Regression model with some regressors depending on other regressors

Question

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?

Answer 1

As the hypothesis of linear model (OLS) are not satisfied...

It is unclear from your description why this would be the case. OLS is actually an estimation method (not a model) used for linear regression models. For a standard multiple linear regression model, it is assumed that the error terms in the model have zero mean, fixed variance, and are uncorrelated. In a Gaussian model it is further assumed that the error terms are normally distributed. Contrary to the remarks you make in the comment section, there is no assumption that the response variable is normally distributed, symmetric, etc. To check the assumptions of a regression model you should be looking at diagnostic plots of the studentised residuals (after fitting the model), not plots of the raw response values.

... So our regressors can be considered somehow nested.

This is unclear from your description. "Nested" variables in a regression occur when one variable is only meaningful conditional on the other taking on some particular value (see e.g., here). The mere fact that there is evidence that one of your regressors is statistically associated with the others does not mean that those regressors are "somehow nested". (The latter simply means there is some multicollinearity in your model.) It appears from your description that $Y_1$ is a grade on a separate test to $Y_2$, in which case these are not nested variables --- i.e., either is perfectly meaningful regardless of the value of the other.

(Incidentally, in mathematical and statistical work, whenever someone says that a thing is "somehow X" or "X, in some sense", that usually means the writer has no clear idea what they mean, and the thing is not in fact X at all. Whenever you find yourself writing something like this, it is a good indicator that you are asserting a result without any clear reasoning. In these cases it is useful to inquire further as to exactly what the "somehow" entails. You can then either make a more specific assertion, or realise that your initial assertion was wrong.)

We have some questions...

It is not possible to tell you how to model your data from the information given. However, I see no issue of nested variables from your description, and the association you are getting between $Y_1$ and the other regressor variables simply means that there is multicollinearity in your model. Unless there is some part of your description that I have misunderstood, I see no reason for any special treatment of $Y_1$ in this analysis.

As a general rule, it is impossible to pre-specify what variables should be included in a regression model just by seeing the names and ranges of those variables. You need to find a model that includes relevant predictors and satisfies model assumptions.