!dtrends and deterministic parameters concentrated out of the likelihood function

Topics: Kalman Filtering
May 27, 2014 at 10:57 PM
Hi folks, I have been doing some research to better understand the 'outOfLik' option for estimating the model using the 'profile' likelihood instead of the 'normal' likelihood.

The sources that I read focused on the disturbance variance parameter and not on the trend/mean parameter. I understand that, in the case of the variance, concentrating it out of the likelihood function means solving analytically for its optimum and plugging this expression back into the likelihood formula. This reduces the dimension of the search involved in the numerical optimization procedure.

What does IRIS do when it concentrates out deterministic trend parameters?

Thanks!
Coordinator
May 27, 2014 at 11:59 PM
If you happen to have the book by Harvey (1991) Forecasting, structural time series models, and the Kalman filter, first read section 3.4.4. i.e. the Rosenberg algorithm. Then turn back to section 3.4.2 i.e. the GLS estimation of params in measurement equations. The latter is exactly the problem of estimating deterministic trends -- simply substitute a vector of ones or a time trend vector for x(t) in equations (3.4.16). You could in theory use the algorithm described in that section 3.4.2 but it's rather clumsy and not very convenient. You can though rewrite the problem into one of fixed elements in the initial state vector, and hence use the algo described in the other section 3.4.4. which is much more elegant (but totally equivalent to the former one). Simply observe that you simply need to create an auxiliary state variable with a transition equation a(t) = a(t-1) (no shocks) whose init cond is fixed but unknown, and then use this auxiliary state as a parameter in the measurement equation.

Let me know if you get stuck on something!

Jaromir
Marked as answer by jaromirbenes on 5/27/2014 at 5:12 PM
May 28, 2014 at 2:28 PM
I do have the book and will check those sections. I will let you know if I get stuck. Thanks!

Hector