GSoC 2016 #4

Time Varying Parameters

Let's consider the following process:

Here yt is an observed process, xt is an exogenous vector, beta are so called time varying parameters that change with time, as (2) equation states. e and v are white noise terms:

Presented model has a name of Time-Varying-Parameter model, and it was a part of my proposal. As you can see, it is non-switching, but it is used to evaluate a good start parameters for switching model likelihood optimization.

TVP and MS-TVP models occurred to be the easiest and the most pleasant items of my proposal. Due to their simplicity I didn't have any difficulties implementing and debugging them. During MLE their parameters occurred to converge to expected values nicely, as well.

TVP: Implementation and testing

TVP model was implemented in upper-level statespace module (tvp.py file), rather then in the regime_switching. Implementation is a concise extension of MLEModel class. I used Kim and Nelson's (1989) modelling changing conditional variance or uncertainty in the U.S. monetary growth ([1], chapter 3.4) as a functional test and for iPython notebook demonstration.

A special thing about TVP is that its MLE results class (TVPResults) has a plot_coefficient method, which can draw a nice plot of time varying parameters, changing with time:

 

Heteroskedastic disturbances

Adding heteroskedastic disturbances to observation equation (1) allows to make a model regime-switching:

where St is a Markov regime process.

MS-TVP: Implementation and testing

TVP model with heteroskedastic disturbances is implemented in switching_tvp.py file of regime_switching module. It is as concise and elegant, as a non-switching analog. I'm going to implement coefficient plotting soon.

I used Kim's (1993) Time-varying-parameter model with heteroskedastic disturbances for U.S. monetary growth uncertainty to perform functional testing. One nice thing about MS-TVP is that it finds a near-correct likelihood maximum from a non-switching start. As you can see in tests.test_switching_tvp.TestKim1993_MLEFitNonswitchingFirst class, I use 0.05% relative tolerance.

What's next?

The remaining part of the summer will be about improving and polishing existing models. Now I am working on adding heteroskedastic disturbances to transition equation (2). As I noted above, I have to add coefficient plotting for a switching model. Other goals are making a MS-TVP notebook demonstration and overall improvement of MS-AR model.

Literature

[1] - "State-space Models With Regime Switching" by Chang-Jin Kim and Charles R. Nelson.