# Nonzero mean variable in a Phillips Curve with infinitely many steady states

I would like to estimate a Phillips Curve with the sum of the parameters on the lag and lead inflation on the RHS is equal to 1.

`pi_t = a*pi(-1) + (1-a)*pi(+1) + b*X + c*d_er + eps_pi`

Where d_er is the log-differenced exchange rate and X is some other variables. The mean of the exchange rate shock is greater than zero due to a greater-than-foreign level of steady state inflation rate. That is:

`d_er = er - er(-1) = pi^foreign - pi^home + eps^er`

Let’s say that I make a shortcut and use this process instead:

`d_er = 0.01 + eps^er`

since I think that this is the exchange rate depreciation rule that does not allow any arbitrage in the model economy. However, plugging er_t into the Phillips Curve rules out any steady state. I know that this is not a purely Dynare related question, but thanks in advance for any help.

I am not sure I understand what is going on. Your equations seems to not be correct. If the foreign inflation rate is bigger and the real interest is the same in both countries, then the difference must be made up by nominal exchange rate changes (appreciation). It is therefore the change in the exchange rate that should show up.
The way you describe it, your model is not stationary, because inflation would explode. I am not aware of any model like this.

@jpfeifer Thank you for the prompt reply and sorry for the confusion. I made some changes to the equations. I mean to imply that it is the change in the exchange rate that has a mean different from zero. And yes, I assume that the real interest rate is the same and fixed in both countries.

Hence, the change in the exchange rate shows up, but still, there is a problem with the steady state in the economy because the inflation rate eventually goes to infinity.

But that cannot be the case. Are you sure your Phillips Curve is correct?

@jpfeifer The Phillips Curve is partly ad-hoc. I assume an indexing mechanism as in Christiano, Eichenbaum, Evans. The ad-hoc part is that I restrict the coefficients to sum up to 1.

Either

• I should not be making that restriction.
• I can make this restriction but cannot have a nonzero mean variable with a nonzero coefficient in the PC
• This specification is still fine but I do not know how to handle this problem of inflation rate diverging to plus or minus infinity

Thanks.