statespace_augmented

The augmented Kalman filter and smoother (DK §5.7): an alternative to the exact initial filter for diffuse states, and the basis of regression estimation (DK §6.2).

With \(\alpha_1 = a + A\delta + R_0\eta_0\) and \(\delta\) unknown, the filter runs once with \(\delta = 0\), giving \(v^*_t, F_t, K_t, P_t\), and carries the effect of \(\delta\) as extra columns:

\[\begin{split}A_1 = A, \quad V^A_t = -Z_t A_t, \quad A_{t+1} = T_t A_t + K_t V^A_t, \\ s = \sum_t V^{A\prime}_t F_t^{-1} v^*_t, \quad S = \sum_t V^{A\prime}_t F_t^{-1} V^A_t,\end{split}\]

so that \(v_t(\delta) = v^*_t + V^A_t \delta\). The generalized least squares estimate is \(\hat\delta = -S^{-1} s\) with variance \(S^{-1}\), and the diffuse log likelihood (DK §7.2.2-7.2.3) is

\[\log L_d = \log L^* + \tfrac12 s' S^{-1} s - \tfrac12 \log|S|.\]

The smoother adds the uncertainty about \(\delta\) to \(V_t\) by the law of total variance. The filter needs \(F_t\) nonsingular over the observed elements; the exact initial filter does not.

augmented_result_t

type :: augmented_result_t
  integer :: k_diffuse                       ! k
  type(filter_result_t) :: filter            ! the delta = 0 filter
  real(dp), allocatable :: A(:, :, :)        ! (m, k, n+1)
  real(dp), allocatable :: VA(:, :, :)       ! (p, k, n)
  real(dp), allocatable :: s(:), S_mat(:, :)
  real(dp), allocatable :: delta(:), delta_cov(:, :)
  real(dp) :: llf, llf_fixed, llf_marginal
end type

llf

Diffuse log likelihood; equal to that of the exact initial filter.

llf_fixed

Log likelihood with \(\delta\) fixed but unknown, concentrated at \(\hat\delta\) (DK §7.2.4): llf + \(\tfrac12 \log|S|\).

llf_marginal

Marginal log likelihood (DK §7.2.6); see marginal_correction in statespace_filter.

augmented_filter

subroutine augmented_filter(rep, res, info)
  type(ssm_rep_t), intent(in) :: rep
  type(augmented_result_t), intent(out) :: res
  integer, intent(out) :: info

info is SS_ERR_NOT_PD if S is singular: the data do not identify \(\delta\).

augmented_smoother

subroutine augmented_smoother(rep, res, alphahat, V, info)
  real(dp), intent(out) :: alphahat(:, :), V(:, :, :)   ! (m, n), (m, m, n)

\[\hat\alpha_t = a^*_t + A_t \hat\delta + P_t (r^*_{t-1} + R^A_{t-1} \hat\delta), \quad V_t = P_t - P_t N_{t-1} P_t + B_t S^{-1} B_t', \quad B_t = A_t + P_t R^A_{t-1}.\]