I want to estimate a Monetaryu Policy Reaction function with GMM.
I have well understood how to estimate a distribution (e.g. the normal distribution, the log-normal distribution, and so on) with GMM. One just sets the condition that the parameter μ is equal to the empirical average, the parameter σ² to the empirical variance, and one sets condition for higher moments, depending on the distribution.
Unfortunately, I could not find so much information on how to use the method for a regression, and more specifically for my case: the estimation of a small open emerging economy, despite the method being widely used.
I assume here a simple reaction function to inflation and output gap, as adding further regressors (exchange rate, oil price, and so on) should not be a problem once I understood how to do.
The standard Taylor rule would be
Where i is the nominal bank rate, r* the estimated natural real rate of interest, π is inflation, π* the inflation target, y the actual gdp growth rate, y* the estimated equilibrium growth rate.
I use a "difference rule" in the sense of Orphanide and Williams (2006): instead of estimating the real rate of interest, I assume that rₜ+πₜ₋₁ is equal to iₜ₋₁. This would reduce the uncertainties connected to the estimation of r. The rule becomes:
Moreover I add further covariates: exchange rate (xr) (if the currency depreciates the cb may want to increase the rate), oil price (oil), and non-performing loans (npl) (if npl increases, the cb may be keen to ease credit). As it is an econometric estimation I add also an intercept.
The resulting econometric model is
A first moment condition is that the error term in average is zero. E[εₜ]=0
Then I could assume that regressor are not correlated with the error term. E[εₜ·yₜ]=0 E[εₜ·πₜ]=0
Now I need at least another condition. I could assume homoskedasticity: E[εₜ²]=σ², but: first I am not sure I want to assume homoskedasticity, because it could not be the case, second I may want further moment conditions to reach overidentification. But how?
The question seems quite trivial, but I could not find any information despite long research...