Out-of-sample predictive checks for Bayesian TVP models

Question

Comparatively new to Bayesian econometrics so apologies if this is a silly question.

I am running a time-varying parameter regression where the parameters are estimated as in Primiceri (2005). My model assumes the time varying coefficients follow a random walk.

Say I have a data sample running from t = 0 - n. I fit the model over the full sample and obtain fitted values for the data (denoted *) as;

y*(t) = a(t) + y(t-1)b(t)

As the state equations are random walks, the optimal prediction of a(t+1) and b(t+1) would be a(t) and b(t) respectively.

This being the case, is the forecast of y(t+1) which uses the parameters as estimated at time t out-of-sample? I.e. if I do:

y*(t+1) = a(t) + y(t)b(t)

Is this equivalent to a one-step, out of sample forecast for y(t+1)? Or, in order to compute out of sample forecasts, do i need to loop the model, updating it with new data at every iteration (e.g. first running it over the sample t=1, then t=1-2, then ... then t=1-n).

Thanks a lot for the clarification!

Primiceri, G.E., 2005. Time varying structural vector autoregressions and monetary policy. The Review of Economic Studies, 72(3), pp.821-852.

Answer 1

You can find the answer in the Garratt et al.(2009)[1], section 4.3.

The author explained why we use a ex-post out-of-sample prediction (which is your first procedure) instead of a real time out-of-sample prediction (which is the loop version).

In brief, the real time one is theoretically ideal but facing many practical issues such as nonstability induced by small sample Bayes estimation. So, in most case we use ex-post one instead.

[1] Garratt A, Koop G, Mise E, et al. Real-time prediction with UK monetary aggregates in the presence of model uncertainty[J]. Journal of Business & Economic Statistics, 2009, 27(4): 480-491.