VMA(1) estimation

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Wei on 7 Aug 2017
Commented: Wei on 11 Aug 2017
I'm trying to estimate a VMA(1) model and I'm wondering if there is a way to do it using the econometrics toolbox without having to convert from a VARMA model? I don't think the toolbox can estimate the MA lags of the VARMA model anyway so not sure how to estimate a VARMA in the first place. If the VMA(1) cannot be directly estimated, can I convert a VAR to VMA using arma2ma? Any help on this is deeply appreciated.
Thanks, Wei

Accepted Answer

Hang Qian
Hang Qian on 11 Aug 2017
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.
Hang Qian
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Wei on 11 Aug 2017
Thanks Hang for your answer. I'm not familiar with the estimation methods you suggested so will have to do some reading on those. Thanks again for your reply. Wei

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