ECMT1020 Introduction to Econometrics 2022S1
Midterm Questions
Probability
1. A random variable X follows the normal distribution: X ⇠ N(3, 27). What is the expected
value of another random variable Y = 10 13X?
Answer: 10 13 ⇥ E(X) = 10 1 = 9.
2. Let X and Y be two correlated random variables with covariance 10. We know that
• the standard deviation of X is 3, and
• the expected values of Y and Y 2 are, respectively, 4 and 97.
What is the standard deviation of a third random variable Z = 3X 2Y ?
Answer: Note that
Var(Z) = Var(3X 2Y ) = Var(3X) + Var(2Y ) 2Cov(3X, 2Y )
= 9Var(X) + 4Var(Y ) 12Cov(X,Y )
= 9⇥ 32 + 4[E(Y 2) (E(Y ))2] 12⇥ 10
= 9⇥ 32 + 4(97 42) 120
= 9⇥ 32 + 4⇥ 81 120 = 285.
Therefore, the standard deviation of Z is Std(Z) =
p
Var(Z) =
p
285 ⇡ 16.88.
3. Suppose X is a random variable indicating the face value when a single unfair dice is thrown.
The probability distribution of X is given by
Pr(X = x) =
7 x
21
, x = 1, 2, 3, 4, 5, 6.
Let Y be another random variable given as Y = 12 3X. What is the covariance of X and Y ?
Answer: The covariance of X and Y is
Cov(X,Y ) = Cov
✓
X,
1
2
3X
◆
= 3Cov(X,X) = 3Var(X).
The variance of X can be computed from the the probability distribution of X using the formula
Var(X) = E(X2) [E(X)]2 = 196
21

✓
56
21
◆2
⇡ 2.22.
So the covariance of X and Y is 6.66.
Statistics
1. Let X be a random variable with variance 1. Suppose we have a random sample for X with only
two observations, X1 and X2. We construct an estimator Z = 0.3X1+0.7X2 for the population
mean of X. Select all correct statement(s).
(a) Z is an unbiased estimator for the population mean of X.
(b) To estimate the population mean of X, Z is less ecient than another estimator Y =
0.5X1 + 0.5X2.
(c) Z is a biased estimator for the population mean of X.
(d) The variance of Z is 0.5.
1

(e) The mean squared error of Z is greater than the variance of Z.
Answer : (a) and (b).
2. Consider the following summary of a data set on the weekly earnings (in hundreds of dollars)
of residents of NSW with a colleague degree:
• Number of observations (n): 25
• Sample mean (X): 20.5
• Sample standard deviation (ˆX): 8
We assume that the weekly earning in the population is normally distributed with unknown
mean and unknown variance.
Please select the correct 95% confidence interval for the mean of the weekly earning (in hundreds
of dollars) in the population.
(a) [17.2, 23.8]
(b) [17.8, 23.2]
(c) [19.8, 21.2]
(d) [20.0, 21.0]
(e) [6.8, 34.2]
(f) [4.0, 37.0]
Answer : (a). X±tn1,0.025 · ˆX = X±tn1,0.025 · ˆXpn = 20.5±t24,0.025 8p25 = 20.5±1.6⇥2.0639.
So the confidence interval is [17.2, 23.8].
3. The sample mean estimator becomes more ecient when sample size increases. (True or false?)
Answer : false
4. A researcher is evaluating whether a job training program had an e↵ect on employment of NSW
veterans in the following half a year. Taking a sample of 30 veterans who took the training,
what should the researcher conclude if there is a mean increase in employment of 15 percent
with 6 percent standard error of the mean?
(a) The program had a significant positive e↵ect on employment at the 10% significance level.
(b) The program had a significant positive e↵ect on employment at the 5% significance level.
(c) The program had a significant positive e↵ect on employment at the 1% significance level.
(d) There is not a significant e↵ect found at any of the 10%, 5%, or 1% significance level.
Answer: (a) and (b) There are 30 1 = 29 degrees of freedom, and the critical values of t test
are
t29,0.05 = 1.699, t29,0.025 = 2.045, t29,0.005 = 2.756,
for 10%, 5%, 1% tests. The t statistic is 12/6 = 2.5. So we reject at both 10% and 5% levels,
but not in the 1% level.
Regression
1. Which of the following is/are not considered as a classical linear regression model?
(a) Y = (1 + 2X)1 + u.
(b) Y = 1 + 2X1 + e3X2 + u.
(c) Y = 1 + 2
p
X + u.
(d) log(Y ) = 1 + 2 log(X) + u.
(e) log(Y ) = 1 + 2X + u.
2
Answer : (a) and (b).
2. We have a sample of two random variables X and Y . The sample means of X and Y are,
respectively, 4 and 10. We run a simple OLS regression of Y on X and obtain the intercept as
30. What is the value of the estimated slope coecient of this regression?
Answer: We know that the OLS estimator of the intercept is
ˆ1 = Y ˆ2X
where ˆ2 is the OLS estimator of the slope coecient. Therefore,
ˆ2 =
Y ˆ1
X
=
10 30
4
= 5.
3. Suppose we run an OLS regression of Y on X, and obtain the slope coecient as 2.4. However,
later the units of measurement of X got changed, and the new measure is 1/12 of the original
one. What is the new slope coecient if we run the OLS regression of Y on X with the new
measurement?
Answer: 2.4⇥ 12 = 28.8. See Exercise 1.13 in textbook.
4. The R2 of an OLS regression is 0.75. The sum of squared residuals is 10. What is the value of
the below quantity?
nX
i=1
(Yˆi Yˆ )2, where Yˆ = 1
n
nX
i=1
Yˆi.
Answer: This quantity is ESS. Given
R2 =
ESS
TSS
=
ESS
ESS+ RSS
,
we have
0.75 =
ESS
ESS+ 10
from which we can solve ESS= 30.
5. For a multiple regression
Y = 1 + 2X2 + 3X3 + u,
the 95% confidence interval for 2 will be
(a) centered at ˆ2.
(b) wider if the sample size becomes smaller.
(c) centered at 2.
(d) wider than the 99% confidence interval.
(e) wider if the standard error of ˆ2 becomes smaller.
Answer : (a) and (b).
6. Select the correct statement(s) about regression analysis.
(a) The residual is a concept related to the fitted model, not the true model.
(b) The di↵erence between the disturbance term and the residual is related to the di↵erence
between the parameters and estimates of the parameters.
(c) We can observe the disturbance term.
(d) We have realizations of the disturbance term.
3
(e) The disturbance term is the di↵erence between the dependent variable and the fitted value
of the dependent variable.
Answer : (a) and (b).
7. Below is the OLS regression output of a simple linear regression
e = 1 + 2g + u
where e denotes the average annual percentage rate of growth of employment, and g denotes
the average annual percentage rate of growth of real GDP, of 31 OECD countries for the period
2002–2007. We know that all the observations for e and g in our sample are positive.
Now suppose we demean the variable g and run an OLS regression of e on the demeaned g.
What do you expect?
(a) The new slope coecient will be 0.2379082.
(b) The new intercept will be greater than 0.4917419.
(c) The new slope coecient will be lower than 0.2379082.
(d) The new slope coecient will be greater than 0.2379082.
(e) The new intercept will be 0.4917419.
(f) The new intercept will be lower than 0.4917419.
Answer: (a) and (b). When the regressor is demeaned, the slope coecient does not change
while the intercept is estimated as the sample mean of the dependent variable. That is, ˆ⇤2 =
ˆ2 = 0.2379082 and ˆ⇤1 = e, where ˆ⇤1 and ˆ⇤2 denote the new intercept and new slope coecient.
Since we know that ˆ1 = e ˆ2g and, given the information in the question, ˆ2 = 0.2379082 > 0
and g > 0 (all observations are positive), it follows that ˆ1 < e = ˆ⇤1 . That is, ˆ⇤1 > ˆ1 =
0.4917419.
8. Consider a simple regression model without intercept:
Yi = Xi + ui, i = 1, . . . , n.
Suppose ˆ is the OLS estimator for . Among the below assumptions of CLRM, which one(s)
didn’t we use when proving that ˆ is an unbiased estimator for ?
(a) X is the fixed regressor.
(b) E(ui) = 0.
(c) The model is correctly specified.
(d) ui are mutually independent.
(e) ui follow the normal distribution.
Answer: (d) and (e).