# Estimate a Monetary Policy Reaction Function with Generalized Method of Moments

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

iₜ=rₜ+πₜ₋₁+λ(π-π)+μ(y-y*)

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:

iₜ=iₜ₋₁+πₜ₋₁+λ(π-π)+μ(y-y*)

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

iₜ=β₀+β₁·iₜ₋₁+β₂·πₜ₋₁+β₄·yₜ₋₁+β₅·xrₜ₋₁+β₆·oilₜ₋₁+β₇·nplₜ₋₁+εₜ

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...

• @Kharion. You may have to edit your notation. Sep 13 '21 at 14:00
• How come you have 4 parameters? I only count 3.
– tdm
Sep 13 '21 at 16:14
• @EB3112 what do you mean? Sep 14 '21 at 12:37
• @tdm you are right, I had first written an equation with the exchange rate, and then I deleted the parameter to keep it short, then I forgot to correct the rest of the text. Sep 14 '21 at 12:38
• Hi @Kharion. Genuinely open to this being an oversight on my part. However, I've not seen the subscript notation you're using beforehand. Sep 14 '21 at 13:53