Kalman Filter Derivation with Non-Diagonal Q Matrix

[spirall]

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

 

[D H]

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.

 

[spirall]

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?