Introduction to Local Level Model and Kalman Filter

Derivation Kalman Filter
Local level model: µt+1 = µt + ηt, yt = µt + εt.
◮ We have Yt = {Yt−1, yt} = {Yt−1, vt} and E(vtyt−j) = 0 for
j = 1, . . . , t − 1;
◮ The lemma is E(x|y, z) = E(x|y) + ΣxzΣ −1
zz z.
In our case, take x = µt+1, y = Yt−1 and
z = vt = (µt − at) + εt;
◮ E(x|y) implies that
E(µt+1|Yt−1) = E(µt|Yt−1) + E(ηt|Yt−1) = at;
◮ Further, Σxz provides the expression
E(µt+1vt) = E(µtvt) + E(ηtvt) = E[(µt − at)(yt − at)] +
E(ηtvt) = E[(µt −at)(µt −at)]+E[(µt −at)εt]+E(ηtvt) = Pt;
◮ Since Σzz = Ft, we can apply lemma and obtain the state
update
at+1 = E(µt+1|Yt−1, yt)
= at + PtF −1
t vt
= at + Ktvt; with Kt = PtF −1
t .