MA model with constraints

[Teerapan]

Dear all,

I would like to estimate a constrained MA(2) model by maximum likelihood estimation. The model has the following structure:

X_t = theta_0*e_t + theta_1*e_{t-1} + theta_2*e_{t-2} with the constraint: 1 = theta_0 + theta_1 + theta_2

where theta_0 through theta_2 denote the regression parameters to be estimated.

I would greatly appreciate any comment on how to estimate this model. Thanks a lot for your help in advance. :D

[startz]

Dear all,

I would like to estimate a constrained MA(2) model by maximum likelihood estimation. The model has the following structure:

X_t = theta_0*e_t + theta_1*e_{t-1} + theta_2*e_{t-2} with the constraint: 1 = theta_0 + theta_1 + theta_2

where theta_0 through theta_2 denote the regression parameters to be estimated.

I would greatly appreciate any comment on how to estimate this model. Thanks a lot for your help in advance. :D

The convention is to set theta_0 equal to 1, and let the other parameters be free. Remember that the variance of e_t is a free parameter. The constraints aren't binding without a statement about that variance.

Suppose the unconstrained estimates you get are theta_1 and theta_2 and v for the residual variance. Then the "constrained" mle estimates are simply 1/(1+theta_1+theta_2), theta_1/(1+theta_1+theta_2), theta_2/(1+theta_1+theta_2), and v*sqrt(1+theta_1+theta_2).

[Teerapan]

[Then the "constrained" mle estimates are simply 1/(1+theta_1+theta_2), theta_1/(1+theta_1+theta_2), theta_2/(1+theta_1+theta_2), and v*sqrt(1+theta_1+theta_2).

Please correct me if I am wrong, 1/(1+theta_1+theta_2), theta_1/(1+theta_1+theta_2), theta_2/(1+theta_1+theta_2) are constrained theta_0, theta_1 and theta_2, respectively, right?

Then, 'v*sqrt(1+theta_1+theta_2)' = variance of the error term?

Also is there any chance that I can set the values of theta_0, theta_1 and theta_2 between 0 and 1?

Thanks a million!! you save my life. I have tried to solve this for DAYSS. :D :D

[startz]

[Then the "constrained" mle estimates are simply 1/(1+theta_1+theta_2), theta_1/(1+theta_1+theta_2), theta_2/(1+theta_1+theta_2), and v*sqrt(1+theta_1+theta_2).

Please correct me if I am wrong, 1/(1+theta_1+theta_2), theta_1/(1+theta_1+theta_2), theta_2/(1+theta_1+theta_2) are constrained theta_0, theta_1 and theta_2, respectively, right?

Then, 'v*sqrt(1+theta_1+theta_2)' = variance of the error term?

Also is there any chance that I can set the values of theta_0, theta_1 and theta_2 between 0 and 1?

Thanks a million!! you save my life. I have tried to solve this for DAYSS. :D :D

You understand correctly.

I don't think there is a way to constrain the MA coefficients, but perhaps someone else has a suggestion.