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Derivation of Kalman filter

  1. Apr 2, 2012 #1
    Hi all,

    I have a standard local level model, but the disturbances are not independent:
    y_t=μ_t+ε_t, μ_t+1=μ_t+η_t, E(ε_t η_t) =/= 0

    In order to derive the Kalman filter, I rewrite this model in state space form
    y_t=Z_t α_t+ε_t, ε_t~NID(0,H_t ),
    α_(t+1)=T_t α_t+R_t η_t, η_t~NID(0,Q_t ),
    α_1~N(a_1,P_1 ),

    α_t=[μ_t
    ξ_t ]

    Z_t=[1 1],
    H_t=0,

    Q_t=[σ_η^2 σ_ξη
    σ_ξη σ_ξ^2 ],

    T_t=[1 0
    0 0],

    R_t=[1 0
    0 1],

    η_t=[η_t
    ξ_(t+1)].

    I wonder whether there is any difference in the derivation of the Kalman filter, since the matrix Q in not diagonal.

    Thank you
     
  2. jcsd
  3. Apr 2, 2012 #2

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    There's nothing in the formulation or derivation of the Kalman filter that requires the noise matrix Q to be diagonal. It just needs to qualify as a covariance matrix. In other words, it needs to be a symmetric positive semidefinite matrix.
     
  4. Apr 3, 2012 #3
    Thank you for reply.

    And what about likelihood estimation of coefficients of system matrices if the transition matrix T depends on some exogenous variables (I suppose this is possible)?

    I have one more question about dynamic factor model. I consider transformation approach (yL_t=A_L y_t) and look for transformation matrix A=[A_L A_H]' such as:
    Ʃ_L=A_L H A´_L
    A_L=CZ´H^-1
    If I use Cholesky decomposition, the matrix C should be lower triangular. But how I choose it?
     
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