trend plus stationary cycle model

[joanm]

Hi there!

This time the question is related to time-varying parameter specification. I included some restrictions to the last model above and things are working much better. Nevertheless, I realised that the assumption of “beta” being constant is not plausible. In fact, the catalan labour productivity was boosting in the 60s, whereas in the 70s and the 80s labour productivity increased but at a slower pace. And from then on, labour productivity behaves countercyclically.

I would like “beta” to be time-varying (in equation 7). So, I thought in two possible strategies:

The first one does not seem plausible to me. It consists on interacting two state variables and including a state variable in the F matrix (the loadings from the W’s to the X’s). I guess that this is not possible by construction.
(7) Employment cycle (t) = delta* Employment cycle (t-1)+beta(t)*GDP cycle(t) +e_ec (t)
(8) beta(t)=beta(t-1)+e_beta(t)

The second one consists on changing the structure of my model and estimating sth like:
Observation equations
(1) GDP (t)=GDPtrend(t)+GDPcycle(t)
(2) ∆Employment (t)=delta*∆Employment(t-1)+beta(t)* ∆GDP(t)+e_e(t)

State equations
(3) GDPtrend(t)=GDPtrend(t-1)+alfa(t)+e_yp(t)
(4) alfa(t)= alfa(t-1)+e_alfa(t)
(5) GDPcycle(t)=ph1*GDPcycle(t-1)+ph2*GDPcycle(t-2)+e_yc(t)
(6) beta(t)=beta(t-1)+e_beta(t)

But, in this case I am dubious about the gains of adding equation (2) and (6) to the Clark model, as there is no modeled relation between GDP and Employment (as the employment equation is specified in first differences). I would appreciate your opinion and some ideas to introduce the productivity dynamics in the Clark model.

[TomDoan]

If beta is time-varying, then you have a non-linear dynamic model (you have states entering multiplicatively). That requires using the extended Kalman filter (EKF) for estimation. An example of doing that in RATS is the Ozbek-Ozlale replication, but that requires a lot of careful attention to get the model to work even in their setting, which is quite a bit simpler than yours. (The published model allows way too much freedom for the time-variation).