Residualised measure of progress/growth

[Efan Fisher]

I am investigating the impact of Latino students' English skills upon entry to university on their attainment upon completion.

Students sit standardised university entry and exit exams, which both include an English, maths and Spanish language component.

I have a dataset of 130,000 students across 383 universities.

I am interested in the following model:

total_exit_score = \beta_0 + \beta_1 entry_english + \beta_2 entry_maths + \beta_2 entry_spanish_language + \beta_i X + \mu

• I have excluded the exit English component from total_exit_score because I am interested in English's impact on the accumulation of other skills, rather than investigating how effective university is at increasing English proficiency.
• All test scores are normalised/standardised
• X is a set of socioeconomic controls (age, gender, household income, parental education etc)
• I also include university fixed effects

I am also interested in considering the impact of students' progress in different components on total_exit_score, but including a measure of progress in English, maths and Spanish introduces significant multicollinearity (entry scores and progress are nearly inversely proportionally correlated).

I am wondering about the possibility of using a residualised measure of progress:

1. Regressing progress in each test score component on the entry score for that component
2. Using the residuals from this regression as a measure of progress that isn't explained by the entry score.

Would this method be valid? If not, how else could I overcome this issue?

Any insight would be greatly appreciated!

[Efan Fisher]

UPDATE

Upon further reading, I now understand the issue with this approach (this useful paper gives the rationale).

Any further comments on possible alternative strategies would be appreciated. Many thanks!