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经济代写-ECON 312
时间：2022-01-23
Paper Code: ECON 312 Page 1 of 15
Paper Code: ECON 312 Department: ULMS
Examiner: Yao Rao Tel. No: 53728
JANUARY EXAMINATIONS 2018
ECON 312 Methods of Economic Investigation II:
Microeconometrics
TIME ALLOWED: Two Hours
INSTRUCTIONS TO CANDIDATES
The use of pre-programmable calculators is not permitted during this exam.
Enter your name and student ID number IN PENCIL on the computer sheet according to the
instructions on that sheet. The digits should be entered in the boxes under “Student ID Number”
and entered by means of horizontal lines in the appropriate boxes underneath, exactly as when
answering questions.
NAME OF STUDENT__________________________________________________________
STUDENT ID NUMBER_______________________USUAL SIGNATURE________________
INSTRUCTIONS TO CANDIDATES
1. Answer All the questions in Section A and Section B.
2. Selected formulas and statistics tables are attached
3. You must show all workings clearly.
4. At the end of the examination, both exam scripts and exam question paper must be handed
in before your leave the exam room.
Paper Code: ECON 312 Page 2 of 15
JANUARY EXAMINATIONS 2018
ECON 312 Methods of Economic Investigation II: Microeconometrics
(Use 5% significance level unless otherwise indicated)
Section A
Question 1
Like credit cards, debit cards are now used extensively by consumers. Vendor prefer
them because when you use a debit card, the amount of your purchase is automatically
deducted from your checking or other designated account. To found out what factors
determine the use of the debit card, we obtained data on 60 customers and considered
the following model:
iiiii uXXXY 433221
Where Y =1 for debit card holder, 0 otherwise; 2X =account balance in dollars; 3X =
number of ATM transactions;
4X =1 if interest is received on the account, 0 otherwise.
The following table summarizes the estimated linear probability model (LPM), Logit
models (standard errors are in parentheses below the estimated coefficients).
Dependant Variable: Debit
Independent variables LPM Logit
Constant 0.3631
(0.1796)
-0.57490
(0.785787)
Balance 0.00028
(0.00015)
0.001248
(0.000697)
ATM -0.0269
(0.208)
-0.120225
(0.093984)
Interest -0.3019
(0.1448)
-1.352086
(0.680988)
R-square 0.1056 0.0804
Paper Code: ECON 312 Page 3 of 15
a) In LPM model, explain whether the signs of explanatory variables (except
intercept) are consistent with your expectation. Are all the variables statistically
significant?
(6 marks)
b) The second customer in the data having $500 balance in the account, and number
of ATM transactions is 3; the account does receive interests. Based on LPM and
Logit estimates, what are the probability, respectively, that this customer will hold
the debit card?
(5 marks)
c) Based on the output table, calculate respectively for LPM and Logit model, the
marginal effect of a one extra dollar deposited in the account on the probability of
the customer described in b) holding the debit card.
(5 marks)
[Total: 16 marks]
Question 2
A Poisson regression is estimated explaining the number of Olympic Games medals
won by various countries as a function of the logarithms of population and gross
domestic product (in 1995 dollars). The estimated coefficients are as follows:
Dependent Variable: MEDALTOT
Method: ML/QML - Poisson Count (Quadratic hill climbing)
Date: 10/13/17 Time: 19:18
Sample: 1 1610 IF YEAR=88
Included observations: 151
Convergence achieved after 6 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
C -15.88746 0.511805 -31.04203 0.0000
LOG(POP) 0.180038 0.032280 5.577348 0.0000
LOG(GDP) 0.576603 0.024722 23.32376 0.0000
R-squared 0.393119 Mean dependent var 4.887417
Adjusted R-squared 0.384918 S.D. dependent var 16.62670
S.E. of regression 13.03985 Akaike info criterion 9.607106
Sum squared resid 25165.58 Schwarz criterion 9.667052
Log likelihood -722.3365 Hannan-Quinn criter. 9.631459
Restr. log likelihood -1586.359 LR statistic 1728.045
Avg. log likelihood -4.783685 Prob(LR statistic) 0.000000
Paper Code: ECON 312 Page 4 of 15
a) Based on the output in the above table, are the signs of explanatory variables
consistent with your expectation? Explain.
(2 marks)
b) In 1988 Canada had GDP=5.19E+11 and a population of 26.9 million, predict
the number of medals that Canada would win.
(3 marks)
c) Calculate the probability that they would win 5 medals or less.
(4 marks)
d) What is the marginal effect of a change in population on the numbers of medals
won? How do you interpret it?
(3 marks)
In addition to population and GDP, the file Olympics contains a dummy variable
PLANNED, which includes nonmarket, typically communist countries and the dummy
variable HOST, which indicates the country hosting the Olympic Games. The Poisson
regression model that adds these two variables to the specification is estimated as in
the following Table:
Dependent Variable: MEDALTOT
Method: ML/QML - Poisson Count (Quadratic hill climbing)
Date: 10/13/17 Time: 19:19
Sample: 1 1610 IF YEAR=92 OR YEAR=96
Included observations: 357
Convergence achieved after 6 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
C -14.84163 0.343789 -43.17070 0.0000
LOG(POP) 0.139775 0.024423 5.723069 0.0000
LOG(GDP) 0.562546 0.017445 32.24643 0.0000
PLANNED 0.623602 0.118473 5.263648 0.0000
HOST 0.119509 0.100505 1.189083 0.2344
R-squared 0.568120 Mean dependent var 4.641457
Adjusted R-squared 0.563213 S.D. dependent var 13.59714
S.E. of regression 8.986338 Akaike info criterion 7.117032
Sum squared resid 28425.50 Schwarz criterion 7.171342
Log likelihood -1265.390 Hannan-Quinn criter. 7.138633
Restr. log likelihood -3167.551 LR statistic 3804.322
Paper Code: ECON 312 Page 5 of 15
Avg. log likelihood -3.544510 Prob(LR statistic) 0.000000
e) In 2000, the GDP (in 1995 US $) of Canada was 6.41256E+11. The Canadian
population in 2000 was 30.689 million. Using these figures, predict the number of
medals won by Canada based on the estimates. Note that the 2000 games were
held in Sydney, Australia.
(4 marks)
[Total: 16 marks]
Question 3
We describe an ordinal Probit model for post-secondary education choice and estimated
a simple model with the dependant variable of interest is PSECHOICE and the
explanatory variable is GRADES (Lower numerical value of GRADES indicate better
performance). We rank the college possibilities as follows:
The following table shows the ordered Probit estimation results:
Dependent Variable: PSECHOICE
Method: ML - Ordered Probit (Quadratic hill climbing)
Date: 10/13/17 Time: 18:52
Sample: 1 1000
Included observations: 1000
Number of ordered indicator values: 3
Convergence achieved after 4 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
GRADES -0.306625 0.019173 -15.99217 0.0000
Limit Points
LIMIT_2:C(2) -2.945600 0.146828 -20.06154 0.0000
LIMIT_3:C(3) -2.089993 0.135768 -15.39385 0.0000
Pseudo R-squared 0.140220 Akaike info criterion 1.757643
Schwarz criterion 1.772367 Log likelihood -875.8217
*
2
*
1 2
*
1
3 (4-year college) if
2 (2-year college) if
1 (no college) if
i
i
i
y
y y
y
Paper Code: ECON 312 Page 6 of 15
Hannan-Quinn criter. 1.763239 Restr. log likelihood -1018.658
LR statistic 285.6716 Avg. log likelihood -0.875822
Prob(LR statistic) 0.000000
a) Based on the estimated results in the above table, comment on the changes for
probabilities of “2-year college” group and “no college” group when GRADES
increases. Explain if they are consistent with your expectations.
(4 marks)
b) Compute the marginal effect of GRADES on the probability that a student with
GRADES=6 attends 2- year college.
(3 marks)
If we now extend the above ordered Probit model by including other variables: FAMINC-
family income in £1000, FAMSIZ-number of family members; and dummy variables
BLACK and PARCOLL=1 if the most educated parent had a college degree, the
estimated results are in the Table below.
Dependent Variable: PSECHOICE
Method: ML - Ordered Probit (Quadratic hill climbing)
Date: 10/13/17 Time: 18:57
Sample: 1 1000
Included observations: 1000
Number of ordered indicator values: 3
Convergence achieved after 4 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
GRADES -0.295292 0.020225 -14.60028 0.0000
FAMINC 0.005252 0.001322 3.973231 0.0001
FAMSIZ -0.024122 0.030185 -0.799135 0.4242
BLACK 0.713131 0.176787 4.033842 0.0001
PARCOLL 0.423623 0.101642 4.167774 0.0000
Limit Points
LIMIT_2:C(6) -2.595845 0.204586 -12.68827 0.0000
LIMIT_3:C(7) -1.694591 0.197136 -8.596029 0.0000
Pseudo R-squared 0.175518 Akaike info criterion 1.693729
Paper Code: ECON 312 Page 7 of 15
Schwarz criterion 1.728084 Log likelihood -839.8647
Hannan-Quinn criter. 1.706786 Restr. log likelihood -1018.658
LR statistic 357.5856 Avg. log likelihood -0.839865
Prob(LR statistic) 0.000000
c) Test the joint significance of the variables added above using a likelihood ratio (LR)
test.
(4 marks)
d) Compute the probability that a Black student from a household of four members,
including a parent who went to college, and household income of £52,000, will attend
a 4-year college if GRADES=6.64.
(4 marks)
[Total: 15 marks]
Question 4
Consider a panel data set on young women for the years 1990 and 1991, with 716
women being interviewed in each year. We are interested in the wage equation that
relates the logarithm of WAGE to years of education (EDUC), working experience
(EXPER), its square EXPER2, and dummy variable BLACK, SOUTH, UNION.
The coefficient estimates for the different parts of the questions are given in the following
Table with the standard errors in parentheses below the estimated coefficients.
Variable 1990 LS 1991 LS Random Effect Fixed Effect
Intercept 0.2268
(0.1881)
0.2216
(0.2227)
0.3086
(0.1610)
1.5468
(0.2522)
EDUC 0.0762
(0.0063)
0.0778
(0.0064)
0.0766
(0.0060)
EXPER 0.0875
(0.0265)
0.0830
(0.0292)
0.0758
(0.0205)
0.0575
(0.0330)
EXPERsquare -0.0020
(0.00096)
-0.00179
(0.00096)
-0.001648
(0.000702)
-0.1234
(0.1102)
BLACK -0.1562
(0.0366)
-0.1309
(0.0327)
-0.1319
(0.0345)
SOUTH -0.1029
(0.0327)
-0.1368
(0.0334)
-0.1350
(0.0303)
-0.3261
(0.1258)
UNION 0.1701
(0.0350)
0.1324
(0.0354)
0.1170
(0.0235)
0.0822
(0.0312)
Paper Code: ECON 312 Page 8 of 15
a) The estimate results (column 2 and 3 in the Table) show the least squares estimates
for each of the years 1990 and 1991. How do the results compare? For these
individual year estimates, what are you assuming about the regression parameter
values across individuals?
(5 marks)
b) Allowing heterogeneity across individuals, the wage equation is modelled as
itititi
ititiiit
eUNIONSOUTHBLACK
EXPEREXPEREDUCWAGE
765
2
4321
)ln(
Explain any differences in assumptions between this model and the models in part a).
(5 marks)
c) Column 5 of the Table reports the results when the model is estimated using a fixed
effects specification. Explain why there are no estimated coefficients for variables
EXPER and BLACK.
(3 marks)
d) Using the t-test statistic
, ,
2 2 1/2
, ,[ ( ) ( ) ]
FE k RE k
FE k RE k
b b
t
se b se b
, test (at 5% significance level) the
difference between the fixed effects and random effects estimates of the coefficients
on EXPER, its square EXPER2, SOUTH and UNION. The t- values are as follows:
exp
exp 2
0.711 ( 0.477)
0.487 ( 0.626)
1.565 ( 0.118)
1.692 ( 0.091)
er
er
south
union
t p value
t p value
t p value
t p value
Do we reject, or fail to reject the null hypothesis that the difference between the
estimates is zero? Which estimate would be better to use, fixed effects or random
effects estimates?
(6 marks)
Paper Code: ECON 312 Page 9 of 15
[Total: 19 marks]
Section B
Question 1
Consider the linear probability model (LPM) iii uXY 21 , where Y takes two
values: 0 and 1. The probabilities are: iii XXY 21)|1Pr(
a) Calculate the mean of error term iu ;
(3 marks)
b) Calculate the variance of error term iu ;
(4 marks)
c) Explain why the linear probability model is not the best model and therefore a Probit
or a Logit model is used.
(5 marks)
[Total: 12 marks]
Question 2
The logarithm of the likelihood function (L) for estimating the population mean and variance for
an i.i.d. normal sample is as follows:
2 2
2
1
1
log(2 ) ( )
2 2
n
i Y
i
n
L Y
a) Derive the maximum likelihood estimators for the mean Y and the variance
2 .
(8 marks)
b) Explain how do they differ, if at all, from the OLS estimators?
(5 marks)
Paper Code: ECON 312 Page 10 of 15
[Total: 13 marks]
Question 3
In a random effects model, define the composite error itiit euv where iu is
uncorrelated with ite and ite have zero mean and constant variance
2
e and are serially
uncorrelated. The random effects iu has zero mean and constant variance
2
u and
serially uncorrelated.
a) Show that 0)( itvE
(2 marks)
b) Find )( itvVar
(3 marks)
c) For st , find ),( isit vvCov
(4 marks)
[Total: 9 marks]
Formula Sheet
1. Probit model: )...(]...Pr[ 12121 kkkk XXXXzp , where
25.0
2
1
)( zez
and
z
u duezZPz
25.0
2
1
][)(
221 )(
)(
x
dx
dt
dt
td
x
p
2. Logit model:
)...(121121 1211
1
)...(]...[
kk XXkkkk e
XXXXLPp
)1( iij pp
x
p
Paper Code: ECON 312 Page 11 of 15
3. Ordered Probit model (for J=3 categories)
)(]1[ 1 xyP
)(
]1[
1 x
x
yP
)()(]2[ 12 xxyP
)]()([
]2[
21 xx
x
yP
)(1]3[ 2 xyP
)(
]3[
2 x
x
yP
4. Count data model
If Y is a Poisson random variable, then its probability function is
W here
and )exp()( 21 xYE , 2
)(
i
i
i
x
yE
5. Multinomial Logit model
Assume there are three alternatives, the probabilities of individual choosing alternatives j = 1,
2, 3 are
1 12 22 13 23
1
, 1
1 exp exp
i
i i
p j
x x
12 22
2
12 22 13 23
exp
, 2
1 exp exp
i
i
i i
x
p j
x x
13 23
3
12 22 13 23
exp
, 3
1 exp exp
i
i
i i
x
p j
x x
, 0,1,2,
!
ye
f y P Y y y
y
1!0 with 1...)2()1(! yyyy
Paper Code: ECON 312 Page 12 of 15
Paper Code: ECON 312 Page 13 of 15
Paper Code: ECON 312 Page 14 of 15
Paper Code: ECON 312 Page 15 of 15
Paper Code: ECON 312 Department: ULMS
Examiner: Yao Rao Tel. No: 53728
JANUARY EXAMINATIONS 2018
ECON 312 Methods of Economic Investigation II:
Microeconometrics
TIME ALLOWED: Two Hours
INSTRUCTIONS TO CANDIDATES
The use of pre-programmable calculators is not permitted during this exam.
Enter your name and student ID number IN PENCIL on the computer sheet according to the
instructions on that sheet. The digits should be entered in the boxes under “Student ID Number”
and entered by means of horizontal lines in the appropriate boxes underneath, exactly as when
answering questions.
NAME OF STUDENT__________________________________________________________
STUDENT ID NUMBER_______________________USUAL SIGNATURE________________
INSTRUCTIONS TO CANDIDATES
1. Answer All the questions in Section A and Section B.
2. Selected formulas and statistics tables are attached
3. You must show all workings clearly.
4. At the end of the examination, both exam scripts and exam question paper must be handed
in before your leave the exam room.
Paper Code: ECON 312 Page 2 of 15
JANUARY EXAMINATIONS 2018
ECON 312 Methods of Economic Investigation II: Microeconometrics
(Use 5% significance level unless otherwise indicated)
Section A
Question 1
Like credit cards, debit cards are now used extensively by consumers. Vendor prefer
them because when you use a debit card, the amount of your purchase is automatically
deducted from your checking or other designated account. To found out what factors
determine the use of the debit card, we obtained data on 60 customers and considered
the following model:
iiiii uXXXY 433221
Where Y =1 for debit card holder, 0 otherwise; 2X =account balance in dollars; 3X =
number of ATM transactions;
4X =1 if interest is received on the account, 0 otherwise.
The following table summarizes the estimated linear probability model (LPM), Logit
models (standard errors are in parentheses below the estimated coefficients).
Dependant Variable: Debit
Independent variables LPM Logit
Constant 0.3631
(0.1796)
-0.57490
(0.785787)
Balance 0.00028
(0.00015)
0.001248
(0.000697)
ATM -0.0269
(0.208)
-0.120225
(0.093984)
Interest -0.3019
(0.1448)
-1.352086
(0.680988)
R-square 0.1056 0.0804
Paper Code: ECON 312 Page 3 of 15
a) In LPM model, explain whether the signs of explanatory variables (except
intercept) are consistent with your expectation. Are all the variables statistically
significant?
(6 marks)
b) The second customer in the data having $500 balance in the account, and number
of ATM transactions is 3; the account does receive interests. Based on LPM and
Logit estimates, what are the probability, respectively, that this customer will hold
the debit card?
(5 marks)
c) Based on the output table, calculate respectively for LPM and Logit model, the
marginal effect of a one extra dollar deposited in the account on the probability of
the customer described in b) holding the debit card.
(5 marks)
[Total: 16 marks]
Question 2
A Poisson regression is estimated explaining the number of Olympic Games medals
won by various countries as a function of the logarithms of population and gross
domestic product (in 1995 dollars). The estimated coefficients are as follows:
Dependent Variable: MEDALTOT
Method: ML/QML - Poisson Count (Quadratic hill climbing)
Date: 10/13/17 Time: 19:18
Sample: 1 1610 IF YEAR=88
Included observations: 151
Convergence achieved after 6 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
C -15.88746 0.511805 -31.04203 0.0000
LOG(POP) 0.180038 0.032280 5.577348 0.0000
LOG(GDP) 0.576603 0.024722 23.32376 0.0000
R-squared 0.393119 Mean dependent var 4.887417
Adjusted R-squared 0.384918 S.D. dependent var 16.62670
S.E. of regression 13.03985 Akaike info criterion 9.607106
Sum squared resid 25165.58 Schwarz criterion 9.667052
Log likelihood -722.3365 Hannan-Quinn criter. 9.631459
Restr. log likelihood -1586.359 LR statistic 1728.045
Avg. log likelihood -4.783685 Prob(LR statistic) 0.000000
Paper Code: ECON 312 Page 4 of 15
a) Based on the output in the above table, are the signs of explanatory variables
consistent with your expectation? Explain.
(2 marks)
b) In 1988 Canada had GDP=5.19E+11 and a population of 26.9 million, predict
the number of medals that Canada would win.
(3 marks)
c) Calculate the probability that they would win 5 medals or less.
(4 marks)
d) What is the marginal effect of a change in population on the numbers of medals
won? How do you interpret it?
(3 marks)
In addition to population and GDP, the file Olympics contains a dummy variable
PLANNED, which includes nonmarket, typically communist countries and the dummy
variable HOST, which indicates the country hosting the Olympic Games. The Poisson
regression model that adds these two variables to the specification is estimated as in
the following Table:
Dependent Variable: MEDALTOT
Method: ML/QML - Poisson Count (Quadratic hill climbing)
Date: 10/13/17 Time: 19:19
Sample: 1 1610 IF YEAR=92 OR YEAR=96
Included observations: 357
Convergence achieved after 6 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
C -14.84163 0.343789 -43.17070 0.0000
LOG(POP) 0.139775 0.024423 5.723069 0.0000
LOG(GDP) 0.562546 0.017445 32.24643 0.0000
PLANNED 0.623602 0.118473 5.263648 0.0000
HOST 0.119509 0.100505 1.189083 0.2344
R-squared 0.568120 Mean dependent var 4.641457
Adjusted R-squared 0.563213 S.D. dependent var 13.59714
S.E. of regression 8.986338 Akaike info criterion 7.117032
Sum squared resid 28425.50 Schwarz criterion 7.171342
Log likelihood -1265.390 Hannan-Quinn criter. 7.138633
Restr. log likelihood -3167.551 LR statistic 3804.322
Paper Code: ECON 312 Page 5 of 15
Avg. log likelihood -3.544510 Prob(LR statistic) 0.000000
e) In 2000, the GDP (in 1995 US $) of Canada was 6.41256E+11. The Canadian
population in 2000 was 30.689 million. Using these figures, predict the number of
medals won by Canada based on the estimates. Note that the 2000 games were
held in Sydney, Australia.
(4 marks)
[Total: 16 marks]
Question 3
We describe an ordinal Probit model for post-secondary education choice and estimated
a simple model with the dependant variable of interest is PSECHOICE and the
explanatory variable is GRADES (Lower numerical value of GRADES indicate better
performance). We rank the college possibilities as follows:
The following table shows the ordered Probit estimation results:
Dependent Variable: PSECHOICE
Method: ML - Ordered Probit (Quadratic hill climbing)
Date: 10/13/17 Time: 18:52
Sample: 1 1000
Included observations: 1000
Number of ordered indicator values: 3
Convergence achieved after 4 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
GRADES -0.306625 0.019173 -15.99217 0.0000
Limit Points
LIMIT_2:C(2) -2.945600 0.146828 -20.06154 0.0000
LIMIT_3:C(3) -2.089993 0.135768 -15.39385 0.0000
Pseudo R-squared 0.140220 Akaike info criterion 1.757643
Schwarz criterion 1.772367 Log likelihood -875.8217
*
2
*
1 2
*
1
3 (4-year college) if
2 (2-year college) if
1 (no college) if
i
i
i
y
y y
y
Paper Code: ECON 312 Page 6 of 15
Hannan-Quinn criter. 1.763239 Restr. log likelihood -1018.658
LR statistic 285.6716 Avg. log likelihood -0.875822
Prob(LR statistic) 0.000000
a) Based on the estimated results in the above table, comment on the changes for
probabilities of “2-year college” group and “no college” group when GRADES
increases. Explain if they are consistent with your expectations.
(4 marks)
b) Compute the marginal effect of GRADES on the probability that a student with
GRADES=6 attends 2- year college.
(3 marks)
If we now extend the above ordered Probit model by including other variables: FAMINC-
family income in £1000, FAMSIZ-number of family members; and dummy variables
BLACK and PARCOLL=1 if the most educated parent had a college degree, the
estimated results are in the Table below.
Dependent Variable: PSECHOICE
Method: ML - Ordered Probit (Quadratic hill climbing)
Date: 10/13/17 Time: 18:57
Sample: 1 1000
Included observations: 1000
Number of ordered indicator values: 3
Convergence achieved after 4 iterations
Covariance matrix computed using second derivatives
Variable Coefficient Std. Error z-Statistic Prob.
GRADES -0.295292 0.020225 -14.60028 0.0000
FAMINC 0.005252 0.001322 3.973231 0.0001
FAMSIZ -0.024122 0.030185 -0.799135 0.4242
BLACK 0.713131 0.176787 4.033842 0.0001
PARCOLL 0.423623 0.101642 4.167774 0.0000
Limit Points
LIMIT_2:C(6) -2.595845 0.204586 -12.68827 0.0000
LIMIT_3:C(7) -1.694591 0.197136 -8.596029 0.0000
Pseudo R-squared 0.175518 Akaike info criterion 1.693729
Paper Code: ECON 312 Page 7 of 15
Schwarz criterion 1.728084 Log likelihood -839.8647
Hannan-Quinn criter. 1.706786 Restr. log likelihood -1018.658
LR statistic 357.5856 Avg. log likelihood -0.839865
Prob(LR statistic) 0.000000
c) Test the joint significance of the variables added above using a likelihood ratio (LR)
test.
(4 marks)
d) Compute the probability that a Black student from a household of four members,
including a parent who went to college, and household income of £52,000, will attend
a 4-year college if GRADES=6.64.
(4 marks)
[Total: 15 marks]
Question 4
Consider a panel data set on young women for the years 1990 and 1991, with 716
women being interviewed in each year. We are interested in the wage equation that
relates the logarithm of WAGE to years of education (EDUC), working experience
(EXPER), its square EXPER2, and dummy variable BLACK, SOUTH, UNION.
The coefficient estimates for the different parts of the questions are given in the following
Table with the standard errors in parentheses below the estimated coefficients.
Variable 1990 LS 1991 LS Random Effect Fixed Effect
Intercept 0.2268
(0.1881)
0.2216
(0.2227)
0.3086
(0.1610)
1.5468
(0.2522)
EDUC 0.0762
(0.0063)
0.0778
(0.0064)
0.0766
(0.0060)
EXPER 0.0875
(0.0265)
0.0830
(0.0292)
0.0758
(0.0205)
0.0575
(0.0330)
EXPERsquare -0.0020
(0.00096)
-0.00179
(0.00096)
-0.001648
(0.000702)
-0.1234
(0.1102)
BLACK -0.1562
(0.0366)
-0.1309
(0.0327)
-0.1319
(0.0345)
SOUTH -0.1029
(0.0327)
-0.1368
(0.0334)
-0.1350
(0.0303)
-0.3261
(0.1258)
UNION 0.1701
(0.0350)
0.1324
(0.0354)
0.1170
(0.0235)
0.0822
(0.0312)
Paper Code: ECON 312 Page 8 of 15
a) The estimate results (column 2 and 3 in the Table) show the least squares estimates
for each of the years 1990 and 1991. How do the results compare? For these
individual year estimates, what are you assuming about the regression parameter
values across individuals?
(5 marks)
b) Allowing heterogeneity across individuals, the wage equation is modelled as
itititi
ititiiit
eUNIONSOUTHBLACK
EXPEREXPEREDUCWAGE
765
2
4321
)ln(
Explain any differences in assumptions between this model and the models in part a).
(5 marks)
c) Column 5 of the Table reports the results when the model is estimated using a fixed
effects specification. Explain why there are no estimated coefficients for variables
EXPER and BLACK.
(3 marks)
d) Using the t-test statistic
, ,
2 2 1/2
, ,[ ( ) ( ) ]
FE k RE k
FE k RE k
b b
t
se b se b
, test (at 5% significance level) the
difference between the fixed effects and random effects estimates of the coefficients
on EXPER, its square EXPER2, SOUTH and UNION. The t- values are as follows:
exp
exp 2
0.711 ( 0.477)
0.487 ( 0.626)
1.565 ( 0.118)
1.692 ( 0.091)
er
er
south
union
t p value
t p value
t p value
t p value
Do we reject, or fail to reject the null hypothesis that the difference between the
estimates is zero? Which estimate would be better to use, fixed effects or random
effects estimates?
(6 marks)
Paper Code: ECON 312 Page 9 of 15
[Total: 19 marks]
Section B
Question 1
Consider the linear probability model (LPM) iii uXY 21 , where Y takes two
values: 0 and 1. The probabilities are: iii XXY 21)|1Pr(
a) Calculate the mean of error term iu ;
(3 marks)
b) Calculate the variance of error term iu ;
(4 marks)
c) Explain why the linear probability model is not the best model and therefore a Probit
or a Logit model is used.
(5 marks)
[Total: 12 marks]
Question 2
The logarithm of the likelihood function (L) for estimating the population mean and variance for
an i.i.d. normal sample is as follows:
2 2
2
1
1
log(2 ) ( )
2 2
n
i Y
i
n
L Y
a) Derive the maximum likelihood estimators for the mean Y and the variance
2 .
(8 marks)
b) Explain how do they differ, if at all, from the OLS estimators?
(5 marks)
Paper Code: ECON 312 Page 10 of 15
[Total: 13 marks]
Question 3
In a random effects model, define the composite error itiit euv where iu is
uncorrelated with ite and ite have zero mean and constant variance
2
e and are serially
uncorrelated. The random effects iu has zero mean and constant variance
2
u and
serially uncorrelated.
a) Show that 0)( itvE
(2 marks)
b) Find )( itvVar
(3 marks)
c) For st , find ),( isit vvCov
(4 marks)
[Total: 9 marks]
Formula Sheet
1. Probit model: )...(]...Pr[ 12121 kkkk XXXXzp , where
25.0
2
1
)( zez
and
z
u duezZPz
25.0
2
1
][)(
221 )(
)(
x
dx
dt
dt
td
x
p
2. Logit model:
)...(121121 1211
1
)...(]...[
kk XXkkkk e
XXXXLPp
)1( iij pp
x
p
Paper Code: ECON 312 Page 11 of 15
3. Ordered Probit model (for J=3 categories)
)(]1[ 1 xyP
)(
]1[
1 x
x
yP
)()(]2[ 12 xxyP
)]()([
]2[
21 xx
x
yP
)(1]3[ 2 xyP
)(
]3[
2 x
x
yP
4. Count data model
If Y is a Poisson random variable, then its probability function is
W here
and )exp()( 21 xYE , 2
)(
i
i
i
x
yE
5. Multinomial Logit model
Assume there are three alternatives, the probabilities of individual choosing alternatives j = 1,
2, 3 are
1 12 22 13 23
1
, 1
1 exp exp
i
i i
p j
x x
12 22
2
12 22 13 23
exp
, 2
1 exp exp
i
i
i i
x
p j
x x
13 23
3
12 22 13 23
exp
, 3
1 exp exp
i
i
i i
x
p j
x x
, 0,1,2,
!
ye
f y P Y y y
y
1!0 with 1...)2()1(! yyyy
Paper Code: ECON 312 Page 12 of 15
Paper Code: ECON 312 Page 13 of 15
Paper Code: ECON 312 Page 14 of 15
Paper Code: ECON 312 Page 15 of 15
