Algorithm 2.2.

Initialize η̂t:t+k−1 and calculate ĥt+1:t+k using (3b). Evaluate d̂s, M̂s, and N̂s using equations (B.1), (B.3) and (B.4) respectively. Compute Gs, Js, and bs, for s = t+2, … , t+k, recursively, as follows: $$\begin{gathered} G_s={\widehat{M}}_s-{\widehat{N}}_s^2{G}_{s-1}^{-1},G_{t+1}={\widehat{M}}_{t+1}, \\ {J}_s=K_{s-1}^{-1}{\widehat{N}}_s,J_{t+1}=0,J_{t+k+1}=0, \\ {b}_s={\widehat{d}}_s-J_s{K}_{t-1}^{-1}{b}_{s-1}{b}_{t+1}={\widehat{d}}_{t+1}, \end{gathered}$$ where $K_s=\sqrt{G_s}$. Define the auxiliary variables ${\widehat{y}}_s={\widehat{\gamma }}_s+G_s^{-1}{b}_s$, where $${\widehat{\gamma }}_s={\widehat{h}}_s+K_s^{-1}{J}_{s+1}{\widehat{h}}_{s+1},s=t+1,\dots ,t+k.$$ Consider the linear Gaussian state-space model $${\widehat{y}}_s=c_s+Z_s{h}_s+H_s{\mathit{\xi }}_s,s=t+1,\dots ,t+k,$$ (10) $$h_{s+1}=\alpha +\varphi h_s+L_s{\mathit{\xi }}_s,s=t,t+1,\dots ,t+k,$$ (11) where ξs ~ 𝒩(0, I2), $c_s=K_s^{-1}{J}_{s+1}\alpha ,Z_s=1+K_s^{-1}{J}_{s+1}\varphi ,H_s=K_s^{-1}[1,J_{s+1}{\sigma }_{\eta }]$, and Ls = [0, ση]. Apply the Kalman filter and a disturbance smoother [25] to the linear Gaussian state space model in equations (10) and (11) and obtain the posterior mean of ηt:t+k−1 (ht+1:t+k) and set η̂t:t+k−1 (ĥt+1:t+k) to this value. Return to Step 2 and repeat the procedure until achieving convergence.