– 3 –
Question 1
. After estimating by OLS a multip
le regression model, the resulting
residuals:
A)
Add up to zero if a constant term
was included in the model.
B)
Are orthogonal to the model regressors only if a constant term was included in
the model.
C)
Have constant variances and null covariances whenever the model errors have
these properties.
Question 2
. In the regression model
01
ttt
YXU
b
b
=
++
,
t=
1,2, …,
n
where
0
Y
=
,
the OLS estimator of
1
b
is:
A)
1
22
ˆ
tt
t
YX
X
nX
b
=
–
å
å
B)
1
2
ˆ
tt
t
YX
X
b
=
å
å
C)
1
ˆ
0
Y
b
==
Question 3
. The statistical significance of a param
eter in a regression model refers to:
A)
The conclusion of testing the null hypothesis that the parameter is equal to zero,
against the alternative that
it is non-zero.
B)
The probability that the OLS estimate of
this parameter is equal to zero.
C)
The interpretation of the sign (posit
ive or negative) of this parameter.
Question 4
. When the matrix
X
in the model
YX U
b
=
+
displays a high degree of
collinearity:
A)
The OLS estimate of
b
is unbiased.
B)
The covariance matrix of
ˆ
OLS
b
cannot be computed because
0
T
XX
=
.
C) The OLS estimate of
b
is NOT efficient.
Question 5.
Consider the model
01
ttt
YXU
b
b
=
++
. We have three observations for the
dependent variable
Y
, which are 2, 4 and 8. After estimating the model by OLS, we
know that
3
2
1
ˆ
80
i
i
Y
=
=
å
. Therefore, the (unadjusted) determination coefficient,
2
R
, is:
A)
0.7857
B)
0.8757
C)
1.0000
– 4 –
Question 6
. Consider the model
12 3
iiii
yxzu
bbb
=
+++
(
1, 2,…, 30
i
=
) which complies
with all the standard hypotheses of
the General Linear Model. If
*
F
stands for the
value of the
F
-statistic to test the global significa
nce of all the slopes in the model,
then the marginal significance (
p-value
) associated with this test is:
A)
*
Pr[ (3, 27)
]
FF
³
B)
*
Pr[ (2, 27)
]
FF
³
C)
*
1 Pr[ (3, 27)
]
FF
³-
Question 7.
Consider the model
12
iii
YXU
bb
=
++
(
1, 2,…, 20)
i
=
, which OLS
residuals are denoted by
ˆ
i
u
(
1, 2,…, 20
i
=
). Assume that the OLS estimation of the
regression (with constant term) of
2
ˆ
i
U
as a function of
i
X
and
2
i
X
(
1, 2,…, 20
i
=
)
yields a
2
R
value of 0.35. If
2
Pr[
(2) 4.61] 0.90
c
=£
and
2
Pr[
(2) 5.99] 0.95
c
=£
, the
null that the model errors (
i
U
) are homoscedastic:
A)
Must be rejected with a 5% signi
ficance, but not with a 10%
B)
Must be rejected both, with a
5% and a 10% significance.
C)
Must be rejected with a 10% significance, but not with a 5%
Question 8.
The test used in the pre
vious question is known as:
A)
Structural change test.
B)
Breusch-Godfrey test.
C)
White test.
Questions 9 to 12 refer to the following case.
We have a sample including: (a) the
scores of 10 students (in the standard 0-10 scale) in the final examination of statistics
(
rfinal
), and in (b) the midterm exam of the same subject (
rmid
). Table 1 provides
some statistics for both variables and Tabl
e 2 shows OLS estimation results of the
simple linear model relating
rfinal
(endogenous) with
rmid
(exogenous). Last, Table 3
provides some results of the OLS estimation
of a model relating variable “difference”
(defined as the difference between
rfinal
and
rmid
) with a constant term.
Table 1:
Sample statistics for
rfinal
and
rmid
Average Median Standard deviation
rfinal
5.5000 5.5000
3.0277
rmid
5.5000 5.5000
3.0277
– 5 –
Table 2:
Model 1: OLS, using observations 1-10
Dependent variable:
rfinal
Coefficien
t
Std. erro
r
T-rati
o
P-valu
e
C
ons
t
0.866667
1.09994
0.7879
0.45345
Rmi
d
0.842424
———
——–
———
Mean dependent var
5.500000
S.D. dependent var
———–
Sum squared resid
23.95152
S.E. of regression
1.730300
R-squared
0.709679
Adjusted
R
-squared
0.673388
F(1,
8)
17.01646
P-value(F)
0.003321
Table 3:
Model 2: OLS, using observations 1-10
Dependent variable:
difference
Coefficien
t
Std. erro
r
Estadístico
t
P-valu
e
Cons
t
——-
0.537484
——
1.00000
Mean dependent var
0.000000
S.D. dependent var
1.699673
Sum squared resid
26.00000
S.E. of regression
1.699673
R-squared
———–
Adjusted
R
-squared
———–
Question 9.
According to the information provid
ed by Tables 1 and 2, the sample
correlation coefficient between
rfinal
and
rmid
is:
A)
0.866667
B)
0.842424
C)
Positive, but we do not have en
ough information to compute it.
Question 10.
According to the information provided by Table 2, the value of the
t
statistic which tests the individual signi
ficance of the parameter associated to
rmid
(use all the available decimals
in your calculations):
A)
Is 4.12510 and the variable rmid is ind
ividually significant at 5% and 10%
significance levels.
B)
Is 1.73030 and the variable
rmid
is individually significant at a 10% significance
level, but not at a 5%.
C)
We do not have enough infor
mation to compute this
t
statistic.
– 6 –
Question 11:
The OLS estimate of the constant and the
2
R
corresponding to the
model in Table 3
A)
…are both equal to zero.
B)
…are both equal to one.
C)
…cannot be computed with the
information in Table 3.
Question 12.
According to the information in
Tables 2 and 3, and knowing that
Pr[ (1,8) 0.68421] 0.43212
F
=³
, the test for the null that the coefficient of
rmid
is equal
to one, against the alternative that it is
different from one (use all the available
decimals in your calculations):
A)
Must be rejected both, with a
5% and a 10% significance.
B)
Must be rejected with a 10% significance, but not with a 5%
C)
Cannot be rejected with a 5% significance.
Questions 13 to 16 refer to the following case. We fitted a regression model relating
the log-price of 546 homess (l_price) as a function of: 1)
lotsize
, the size of the lot in
square meters, 2)
bedrooms
, number of bedrooms, 3)
bathrms
, number of bathrooms,
4)
recroom
, dummy variable which value is 1 if
the home has a games room and zero
otherwise, 5)
aircon
, dummy variable which value
is 1 if the home has air
conditioning and zero otherwise, 6)
prefarea
, dummy variable which value is 1 if the
home is located in an upscale
neighborhood and zero otherwise, and 7)
garagepl
,
number of parking lots. Table 4 shows
the OLS results for this model.
Table 4:
Model: OLS, using observations 1-546
Dependent variable: l_price
Coefficien
t
Std. erro
r
T-rati
o
P-valu
e
const
10.1586
0.0464674
218.6185
lotsize
5.25425e-05
5.17511e-06
10.1529
bedrooms
0.0689575
0.0147725
4.6680
bathrms
0.204855
0.0220184
9.3038
recroom
0.114989
0.0269212
4.2713
0.00002
airco
0.205614
0.0225499
9.1182
prefarea
0.156268
0.0245518
6.3648
garagepl
0.0572742
0.0125616
4.5595
Mean dependent var
11.05896
S.D. dependent var
0.371985
Sum squared resid
———-
S.E. of regression
0.233669
– 7 –
R-squared
0.610473
Adjusted
R
-squared
0.605405
F(7, 538)
120.4519
P-value(F)
7.6e-106
Question 13:
According to the information in
Table 4, the sum of squared residuals
for this model is (use all the avail
able decimals in y
our calculations):
A)
23.36690
B)
23.09111
C)
29.37545
Question 14:
According to the information
in Table 4, choose the CORRECT
statement (you can round your calcul
ations up to two decimals):
A)
An increase of two bathrooms increases
the expected price by 20.49% approx.
B)
An increase of one room decreases th
e expected price
by 6.90% approx.
C)
An additional parking lot increases th
e expected price by 5.73% approx.
Question 15:
According to the results in Table 4, the expected difference between the
price of a home with air conditioning in
comparison with the price of another one
which does not, being equal the othe
r characteristics, is approximately:
A)
20.5614% and is significant both,
with a 5% and 10% significance.
B)
0.205614 monetary units and is significant bo
th, with a 5% and
10% significance.
C)
20.5614% and is significant with a 10%
significance, but not with a 5%
Question 16:
Knowing that: (a) the covariance between the OLS coefficient associated
to
bathrms
and the OLS coefficient associated to
airco
is zero, and (b)
Pr[ (1, 538) 3.8588] 0.05
F
=³
, the null that both parameters are equal, against the
alternative that they are
not equal (use all the avai
lable decimals in your
calculations):
A)
Cannot be rejected with a 5% significance
B)
Must be rejected with a 5% significance
C)
We do not have enough informati
on to test this hypothesis.
Question 17.
Choose which of the following
statements are TRUE and FALSE:
1. An influential observation always has a large residual.
2. The OLS residual plots and histogram
are valid instruments to detect outliers.
– 8 –
3. Heteroscedasticity and autocorrelation ar
e frequent problems when modeling cross-
section and time series data, respectively.
4. A monthly time series with
seasonality can be
mean-stationary.
5. The transformations inducing stability in
the level and dispersion of a time series
are, respectively, differe
ncing and log-transform.
A)
True: 2, 3 and 4. False: 1 and 5
B)
True: 2, 4 and 5. False: 1 and 3
C)
True: 2, 3 and 5.
False: 1 and 4
Questions 18 to 20 refer to the following statement.
Figure 1
displays the quarterly
series of (log) real consumption, denoted by
c
, and (log) real income, denoted by
y
,
from 1960 1
st
quarter to 2009 4
th
quarter.
Figure 2
displays the first-order difference
of both series, that is,
1
_
ttt
dc c c c
–
ºÑ=-
and
1
_
ttt
dy y y y
–
ºÑ=-
.
Figure 1:
Figure 2:
– 9 –
Table 5
displays the results of the OLS estimation of a linear regression relating (log)
consumption with (log) income. Last,
Table 6
summarizes the OLS results for a
regression relating the variables in
first-order differences, that is,
_
dc
as a function
of
_
d
y
Table 5
Model: OLS, using observations 1960:1-2009:4 (T = 200)
Dependent variable: c
Coefficien
t
Std. erro
r
T-rati
o
P-valu
e
const
-0.404163
0.0250534
-16.1320
y
1.03529
0.00294698
351.3050
Mean dependent var
8.383637
S.D. dependent var
0.490502
R-squared
0.998398
Adjusted
R
-squared
0.998390
Rho
0.889723
Durbin-Watson
0.202570
Table 6
Model : OLS, using observations 1960:2-2009:4 (T = 199)
Dependent variable: d_c
Coefficien
t
Std. erro
r
T-rati
o
P-valu
e
const
0.00554085
0.000601578
9.2105
***
d_y
0.339905
0.0494111
6.8791
***
Mean dependent var
0.008330
S.D. dependent var
0.006965
R-squared
0.193688
Adjusted
R
-squared
0.189595
Rho
0.053566
Durbin-Watson
1.888731
Question 18.
According to Figures 1 and 2:
A)
The variable
c
is stationary in the mean but the variable
y
is not.
B)
The quarterly log rates of consumption
and income are stationary in the mean.
C)
The variable
y
is stationary in the mean but the variable
c
is not.
Question 19
. The value of the Breusch-Godfrey statistic, to test the null of no
autocorrelation in the errors of the model
in Table 5, against the alternative that
there is a first-order autocorrelation, is equal to 747.37 with a
p
-value = 0.00000. The
value of this statistic to test the same hypot
hesis on the errors of the model in Table
6 is equal to 0.633, with a
p
-value = 0.427. Accordingly:
– 10 –
A)
The null cannot be rejected with a 5%
significance in any of these models.
B)
The null can be rejected in both
models with a 5% significance.
C)
If we set the significance level at 5%, the null would be rejected for the model in
Table 5 and not rejected for
the model in Table 6.
Question 20
. Choose which of the following statements are TRUE and which are
FALSE:
1. The model in Table 5 s
hould be preferred to the m
odel in Table 6 because the
2
R
is much larger in the former.
2. The OLS estimator for the parameters in
Table 5 is biased and inefficient.
3. According to the model in Table 6,
if the quarterly variation rate of income
increases by 1 percent point, the quarter
ly variation rate of consumption would
increase by 0.34 percentage points approximately.
4. The average quarterly variation rate of consumption is 0.83% in the sample
employed.
A) True: 1 and 2. False: 3 and 4
B) True: 3 and 4. False: 1 and 2
C) True: 1 and 3. False: 2 and 4
– 11 –
OPERACIONES
– 12 –
ECONOMETRICS – FINAL EXAM, 3rd YEAR (GECO & GADE)
May 26, 2015 – 12:00
First family name:
Second family Name:
Name: ECO/ADE:
DNI/ID: Instructor:
Mobile: E-mail:
Question 1
A B C Blank
Question 2
A B C
Blank
Question 3
A
B C
Blank
Question 4
A
B C
Blank
Question 5
A
B C
Blank
Question 6
A
B C
Blank
Question 7
A
B C
Blank
Question 8
A
B
C
Blank
Question 9
A
B C
Blank
Question 10
A
B C
Blank
Question 11
A
B C
Blank
Question 12
A
B
C
Blank
Question 13
A
B
C
Blank
Question 14
A
B
C
Blank
Question 15
A
B C
Blank
Question 16
A
B C
Blank
Question 17
A
B
C
Blank
Question 18
A
B C
Blank
Question 19
A
B
C
Blank
Question 20
A
B C
Blank
Correct Incorrect Blank Final grade
