WISE International Masters
ECONOMETRICS
Instructor: Brett Graham
INSTRUCTIONS TO STUDENTS
1 The time allowed for this examination paper is 2 hours.
2 This examination paper contains 29 questions. You are REQUIRED to answer ALL questions. No marks will be deducted for wrong answers.
3 For each multiple choice question there is one and ONLY ONE suitable answer.
4 All numerical answers should be rounded to 3 decimal places. I will accept every answer to
within 0.002 of the correct answer. All probabilities should be expressed in decimal form.
5 This examination paper contains 9 pages including this instruction sheet, an answer sheet
for the first 28 questions, an answer sheet for question 29 and a blank page at the end of
the exam.
6 This is a closed-book examination. You are allowed to bring one handwritten 105mm by
75mm piece of paper to the exam. You are also allowed to use a financial calculator.
7 You are required to return all examination materials at the end of the examination.
8 Where required, please use the following critical values,
Tail-end Probabilities of the Normal Distribution
z
0.994 1.405 1.751 2.054 2.326 2.576
Pr(Z ≥ z) (%)
16
8
4
2
1
0.5
5% Significance Level Critical Values of the χ2m Distribution
Degrees of Freedom
1
2
3
4
5
6
7
8
9
Critical Value
3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92
Use the following information to answer the next 3 questions (1-3):
The random variable X has a Bernoulli distribution with Pr(X = 1) = 0.9. You randomly
sample 2 times from this distribution and calculate the sample mean X̄.
1. What is Pr(X̄ = 0)?
(1)
2. What is E(X̄)?
(1)
3. What is E(X̄ 2 )?
(1)
Use the following information to answer the next 4 questions (4-7):
A survey of 800 registered voters is conducted, and the voters are asked to choose between
candidate A and candidate B. Let p denote the fraction of voters in the population who prefer
candidate A, and let p̂ denote the fraction of voters in the sample who prefer candidate A. In
the survey p̂ = 0.5.
4. What is the test statistic associated with H0 : p = 0.48 vs. H1 : p > 0.48?
(1)
5. What is the upper confidence limit of the 92% confidence interval for p?
(1)
6. Suppose that the survey is carried out 8 times, using independently selected voters in each
survey. Suppose the statistical hypothesis H0 : p = 0.48 vs. H1 : p > 0.48 is tested each time
using the 8% significance level. If the true fraction of voters in the population who prefer
candidate A equals 0.48, what is the expected proportion of these surveys that will reject
the null hypothesis in favor of the alternative?
(1)
7. Suppose that the survey is carried out 8 times, using independently selected voters in each
survey and the 84% confidence interval is calculated each time. What is the probability that
the true value of p is contained in all of these confidence intervals?
(1)
Use the following information to answer the next 5 questions (8-12):
Suppose that a random sample of 220 twenty-year-old men is selected from a population and
that these men’s height and weight are recorded. A regression of weight on height yields
Wd
eight = −11.086
(8.688)
+
42.949 Height, R2 = 0.253, SER = 6.833,
(4.955)
where W eight is measured in kilograms and Height is measured in meters.
8. John, who is 1.65 meters tall and weighs 75 kilograms is one of the randomly sampled men.
What is the regression’s weight prediction for John?
(1)
9. What is the residual associated with John?
(1)
Page 2
10. A man has a late growth spurt and grows 2.72 centimeters over the course of a year. What
is the regression’s prediction for the increase in this man’s weight?
(1)
11. Suppose that instead of measuring weight and height in kilograms and meters, these variables
are measured in grams and centimeters. What is the regression estimate of the coefficient on
Height from this new gram-centimeter regression?
(1)
12. Suppose that instead of measuring weight and height in kilograms and meters, these variables
are measured in grams and centimeters. What is the regression SER from this new gramcentimeter regression?
(1)
Use the following information to answer the next 3 questions (13-15):
A random sample contains nR = 150 individuals who live in a rural area and nU = 190 individuals
who live in an urban
The sample mean of years of education of those individuals from a
Parea.
R
YR,i ) is 9.7 years, and the sample standard deviation of individuals from
rural area (ȲR = n1R ni=1
a rural area, sR , is 1.3 years. The corresponding values for those individuals from an urban area
are ȲU = 12.5 years and sU = 4.7 years. Let U rban denote an indicator variable that is equal to
1 for individuals from an urban area and 0 otherwise, and suppose that all 340 observations are
used to estimate the regression line Ŷ = β0 + β1 U rban.
13. What is the OLS estimate of β1 ?
(1)
14. What is the standard error of the OLS estimate of β1 ?
(1)
15. What is the value of the test statistic to test if individuals from rural and urban areas have
different levels of education?
(1)
16. Denote the standard normal cumulative distribution function as Φ. The p-value for a onesided left-tail test where the test statistic equals tact is given by
a. Φ(tact ) b. 2Φ(tact ) c. 1 − Φ(tact ) d. 2(1 − Φ(tact )) e. α (normally 0.05)
(1)
17. In multiple linear regression which of the following is not a random variable?
a. β̂0 b. µu c. X d. Ȳ
e. Ŷ
(1)
18. The effect of decreasing the student-teacher ratio by one is estimated to result in an improvement of the districtwide score by 2.28 with a standard error of 0.52. Construct a 96%
confidence interval for the predicted effect of increasing the student-teacher ratio by 4.
(1)
Page 3
19. Consider the following statistics that can be calculated from multiple regression analysis.
(1)
(I) R2
(II) F -test for a joint hypothesis.
(III) SER
(IV) t-test for an individual hypothesis.
Which of the above statistics can be used for model selection?
a. (II) only
b. (I), (II) and (III)
c. (I), (II) and (IV)
d. (I), (III) and (IV)
e. (I), (II), (III) and (IV)
Use the following information to answer the next 2 questions (20-21):
Let X and Z be independently distributed random variables where X ∼ N (−3, 7) and Z ∼
N (−6, 9), and let Y = 3X 2 − 4Z 3 .
20. What is the E(Y )?
(1)
21. What is the E(Y |X = 8)?
(1)
22. A pizza delivery store is considering offering customers a discount on any delivery that takes
more than half an hour. They will only offer the discount if less than 10% of current deliveries
take more than half an hour. From a random sample of 208 current deliveries, 15 take more
than half an hour. What is the value of the test statistic?
(1)
23. For the simple regression model Yi = β0 + β1 Xi + ui you have been given the following data:
(1)
540
X
Yi = −1056.8;
i=1
540
X
i=1
Xi Yi = −4323.6;
540
X
Xi = 2124.2;
i=1
540
X
Xi2
= 9082.8;
i=1
Calculate the regression slope and the intercept.
Page 4
540
X
i=1
Yi2 = 50783.7.
24. You regress Y on X1 and X2 . To test the joint hypothesis that β1 = β2 = 0, you reject
β̂1
β̂2
the null if either |t1 | > 1.751 or |t2 | > 1.751 (or both), where t1 = S.E.(
and t2 = S.E.(
.
β̂ )
β̂ )
1
(1)
2
Assuming that β̂1 and β̂2 are independent, what is the significance level of this test?
25. The random variables Yi , i = 1, . . . , n are i.i.d. and have a Bernoulli distribution with p = 0.9.
Let Ȳ denote the sample mean. If n = 500, which value is exactly 8% of sample means larger,
i.e. find Ȳ0.08 where Pr(Ȳ > Ȳ0.08 ) = 0.08.
(1)
26. The central limit theorem
a. can only be applied if E(u|X) = 0.
(1)
b. only holds in the presence of the law of large numbers.
c. postulates that the sample mean Ȳ is a consistent estimator of the population
mean µY .
d. states conditions under which the distribution of a variable involving the sum of
Y1 , . . . , Yn i.i.d. variables becomes the standard normal distribution.
p
e. states that Ȳ → µY .
27. The power of the test
a. is the probability that the test incorrectly rejects the null hypothesis when the null
is true.
b. is the probability that the test correctly rejects the null hypothesis when the null
is true.
c. is the probability that the test correctly rejects the null hypothesis when the alternative is true.
d. is the probability that the test incorrectly does not reject the null hypothesis when
the alternative is true.
e. is one minus the size of the test.
Page 5
(1)
28. Under the four least squares assumptions for the multiple linear regression problem, the OLS
estimators for the slopes and intercept
a. are BLUE.
(1)
b. are efficient among all linear estimators of the slope and intercept.
c. are unbiased and consistent.
d. have a normal distribution in small samples as long as the errors are homoskedastic.
e. have an exact normal distribution for n > 25.
Long Answers
29. Consider the following regression model in matrix form
Y = Xβ + U
where


 
1 X1
Y1
u1
1 X 2 
β0

 .. 
 .. 

.
Y =  .  , X =  .. ..  , U =  .  , β =
β1
. . 
Yn
un
1 Xn


(a) What is X T X?
(b) What is (X T X)−1 ? (Hint: If A =
(3)
a b
c d
and ad 6= bc then A−1 =
1
(ad−bc)
d −b
.)
−c a
(c) Use the matrix representations of X and U to express β̂ − β in matrix form.
Page 6
(3)
(4)
Answer Sheet
Econometrics Mid-term Exam
Question
Points
1
Answer
Name:
Question
Points
1
13
1
2
1
14
1
3
1
15
1
4
1
16
1
5
1
17
1
6
1
18
1
7
1
19
1
8
1
20
1
9
1
21
1
10
1
22
1
11
1
23
1
12
1
24
1
Total:
12
25
1
26
1
27
1
28
1
Total:
16
Answer
Answer Sheet
Econometrics Mid-term Exam
Question 29 (10 points)
Page 8
Name: