How to Estimate a TVP Regression by Gibbs Sampling

Guide 2 of 3 in the series Time-Varying Parameters

Based on Chan and Jeliazkov (2009)
[ Journal Version | Working Paper | Code | Full guide on GitHub ]

We estimate the time-varying parameter (TVP) regression of How to Draw Time-Varying Regression Coefficients by Gibbs sampling, on the data of that guide. That guide holds the error variance σ², the variances ω_i² of the random walks and the initial coefficients β₀ at their true values. Here we estimate them along with the coefficients.

A Gibbs sampler draws the unknowns of a model in blocks and repeats. Each block is drawn from its distribution given the data and the current values of the other blocks. How to Estimate a State Space Model by Gibbs Sampling builds one for the local level model. Each iteration of our sampler draws the path of the coefficients as in the first guide, and then the parameters, each from a standard distribution.

The steps

1. Split the Unknowns into Blocks
2. Derive the Conditional Distributions
3. Write the Loop
4. Check the Chain

Figure 1: Top, the 20,000 draws of σ², ω₁² and ω₂² kept after the burn-in, in the order the sampler makes them. Bottom, their histograms, the posterior densities computed on a grid without the sampler (solid lines; see How We Check It) and the true values (dashed lines).

Try it

In MATLAB, from the root of a copy of statespace-toolkit, run the script that prints every number and draws every figure of the guide:

run guides/tvp_gibbs/guide.m

In this series

Previous: How to Draw Time-Varying Regression Coefficients
Next: How to Tell Whether a Coefficient Varies over Time