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EC252 AUTUMN MIDTERM PRACTICE TEST, 2021
1. Answer each of the following questions based on the estimated relations given in each case:
(a) [6 marks] What is the predicted change in y given a 2 unit change in x?
y? = 3.2 ? 6.4 x
(b) [6 marks] What is the predicted change in y given a 1 percent change in x?
y? = ?9.1 + 81.2 log(x)
(c) [6 marks] What is the predicted percentage change in y given a 1 unit change in x?
= ?3.6 + 0.022 x
log(y)
(d) [7 marks] What is the predicted percentage change in y given a 5 percent change in x?
= 0.22 ? 2.1 log(x)
log(y)
2. You are given the multiple regression model
y = ?0 + ?1 x1 + ?2 x2 + ?3 x3 + ?4 x4 + u,
(1)
where u is an unobserved error term.
(a) [5 marks] Classify the following as observable or unobservable: y, x1 , x4 , ?0 , ?3 , x3 .
(b) [5 marks] Is the following statement true of false?
“Adding another variable to the right hand side of (1) can never increase the explanatory power of the model.”
Explain your answer briefly.
(c) You are interested in testing the null hypothesis
H0 : ?1 = 3
against the alternative
H1 : ?1 > 3.
(i) [3 marks] What is the name of the test you would use to test this hypothesis?
(ii) [5 marks] Give the formula of the test statistic, for a sample of size n.
(iii) [7 marks] Define all components of the test statistic in part (ii) above.
3.
(a) Answer the questions below
(i) [5 marks] Suppose that Y is distributed as t432 . Find P (Y ? 1.96).
(ii) [5 marks] If W is an estimator of some parameter ?, is it generally true that M SE(W ) =
var(W )? If not, then when is it true?
(b) Consider the general multiple regression model
y = ?0 + ?1 x1 + ?2 x2 + . . . + ?k xk + u,
(2)
where u is an unobserved error term. Suppose that var(ux1 , x2 , . . . , xk ) = ? 2 .
(i) [5 marks] Name one method for obtaining estimates of the ?i , i = 0, . . . , k. No
explanations necessary.
(ii) [10 marks] You are given the following two estimates for ? 2 :
s21 =
SSR
SSR
and s22 =
,
n
n?k?1
where n denotes sample size and SSR denotes the sum of squared residuals. Which
one provides an unbiased estimate of ? 2 ? Explain briefly.
4.
(a) [6 marks] State the Classical Linear Model (CLM) assumptions. Remember, there are
six of them and we called them MLR.1 through MLR.6.
(b) You are investigating the effects of smoking on stamina (measured by the variable stam)
for a sample of sportspersons who are all smokers. You find the following ordinary least
squares (OLS) regression results in a published paper:
d = ?8.65 ? 3.54 cigs + 1.98 train
stam
(0.43)
(0.46)
(0.30)
n = 103, R2 = 0.162.
where cigs is daily cigarette consumption, and train is daily training time, n is the sample
size, and R2 is the coefficient of determination.
(i) [6 marks] Interpret the intercept estimate. If you think this is plausible, explain why,
and if you think this is not plausible, explain why not. Explanations can be very
brief and still earn full marks.
(ii) [10 marks] Test the null hypothesis that daily cigarette consumption has no impact
on stamina against the twosided alternative that cigarette consumption does have
an impact at the 1% significance level. The critical value for the test can be found
from the table on the last page of the test paper.
(iii) [3 marks] What does the value of R2 tell us?
Percentage points of Students tdistribution with ? degrees of freedom
Area ? under righthand tail
@
?@@
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
25
30
40
60
100
?
0.25
0.10
.05
.025
.01
.005
1.000
0.816
0.765
0.741
0.727
0.718
0.711
0.706
0.703
0.700
0.697
0.695
0.694
0.692
0.691
0.690
0.689
0.688
0.688
0.687
0.684
0.683
0.681
0.679
0.677
0.674
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.316
1.310
1.303
1.296
1.290
1.282
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.708
1.697
1.684
1.671
1.660
1.645
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.060
2.042
2.021
2.000
1.984
1.960
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.485
2.457
2.423
2.390
2.364
2.326
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.787
2.750
2.704
2.660
2.626
2.576
EC2525/7AU/ZA
1
UNIVERSITY OF ESSEX
Autumn MidTerm Test, November 2021
INTRODUCTION TO ECONOMETRIC METHODS
Time allowed: 2 hours. Please see your exam timetable or check on FASER for the deadline to
upload your answer.
The time allocated for this assessment includes time for you to download this question paper
and to upload your answers to FASER. Allow at least 30 minutes at the end of your exam
time to upload your work.
The times shown on your timetable are in UK time. Please check online for a conversion to your
local time if you will be undertaking your assessment outside the United Kingdom.
Candidates are permitted to use:
Calculator, Textbook(s), Lecture materials, Statistical table of their choice
This paper consists of 4 questions.
Candidates must answer all 4 questions.
All questions carry equal weight.
Write your Student Registration Number at the top of each page of your answer papers. All answers
must be written in the document(s) uploaded to FASER.
If you write your answers by hand please use a black pen and make sure the photos and/or scans in your
uploaded document are comfortably readable and arranged in sequential order.
If you think there is an error in the wording of any question or the instructions are not clear, contact
the module lecturer immediately via email: a.gupta@essex.ac.uk A response will be sent to your
Essex Email Account (click!).
If you have a technical problem with FASER, please go to the IT Helpdesk (click!) to find contact details
of the teams that can help you.
If you type your answers on a computer please save your work throughout the exam to avoid losing your
work.
It is forbidden to communicate with any other candidate in any way during this assessment. Your response must be your own work. Procedures are in place to detect plagiarism and collusion.
EC2525/7AU/ZA
2
1. Answer each of the following questions based on the estimated relations given in each case:
(a) [6 marks] What is the predicted change in y given a 1 unit change in x?
y? = 2.3 + 1.1 x.
(b) [6 marks] What is the predicted change in y given a 4 percent change in x?
y? = ?1 + 4 log(x).
(c) [6 marks] What is the predicted percentage change in y given a 1 unit change in x?
= 1.2 x.
log(y)
(d) [7 marks] What is the predicted percentage change in y given a 3 percent change in x?
= 3 + 1.9 log(x).
log(y)
2. You are given the multiple regression model
y = ?0 + ?1 x1 + ?2 x2 + ?3 x3 + u,
(1)
where u is an unobserved error term and you have a sample of size n.
(a) [5 marks] Is the following statement true or false?
“Adding another variable to the right hand side of (1) can never decrease the explanatory
power of the model.”
Explain your answer briefly.
(b) [5 marks] Is u observed by the econometrician? Explain.
(c) You are interested in testing the null hypothesis
H0 : ?2 = 0 and ?3 = 0
against the alternative
H1 : ?2 6= 0 or ?3 6= 0.
(i) [5 marks] Can you test this hypothesis with a t test? Explain briefly.
(ii) [10 marks] Suppose instead that you wish to test the null hypothesis
H0 : ?2 = 0
against the alternative
H1 : ?2 < 0.
Propose a test for this null and alternative hypothesis, writing down the formula of
the test statistic along with its distribution under H0 . Define clearly each component
of the statistic.
EC2525/7AU/ZA
3
3.
(a) Answer the questions below
(i) [5 marks] Suppose that Y is distributed as t12 . Find P (Y ? 1.782).
(ii) [5 marks] If W is an estimator of some parameter ?, define the mean squared error
(MSE) of W .
(b) Consider the general multiple regression model
y = ?0 + ?1 x1 + ?2 x2 + . . . + ?7 x7 + u,
(2)
where u is an unobserved error term. Suppose that var(ux1 , x2 , . . . , x7 ) = ? 2 .
(i) [5 marks] Name one method for obtaining estimates of the ?i , i = 0, . . . , 7. No
explanations necessary.
(ii) [10 marks] You are given the following estimate for ? 2 :
s2 =
SSR
,
n
where n denotes sample size and SSR denotes the sum of squared residuals. Does
this provide an unbiased estimate of ? 2 ? Explain briefly.
4.
(a) You are investigating the effect of advertising on sales (measured by the variable sales)
for a sample of car manufacturers. You find the following ordinary least squares (OLS)
regression results in a published paper:
d =
sales
11.37 + 1.32 adv + 1.32 models
(0.08)
(0.14)
(0.15)
n = 63, R2 = 0.15.
where adv is spending on advertising, models is the number of car models marketed by
a manufacturer, n is the sample size, and R2 is the coefficient of determination.
(i) [6 marks] Interpret the intercept estimate. If you think this is plausible, explain why,
and if you think this is not plausible, explain why not. Explanations can be very
brief and still earn full marks.
(ii) [15 marks] Test the null hypothesis that advertising has no impact on sales against
the twosided alternative that it does have an impact at 10% significance level. The
critical value for the test can be found from the table on the last page of the test
paper, or you can use a statistical table of your choice.
(iii) [4 marks] What does the value of R2 tell us?
EC2525/7AU/ZA
4
Percentage points of Students tdistribution with ? degrees of freedom
Area ? under righthand tail
@
0.25
0.10
.05
.025
.01
.005
1
1.000
3.078
6.314
12.706
31.821
63.657
2
0.816
1.886
2.920
4.303
6.965
9.925
3
0.765
1.638
2.353
3.182
4.541
5.841
4
0.741
1.533
2.132
2.776
3.747
4.604
5
0.727
1.476
2.015
2.571
3.365
4.032
6
0.718
1.440
1.943
2.447
3.143
3.707
7
0.711
1.415
1.895
2.365
2.998
3.499
8
0.706
1.397
1.860
2.306
2.896
3.355
9
0.703
1.383
1.833
2.262
2.821
3.250
10
0.700
1.372
1.812
2.228
2.764
3.169
11
0.697
1.363
1.796
2.201
2.718
3.106
12
0.695
1.356
1.782
2.179
2.681
3.055
13
0.694
1.350
1.771
2.160
2.650
3.012
14
0.692
1.345
1.761
2.145
2.624
2.977
15
0.691
1.341
1.753
2.131
2.602
2.947
16
0.690
1.337
1.746
2.120
2.583
2.921
17
0.689
1.333
1.740
2.110
2.567
2.898
18
0.688
1.330
1.734
2.101
2.552
2.878
19
0.688
1.328
1.729
2.093
2.539
2.861
20
0.687
1.325
1.725
2.086
2.528
2.845
25
0.684
1.316
1.708
2.060
2.485
2.787
30
0.683
1.310
1.697
2.042
2.457
2.750
40
0.681
1.303
1.684
2.021
2.423
2.704
60
0.679
1.296
1.671
2.000
2.390
2.660
100
0.677
1.290
1.660
1.984
2.364
2.626
?
0.674
1.282
1.645
1.960
2.326
2.576
?@@
EC2525/7AU/ZA
1
UNIVERSITY OF ESSEX
Autumn MidTerm Test, November 2021
INTRODUCTION TO ECONOMETRIC METHODS
Time allowed: 2 hours. Please see your exam timetable or check on FASER for the deadline to
upload your answer.
The time allocated for this assessment includes time for you to download this question paper
and to upload your answers to FASER. Allow at least 30 minutes at the end of your exam
time to upload your work.
The times shown on your timetable are in UK time. Please check online for a conversion to your
local time if you will be undertaking your assessment outside the United Kingdom.
Candidates are permitted to use:
Calculator, Textbook(s), Lecture materials, Statistical table of their choice
This paper consists of 4 questions.
Candidates must answer all 4 questions.
All questions carry equal weight.
Write your Student Registration Number at the top of each page of your answer papers. All answers
must be written in the document(s) uploaded to FASER.
If you write your answers by hand please use a black pen and make sure the photos and/or scans in your
uploaded document are comfortably readable and arranged in sequential order.
If you think there is an error in the wording of any question or the instructions are not clear, contact
the module lecturer immediately via email: a.gupta@essex.ac.uk A response will be sent to your
Essex Email Account (click!).
If you have a technical problem with FASER, please go to the IT Helpdesk (click!) to find contact details
of the teams that can help you.
If you type your answers on a computer please save your work throughout the exam to avoid losing your
work.
It is forbidden to communicate with any other candidate in any way during this assessment. Your response must be your own work. Procedures are in place to detect plagiarism and collusion.
EC2525/7AU/ZA
2
1. Answer each of the following questions based on the estimated relations given in each case:
(a) [6 marks] What is the predicted change in y given a 1 unit change in x?
y? = 2.3 + 1.1 x.
(b) [6 marks] What is the predicted change in y given a 4 percent change in x?
y? = ?1 + 4 log(x).
(c) [6 marks] What is the predicted percentage change in y given a 1 unit change in x?
= 1.2 x.
log(y)
(d) [7 marks] What is the predicted percentage change in y given a 3 percent change in x?
= 3 + 1.9 log(x).
log(y)
2. You are given the multiple regression model
y = ?0 + ?1 x1 + ?2 x2 + ?3 x3 + u,
(1)
where u is an unobserved error term and you have a sample of size n.
(a) [5 marks] Is the following statement true or false?
"Adding another variable to the right hand side of (1) can never decrease the explanatory
power of the model."
Explain your answer briefly.
(b) [5 marks] Is u observed by the econometrician? Explain.
(c) You are interested in testing the null hypothesis
H0 : ?2 = 0 and ?3 = 0
against the alternative
H1 : ?2 6= 0 or ?3 6= 0.
(i) [5 marks] Can you test this hypothesis with a t test? Explain briefly.
(ii) [10 marks] Suppose instead that you wish to test the null hypothesis
H0 : ?2 = 0
against the alternative
H1 : ?2 < 0.
Propose a test for this null and alternative hypothesis, writing down the formula of
the test statistic along with its distribution under H0 . Define clearly each component
of the statistic.
EC2525/7AU/ZA
3
3.
(a) Answer the questions below
(i) [5 marks] Suppose that Y is distributed as t12 . Find P (Y ? 1.782).
(ii) [5 marks] If W is an estimator of some parameter ?, define the mean squared error
(MSE) of W .
(b) Consider the general multiple regression model
y = ?0 + ?1 x1 + ?2 x2 + . . . + ?7 x7 + u,
(2)
where u is an unobserved error term. Suppose that var(ux1 , x2 , . . . , x7 ) = ? 2 .
(i) [5 marks] Name one method for obtaining estimates of the ?i , i = 0, . . . , 7. No
explanations necessary.
(ii) [10 marks] You are given the following estimate for ? 2 :
s2 =
SSR
,
n
where n denotes sample size and SSR denotes the sum of squared residuals. Does
this provide an unbiased estimate of ? 2 ? Explain briefly.
4.
(a) You are investigating the effect of advertising on sales (measured by the variable sales)
for a sample of car manufacturers. You find the following ordinary least squares (OLS)
regression results in a published paper:
d =
sales
11.37 + 1.32 adv + 1.32 models
(0.08)
(0.14)
(0.15)
n = 63, R2 = 0.15.
where adv is spending on advertising, models is the number of car models marketed by
a manufacturer, n is the sample size, and R2 is the coefficient of determination.
(i) [6 marks] Interpret the intercept estimate. If you think this is plausible, explain why,
and if you think this is not plausible, explain why not. Explanations can be very
brief and still earn full marks.
(ii) [15 marks] Test the null hypothesis that advertising has no impact on sales against
the twosided alternative that it does have an impact at 10% significance level. The
critical value for the test can be found from the table on the last page of the test
paper, or you can use a statistical table of your choice.
(iii) [4 marks] What does the value of R2 tell us?
EC2525/7AU/ZA
4
Percentage points of Students tdistribution with ? degrees of freedom
Area ? under righthand tail
@
0.25
0.10
.05
.025
.01
.005
1
1.000
3.078
6.314
12.706
31.821
63.657
2
0.816
1.886
2.920
4.303
6.965
9.925
3
0.765
1.638
2.353
3.182
4.541
5.841
4
0.741
1.533
2.132
2.776
3.747
4.604
5
0.727
1.476
2.015
2.571
3.365
4.032
6
0.718
1.440
1.943
2.447
3.143
3.707
7
0.711
1.415
1.895
2.365
2.998
3.499
8
0.706
1.397
1.860
2.306
2.896
3.355
9
0.703
1.383
1.833
2.262
2.821
3.250
10
0.700
1.372
1.812
2.228
2.764
3.169
11
0.697
1.363
1.796
2.201
2.718
3.106
12
0.695
1.356
1.782
2.179
2.681
3.055
13
0.694
1.350
1.771
2.160
2.650
3.012
14
0.692
1.345
1.761
2.145
2.624
2.977
15
0.691
1.341
1.753
2.131
2.602
2.947
16
0.690
1.337
1.746
2.120
2.583
2.921
17
0.689
1.333
1.740
2.110
2.567
2.898
18
0.688
1.330
1.734
2.101
2.552
2.878
19
0.688
1.328
1.729
2.093
2.539
2.861
20
0.687
1.325
1.725
2.086
2.528
2.845
25
0.684
1.316
1.708
2.060
2.485
2.787
30
0.683
1.310
1.697
2.042
2.457
2.750
40
0.681
1.303
1.684
2.021
2.423
2.704
60
0.679
1.296
1.671
2.000
2.390
2.660
100
0.677
1.290
1.660
1.984
2.364
2.626
?
0.674
1.282
1.645
1.960
2.326
2.576
?@@
1:52
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