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###### 日期：2020-05-09 10:07

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.

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 ?α1 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).

Yˉ = 367.693 Xˉ1 = 402.760 Xˉ2 = 8.0

X(Yi ? Yˉ )2 = 66042.269 X(X1i ? Xˉ1)2 = 84855.096X(X2i ? Xˉ2)

2 = 280.000 X(Yi ? Yˉ )(X1i ? Xˉ1) = 74778.346X(Yi ? Yˉ )(X2i ? Xˉ2) = 4250.900 X(X1i ? Xˉ1)(X2i ? Xˉ2) = 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

β?2 =PnPi=1 XiYini=1 X2i

(a) (5 points) Demonstrate that β?

2 may be decomposed as

β?2 = β2 +Xni=1αiui

and calculate the suitable αi.

(b) (5 points) Show that β?2 is an unbiased estimator for β2

(c) (5 points) Show that V ar(β?2) = Pα2

i V ar(ui) and therefore, V ar(β?2) = 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

2 from the linear model Yi = β2Xi + ui, he argued

that β?2 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, ? u?) 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 β? and β?.

(b) (5 points) Find the relationship between ?u 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, ? and obtain the

formula for OLS estimator β? and the residual vector ?u.

(e) (5 points) Write down the classical assumptions in the linear regression model.