Dear list users,
here there is a question about Dynamic Linear Models.
Let me consider the usual DLM. For simplicity let me consider only the seasonal component, with 4 seasonal data per year.
The sum of the seasonal components must sum to zero, and therefore the last component can be ignored:
transpose(theta_t) = (s_t, s_t-1, s_t-2)
The rotation matrix G becomes
-1 -1 -1
G = 1 0 0
0 1 0
(It is also reasonable suppose that the system variance adds an error only in the first variable):
omega 0 0
V[omega]= 0 0 0
0 0 0
The full conditional distribution (fcd) for theta_t is
f(theta_t | theta_t-1,theta_t+1,y_t)
which is a multivariate normal distribution.
If I want to sample theta_t from its fcd, it happens that its variance, is zero. This independently from the variance matrix which is used.
This means that given theta_t-1 and theta_t+1, theta_t is fully determined.
The impossibility of sampling theta_t is for me totally unreasonable.
These problems are avoided if the notations of West&Harrison (Bayesian Forecasting and DLM, 1997) are used:
transpose(theta_t) = (s_t, s_t+1, s_t+2, s_t+3)
and
0 1 0 0
G = 0 0 1 0
0 0 0 1
1 0 0 0
Why in this second case the variance of the fcd of theta_t is not zero and a value for theta_t can be sampled?
Why in the first case sampling is not permitted?
Which is the difference?
Any help will be highly appreciated, really.
Stefano
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