 The difference of the risk premium function

Hallo every one, I have confusion about the specification of the risk premium function in small open economy DSGE model as following

On the page 16, the risk premium function, `risk(t)`, is defined as

`risk(t) = exp[ - Psi*( A(t) - barA ) + phi(t) ]`

Then the log-linear form of the UIP is derived on page. 16 as

`R(t) - R*(t) = E_(t) S(t+1) - S(t) - Psi*a(t) + phi(t)`

So based on the above UIP, we see that a positive shock to risk premium would rises domestic interest rate and real exchange rate depreciation.

1. Second paper, Justiniano and Preston (2010): https://onlinelibrary.wiley.com/doi/abs/10.1002/jae.1153>

On the page of 96, the risk premium function, `risk(t)`, is defined as

`risk(t) = exp[ - Psi*( A(t) + phi(t) ) ]`

Then the log-linear form of the UIP is derived on page. 101 as

`[ i(t) - E_(t)pi(t+1) ] - [ i*(t) - E_(t) pi*(t+1) ] =E_(t)q(t+1) - q(t) - Psi* a(t) - phi(t)`

So based on the above UIP, we see that a positive shock to risk premium would decline domestic interest rate and real exchange rate appreciation.

In short, so we can see the big difference of specification btw these two paper in that based on Adolson et al.(2007): a positive shock to risk premium would rises domestic interest rate and real exchange rate depreciation.

but according to Justiniano and Preston (2010): a positive shock to risk premium would decline domestic interest rate and real exchange rate appreciation.

But can someone give me some explanations for that: why Justiniano and Preston (2010) specify a strictly decreasing function in the risk premium shock ? Meanwhile, Adolfson (2007) specify a strictly increasing function in the risk premium shock?

I am personally satisfy with the specification of Adolfson (2007), for example, a positive risk premium will increase domestic interest rate (see the Adolfson (2007) specification of the UIP above). Intuitively, home country is now net borrower, so they must pay more interest rate to foreigner. This meaning is identical to the specification of an influential study of Schmitt-Grohé and Uribe (2003) at https://www.sciencedirect.com/science/article/pii/S0022199602000569

Thus, These two different specification of Adolfson (2007) and Justiniano and Preston (2010) result the different responsiveness of domestic interest rate and exchange rate

Thank you a lot

I think you are confusing something here. Where a shock is a positive shock to the risk premium depends on whether the shock causes the risk premium to increase. Given the different sign on `phi(t)`, a “positive shock” to the risk premium corresponds to an increase in phi in the one specification and a decrease in phi in the other one.

Dear Prof. Pfeifer

Thank you a lot. I now understand how to define a positive shock? These two specifications above havd no difference at all in economiccal meaning