statespace_collapse

Collapsing large observation vectors (DK §6.5; Jungbacker and Koopman 2015).

When p is large relative to m, each \(y_t\) is replaced by its generalized least squares projection on the state:

\[\bar y_t = (Z' H^{-1} Z)^{-1} Z' H^{-1} (y_t - d_t) = \alpha_t + \bar\varepsilon_t, \qquad Var(\bar\varepsilon_t) = (Z' H^{-1} Z)^{-1},\]

over the observed elements. The residual \(e_t = y_t - d_t - Z \bar y_t\) is independent of \(\bar y_t\) and of the states, so the collapsed model (observation \(\bar y_t\), Z = I) gives the same filtered and smoothed states, and

\[\log L = \log L^* + \sum_t \Big[ -\tfrac{p_t - m}{2} \log 2\pi - \tfrac12 \log \tfrac{|H_t|}{|\bar H_t|} - \tfrac12 e_t' H_t^{-1} e_t \Big].\]

collapse_observations

subroutine collapse_observations(rep, crep, llf_adjust, info)
  type(ssm_rep_t), intent(in) :: rep
  type(ssm_rep_t), intent(out) :: crep
  real(dp), intent(out) :: llf_adjust(:)   ! (n)
  integer, intent(out) :: info

Build the collapsed representation and the per-period adjustment, so that loglike(rep) = loglike(crep) + sum(llf_adjust). Every period with observations needs the observed block of \(H_t\) nonsingular and the observed rows of \(Z_t\) of full column rank m; otherwise info is SS_ERR_NOT_PD.