Dealing with indeterminacy by auxiliary equation(s)

Dear all,

I am trying to do Bayesian estimation of DSGE model for small open economy.

I am interested in the estimation of the coefficients of monetary policy rule both for determinacy region and indeterminacy region. The model includes 7 forward looking variables.(Following the paper: ‘A Generalized Approach to Indeterminacy in Linear Rational Expectations Models’ by Francesco Bianchi Giovanni Nicolo for adding auxiliary equations)

As proposed I have added 7 auxiliary equations to the model block:
w_{i,t}=(1/alpha_i)*w_{i,t-1}+v_{i,t}+nu_{i,t} i=1,2…7;

w_{i,t} is auxiliary variable, v_{i,t} is sunspot shock, nu_{i,t} is forecast error.

For estimation of alpha_i unit distribution for the interval (0,2) is chosen which gives the auxiliary equation equal probability of being diverging(explosive) or converging.

1. When I set prior of alpha(s) for the interval (0,2) it says BK condition is not satisfied.What could be the reason?

2.1 If change it to (0,2.0001) it is working but the mode check graph shows that the model is not satisfied for the values less than 1 meaning that added auxiliary block is not being explosive. Does this mean that the data strongly supports determinacy?
2.2 Another problem is that mode check graphs of the parameters of interest still have the region of indeterminacy despite of added auxiliary block.

Any help or advice is highly appreciated. Thank you in advance!

I am not sure I get the problem. For 1., at which point are you evaluating the model? The prior mean? If yes, that may explain it, because with the bigger interval, the prior mean is now bigger than 1.
The other thing may be a problem of the interpretation of the mode_check plots. The plots vary one parameter at a time. If the model is BK stable at the mode, then altering any of the roots will of course lead to BK violations.

Again, the mode_check plots vary one parameter at a time. But if psi_i moves to a region of indeterminacy, one of the alphas would need to simultaneously change to restore the BK conditions in your model. But for the plots, you only vary psi_i.

Thank you professor for the response,
I understood in terms of mode check plots.

But, as mentioned above I am trying to estimate the coefficients for both determinacy and indeterminacy region. As I checked the trace plots, alpha(s) are realizing only on the half of the (0,2) interval.(Not providing explosive roots at all). I am not sure if I am being able to estimate for both regions. What do you advise?

Thank you

Have you tried starting the MCMC in the indeterminacy region?

What I did to start MCMC in the indeterminacy region is as follows:

I revised prior means of psi_pi(to inderminacy region which is close to boundary) and one of the alphas(to the one that recovers BK condition).

In this case the alpha is being stuck in the starting region. ( alpha and psi_pi are being stuck in that regions without jumping)

The issue may be the probability of a move. With 7 auxiliary variables, you can shift 7 roots, but typically, there is only 1 combination that will not be rejected. The probability of a move in the right direction may simply be low.

Could it be better to use only one auxiliary equation to improve the probability? If yes, I have tried that. But let me repeat that and make sure once more.

I have checked once more with one auxiliary equation the result is unchanged. Is it because of the nature of the model or something related to the MCMC? What could be possible roots of the problem?

I have not worked with their strategy, so I cannot answer. Did you try a simple prior simulation for your model? Do random draws from the prior cover both regions?

Thank you professor,

I will try that