Hi Wei,
In many cases, the VMA coefficients cannot be reliably estimated, unless there is a large sample with high quality data. If we really want to estimate those coefficients, one approach is the moment estimator. With a few constraints on the coefficients, the sample moments of y(t) and y(t-1) may give a reasonable estimator for those coefficients by solving equations. In that case, the Optimization Toolbox will be helpful for solving equations and minimizing the loss functions.
Another method is to cast the VMA model in the state-space form, and then use the Kalman filter for the likelihood function, which can then be maximized numerically. Sometimes it works for low-dimension VMA models if the starting values are reasonable. The state variables can be the stacked disturbances in a few recent periods, and the observations are linear with respect to those disturbances in a VMA model. For example, for a VMA(1), the states can be e(t) and e(t-1), with the transition matrix like
A = [ [zeros(dim),zeros(dim)]; [eye(dim), zeros(dim)]];
B = [chol(Sigma)’;zeros(dim)];
The observation matrix is just the MA system, say
C = [eye(dim),Psi];
As there are no additional noises in the observation matrix, the coefficient D can be omitted. With A, B, C, D matrices specified as inputs, the state-space model is ready. The MA coefficients Psi and the covariance matrix Sigma, can be estimated by numerical maximum likelihood.
Best,
Hang Qian
