Filter free estimation

[matyasfarkas]

Implementation of the filter-free DSGE estimation in MacroModelling.jl

[codecov-commenter]

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[thorek1]

@matyasfarkas
Check out the latest version of the filter free file. It works: recovers the shocks and true parameter values and is fast. It breaks down for other setups but let's work from this and generalise step by step.

Suggested next steps (with the setup that works):

• make the filter free part inside the Turing model a function similar to the Kalman one
• generalise that function to work with all solution methods (2nd, 3rd order)
• include Kalman filter in that function

[thorek1]

I continued playing around with it and there is an issue with the gradient size of the shocks relative to the parameters. You need the ones for the shocks big enough so that he gets the shocks right but if they are too big he doesn't move around for the parameters at all. Todos:

• check how that works with the diffstatespacemodels implementation
• check whether we can use abs(x)^n n>2 as a solution or if abs(x) < epsilon then 0 else abs(x)^n

[thorek1]

diffstatespacemodels uses the MvNormal and they also do not recover the latent shocks up to high precision. using other loss functions instead of the MvNormal come down to the same thing and a re a dead end. the std is the key parameter balancing, converge (too small -> no convergence), speed (smaller -> slower), and accuracy (smaller -> more accurate estimates of latent shocks and shock size)

[thorek1]

I wrote an issue for MuseInference to help us get it working agian. In the meantime, I would suggest to try Gibbs(hamiltonian sampler for model parameters, variational inference for latent states). I hope this brings down inference time, as it is now prohibitive.

The approach is suggested as an alternative in the MuseInference paper and easy to implement in Turing: sample(model, Gibbs(HMC(0.01, 5, :s), ADVI(10, 1000, :m)), 1000).

Here is some theory of combining samplers and some more info about variational inference. I think this could serve as a second best with the first best potentially being MuseInference.

[thorek1]

in hindsight, maybe the trick is really to break the correlation structure between the latent states and the parameters because for small Omega that seems to be where he got stuck

[thorek1]

after some investigation we figured that MuseInference works if Julia is single threaded. it errors with more than one thread

[thorek1]

I experimented a bit with Turing and using a Gibbs sampler to separate the sampling of the latent states and the parameters. My experiments showed that this doesn't really help in making the sampling more efficient. While playing around I came up with another idea: instead of integrating out the latent states, minimise the square sum of the shock sizes which have the model fit the data exactly. another issue is the initial value. instead of figuring out the likelihood given the ergodic distribution I would add more shocks before the time series starts and have an initial value = NSSS at t - x plus x shocks which bring the system to the actual initial value . I think this is a neat trick but a minor issue with this technique is that additionally you need to take derivatives wrt shocks.

[thorek1]

another thing I figured out is that any symmetric distribution is approximated by minimal sum of squares. in that sense estimating the degrees of freedom of the student T distribution with a uniform prior was at best confusing for the sampler.

[thorek1]

after some thinking: easiest would be to use UKF. see an implementation here