Continuing with the previous problem, consider the evaluation of the Hessian matrix and the numerical evaluation of the asymptotic variance–

covariance matrix of the parameter estimates. The information matrix satisfies E



−∂2 lnLY (Θ)

∂Θ ∂Θ0



= E

(∂ lnLY (Θ)

∂Θ  ∂ lnLY (Θ)

∂Θ 0

)

;

see Anderson (1984, Section 4.4), for example. Show the (i, j)-th element of the information matrix, say, Iij (Θ) = E 

−∂2 lnLY (Θ)/∂Θi ∂Θj

, is Iij (Θ) = Xn t=1 E

n

∂i0 t Σ−1 t ∂jt +

1 2

tr

Σ−1 t ∂iΣt Σ−1 t ∂jΣt

+

1 4

tr

Σ−1 t ∂iΣt

tr

Σ−1 t ∂jΣt

o

.

Consequently, an approximate Hessian matrix can be obtained from the sample by dropping the expectation, E, in the above result and using only the recursions needed to calculate the gradient vect