# SOE Risk Premium and Unit Root

Hi.
I’m trying to replicate this paper below (Kam et al), that is a version of Justiniano and Preston (2010) and similar to Monacelli (2005):

I have some questions:

1. I have an unit root and I suspect it comes from the uip equation, but I couldn’t get rid of that. Any suggestions? (see code)

2. In Monacelli (2005) and in Gali and Monacelli (2005) the UIP equation can be obtained throught other equations in the model like Euler for local and world economies and International risk sharing condition. Can we obtain the International risk sharing equation from the other equation in Kam et al and Justiniano and Preston? I think it is possible since that equation does not appear in Kam et al and Justiniano and Preston.

3. I think in the UIP and International Risk Sharing as equilibrium equations of the model. So the fact that they are not present as equilibrium equations simultaneously is a consequence of some type of Walras law or that they are versions of the same thing?

KAM.mod (4.7 KB)

I am not sure I fully understand. As outlined in the paper, the UIP/real interest parity condition is obtained from the real interest rate definition (32) and the risk sharing condition. So it’s implied by other conditions.

I am not particularly bothered by the unit root, but I find it strange that the model does not run with the equations given in the paper. One source of the problem can be that complete markets are assumed. That implies that equation (15) holds at all times, not just in expectations. But then they use a version in expectations (16) instead.

Thanks for your response, professor Pfeifer.

About my point 2, I meant that in some papers using this structure, like Monacelli (2005), the International Risk sharing is used as one of the equilibrium equations instead the UIP condition. He argued that UIP condition will be redundant since it can be obtained combining Euler for local and world economies and International risk sharing. The same happens in Kam et al: using Euler for both local and world economies (equation 35) taking first differences and combining with International risk sharing (equation 14), we can get the UIP condition (using also that y*=c*). The way you describe (using (14),(16), (32)) also works.

So, my question is: is it equivalent to use the UIP instead International Risk Sharing? If it is, I suppose the reciprocal process could be done, i.e., starting with the UIP one can obtain the International Risk Sharing (using some other equations). But I couldn’t do this reverse calculation.
In addition, I suspect that it is not equivalent, although I am not sure (at least looking at the theoric model). Looking at the dynare model, if I replace the UIP condition (38) by the International Risk Sharing (equation (14)), the rank condition is satisfied (see program KAM_2.mod). But in this version I don’t have the more direct intuition of the risk premium shock as if I use the UIP condition.

About you comment, I didn’t understand why there is no concern about the unit root.
Also, I don’t understand exactly what should be the implication of complete markets in this context. I understand that a relation like (16) must holds (with expectations). But I am not sure if complete markets have to implies in a weaker version like (15). I believe it is the same case in Gali and Monacelli:

They show that under complete markets the equation above 19 holds (with expectations). And this one implies in (19) that is the same as (16) in Kam.

I am not an expert in these open economy models. But UIP is weaker than complete markets, because complete markets implies that UIP holds, while UIP does not imply complete markets.
If you look at the replication codes for Gali/Monacelli, the UIP condition is not one of the required equations: https://github.com/JohannesPfeifer/DSGE_mod/blob/master/Gali_Monacelli_2005/Gali_Monacelli_2005.mod

I see what you mean.
Just one more question: do you see any obvious variable that causes the unit root or, more probably, it is not the case of an unit root and maybe there is another problem?
Thanks again, professor.

No. But it’s often easier to debug the model when it runs. Because then you see which variables are affected by the unit root.

Ok. Thank you again, professor Pfeifer.