IV support

[s3alfisc]

Required Steps:

• allow three-part formulas
• implement IV estimation & inference. Start with just-identified models.
• update summary() and tidy() methods. For tidy(), simply add index "stage1" and "stage2". Split summary() into two parts. Check out what fixest does.

Nice to have's (for now):

• common IV diagnostics
• AR confidence intervals

In Practice:

• implement IV estimator (Z'X)^(-1) Z'Y.
• always create X and Z. For OLS, set X = Z
• pass everything through, done (at least for models with only one endog variable)
• after this is implemented, allow for over identified models. This requires implementation of the 2SLS estimator + updates to inference procedures

[s3alfisc]

Some design considerations for internal processing:

Assume the following model formula: fml = "Y + Y2~ X1 | csw0(X3, X4) .

Currently, information on model variables is e.g. encoded in self.var_dict in the following way

>>> self.var_dict
{'0': ['Y', 'Y2', 'X1'], 'X3': ['Y', 'Y2', 'X1'], 'X3+X4': ['Y', 'Y2', 'X1']}

and in self.fml_dict as

>>> self.fml_dict
{'0': ['Y~X1', 'Y2~X1'], 'X3': ['Y~X1', 'Y2~X1'], 'X3+X4': ['Y~X1', 'Y2~X1']}

With instruments, the formula might look as fml = "Y + Y2~ X1 | csw0(X3, X4) | X1 ~ Z1". It is likely preferable to use a "clearer" data structure to save the information above. E.g.

{'0': [{"depvar": "Y", x_vars = ["X1"], z_vars = ["Z1"]}, {{"depvar": "Y2", x_vars = ["X1"], z_vars = ["Z1"]}}, "X3":  ...].

[s3alfisc]

Status:

• In dev branch, IV-Estimator is implemented + associated SEs (iid, HC1-3, CRV1).
• some tests against fixest are implemented & pass
• still to be done: error handling + tests for multiple estimations

Also, formula syntax currently deviates from fixest. For models without covariates, fixest syntax is

feols(Y ~ 1 | fe | endogvar ~ instrument)

while pyfixest is

Fixest(df).feols("Y ~ endogvar | fe | endogvar ~ instrument"). 

After tackling all points above, I will merge the PR and implement the 2SLS estimator in a separate PR.

[s3alfisc]

Basic IV support added with version 0.5.

[s3alfisc]

2SLS estimator now implemented in the dev branch.

[s3alfisc]

To do:

• currently, there is a minor deviation to fixest syntax. For models without any covariates, fixest syntax is feols(depvar ~ 1 | fe | endogvar ~ instruments), while pyfixest requires feols(depvar ~ endogvar | fe | endogvar ~ instruments). Align pyfixest syntax with fixest.
• multiple estimation syntax does not work in the first part of the three-part formula, i.e. depvar ~ csw(X1, X2) | fe | endogvar ~ instrument leads to a bug
• add tests for IV with multiple estimation
• add F-test (for first stage and in general)
• some documentation improvements are definitely possible