Hi there, I estimate a model using FE and also FE2SLS and I would like to compare the FE and FE2SLS estimates to test for endogeneity.

let y1 be the dependent variable, y2, y3 endogenous variables, z's exogenous variables with L > M and of course at least two instruments from outside the model.

How would I go about this?

I would presume to do the following: estimate the reduced forms using FE and take the residuals. Then augment the original equation that I seek to estimate and test for their joint significance:

xtreg y2 z1 z2 ... zL, fe predict v2hat, resid xtreg y3 z1 z2 ... zL, fe predict v3hat, resid xtreg y1 y2 v2hat v3hat z1 z2 ... zM, fe vce(cluster id) test v2hat v3hat
The problem I encounter is that I do not know what resid is giving me and I think I need them to be the fixed effects residuals: (\tilde{y} - \tilde{x}\hat{\beta}).

I have a copy of Wooldride's Econometric Analysis of Cross Section and Panel Data (second edition). His example 11.1 compares FE and FEIV estimates but I am unable to obtain the FE residuals from the reduced form equation. When I use predict v2hat, resid after estimation of the reduced form and include these in the original equation I do not get the same t-statistic as him on vhat (Chapter 11, page 358).

Thank you!