Good evening everyone,
I hope you all are safe amid the current situation.
Currently I am working on an SOE NK DSGE model a la Gali 2005 (not-linearized) which is working perfectly fine. I am seeking to add foreign debt holdings to the household and augmented the budget constraint by foreign debt holdings, which the HH has to pay with a debt elastic interest rate a la Schmitt-Grohe; and added the FOC for foreign bonds. I impose the HH to have a SS foreign debt holding; and the household balances this in SS by saving in domestic bonds in equal amount.
Case 1: The timing in the budget constraint is paying (receiving) interest on debt (savings), while the interest rate and amount of bonds have been determined one period earlier, i.e. Domestic: (1+r(-1)) * d(-1)), Foreign: (1+r*(-1)) d*(-1)) * real FX rate. The amount of domestic and foreign debt/savings for the next period is chosen today. In the FOCs for domestic and foreign bonds, the time subscript is then +1 ahead. Now, when using a CPI/domestic taylor rule, the model works fine like that.
Case 2: Once I try to implement an exchange rate peg, i.e. exchange rate growth to 1 and delete the taylor rule, the rank condition fails to hold. Since a couple of days I scanned the forum here and thought about timing issues. Once I adjust the timing in the budget constraint for ONLY the domestic bonds +1 ahead, i.e. Domestic: (1+r)) * d(-1)) and in FOC domestic bond also +1 ahead, while keeping foreign interest rate timing the same, the rank condition is satisfied, and the model runs.
My question: Which timing is correct? Usually, I would say Case 1 is correct and inline with my equations, but the exchange rate peg doesn’t work - maybe I miss something in comprehending the timing here. I am struggling with this for several days and tried many different solutions, and am not quite sure if any of that is correct. I would be grateful for some advice and would appreciate any help/suggestions.