$1)$ Multicollinearity

This question is based on the Boston mortgage study. If you have the variables P$/$I ratio $($debt payments divided by income$)$ and income included in your specification, what other variable would cause there to be perfect multicollinearity if added to your model?

$2)$ DID

Aydan, Brita, Carlos, and Dinella are in a class where they are learning to type. The students$$ typing speed $($in words per minute, WPM$)$ is measured after the first $3$ weeks of the class. Aydan and Brita are given access to a new artificial$-$intelligence $($AI$)-$powered software that is supposed to help increase typing speed and they use that program in class after the $4$th week of class while Carlos and Dinella keep using the original typing program. Use the table to generate a $4-$number summary to calculate the DID effect of the AI$-$powered software.

$($Table didn't format$)$ Lower scores$=$before higher$=$after$)$

Before

After

Aydan

$65$

$85$

Brita

$73$

$92$

Carlos

$54$

$73$

Dinella

$80$

$91$

$3)$ Fixed Effects

Suppose that you are trying to study the effects of a tutoring program in students of $3$ different sections of a class. A randomly selected half of the students in each course were offered free tutoring and you would like to see if those students get higher test scores. The teachers of the $3$ sections coordinated with each other and gave the exact same tests $($midterms $1-3$ and a final exam$).$

You have data on each student$$s tutoring status and scores on each of the $4$ exams.

a$)$ Each section is taught by a different instructor, so you think that you should include fixed effects in the regression to help control for those differences. Explain why you can or cannot use entity fixed effects for the different sections.

b$)$ If you were going to include time fixed effects, how many time periods would there be$?$ Why?

c$)$ Write a specification with the proper subscripts where you try to predict the score each student earns on each test using the information that you have.