Example 13.4 Diffuse Kalman Filtering

The nonstationary SSM is simulated to analyze the diffuse Kalman filter call KALDFF. The transition equation is generated by using the following formula:

where . The transition equation is nonstationary since the transition matrix has one unit root. Here is the code:

proc iml;
z_1 = 0; z_2 = 0;
do i = 1 to 30;
   z = 1.5*z_1 - .5*z_2 + rannor(1234567);
   z_2 = z_1;
   z_1 = z;
   x =  z + .8*rannor(1234578);
   if ( i > 10 ) then y = y // x;
end;

The KALDFF and KALCVF calls produce one-step prediction, and the result shows that two predictions coincide after the fifth observation (Output 13.4.1). Here is the code:

      t = nrow(y);
      h = { 1 0 };
      f = { 1.5 -.5, 1 0 };
      rt = .64;
      vt = diag({1 0});
      ny = nrow(h);
      nz = ncol(h);
      nb = nz;
      nd = nz;
      a  = j(nz,1,0);
      b  = j(ny,1,0);
      int = j(ny+nz,nb,0);
      coef = f // h;
      var = ( vt || j(nz,ny,0) ) //
            ( j(ny,nz,0) || rt );
      intd = j(nz+nb,1,0);
      coefd = i(nz) // j(nb,nd,0);
      at = j(t*nz,nd+1,0);
      mt = j(t*nz,nz,0);
      qt = j(t*(nd+1),nd+1,0);
      n0 = -1;
      call kaldff(kaldff_p,dvpred,initial,s2,y,0,int,
                  coef,var,intd,coefd,n0,at,mt,qt);
      call kalcvf(kalcvf_p,vpred,filt,vfilt,y,0,a,f,b,h,var);
      print kalcvf_p kaldff_p;

The likelihood function for the diffuse Kalman filter under the finite initial covariance matrix is written

where is the dimension of the matrix . The likelihood function for the diffuse Kalman filter under the diffuse initial covariance matrix is computed as , where the matrix is the upper matrix of . Output 13.4.2 displays the log likelihood and the diffuse log likelihood. Here is the code:

      d = 0;
      do i = 1 to t;
         dt = h*mt[(i-1)*nz+1:i*nz,]*h` + rt;
         d = d + log(det(dt));
      end;
      s = qt[(t-1)*(nd+1)+1:t*(nd+1)-1,1:nd];
      log_l = -(t*log(s2) + d)/2;
      dff_logl = log_l - log(det(s))/2;
      print log_l dff_logl;