C'*K1' - ...
                    E*S*dCi(j*sC+1:i*sC,:)'*K1' - ESCt*dK1i';

           for k=1:i
               l = k-1;
               Im(i,k) = Im(i,k) + 0.5*trace( (dB1i(j*m+1:i*m,:)'*iB1)*(dB1i(l*m+1:k*m,:)'*iB1) ) + ...
                                             (dz1i(j*m+1:i*m,:)'*iB1*dz1i(l*m+1:k*m,:));
           end
        %
        else
        %
           dK1i(j*p+1:i*p,:) = (dPhii(j*p+1:i*p,:)*P0*H' + Phi*dP0i(j*p+1:i*p,:)*H' + ...
                 Phi*P0*dHi(j*m+1:i*m,:)' + ...
                 dEi(j*sE+1:i*sE,:)*S*C' + E*dSi(j*sS+1:i*sS,:)*C' + ...
                 E*S*dCi(j*sC+1:i*sC,:)' - K1*dB1i(j*m+1:i*m,:))*iB1;

           dPhibi= dPhii(j*p+1:i*p,:) - dK1i(j*p+1:i*p,:)*H - K1*dHi(j*m+1:i*m,:);

           if MV
              if r
                 dz1i(j*m+1:i*m,:) = -dHi(j*m+1:i*m,:)*x0 -H*dx0i(j*p+1:i*p,:) - ...
                                      dDi(j*sD+1:i*sD,:)*z(t,m+1:m+r)';
 
                 dGambi  = dGami(j*sGam+1:i*sGam,:) - dK1i(j*p+1:i*p,:)*D - K1*dDi(j*sD+1:i*sD,:);
 
                 dx0i(j*p+1:i*p,:) = dPhibi*x0 + Phib*dx0i(j*p+1:i*p,:) + ...
                                     dGambi*z(t,m+1:m+r)' + dK1i(j*p+1:i*p,:)*z(t,1:m)';
              else
                 dz1i(j*m+1:i*m,:) = -dHi(j*m+1:i*m,:)*x0 -H*dx0i(j*p+1:i*p,:);

                 dx0i(j*p+1:i*p,:) = dPhibi*x0 + Phib*dx0i(j*p+1:i*p,:) + ...
                                     dK1i(j*p+1:i*p,:)*z(t,1:m)';
              end
           end

           dP0i(j*p+1:i*p,:) = dPhibi*P0*Phib' + Phib*dP0i(j*p+1:i*p,:)*Phib' + Phib*P0*dPhibi' + ...
                    dEi(j*sE+1:i*sE,:)*Q*E' + E*dQi(j*sQ+1:i*sQ,:)*E' + ...
                    E*Q*dEi(j*sE+1:i*sE,:)' + dK1i(j*p+1:i*p,:)*CRCt*K1' + ...
                    K1*dCi(j*sC+1:i*sC,:)*R*C'*K1' + K1*C*dRi(j*sR+1:i*sR,:)*C'*K1' + ...
                    K1*C*R*dCi(j*sC+1:i*sC,:)'*K1' + K1*CRCt*dK1i(j*p+1:i*p,:)' - ...
                    dK1i(j*p+1:i*p,:)*ESCt' - K1*dCi(j*sC+1:i*sC,:)*S'*E' - ...
                    K1*C*dSi(j*sS+1:i*sS,:)'*E' - K1*C*S'*dEi(j*sE+1:i*sE,:)' - ...
                    dEi(j*sE+1:i*sE,:)*S*C'*K1' - E*dSi(j*sS+1:i*sS,:)*C'*K1' - ...
                    E*S*dCi(j*sC+1:i*sC,:)'*K1' - ESCt*dK1i(j*p+1:i*p,:)';

           Phibi = dPhii(j*p+1:i*p,:) - K1*dHi(j*m+1:i*m,:);

           if r & ~MV
              dzc(j*m+1:i*m,:) = -dHi(j*m+1:i*m,:)*xc -H*dxc(j*p+1:i*p,:) - ...
                                  dDi(j*sD+1:i*sD,:)*z(t,m+1:m+r)';

              dxc(j*p+1:i*p,:) = Phibi*xc + Phib*dxc(j*p+1:i*p,:) + ...
                                 (dGami(j*sGam+1:i*sGam,:) - K1*dDi(j*sD+1:i*sD,:))*z(t,m+1:m+r)';
           end

           for k=1:i
           %
               l = k-1;
               ptri = (j*i/2 + l)*p + 1;
               ptrf = ptri + p - 1;

               Phibj = dPhii(l*p+1:k*p,:) - K1*dHi(l*m+1:k*m,:);
 
               Bij = dHi(j*m+1:i*m,:)*Pc11*dHi(l*m+1:k*m,:)' + ...
                     H * Pc21(j*p+1:i*p,:)*dHi(l*m+1:k*m,:)' + ...
                     dHi(j*m+1:i*m,:)*Pc21(l*p+1:k*p,:)'*H' + ...
                     H*Pc23(ptri:ptrf,:)*H';

               Pc23(ptri:ptrf,:) = Phibi*Pc11*Phibj' + Phib*Pc21(j*p+1:i*p,:)*Phibj' + ...
                         Phibi*Pc21(l*p+1:k*p,:)'*Phib' + Phib*Pc23(ptri:ptrf,:)*Phib' + ...
                         dK1i(j*p+1:i*p,:)*B1*dK1i(l*p+1:k*p,:)';

               if r & ~MV
                  Im(i,k) = Im(i,k) + 0.5*trace( (dB1i(j*m+1:i*m,:)'*iB1)*(dB1i(l*m+1:k*m,:)'*iB1) ) + ...
                                          trace(iB1*(Bij + dzc(j*m+1:i*m,:)*dzc(l*m+1:k*m,:)'));
               else
                  Im(i,k) = Im(i,k) + 0.5*trace( (dB1i(j*m+1:i*m,:)'*iB1)*(dB1i(l*m+1:k*m,:)'*iB1) ) + ...
                                          trace(iB1*Bij);
               end
           %
           end
        %
        end

        if MV
           dHPhi = dHi(j*m+1:i*m,:)*Phibb0 + H*dPhibb0(j*p+1:i*p,:);
           dWW(j*p+1:i*p,:) = dWW(j*p+1:i*p,:) + dHPhi'*iB1*HPhi - HPhi'*iB1*dB1i(j*m+1:i*m,:)*iB1*HPhi + HPhi'*iB1*dHPhi;
           dWZ(j*p+1:i*p,:) = dWZ(j*p+1:i*p,:) + dHPhi'*iB1*z1 - HPhi'*iB1*dB1i(j*m+1:i*m,:)*iB1*z1 + HPhi'*iB1*dz1i(j*m+1:i*m,:);
           dPhibb0(j*p+1:i*p,:) = dPhibi*Phibb0 + Phib*dPhibb0(j*p+1:i*p,:);
        end
    %
    end  % i

    if MV
       WW  = WW + HPhi'*iB1*HPhi;
       WZ  = WZ + HPhi'*iB1*z1;
       Phibb0 = Phib*Phibb0;
    end

    if aprox
       if r
          x0 = Phib*x0 + Gamb*z(t,m+1:m+r)' + K1*z(t,1:m)';
       else
          x0 = Phi*x0 + K1*(z(t,:)' - H*x0);
       end
    else
    %
       if MV
          if r
             x0 = Phib*x0 + Gamb*z(t,m+1:m+r)' + K1*z(t,1:m)';
          else
             x0 = Phi*x0 + K1*(z(t,:)' - H*x0);
          end
       end

       for i=1:np
           j = i-1;
           Phibi = dPhii(j*p+1:i*p,:) - K1*dHi(j*m+1:i*m,:);
           Pc21(j*p+1:i*p,:) = Phibi*Pc11*Phi' + Phib*Pc21(j*p+1:i*p,:)*Phi' + dK1i(j*p+1:i*p,:)*B1*K1';
       end

       Pc11 = Phi*Pc11*Phi' + K1*B1*K1';

       if r & ~MV
          xc = Phi*xc + Gam*z(t,m+1:m+r)';
       end
    %
    end

    P0 = Phib*P0*Phib' + EQEt + K1*CRCt*K1' - ...
         K1*ESCt' - ESCt*K1';
%
end % t


if MV
%
   WW = pinv(WW, zeps);
   xc = WW*WZ;
   for i = 1:np
       j = i-1;
       dxc(j*p+1:i*p,:) = WW*(dWZ(j*p+1:i*p,:)-dWW(j*p+1:i*p,:)*xc);
   end
   xc = xc + x00;
   P0 = P00;

   for t = 1:n
   %
       B1   = H*P0*H' + CRCt;
       if scaleb, U = cholp(B1, abs(B1));
       else       U = cholp(B1); end
       iB1 = (eye(size(U))/U)/U';

       K1  = (Phi*P0*H' + ESCt)*iB1;
       Phib  = Phi - K1*H;
       if r, Gamb   = Gam  - K1*D; end

       for i = 1:np
       %        
           j = i-1;

           Phibi = dPhii(j*p+1:i*p,:) - K1*dHi(j*m+1:i*m,:);

           if r
              dzc(j*m+1:i*m,:) = -dHi(j*m+1:i*m,:)*xc -H*dxc(j*p+1:i*p,:) - ...
                                  dDi(j*sD+1:i*sD,:)*z(t,m+1:m+r)';

              dxc(j*p+1:i*p,:) = Phibi*xc + Phib*dxc(j*p+1:i*p,:) + ...
                                 (dGami(j*sGam+1:i*sGam,:) - K1*dDi(j*sD+1:i*sD,:))*z(t,m+1:m+r)';
           else
              dzc(j*m+1:i*m,:) = -dHi(j*m+1:i*m,:)*xc -H*dxc(j*p+1:i*p,:);
              dxc(j*p+1:i*p,:) = Phibi*xc + Phib*dxc(j*p+1:i*p,:);
           end

           for k = 1:i
               l = k-1;
               Im(i,k) = Im(i,k) + dzc(l*m+1:k*m,:)'*iB1*dzc(j*m+1:i*m,:);
           end
       %
       end

       if r
          xc = Phi*xc + Gam*z(t,m+1:m+r)';
       else
          xc = Phi*xc;
       end

       P0 = Phib*P0*Phib' + EQEt + K1*CRCt*K1' - ...
            K1*ESCt' - ESCt*K1';
   %
   end
%
end

Im = Im + tril(Im,-1)';

if min(eig(Im)) <= 0, e4warn(8); end

varm = pinv(Im);

dts   = sqrt(diag(varm));
corrm = diag(1./dts)*varm*diag(1./dts);
E4OPTION(1) = oldfil;
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