[R] dlm package: how to specify state space model?

[Giovanni Petris]

Dear Christian,

It is not totally clear to me which variables in your model you can
observe and which are unobservable, so I will assume that g_t is
observable and u_t, un_t and r_t are not.

You can take the state vector b_t = (one_t, u_t, r_t, un_t)', where
one_t = 1 (more on that below). Then you need

F = (x[1], x[2], x[3], x[2])
and
G = diag(1, 1, 1, x[4])

Since you did not specify the dynamics of u_t and r_t, I am assuming
they follow a random walk. If they are observable, then you can just add
one row to the F matrix for each one of them.

Since you want one_t to be 1 at any time, you should specify in the
prior mean m0[1] = 1 and C0[1,1] = epsilon (in theory this should be
zero, but in order for dlm to work you need a nonsingular C0: taking a
tiny epsilon, 1e-7 say, usually does the job). Similarly, you don't want
one_t to change from one time to the next, so you need to specify W[1,1]
= 0. Other entries in W can be estimated by maximum likelihood.

If you take any of u_t and r_t to be observable, you will also need to
set the corresponding value in the V matrix to a tiny epsilon, for the
same reason that V must be nonsingular. Or, even better, you can take
the observation variance for u_t and/or r_t to be a parameter and you
will end up with a random walk plus noise model (local level model) for
u_t (and/or r_t).

HTH,
Giovanni Petris


On Tue, 2010-10-05 at 22:30 -0400, Christian Schoder wrote:
> Dear r-users!
> 
> I have another question regarding the dlm package and I would be very
> happy if someone could give me a hint!
> 
> I am using the dlm package to get estimates for an endogenous rate of
> capacity utilization over time. The general form of a state space model
> is
> 
> (1) b_t = G * b_t-1 + w_t    w_t ~ N(0,W)
> 
> (2) y_t= A' * x_t + H' * b_t + v_t     v_t ~ N(0,V)
> 
> (Hamilton 1984: 372)
> 
> The investment function I would like to use for estimating my endogenous
> capacity utilization rate looks like
> 
> (3) g_t = x[1] + x[2]*(u_t-un_t) + x[3]*r + v_t
> 
> where g_t is the investment rate, r_t is the profit rate, u_t is the
> actual utilization rate and un_t is the 'normal' utilization rate which
> I take as endogenous (=time varying). x[i] are parameters. I'm
> particularly interested in this endogenous normal utilization rate. How
> can I specify a state space model which allows me to estimate it and is
> consistent with the structure of the state space models in the dlm
> package?
> 
> In the form found in Hamilton my system would look like
> 
> (4) un_t = x[4] * un_t-1 + w_t      w_t ~ N(0,W)
> 
> (5) g_t = (x[1],x[2],x[3]) * (1,u_t,r_t)' + x[2] * un_t + v_t    v_t ~
> N(0,V)
> 
> which theoretically can be estimated even with the restriction that the
> parameters of u_t and un_t have opposite signs, but are otherwise equal.
> But how can I do this with the plm package which requires a model of the
> following form:
> 
> (6) b_t = G * b_t-1 + w_t    w_t ~ N(0,W)
> 
> (7) y_t = F * b_t + v_t     v_t ~ N(0,V)
> 
> How can I write my model in the form of (6) and (7) such that my state
> vector includes un_t and I can get estimates for the normal rate of
> capacity utilization??
> 
> I would be very grateful for any help, cause I've been sitting on this
> issue for a while!
> 
> Christian
> 
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