- The data is empirically observed data. The likelihood is the likelihood of observing the actual data realizations given the current model (this is an implicit conditioning you are missing) and the current parameter vector. If you forget about Bayesian estimation for a second, the goal of maximum likelihood estimation would be to find the values for the parameters of the model that make observing the empirical data the most likely. Consider an AR1-process
where epsilon is standard normal. The goal is to find rho and sigma_eps. Given any draw for theta=[rho;sigma_eps] the likelihood of observing x can be evaluated as
To evaluate that function, you need the parameter vector theta.
- The only restriction is that you cannot have fewer shocks than observables as the model would be stochastically singular. Having more shocks is generally not an issue. Estimation means finding the maximum/mode of the posterior. Usually you do that by finding a point where the first derivative is 0 (first order condition). This point should be the maximum. But there might be cases for various reasons where the derivative of the likelihood function w.r.t a particular parameter is 0 not only at the maximum but for the whole parameter range. In that case the maximum cannot be found and the parameter is not identified (i.e. the zero derivative does not only hold at the true value). This can be easily checked by looking at the Jacobian of the model. The bigger problem is weak identification where the likelihood function is not horizontal in a particular parameter direction but where it is very flat. In that case it will be hard to find the actual maximum. You might want to take a look at the references pointed out here [Quick Help)