We want to investigate which variables determine the final grade in a University exam (say Y_2), which can assume integer values between 18 and 32.
We think that Y_2 depends on: 1) Personal variables related to the student: sex, family, income, city, ecc., say X_1, X_2,...X_n which can be both categorical or continuous; 2) variables related to the previous school attended: type, final grade, ecc. say Z_1, Z_2,... Z_m which can be both categorical or continuous; 3) grade of the admission (assessment) test (part of mathematics), say Y_1, which can assume integer values between 5 and 8.
As the hypothesis of linear model (OLS) are not satisfied, we are using two robust models: M-estimation and quantile regession.
We have determined that Y_1 depends on some regressors among X_1, X_2,...X_n and Z_1, Z_2,... Z_m. So our regressors can be considered somehow nested.
We have then analized Y_2 regressed on all the independent variables, excluding Y_1. We have determined that Y_1 and Y_2 have some regressors in common (as the type of secondary school) and some regressors which influence only one of them (i.e. sex influences Y_1 but not Y_2).
We have some questions:
1) When we introduce Y_1 among the regressors of Y_2, which of the X_1, X_2,...X_n and Z_1, Z_2,... Z_m variables should we consider? Only the common ones, the ones which influence Y_2, the ones which influence Y_1 or Y_2, or all the previous variables?
2) The variables which influence only Y_1 should be considered in the Y_2 regession model only as cross-effects (Y_1:X_i)?
3) Can you suggest any other model which can be used to study this kind of data?