First Step: I've collected inflation and interest rate data for 25 countries in a monthly frequency for the last 20 years. I run a simple linear of inflation (right side, X) on interest rate (left side, y) for each country.
At the end of this step, I have 25 coefficients for inflation - one for each country - and, of course, the standard errors of these measures.
Second Step: I want to run a regression of the coefficients taken of the first step on specific data sets I have for the countries (all including 25 observations, 1 for each country). As far as I know, this second step should be taken with precaution, because I'll be using estimated variables as dependent variables (Y).
What is the specific treatment and care that I should have on the second step?
I would check for autocorrelation and trends in the first step. It it is highly unlikely that the interest rates were a random sample - so your regression will probably be off . I am no expert in time series but I guess first you will beed to separate the random component of the inflation and only analyse that. I would then do the analysis in one shot with a continuous and a discrete IV (interest rate and country)
To be honest, I don't think I perfectly get your point. What do you mean by my regression will be off on the first step? And then, on the second step I also don't get exactly the treatment you explained. Can you please detail it to me a bit more?
Actually, I had more in mind a treatment like WLS or GLS for my second regression. Don't you think it is a good choice?