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Appendix A. The forward filtering backward smoothing algorithm.

1) Forward filtering algorithm: The posterior and one-step forecast distribution in the NDLM can be calculated as follows:

a) Posterior at time t - 1: for some mean m_t-1 and variance C_t-1, x_t-1|D_t-1 N[m_t-1, C_t-1].

b) Prior at time t: x_t|D_t-1 N[a_t, R_t], where a_t = gm_t-1 and R_t = g²C_t-1 + W.

c) One-step forecast: y_t|D_t-1 N[s_t, Q_t], where s_t = fa_t and Q_t = f²R_t = V.

d) Posterior at time t: x_t|D_t N[m_t, C_t], with m_t = a_t + A_te_t and C_t = R_t - A²_tQ_t, where A_t = R_tF/Q_t and e_t = y_t - s_t.

2) Backward smoothing algorithm: Given that p(x_t|D_t) N[m_t, C_t], for all k such that 1 k t, the filtering marginal distributions are

x_t-k|D_t N[a_t(-k), R_t(-k)]

where B_t = C_tgR^-1_t+1, a_t(-k) = m_t-k + B_t-k[a_t(-k + 1) - a_t-k+1], and R_t(-k) = C_t-k+B_t-k[R_t(-k+1) - R_t-k+1]B'_t-k

with starting values a_t(0) = m_t and R_t(0) = C_t, and where a_t-k(1) = a_t-k+1 and R_t-k(1) = R_t-k+1.