From a linear state-space perspective,

y_t=M_1(\theta) s_t+M_0(\theta)+\epsilon_t

s_t=N_1(\theta) s_{t-1}+u_t

To compute the likelihood,when the error terms are Gaussian, we use a Kalman filter.

In a frequentist view, we usually estimate the ‘structural’ parameters using maximum likelihood estimators(MLE).

However, in the programs like Sims(2002), to solve the transition equation above, one needs to introduce some numerical parameter values as input, and then the program will compute a * numerical* solution for the transition equation. This doesn’t seem to allow the use of a numerical optimisation routine to find the MLE, since to maximise the likelihood we would need a function not yet evaluated in the parameters.

How does the frequentist proceed then?

P.S.: I’ve been highlighting the expression ‘frequentist’, since with a Bayesian perspective, I don’t think the problem exists. Because with the use of MCMC algorithms, we’re always using numerical parameter values which will be used in the histogram for the posterior distribution.