[Gretl-users] Re: Kalman Filter together with maximum likelihood - clarification about the Variance-Covariance things
On 24/05/2024 11:04, Sven Schreiber wrote:
Am 23.05.2024 um 18:15 schrieb Alecos Papadopoulos:
> I realized that I am a bit confused as to what happens with the
> various VCV matrices when Kalman Filter is used together with maximum
> likelihood. Assume a time-invariant state space model.
>
> What I understand is the following:
>
> The Kalman Filter will run as usual and will provide the various VCV
> estimates as described in Ch. 36 of the user's guide, based, in the
> end, on the final ML estimates to obtain the residual series (the etas
> and the epsilons).
>
> But at the same time, we may have asked the mle to estimate also the
> elements of the VCV matrices, (see eg. the ARMA estimation example
> Listing 36.1). This will give us another, different estimate of the
> VCV matrices... or will this agree with the KF estimates above, due
> perhaps to the fact that this is Normal mle that we necessarily assume?
>
Hi, I'm looking at listing 36.1 but I'm not exactly sure what you mean,
sorry. Are you talking about sigma there?
Apart from that, jack can probably comment some more, but I think he is
currently unavailable.
Sorry for the delay, I'm just back. Like Sven, I'm not sure I understand
what you mean. There are several VCVs that one may be interested to in
the context of a state space model (SSM):
1) the VCVs of the stochastic shocks to the SSM, that are referred to as
\epsilon_t and \eta_t in the Guide, that is the matrices we call \Sigma
and Omega. These can be time varying, but in most models used in
practice are assumed to be fixed. The parameter called "sigma" in script
36.1 is just, well, \Sigma.
2) the VCVs of the estimated parameters: these are not part of the SSM,
but come as a by-product of the estimation process. If you use the mle
command, this is what you get via the accessor $vcv.
3a) the (time-varying) VCVs of the predicted states, that is a measure
of the uncertainty we have in the measurement of the unobservable states
(\alpha_t in the Guide). These become available as the "stvar" bundle
element after running kfilter() and/or ksmooth().
3b) the (time-varying) VCVs of the smoothed disturbances, that is a
measure of the uncertainty we have when we predict \epsilon_ and \eta_t.
Read section 36.7 on this: the point is quite subtle.
Hope this helps.
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Riccardo (Jack) Lucchetti
Dipartimento di Scienze Economiche e Sociali (DiSES)
Università Politecnica delle Marche
(formerly known as Università di Ancona)
r.lucchetti(a)univpm.it
http://www2.econ.univpm.it/servizi/hpp/lucchetti
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