In 6.3, we discussed that it is possible to obtain a recursion for the gradient vector, lnLY ()/. Assume the model is given by

In §6.3, we discussed that it is possible to obtain a recursion for the gradient vector, −∂ lnLY (Θ)/∂Θ. Assume the model is given by (6.1) and

(6.2) and At is a known design matrix that does not depend on Θ, in which case Property 6.1 applies. For the gradient vector, show

∂ lnLY (Θ)/∂Θi = Xn t=1



0 tΣ−1 t

∂t

∂Θi

− 1 2

0 tΣ−1 t

∂Σt

∂Θi

Σ−1 t t

+

1 2

tr 

Σ−1 t

∂Σt

∂Θi

 , where the dependence of the innovation values on Θ is understood. In addition, with the general definition ∂ig = ∂g(Θ)/∂Θi, show the following recursions, for t = 2, . . . , n apply:

(i) ∂it = −At ∂ixt−1 t ,

(ii) ∂ixt−1 t = ∂iΦ xt−2 t−1 + Φ ∂ixt−2 t−1 + ∂iKt−1 t−1 + Kt−1 ∂it−1,

(iii) ∂iΣt = At ∂iPt−1 t A0 t + ∂iR,

(iv) ∂iKt =

∂iΦ Pt−1 t A0 t + Φ ∂iPt−1 t A0 t − Kt ∂iΣt

Σ−1 t ,

(v) ∂iPt−1 t = ∂iΦ Pt−2 t−1 Φ0 + Φ ∂iPt−2 t−1 Φ0 + Φ Pt−2 t−1 ∂iΦ0 + ∂iQ,

− ∂iKt−1 ΣtK0 t−1 − Kt−1 ∂iΣt K0 t−1 − Kt−1Σt ∂iK0 t−1, using the fact that Pt−1 t = ΦPt−2 t−1 Φ0 + Q − Kt−1ΣtK0 t−1.