Introduction to Local Level Model and Kalman Filter

Smoothing
◮ The filter calculates the mean and variance conditional on Yt;
◮ The Kalman smoother calculates the mean and variance
conditional on the full set of observations Yn;
◮ After the filtered estimates are calculated, the smoothing
recursion starts at the last observations and runs until the
first.
ˆµt = E(µt|Yn), Vt = var(µt|Yn),
rt = weighted sum of future innovations, Nt = var(rt),
Lt = 1 − Kt.
Starting with rn = 0, Nn = 0, the smoothing recursions are given
by
rt−1 = F −1
t vt + Ltrt, Nt−1 = F −1
t + L 2 t Nt,
ˆµt = at + Ptrt−1, Vt = Pt − P 2 t Nt−1.