IV with multiple endogenous variables

[leostimpfle]

What is an instrumental variable formula with multiple endogenous variables expected to do? I don't think this documented anywhere. The fixest documentation explicitly mentions the possibility to include multiple endogenous variables but doesn't appear to say much else:

To include several endogenous regressors, just use "+", like in: fml = y ~ x1 | x_endo1 + x_end2 ~ x_inst1 + x_inst2

In Pyfixest the formula Y ~ 1 | X1 + X2 ~ Z1 + Z2 is currently parsed as Y~X1+X2|X1~Z1+Z2. Is this correct?

from pyfixest.estimation.FormulaParser import FixestFormulaParser
fml = "Y ~ 1 | X1 + X2 ~ Z1 + Z2"
FixestFormulaParser(fml).FixestFormulaDict["0"][0].fml

---

Estimates are close but not identical between pyfixest and fixest:

pyfixest

data = pf.get_data(N=1_000, seed=0, model="Feols")
fml = "Y ~ 1 | X1 + X2 ~ Z1 + Z2"
py = pf.feols(fml=fml, data=data).tidy()

             Estimate  Std. Error    t value  Pr(>|t|)      2.5%     97.5%
Coefficient                                                               
Intercept    1.304583    0.141534   9.217428  0.000000  1.026843  1.582323
X1           0.358530    0.124630   2.876750  0.004104  0.113962  0.603098
X2          -0.305560    0.022170 -13.782747  0.000000 -0.349065 -0.262055

fixest

import rpy2.robjects as ro
from rpy2.robjects import pandas2ri
from rpy2.robjects.packages import importr

pandas2ri.activate()
fixest = importr("fixest")
broom = importr("broom")


r = fixest.feols(
    ro.Formula(fml),
    data=data,
    se="hetero",
)
pd.DataFrame(broom.tidy_fixest(r, conf_int=ro.BoolVector([True])))

             0         1         2         3         4         5         6
0  (Intercept)  1.304262  0.134144  9.722851       0.0  1.041025    1.5675
1       fit_X1  0.358995  0.123502  2.906808  0.003732  0.116642  0.601349
2       fit_X2 -0.302835    0.0232 -13.05321       0.0 -0.348362 -0.257309

[s3alfisc]

In Pyfixest the formula Y ~ 1 | X1 + X2 ~ Z1 + Z2 is currently parsed as YX1+X2|X1Z1+Z2. Is this correct?

No, this is not correct and should throw an error. My expectation was that this was happening? This is exactly the issue I thought we needed to tackle in this issue - adjust the parser to handle IV with more than one endog variables correctly. #791

The problem with this parsing Y~X1+X2|X1~Z1+Z2 is that X2 is neglected as endogenous variable and assumed to be exogenous - we instrument only X1 with Z1 + Z2.

In principle, it should be possible to fit a model with two endogenous variables and two instruments with the standard 2SLS estimator inv(Z'X)Z'Y, with a more general solution for the overidentified case (which is implemented for the IV fit). The fitting process should "just work", provided we pass the correct endogenous and exogenous matrices.