Unobserved Component model with Stochastic volatility

[ac_1]

I would like to set-up a benchmark Unobserved Component model with Stochastic volatility (UC-SV), and forecast from it (in a Bayesian setting), but not certain how to do it? Please help.

This is as in Stock and Watson (2007) "Why has US Inflation Become Harder to forecast" JMCB,39, pp3-33; but recently used in Bauwens Koop Korobolis and Rombouts (2011) "The Contribution of Structural Break Models to Forecasting Macroeconomic Time series", Working Paper, University of Strathclyde, pp1-30.

A time-varying trend (no AR dynamics) of the form:

y(t)=μ(t)+σ(ϵ,t) ε(t)

log(σ(ϵ,t))= log(σ(ϵ,t-1))+v(t)


μ(t)=μ(t-1)+σ(γ,t) γ(t)

log(σ(γ,t))= log(σ(γ,t-1))+w(t)


where

(ε(t),γ(t)) ~N(0,I(2))

u(t) ~N(0,γ(1))

v(t) ~N(0,γ(2))

Set γ(1)= γ(2)=0.2


Using the following Priors to forecast

μ(0)~N(m)(0,4)

log(σ(ϵ,0))~N(0,1)

log(σ(γ,t))~N(0,1)

inverse(B(0))~Gamma(1,0.1)

inverse(γ)~Gamma(1,0.1)


This would presumably with the RAT's DLM command ?

[TomDoan]

The simulation of that process has three parts, only one of which can be done with DLM.

1. Simulate mu given y and the sigma processes, using DLM with TYPE=CSIM.
2. Simulate sigma(epsilon) treating y(t)-mu(t) as observed, using the techniques from the Jacquier, Polson, Rossi replication http://www.estima.com/forum/viewtopic.php?f=8&t=628. The rejection method is the more efficient of the two.
3. Simulate sigma(gamma) treating mu(t)-mu(t-1) as observed, as in step 2.

DLM can be used for a single layer stochastic volatity model using well-known approximation techniques (converting the observable to log squares), but can't do anything to handle the SV part once you add the time-varying mean. I'm surprised that they didn't have a technical appendix on that.

[ac_1]

The simulation of that process has three parts, only one of which can be done with DLM.

1. Simulate mu given y and the sigma processes, using DLM with TYPE=CSIM.
2. Simulate sigma(epsilon) treating y(t)-mu(t) as observed, using the techniques from the Jacquier, Polson, Rossi replication http://www.estima.com/forum/viewtopic.php?f=8&t=628. The rejection method is the more efficient of the two.
3. Simulate sigma(gamma) treating mu(t)-mu(t-1) as observed, as in step 2.

DLM can be used for a single layer stochastic volatity model using well-known approximation techniques (converting the observable to log squares), but can't do anything to handle the SV part once you add the time-varying mean. I'm surprised that they didn't have a technical appendix on that.

Okay, this is somewhat more involved than I initially thought, especially for a benchmark model! I need to read the Jacquier, Polson, Rossi paper and RATs program.