I'm trying to implement a Kalman Filter for the parameter estimation of a linear gaussian two factor model in Matlab. (Schwartz Smith model for commodity prices) In other words: I try to compute the log-likelihood of the parameters.
My model:
$X_t = A X_{t-1} + \epsilon_X$ , with X beeing two-dimensional.
$Y_t = \begin{pmatrix} 1 \\ 1 \end{pmatrix}^T X_t + \epsilon_Y$ , with Y beeing one-dimensional.
$A$ is time invariant and only depends on the parameters $\theta$, which I would like to determine.
My question:
Whats the variance of $\epsilon_Y$ ? I know, that it normally represents the noise of the measurement process, but I don't know what would be the equivalent in an economic context?
I managed to implement the filter and the results for experimenting with the variance are (mean = 0):
- For high variance: High log-likelihood in unreasonable areas far away from the real values.
- For very (!) small variance: Reasonable results close to real values, but extremly small log-likelihoods around $-10^{12}$. I fear, that this will cause numerical issues in the upcoming maximum likelihood estimation. (I plan to use Metropolis Hastings)
Any help would be appreciated! Thanks!