I have a question regarding the NKPC. I would like to know if it is possible to derive the NKPC from a sticky prices model, without making assumptions regarding the production function firms face. I am trying to estimate an open economy NKPC. As far as I am concerned, much of the extensions of the NKPC for the open economy involve substituting the commonly used Cobb-Douglas function for another one which explicitly incorporates imported goods. Some examples of this approach are Galí and López-Salido, Balakrishnan and López-Salido, and Rumler. However, almost all of the the derivations of the NKPC assume that firms face a Cobb-Douglass production function, as you can see in Gali and Gertler, Galí and López-Salido, Balakrishnan and López-Salido, Romer, and Galí. Wouldn´t there be a contradicction if (this is the approach followed by Balakrishnan and López-Salido), one assumes that firms face a Cobb-Douglas function but at the aggregate level one assumes a different production function, such as a CES production function that incorporates imported inputs? I am having trouble with reconciling the assumption of different production functions. Because of this apparent contradiction, I thought that, if it is posibble to derive the NKPC from a sticky prices model without assuming a Cobb-Douglass production function (or any other production function for that matter), I could easily bypass the problem. Hope I made my case clear. Thank you in advance!
P.S. I know that Rumler derives a NKPC assuming just a CES production function. However, I havent been able to understand exactly how he estimated the coefficients of his model nor what variables he actually used. For an undergradute such as me, estimating such a model seems to be out of reach. However, if someone is willing to give me a hand, I would really appreciate it. Thank you!