>> unrelated suggestion to also test if the pooling is actually adequate.
>
> Running the panel diagnostics informed me that
>
> Residual variance: 1.35107e+06/(822 - 60) = 1773.06
> Joint significance of differing group means:
> F(46, 762) = -1.99828e-14 with p-value 1.79769e+308

What version of gretl is this? (That p-value means "NA", as it obviously
has to be if the F-statistic is numerically negative, but I thought we'd
purged all cases of printing NA as if it were a number.)

> I can't properly run IV2SLS models in -gretl- because I only have one IV
> and the main model has 96 predictors, so it responds that I don't have
> enough instruments.

That sounds like a non-sequitur: for one endogeous variable you need one
additional instrument, regardless of how many exogenous regressors are
included in the model. Have you looked up the syntax of gretl's "tsls"?
It's

tsls <dependent-var> <regressors> ; <instruments>

where you repeat any exogenous regressors as (their own) instruments in
the second list, behind the semicolon. For example

tsls y const x1 x2 n ; const x1 x2 z

where 'n' represents an endogenous regressor and 'z' the associated
instrument

> So, what to do in order to test the endogeneity of my regressor in a
> 'standard' model?

Well, what sort of test do you have in mind, other than running an IV
model and comparing the results with OLS via a Hausman test? No
residual-based test is available since the residuals are by construction
orthogonal to all the regressors. (You get a Hausman test automatically
when you use the "tsls" command in gretl.