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 ?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

Answer the following questions below.

(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

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 β?

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.

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