Student Name:
Economics 4818 - Introduction to Econometrics - Fall 2007
Final Exam - Answers
SHOW ALL WORK!
Evaluation: Problems: 3, 4C, 5C and 5F are worth 4 points. All other
questions are worth 3 points.
1. Answer the following questions:
A) What is the consequence of specifying a model with a varable in log form, if in the
population model, the variable is in level form?
This is a case of Functional Form Misspeci…cation, which causes the OLS
estimators to be biased.
B) Suppose the true income-consumption model is cons = 0 + 1 inc + 2 inc2 + u.
What is the consequence, if we estimate the model without the quadratic term inc2 ?
This is another case of Functional Form Misspeci…cation, which causes the
OLS estimators to be biased. It is also a case of omitted variable.
C) Why does the simple regression model y = 0 + 1 x + u typically fail to uncover
the ceteris paribus e¤ect of x on y?
There are typically unobserved factors of y, which are correlated with x, and
enter into the error term, causing the OLS estimators to be biased.
D) Is a regression with a low R2 useless? Explain. What does a low R2 imply about
the speci…ed regression model?
Not necessarily. However, the low R2 implies that the included regressors
do not explain much of the variation in y. That is, there are important omitted
factors.
E) What would be the likely sign of the bias of the coe¢ cient on IQ if we omitt edu
from the model: log (wage) = 0 + 1 IQ + 2 edu + 3 tenure + u?
Bias f1 = 2 e1
2 is expected to be > 0
e1 = Corr (IQ; edu) is expected to be > 0
Bias f1 is expected to be > 0
F) What is the e¤ect of increasing the sample size on se cj ? Explain.
1
Increasing
r the sample size decreases the error variance V ar (u) =
2
se cj = SST b1 R2 will decrease, where b2 is the estimator of 2 .
j(
j)
2
and so
G) What is the e¤ect of increasing the sample size on bias cj ? Explain.
The bias in c remains as the sample size increases.
j
2. Suppose you have estimated the following linear probability model explaining 401(k)
eligibility in terms of of income, age, and gender:
\ =
e401k
:000062inc2 + :0265age
:506 + :0124inc
(:081) (:0006)
n = 9; 275;
(:000005)
(:0039)
:00031age2
(:00005)
:0035male
(:0121)
R2 = :094
where e401k is a binary variable for eligibility in a 401(k) plan (e401k = 1 if eligble for
401(k), and = 0 otherwise), inc denotes family annual income (in $1; 000), age denotes
the individual’s age (in years), and male is a binary variable for gender (male = 1 if male
individual, and = 0 otherwise).
A) Based on the reported regression results, would you say that 401(k) eligibility depends
on income in a statistically signi…cant way? Explain. (show test statistic, critical value,
rejection rule, etc.)
401(k) eligibility clearly depends on income. Each of the two terms involving
inc have very signi…cant t statistics.
B) What is the estimated e¤ect of income on 401(k) eligibility? (Be speci…c. Use the
estimated coe¢ cients). Interpret your result.
\
@ e401k
2 (:000062) inc
@ inc = :0124
C) Holding other factors …xed, for an individual with family income of $200; 000, if
annual income increases by $1; 000, what happens to the probability of 401(k) eligibility?
\
@ e401k
2 (:000062) (200) = 0:012 4
@ inc = :0124
So, for an individual with family income of $200; 000, if annual income increases by $1; 000 (inc increases by 1), the estimated 401(k) eligibility decreases
by 0:012 4.
D) Interpret the coe¢ cient on male.
The di¤erence in the predicted 401(k) eligibility between male and female
individuals.
2
E) Is there statistically signi…cant evidence of gender discrimination? (show test statistic,
critical value, rejection rule, etc.)
:0035
t male = :0121
= 0:289 26
t5% = 1:96
Reject H0 : male = 0 if j 0:289 26j > 1:96. So, cannot reject H0 or male is not
statistically di¤erent from 0. Therefore, there isn’t su¢ cient evidence of gender
discrimination.
F) We add the variable pira, a binary variable representing whether a family member
has an individual retirement account (IRA) (pira = 1 if a family member has an IRA,
and = 0 otherwise), as an explanatory variable to the model. Its estimated coe¢ cient is
[
value = :105. Is that estimated coe¢ cient statistically
pira = :0198 with a two-sided p
signi…cant at the 1%; 5% or 10% level?
No, pira is not statistically signi…cant even at the 10% level.
G) Now, we specify the following version of the above model:
e401k =
0
+
+
1 inc
+
2
2 inc
6 black_male
+
+
3 age
+
4 age
7 black_f emale
2
+
8 nonblack_f emale
+u
What is the interpretation of 6 ?
It is the predicted di¤erence in e401k for black male and nonblack male (the
base group).
H) Refering to the model in part G, what is the interpretation of ( 6
7 )?
It is the predicted di¤erence in e401k for black male and black female.
I) Write the model that you would estimate to determine whether (
cally signi…cant.
Choose black_f emale as base group.
6
7)
is statisti-
J) Using the estimation results from the original model
\ =
e401k
:506 + :0124inc
(:081) (:0006)
:000062inc2 + :0265age
(:000005)
(:0039)
:00031age2
(:00005)
:0035male
(:0121)
predict the 401(k) eligibility of a person with the following characteristics: inc = 100;
age = 30 , and f emale. Does the predicted probability make sense? What is the reason
for obtaining this result? Explain what exactly is going on.
3
\ = :506 + :0124 (100) :000062 (100)2 + :0265 (30)
e401k
0:63
Yes, it is in the range [0; 1].
:00031 (30)2
:0035 (0) =
3. Suppose you want to estimate the following model
y=
0
+
1 x1
+
2 x2
+
3 x3
+u
Describe how you would test for heteroskedasticity using the special case of the White
test? Describe each step, state the null and the alternative hypotheses, the test statitsic and
the rejection rule.
(i) Estimate y = 0 + 1 x1 + 2 x2 + 3 x3 + u and obtain the resduals u
b
2
2
(ii) Estimate u
b = 0 + 1 yb + 2 yb + u
(iii) Test H0 :
1
= 0;
2
R2 2 =k
= 0; F =
u
b
1 R2 2 =(n k 1)
u
b
4. Suppose you want to estimate the following capital asset pricing model for some
company:
Rtf =
0
+
m
1 Rt
+ ut
where Rtf is the company’s return at time t and Rtm is the market return at time t.
A) How would you augment this model to test whether the announcement of the retirement of the company’s CEO has had a signi…cant efect on the company’s stock return?
What will be the null and the alternative hypothesis, what test would you use?
Rtf =
0
+
m
1 Rt
+
2 dt
+ ut
Include a dummy variable, say dt , for the day of and a few days after the
announcement. Test H0 : 2 = 0 against H1 : 2 6= 0. using a t-test.
B) How would you further augment the model to test whether there has been insider
trading on the information of the CEO retirement?
Rtf =
0
+
m
1 Rt
+
2 dt
+
3 bt
+ ut
Include another dummy variable, say bt , for a few days before the announcement. Test H0 : 3 = 0 against H1 : 3 6= 0 using a t-test.
4
C) Assuming that Rtm is strictly exogenous, how would you test whether the errors in
the model Rtf = 0 + 1 Rtm + ut are serially correlated? Describe each step, the null and
the alternative hypotheses, the test statistic and the rejection rule.
(i) Estimate Rtf = 0 + 1 Rtm + ut and obtain the residuals ubt
(ii) Estimate ubt = ud
t 1 + et and obtain b
(iii) Test H0 : = 0 against H1 : 6= 0 using a t-test.
5. Suppose you have esimated the following model of fertility and personal exemption:
dr = 92:05 + :089pet
gf
t
:0040pet
(3:33) (:126)
n = 68;
+ :0074pet
1
(:1531)
2
(:1651)
21:34ww2t
31:08pillt
(11:54)
(3:90)
2
R = :537
where gf r births per 1; 000 women in child bearing age, and pe is average real dollar
value of the personal tax exemption.
A) Interpret the coe¢ cient on pet . Be speci…c.
It is the impact multiplier. It implies that a one-dollar increase in pet at time
t, is predicted to increase fertility at time t by :089 births per 1; 000 women.
B) What is the long-run e¤ect of pe on gf r?
(:089 :0040 + :0074) = 0:092 4
C) How would you test whether the long-run e¤ect is statistically signi…cant?
Let’s denote the coe¢ cients of pet , pet 1 and pet 2 by 1 , 2 and 3 , respectively. De…ne a new parameter
= 1 + 2 + 3 , which equals the long-run
multiplier. Solve for 1 :
1
=
(
2
+
3) :
Plug the result in the original regression and rearrange:
gf rt =
0
+
1 pet
+
gf rt =
0
+[
(
2
gf rt =
0
+ pet +
2 pet 1
+
+
3 )] pet
2 (pet 1
3 pet 2
+
+
2 pet 1
pet ) +
4 ww2t
+
3 pet 2
3 (pet 2
Use a t test for the statistical signi…cance of
5
+
5 pillt
+
+ ut
4 ww2t
+
5 pillt
+
+ ut
pet ) +
4 ww2t
5 pillt
+ ut
(H0 :
= 0 against H1 :
6= 0):
D) How would you augment the model if you suspect that some of the variables exhibit
a linear time trend?
Add a linear trend regressor, t:
gf rt =
0
+
1 pet
+
2 pet 1
+
3 pet 2
+
4 ww2t
+
5 pillt
+
6t
+ ut
E) Suppose that gf r and pe both have upward time trends, what would be a potential
consequence of not including a time trend in the regression model? What is this phenomenon
called?
Spurious correlation between gf r and pe.
F) Suppose that gf r and pe both have upward time trends. How would you augment
the model to obtain a goodness-of-…t measure, which is not a¤ected by the presence of time
trend. Show all work.
::
(i) Regress gf r = 0 + 1 t + et and obtain the residuals gf r, which are the
detrended gf r.
::
(ii) Replace gf r with gf r in the original regression. The R2 from the last
regression is not a¤ected by the trending of gf r.
6. Suppose you have tested a model of rent rates and student population in a college
town
log\
(rent) = 1:39 + :066 log (pop) + :507 log (avginc) + :0056pctstu + u
(:844)
n = 64;
(:039)
(:081)
(:0017)
R2 = :458
where rent is the average monthly rent paid on rental units in a college town in the United
States, pop denotes the total city population, avginc denotes the average city income, and
pctstu denotes the student population as a percentage of the total population.
A) Suppose you want to test H0 : log(avginc) = 0:5 against H1 : log(avginc) 6= 0:5.
Construct the 95% con…dence interval for the parameter log(avginc) .
The 5% critical value for a two-tailed test with df = 64 4 = 60 is 2:00. The
95% con…dence interval for 2 is: :507 2 (:081) = [0:345 ; 0:669] :
B) Test the hypothesis in part A using the calculated con…dence interval for log(avginc) .
Since :5 is within the 95% con…dence interval for 2 , then we fail to reject
H0 : 2 = 0:5 against H1 : 2 6= 0:5 at the 5% level.
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C) Write down the t statistic for testing H0 : log(pop) = 2 pctstu against H1 :
2 pctstu at the 5% level. (Do not try to calculate or simplify.)
t=
log(pop)
>
c c
1
3
se c1 c3
D) Write the rejection rule for the test in part C. Make sure to report the critical value
for the test. (Do not try to perform the test.)
t < 1:671
E) Test the joint signi…cance of log(pop) , 2 pctstu and log(avginc) at the 5% level, that
is, test H0 : log(pop) = 0; 2 pctstu = 0 and log(avginc) = 0:
This is a test for overall regression signi…cance, that is, we test whether all
slope coe¢ cients are equal to 0 and so Rr2 = 0.
2
:458=3
u =k
F = (1 R2R)=(n
= (1 :458)=(64
4) = 16: 9 > t5%
k 1)
u
Reject H0 . The coe¢ cients log(pop) , 2 pctstu and log(avginc) are jointly signi…cant at the 5% level.
7