Introductory Econometrics
Lecture 2: Introduction to linear regression model
Suggested Solution
by Hieu Nguyen
Fall 2024
1.
Derive the ‘summation formula’ for the OLS estimator ˆβ1 in the linear regression
model:
yi = β0 + β1xi + εi.
2.
Suppose you are a university director considering canceling the entrance exams
and searching for new criteria for selecting good students for your school.
Assume you consider using the grades from high school (in the US called the
American College Testing [ACT] scores, 1–36).
To decide, some data about current students seem to be helpful. Let us say
you have a sample of 15 students. The following table contains the ACT scores
and the GPA (Grade Point Average, 0–4) for these 15 students. Grade Point
Average represents the student’s performance at the university; it is based on a
four-point scale and has been rounded to one digit after the decimal.
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Student GPA ACT
1 2.8 21
2 3.4 24
3 3.0 26
4 3.5 27
5 3.6 29
6 3.0 25
7 2.7 25
8 3.7 30
9 3.2 23
10 3.0 28
11 3.5 30
12 2.5 20
13 3.8 32
14 2.6 26
15 2.7 23
(a)
Estimate the relationship between GPA and ACT using OLS; that is, obtain
the intercept and slope estimates in the equation: GPA = β0 + β1ACT + ε. Use
Excel for the computation:
1. Compute the intercept and slope estimates using the summation formulas
for ˆβ0 and ˆβ1.
2. Compute the intercept and slope estimates using the matrix formula for
ˆβ.
(b)
Comment on the direction of the relationship. Does the intercept have a meaningful
interpretation here? Explain. How much higher is the GPA predicted if
the ACT scores increase by 5 points?
(c)
Find and list the fitted values and the residuals of the model.
(d)
What is the predicted value of GPA when ACT = 20?
3. BONUS EXERCISE (very similar to 2.) to take home
and master OLS in Excel:
This exercise illustrates the crucial distinction between the stochastic error term
and the residual. Usually, we can never observe the error term, but we can get
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around this difficulty if we assume values for the true parameters. Calculate
values of the error term and the residual for each of the following six observations
given that the true β0 equals 0, the true β1 equals 1.5.
yi xi
1 2 1
2 6 4
3 3 2
4 8 5
5 5 3
6 4 4
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