Econometrics
Mid-Term
A
April 2008
João Valle e Azevedo
António José Morgado
Tiago Vieira
Time for completion: 70 min
For each question, identify the correct answer.
For each question, there is one and only one correct answer.
A correct answer is worth 1 point; an incorrect answer shall be attributed 0 points.
Mark your choice on the answer sheet provided at the end of this examination paper.
EITHER USE A PENCIL OR DO NOT MAKE CORRECTIONS.
Identify your answer sheet with your name and student number
1. In a multiple linear regression where the Gauss-Markov assumptions hold, why can
you interpret each coe¢ cient as a ceteris paribus e¤ect?
(a) Because the Ordinary Least Squares (OLS) estimator of the coe¢ cient on
variable xj is based on the covariance between the dependent variable and the
variable xj after it is purged from the e¤ects of other regressors.
(b) Because the regressors are not correlated with the error term
(c) Because the model is linear
(d) Because the Romans said so and all the roads lead to Rome
(e) None of the answers above is correct
2. Of the following assumptions, which one(s) is (are) NOT necessary to guarantee
unbiasedness of the OLS estimator in a multiple linear regression context?
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Linearity of the model in the parameters
Zero conditional mean of the error term
Absence of multicollinearity
Homoskedasticity of the error term
Random sampling
Both c) and d) above
Both d) and e) above
3. In a random sample:
(a) All the individuals or units from the population have the same probability of
being chosen
(b) The observed values of the independent variables are all di¤erent
(c) The observed values of the dependent variable are all di¤erent
(d) Both a) and b) above
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4. In a multiple linear regression, what means heteroskedasticity?
(a) All error terms ui have the same variance across all i = 1; 2; :::; n
(b) The regressors have each the same variance across all random observations
i = 1; 2; :::; n
(c) Two error terms ui and uj with j 6= i have always a di¤erent variance for
i = 1; 2; :::; n
(d) The error terms ui do not have the same variance across all i = 1; 2; :::; n
(e) The regressors and the error term are linearly independent
\
5. In the estimated model log
(qi ) = 502; 57 0:9 log (pi ) + 0:6 log (psi ) + 0:3 log (yi ) ;
where p is the price and q is the demanded quantity of a certain good, ps is the
price of a substitute good and y is disposable income, what is the meaning of
the coe¢ cient on p? (Assume that the Gauss-Markov assumptions hold in the
theoretical model)
(a) If the price increases by 1%, the demanded quantity will be 0.009% lower on
average, ceteris paribus
(b) If the price increases by 1%, the demanded quantity will be 90% lower on
average, ceteris paribus
(c) If the price increases by 1%, the demanded quantity will be 0.9% lower on
average, ceteris paribus
(d) None of the answers above is correct
\
6. In the estimated model log
(qi ) = 502; 57 0:9 log (pi ) + 0:6 log (psi ) + 0:3 log (yi ) ;
where p is the price and q is the demanded quantity of a certain good, ps is the
price of a substitute good and y is disposable income, what is the meaning of the
coe¢ cient on ps?
(a) It is the cross-price elasticity of demand in relation to the substitute good and
it bears the expected sign.
(b) It is the cross-price elasticity of demand in relation to the substitute good but
it does not bear the expected sign.
(c) It is the semi-elasticity of demand in relation to a substitute good and it bears
the expected sign.
(d) It is the semi-elasticity of demand in relation to a substitute good but it does
not bear the expected sign.
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7. In the model GP A = 0 + 1 study + 2 leisure + 3 sleep + 4 work + u where each
regressor is the amount of time (hours), per week, a student spends in each one of the
named activities and where the time alocation for each activity is explaining Grade
Point Average, what assumption is necessarily violated if the weekly endowment
of time (168 hours) is entirely spent either studying, or sleeping, or working or in
leisure activities?
(a) Linearity of the model
(b) Absence of multicollinearity
(c) Zero conditional mean of the error term
(d) Homoskedasticity
(e) Both a) and b) above
(f) Both b) and c) above
(g) Both b) and c) as well as d) above
8. Let R2 be the R-squared of a regression (that includes a constant), SST be the
total sum of squares of the dependent variable and df be the degrees of freedom.The
estimator of the error variance, b2 , can be written as:
(a)
(1 R2 )SST
df
(b)
R2 :SST
df
(c)
R2
SST (1 R2 )df
(d) None of the answers above is correct
9. Take an observed (that is, estimated) 95% con…dence interval for a parameter of a
multiple linear regression. Then:
(a) With a probability of 95%, the true parameter value lies in the con…dence
interval
(b) We cannot assign a probability to the event that the true parameter value lies
inside that interval
(c) With a probability of 5% we reject the null that Portugal will win the Euro
2008
(d) With a probability of 5% the true parameter lies outside the interval
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To answer questions 10 and 11 consider the following Eviews output of the model,
where wage is the wage in euros, educ is the level of education measured in years
and hours is the average weekly hours of work:
wagei =
0
+
1 educi
+
2 hoursi
+
3 agei
(delete this term) + ui
10. What can you say about the estimated coe¢ cient of the variable educ? (consider
a one-sided alternative for testing signi…cance of the parameters)
(a) For each additional year of education, wage is predicted to increase by 60:88
euros, on average, ceteris paribus. But it is not statistically signi…cant at a
5% level of signi…cance.
(b) For each additional year of education, wage is predicted to increase by 60:88
euros, on average, ceteris paribus. And it is statistically signi…cant at a 5%
level of signi…cance.
(c) For each additional year of education, wage is predicted to increase by 60:88%,
on average, ceteris paribus. And it is statistically signi…cant at a 5% level of
signi…cance.
(d) It is statistically signi…cant at a 5% level of signi…cance but it is not signi…cant
at 1% level of signi…cance.
11. What is the F-statistic for the overall signi…cance of the model
(a) 24.832
(b) 56.747
(c) 56.808
(d) We have not enough information to answer this question, we would need to
estimate the restricted model.
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12. Consider the model: log(pricei ) = 0 + 1 scorei + 2 breederi + ui , where price
is the price of an adult Lusitano Horse, score is the grade given by a juri (higher
score means higher quality of the horse) and breeder is the reputation of the horse
breeder. Because reputation of the breeder is di¢ cult to measure we decided to
estimate the model omitting the variable breeder. What bias can you expect in the
score coe¢ cient, assuming breeder reputation is positively correlated with score?
(a) The estimated coe¢ cient of score will be biased downwards, perhaps even
becoming negative.
(b) The estimated coe¢ cient of score will be biased upwards.
(c) The estimated coe¢ cient of score will be biased downwards, but cannot become negative.
(d) There will be no bias in the coe¢ cient on score.
(e) None of the answers above is correct.
\ i ) = 5:84+0:21scorei +0:13breederi . What
13. Consider the estimated model: log(price
is the interpretation of the estimated coe¢ cient on score?
(a) Each additional grade point increases the horse’s price by 21%, on average,
ceteris paribus.
(b) Each additional grade point increases the horse’s price by 21 units, on average,
ceteris paribus.
(c) Each additional grade point increases the horse’s price by 0:21%, on average,
ceteris paribus.
(d) Each additional grade point increases the horse’s price by 0:21 units, on average, ceteris paribus.
(e) None of the answers above is correct
14. Consider the model: yi = 0 + 1 xi1 + 2 xi2 + 3 xi3 + ui , satisfying the GaussMarkov Assumptions. Assume that x1 and x2 are positively correlated and 2 is
negative. What is the bias of the estimator of 1 if we estimate this model omitting
x2 using OLS?
(a) The estimator will be biased upwards .
(b) The estimator will be biased downwards .
(c) Only if we assume that x1 and x3 are not correlated, we can infer about the
sign of this bias and it will be negative.
(d) There will be no bias.
(e) None of the answers above is correct
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15. Consider that the true model for a population is yi = 0 + 1 xi + ui , and assume
that all the Classical Linear Model Assumptions hold. Now consider the following
estimator ~ 1 = yxnn yz2 . What should z be equal to so that ~ 1 is an unbiased estimator
for 1 ?
(a) xn
(b) x2
(c) yn
(d) 1
(e) None, because it will be always biased
16. In the simple linear regression model for cross-sectional data, the zero conditional
mean assumption, stating that E[ujx] = 0, implies:
(a) E[xju] = 0
(b) x and u are uncorrelated
(c) E[u] = 0
(d) Both b) and c) above are correct
(e) Both a) and c) above are correct
(f) Both a) and b) as well as c) above are correct
17. Given the variance of the OLS estimators in matrix form:
(a) It is only possible to know the variance of the individual estimators
(b) It is possible to know the variance of the individual estimators as well as their
covariances
(c) It is possible to derive the variance of a sum of individual estimators
(d) It is possible to derive the variance of a linear combination of estimators
(e) Both b) and c) above are correct
(f) Both b) and c) as well as d) above are correct
(g) None of the answers above is correct
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18. In testing multiple exclusion restrictions in the multiple regression model under the
Classical assumptions, we are more likely to reject the null that some coe¢ cients
are zero if:
(a) The R-squared of the unrestricted model is large relative to the R-squared of
the restricted model
(b) The R-squared of the unrestricted model is small relative to the R-squared of
the restricted model
(c) The total sum of squares, SST , is large
(d) The intercept parameter is greater than the signi…cance level
(e) Both a) and c) above
(f) Both b) and c) above
19. Consider the following linear regression models:
Model 1: yi = 0 + 1 xi1 + 2 xi2 + ui and Model 2: yi = 0 + 1 xi1 + ui . Consider
the OLS estimates b1 (for Model 1) and e1 (for Model 2 ), computed using the
same sample observations (Obviously, for e1 we do not use the observations on x2 ).
Then b1 = e1 , necessarily, if:
(a) x1 and x2 are uncorrelated in the population
(b) x1 and x2 are uncorrelated in the sample
(c) the estimated e¤ect b2 (for Model 1) is equal to zero
(d)
2
=0
(e) Both a) and c) above are correct
(f) Both b) and c) above are correct
(g) Both a) and d) above are correct
(h) Both b) and d) above are correct
(i) Both a), b), c) and d) above are correct
20. Under the Gauss-Markov assumptions in the multiple linear regression model:
(a) The OLS estimator is Consistent for the model parameters
(b) The OLS estimator is necessarily Normally distributed
(c) The OLS estimator is asymptotically Normally distributed
(d) The OLS estimator is an LM statistic for testing multiple exclusion restrictions
(e) Both a) and b) above are correct
(f) Both a) and c) above are correct
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ANSWER SHEET A
Name:
Number:
Mark your answers with an X in the table below.
Any answers outside the table will not be considered.
EITHER USE A PENCIL OR DO NOT MAKE CORRECTIONS
Question a) b) c) d) e) f) g) h) i)
1.
X
2.
X
3.
X
4.
X
5.
X
6.
X
7.
X
8.
X
9.
X
10.
X
11.
X
12.
X
13.
X
14.
X
15.
X
16.
X
17.
X
18.
X
19.
X
20.
X
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