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Department of Economics
Midterm Take Home Exam
Instruction:
• This is a 6-hour open book exam. You can use the course-related materials you want
to use including course lecture notes, textbooks, and handouts. But, you may NOT use
other external resources including internet for the exam.
• You must take this exam completely alone, and discussing this exam with others are
NOT allowed.
• You must make your answer sheets by yourself and checking your exam answers with
any person are NOT allowed.
• Total points in this exam is 120.
1. (Total 25 points) Consider the following regression results.
Variable Coefficient s.e t-test
Constant 0.203311 0.0976 -
X 0.656040 - 3.35
n = 19 R2 =0.397
It was also found that ESS = 0.0358. Suppose that the model Y = β0+β1X + satisfies
the usual regression assumptions.
(a) (5 points) Fill in the missing numbers (-).
(b) (5 points) Compute V ar\(Y ) and sample correlation coefficient,rX,Y .
(c) (5 points) Construct the 95% confidence interval for the slope of the true regression
line, β1.
(d) (5 points) Test the hypothesis: H0 : β1 = 1 versus H1 : β1 < 1 at the 5%
significance level.
(e) (5 points) I reversed Y and X in the above regression like below.
X = α0 + α1Y + u
Compute ˆα1 and R2.
2. (Total 15 points) Table below shows the final scores Y and the scores in two quizzes X1
and X2, for 3 students.
(a) (5 points) Estimate the coefficients of the following regressions
(2)
Yi = β0 + β1X1i + β2X2i + ui (3)
To see the relationship between X’s, I set up two more regressions:
X1i = α0 + α2X2i + ei, then discuss how we
can interpret the regression coefficients in a multiple regression equation.
(c) (5 points) This time, I fit the model (3) with 15 students data. From the following
data, estimate the regression coefficients using the fact in (b).
Y¯ = 367.693 X¯1 = 402.760 X¯2 = 8.0
X(Yi − Y¯ )2 = 66042.269 X(X1i − X¯1)2 = 84855.096X(X2i − X¯2)
2 = 280.000 X(Yi − Y¯ )(X1i − X¯1) = 74778.346X(Yi − Y¯ )(X2i − X¯2) = 4250.900 X(X1i − X¯1)(X2i − X¯2) = 4796.000
n = 15
3. (Total 25 points) Suppose that a variable Y is determined by X, and the true relationship
between the two variable is known as
Yi = β2Xi + ui
ui ∼ iidN(0, σ2
). So it can be said that when we are given with n number of observations,
the OLS estimator of β2 is
βˆ2 =PnPi=1 XiYini=1 X2i
Answer the following questions below.
(a) (5 points) Demonstrate that βˆ
2 may be decomposed as
βˆ2 = β2 +Xni=1αiui
and calculate the suitable αi.
(b) (5 points) Show that βˆ2 is an unbiased estimator for β2
(c) (5 points) Show that V ar(βˆ2) = Pα2
i V ar(ui) and therefore, V ar(βˆ2) = σ2 PX2i
(d) (5 points) Explain in your own word; how would answers of question (a), (b), and
(c) would be affected if ui had variance σ2i
instead of σ2.
(e) (5 points) Now let’s say that the initial model specification was incorrect; after all,
the true model turned out to be
Xi = γ2Yi + ei
After the researcher realized his mistake, he asserted that the real relationship
could be transformed into
Since he’d already estimated βˆ
2 from the linear model Yi = β2Xi + ui, he argued
that βˆ2 can be a good estimator for 1γ2. Is this assertion true? Can one verify that
this estimator is unbiased?
4. (Total 10 points) Consider
yi = β1 + β2xi + ui for i = 1, · · · , n
(b) (5 points) Denote ub+ as the residual from the regression y+ on X+, and ub as the
residual from the original regression. Compare ub
+ and u. b
5. (Total 10 points) Let Y be n × 1, X be n × k (rank k), and Z = XB, where B is k × k
with rank k. Let (β, ˆ uˆ) denote the OLS coefficients and residuals from regression of y
on X. Similarly, let (β, ˜ u˜) denote the OLS coefficients and residuals from regression of
y on Z.
(a) (5 points) Find the relationship between βˆ and β˜.
(b) (5 points) Find the relationship between ˆu and ˜u.
6. (Total 10 points) You want to regress a GDP variable on time index as follows,
where the regression error et satisfies the classical assumptions.
(a) (5 points) Obtain the OLS estimator β. b
(b) (5 points) Is βb consistent? Explain.
5
7. (Total 25 points) Consider the following multiple regression model
yi = β1 + β2X2i + · · · + βkXki + ui for i = 1, 2, ..., n.
Denote
(a) (5 points) Construct the objective function for OLS estimator β, ˆ and obtain the
formula for OLS estimator βˆ and the residual vector ˆu.
(e) (5 points) Write down the classical assumptions in the linear regression model.

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