[Gretl-users] Re: Gretl User's guide State Space modeling - clarification
On Mon, 18 Mar 2024, Alecos Papadopoulos wrote:
On page 349 of gretl's user guide the state space model is
defined
as
y_t = Z_t \alpha_t + \epsilon_t (36.1)
\alpha_{t+1} = T_t \alpha_t + \eta_t (36.2)
Later, when presenting the state-space GUI (p. 369) we read that
it supports models of the form
y_t = Z alpha_t + \epsilon_t (36.6)
\alpha_t = T \alpha_{t-1} + R \eta_t (36.7)
My only question is about the time indices on the state vector in
the transition equation.
In the first model (p. 349), the starting values for the state
variables (alpha) should be for time t, and will also affect the
first time point of the measurement equation. In the second
formulation (p. 369), these starting values should be for time
(t-1), they should lead to alpha values for time t through the
transition equation alone, which will then affect the first time
point of the measurement equation.
Which of the two does gretl do?
Gretl does the first of these. The first state value lines up with
the first observation in both filtering and simulation. But I'd like
to hear from Jack in case I'm missing something.
Besides your point about initial values, it seems to me there's
another question here. That is, for the state in any given period,
is its disturbance component dated to the prior period (the first
case above) or to the current period (the second case)? While this
may just be a matter of convention, it could make a difference when
interpreting results -- you'd want to know of any given software
which convention it's using.
I'm looking through the various state-space articles on my HD and
the first convention is used in most cases (the doc for SsfPack and
KFAS, all de Jong's papers, most articles by Koopman, the Durbin and
Koopman book). The second one I'm seeing in a couple of papers that
have Koopman and Shephard as co-authors.
Allin Cottrell