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:

x_t+1=Ax_t+Cw_t+1 y_t=Gx_t+v_t 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,

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

y_t is a measurement of x with scalings G and measurement errors v_t,

w_t are innovations to the hidden x_t process,

Ew_t+1w_t'=1 by normalization,

Ev_tv_t=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 S_t and K_t 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:

z_t+1=Az_t+Ka_t y_t=Gz_t+a_t where z_t is defined to be E_t-1x_t,

a_t is defined to be y_t-E_t-1y_t,

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

The definition of those two matrices S_t and K_t is itself most of the definition of the Kalman filter:

K_t=AS_tG'(GS_tG'+R)^-1 S_t+1=(A-K_tG)S_t(A-K_tG)'+CC'+K_t RK_t'

K_t is called the Kalman gain. (Econterms)

Terms related to The Kalman Filter:

• Kalman gain
• OLS