Interpretation of Multiple Regressions Worksheet

Week 6This week, we will be looking at the interpretation of multiple regressions (interpretation weighs
more than the coding in R for this discussion). A multiple regression has more than one
independent variable. That is, two or more X variables are used to explain Y at the same time. A
simple regression, on the other hand, has only one X variable. The data we are going to use is
named “gpa”. It is an R dataset, so, you can open it directly in RStudio using the “open’ icon.
We will focus on three variables: colGPA (college GPA), hsGPA (high school GPA), and ACT.
When you run a multiple regression, say, 𝑌 = 𝛽̂0 + 𝛽̂1 𝑋1 + 𝛽̂2 𝑋2 + 𝑢̂, you try to explain or
predict Y using the two X’s at the same time. More specifically, 𝛽̂1 is the effect of 𝑋1 on Y
holding 𝑋2 constant, and 𝛽̂2 is the effect of 𝑋2 on Y holding 𝑋1 constant (holding all else constant
or equal is also called ceteris paribus). The residual 𝑢̂ captures the part of Y that is not explained
by 𝑋1or 𝑋2.
Do the following:
1. Regress colGPA on hsGPA and ACT, and present the regression output table from
RStudio. This is one regression, not two separate regressions.
2. The coefficient on hsGPA is 0.453456 (round it to 0.45). Interpret this number (explain
what it means in words in this regression context). Hint: your interpretation must be a
ceteris paribus interpretation.
3. Run hsGPA on ACT, and then predict the residuals from this regression and call the
residual column e (we practiced this before).
4. Now, run colGPA on e. Present the output table from this regression. The coefficient on e
should be equal to 0.45346, the same as the one from the multiple regression you did in
part 1.
5. This is no coincidence. Discuss with the class why and how these two simple regressions
combined can produce the same results as from the multiple regression. Provide photos or
screenshots of your work. Hint: Explain what ceteris paribus really mean and what
residuals e captures. Do some research if necessary. You are asked to start participating
by end of day Friday.