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The Kalman Filter <Back to Last Page>     <Full Glossary>

Definition of The Kalman Filter: The Kalman filter is an algorithm for sequentially updating a linear projection for a dynamic system that is in state-space representation.

Application of the Kalman filter transforms a system of the following two-equation kind into a more solvable form:

xt+1=Axt+Cwt+1 yt=Gxt+vt in which:

A, C, and G are matrices known as functions of a parameter q about which inference is desired (this is the PROBLEM to be solved),

t is an whole number, usually indexing time,

xt is a true state variable, hidden from the econometrician,

yt is a measurement of x with scalings G and measurement errors vt,

wt are innovations to the hidden xt process,

Ewt+1wt'=1 by normalization,

Evtvt=R, an unknown matrix, estimation of which is necessary but ancillary to the problem of interest which is to get an estimate of q. The Kalman filter defines two matrices St and Kt such that the system described above can be transformed into the one below, in which estimation and inference about q and R is more straightforward, possibly even by OLS:

zt+1=Azt+Kat yt=Gzt+at where zt is defined to be Et-1xt,

at is defined to be yt-Et-1yt,

K is defined to be lim Kt as t goes to infinity.

The definition of those two matrices St and Kt is itself most of the definition of the Kalman filter:

Kt=AStG'(GStG'+R)-1 St+1=(A-KtG)St(A-KtG)'+CC'+Kt RKt'

Kt is called the Kalman gain. (Econterms)

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