Estimated std of shock covers very wide range

Dear Johannes,

Sorry, Johannes, this question is long…Hope you could have a look at your convenience.

I set up and estimate an indeterminate model with specified trend with permanent Tech, preference, sunspot and measurement error to output and investment, totally 5 shocks.
using growth rates of y c I h , totally 4 data. 5 shocks >4 observable variables to rule out stochastic singular.

I use mode_compute=6 several times to find the mode, until the log posterior does not change(but hessian matrix is still negative definite at mode if I run mode_compute=9 or 4 here). Then I run MCMC 4 MILLION draws and drop the first half as the burn-in.

I also use endogenous_prior command.

Results show:

prior mean post. mean 90% HPD interval prior pstdev

miu 1.400 1.4489 1.4479 1.4499 beta 0.0200
rhoz 0.500 0.9975 0.9958 0.9992 beta 0.2000
rhod 0.500 0.0033 0.0004 0.0061 beta 0.2000

standard deviation of shocks
prior mean post. mean 90% HPD interval prior pstdev

sunspot 0.001 0.0042 0.0039 0.0045 invg Inf
e_z 0.001 0.0632 0.0225 0.1114 invg Inf
e_d 0.001 0.0068 0.0064 0.0071 invg Inf
i_Me 0.004 0.0079 0.0079 0.0079 unif 0.0023
y_Me 0.002 0.0034 0.0034 0.0034 unif 0.0010[/quote]

rhoz is persistence of preference shock, while rhod is persistence of permanent technology shock. Measurement errors hit upper bound otherwise the variance decomposition would be too large.

Please find attached posterior plots.

This result has **very good model dynamics **compared to growth rates data(including std, autocorrelation coefficient , correlation with output growth rates) . However, the 90% HPD interval for e_z preference shock is TOO WIDE so that the posterior plot is almost a flat line, and std of e_z is very large compared with other shocks. I have never seen such problem before, do you have any advice on why this happens?

Many thanks in advance.

growthtrendredeem_PriorsAndPosteriors1.pdf (61.5 KB)

The concentrated prior makes this hard to judge in the plot. Please look at the convergence diagnostics and trace_plots to see whether the MCMC correctly sample the posterior. My guess is no.

Thank you Johannes.

Convergence seems OK since I run several Million draws. I find that trace plot of this shock(both std and persistent) are rather bad, as attached figure shows, in the middle it jumps, while the other estimated parameters trace plot are all flat. So it means that the distribution of this shock is not identified ? However, no matter what mode-finders I use, how many MCMC draw I run (have already tried up to 5 Million), I finally always get to the same mode and same mean.

Could you please give me any advice to deal with this problem?

Many thanks,
TracePlot_SE_e_z.pdf (116 KB)

No. If the persistence and standard deviation of a shock jump at the same time it means your posterior is bimodal. This can easily happen in short series because for AR1-processes the unconditional variance is given by

and it is often hard to disentangle the two parameters.

From my perspective, the trace plot is nothing to worry about. However, you should be careful in that the mean may not be a good representation of the posterior, while the median or mode may be.