SOURCE. Harvey, A.C. (1989), Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press; doi:10.1017/CBO9781107049994. The basic structural model of Section 2.3 is y_t = mu_t + gamma_t + eps_t, mu_t = mu_{t-1} + beta_{t-1} + eta_t, beta_t = beta_{t-1} + zeta_t, gamma_t = -sum_{j=1}^{s-1} gamma_{t-j} + omega_t. It is put in state space form (Sec. 3.1) and run through the Kalman filter (Sec. 3.2): a' = T a, P' = T P T' + Q, v = y - Z a', F = Z P' Z' + 1, K = P' Z'/F, a = a' + K v, P = P' - K Z P'.
Details
Disturbance variances are ratios q to sigma^2, so sigma^2 concentrates out exactly (Sec. 3.4): sigma2 = sum_{t>d} v_t^2/F_t / (T-d) and logL = -(T-d)/2 (log 2pi + 1 + log sigma2) - 1/2 sum_{t>d} log F_t.
DIFFUSE PRIOR. Sec. 3.3.4's limit of an infinite prior variance is approximated by P_0 = kappa I with kappa large, and by dropping the first d prediction errors, d = state dimension.
SCOPE. Level, slope, seasonal and irregular only. The stochastic cycle of Sec. 2.3.4 is NOT implemented – it needs two further states plus a damping and a frequency parameter. That is this implementation's scope choice, stated rather than attributed. Likewise the variance ratios are chosen by exhaustive search over a fixed lattice, not by numerical optimisation, so both language arms land on identical numbers.
NOT read from the book's own page images – Harvey (1989) is not in the local corpus. Anchored on the closed-form local-level case.
References
Harvey, A.C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press. doi:10.1017/CBO9781107049994.
Examples
Unobts(c(1, 2, 3, 4, 5, 6), "level")$sigma2
#> [1] 0.7555556