Instrumental variable and GLM

[Deboshmita Brahma]

I am currently using a non-linear model where one variable, X, is endogenous represented by (1)

Y= f(X, P, Q, R, S)----(1)

where P, Q, R and S are other covariates determining Y in (1)
So I wish to use an instrument Z, which will account for endogeneity between X and Y.
I have made a linear relationship between X and Z, as given below:

X= a+ b*Z + c* P + d*Q + e -----(2)
Here, I didn't account for R and S in equation (2) because they are not related to X.

My queries are as follows:
i. If I don't include R and S in (2), will that create any problem related to identification in the model above (equations 1 & 2 together)?
ii. Please also suggest a method of calculating over-identification manually in the model above.

[Fei Wang]

Deboshmita, the first-stage regression (Eq. (2)) needs to control for ALL covariates of Eq. (1). Further information would be helpful for suggestion on models and methods: e.g., what are the nature of Y and X, continuous or discrete?

[Deboshmita Brahma]

Y is a continuous variable with a many 0 observation. It is over-dispersed so I cannot use Tobit model. X is a categorical variable,which is household level information. Z, P and Q are also household level information, such as religion, caste, etc. The reason why I excluded R and S is because they are individual level characteristics.