# data留学生作业代做、Java，Python，c/c++程序语言作业代写、代写program作业 代写留学生Prolog|调试Matlab程序

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
(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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