Support "Background" Covariates for Gelbach Decomposition / `.decompose()` method

[s3alfisc]

Good first issue

I've labeled this a "good first issue" because it shouldn't be too difficult to implement, still the changes are somewhat intricate and it is not necessarily a "quick win".

Context

Currently, we only support regression decompositions without any background covariates in the decompose class.

Hence, the short regression is always defined as depvar ~ param and it is not possible to always additionally control for any other variables.

In math, $X1$ always consists of an intercept and one additional covariate.

We would like to allow users to add background variables always contained in $X1$.

To do so, we have to add an argument to Feols.decompose(), which we might call xfml. The decompose API might then look like this:

import pyfixest as pf 

fit = pf.feols("Y ~ X1 + X2 + C(f1)", data = pf.get_data())
fit.decompose(param = "X1", xfml = "X2")

Now, the "short" regression is defined as Y ~ X1 + X2 and the "long" regression becomes Y ~ X1 + X2 + C(f1).

Without providing a xfml argument, the short regression is

fit.decompose(param = "X1")

Y ~ X1 and the long regression as above.

To Do

• We need to add a xfml argument to the decompose() method of Feols and add appropriate checks and parsing.
• We need to update the Decomposition class so that we pass X1 including the covariates in xfml here:

  pyfixest/pyfixest/estimation/decomposition.py

  Line 107 in 1ceac58

  results = self.compute_gelbach(

• We need to add unit test against Stata, might have to as @leostimpfle to help us out once again with running some Stata code =)