replicating a multi-stage 2sls model with interacted endogenous terms by hand

[France Maliwat]

Dear all, a pleasant day.

I am attempting to replicate a multi-stage 2sls model manually. The endogenous variables have interaction terms. I run the command in Stata using:

reg3 (S Y YxB1 YxB2 C CxB1 CxB2) (Y Z ZxB1 ZxB2 $X) (YxB1 Z ZxB1 ZxB2 $X) (YxB2 Z ZxB1 ZxB2 $X) (C Y YxB1 YxB2 $X) (CxB1 Y YxB1 YxB2 $X) (e2: CxB2 Y YxB1 YxB2 $X), 2sls

where B1 and B2 are binary interaction variables. $X is just a vector of control variables.

(From my readings here, I understand that I should run separate first-stage regressions for the endogenous variable and the same variable times the interactions, using the instrumental variable times the interactions.)

I am trying to replicate this by doing:

reg Y Z ZxB1 ZxB2 $X
predict Y_hat

reg YxB1 Z ZxB1 ZxB2 $X
predict YxB1_hat

reg YxB2 Z ZxB1 ZxB2 $X
predict YxB2_hat

reg C Y_hat YxB1_hat YxB2_hat $X
predict C_hat

reg CxB1 Y_hat YxB1_hat YxB2_hat $X
predict CxB1_hat

reg CxB2 Y_hat YxB1_hat YxB2_hat $X
predict CxB2_hat

reg S Y_hat YxB1_hat YxB2_hat C_hat CxB1_hat CxB2_hat

I am getting the same coefficient estimates in both methods until the last stage where I regress S on the all of the hatted variables.

Would anyone know why I am not getting the same coefficients for the last-stage regression?

Curiously, when I do a variation of this without interactions, all of the coefficient estimates in each stage are the same.

Thank you very kindly.