arord.m

function [sbc, fpe, logdp, np] = arord(R, m, mcor, ne, pmin, pmax)
%ARORD	Evaluates criteria for selecting the order of an AR model.
%
%  [SBC,FPE]=ARORD(R,m,mcor,ne,pmin,pmax) returns approximate values
%  of the order selection criteria SBC and FPE for models of order
%  pmin:pmax. The input matrix R is the upper triangular factor in the
%  QR factorization of the AR model; m is the dimension of the state
%  vectors; the flag mcor indicates whether or not an intercept vector
%  is being fitted; and ne is the number of block equations of size m
%  used in the estimation. The returned values of the order selection
%  criteria are approximate in that in evaluating a selection
%  criterion for an AR model of order p < pmax, pmax-p initial values
%  of the given time series are ignored.
%
%  ARORD is called by ARFIT. 
%	
%  See also ARFIT, ARQR.

%  For testing purposes, ARORD also returns the vectors logdp and np,
%  containing the logarithms of the determinants of the (scaled)
%  covariance matrix estimates and the number of parameter vectors at
%  each order pmin:pmax.

%  Modified 17-Dec-99
%  Author: Tapio Schneider
%          tapio@gps.caltech.edu
  
  imax 	  = pmax-pmin+1;        % maximum index of output vectors
  
  % initialize output vectors
  sbc     = zeros(1, imax);     % Schwarz's Bayesian Criterion
  fpe     = zeros(1, imax);     % log of Akaike's Final Prediction Error
  logdp   = zeros(1, imax);     % determinant of (scaled) covariance matrix
  np      = zeros(1, imax);     % number of parameter vectors of length m
  np(imax)= m*pmax+mcor;

  % Get lower right triangle R22 of R: 
  %
  %   | R11  R12 |
  % R=|          |
  %   | 0    R22 |
  %
  R22     = R(np(imax)+1 : np(imax)+m, np(imax)+1 : np(imax)+m);

  % From R22, get inverse of residual cross-product matrix for model
  % of order pmax
  invR22  = inv(R22);
  Mp      = invR22*invR22';
  
  % For order selection, get determinant of residual cross-product matrix
  %       logdp = log det(residual cross-product matrix)
  logdp(imax) = 2.*log(abs(prod(diag(R22))));

  % Compute approximate order selection criteria for models of 
  % order pmin:pmax
  i = imax;
  for p = pmax:-1:pmin
    np(i)      = m*p + mcor;	% number of parameter vectors of length m
   if p < pmax
      % Downdate determinant of residual cross-product matrix
      % Rp: Part of R to be added to Cholesky factor of covariance matrix
      Rp       = R(np(i)+1:np(i)+m, np(imax)+1:np(imax)+m);

      % Get Mp, the downdated inverse of the residual cross-product
      % matrix, using the Woodbury formula
      L        = chol(eye(m) + Rp*Mp*Rp')';
      N        = L \ Rp*Mp;
      Mp       = Mp - N'*N;

      % Get downdated logarithm of determinant
      logdp(i) = logdp(i+1) + 2.* log(abs(prod(diag(L))));
   end

   % Schwarz's Bayesian Criterion
   sbc(i) = logdp(i)/m - log(ne) * (ne-np(i))/ne;

   % logarithm of Akaike's Final Prediction Error
   fpe(i) = logdp(i)/m - log(ne*(ne-np(i))/(ne+np(i)));

   % Modified Schwarz criterion (MSC):
   % msc(i) = logdp(i)/m - (log(ne) - 2.5) * (1 - 2.5*np(i)/(ne-np(i)));

   i      = i-1;                % go to next lower order
end