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ECO5185 Midterm Answers; Fall 2025.
Question 1
a. Briefly explain (in words) the idea/concept of each of the following, and briefly explain their
implication for econometric analyses
(a) random sample
Answer: A random sample is a sample whose draws are independent and identically
distributed. It means that for the simple linear regression model A3, i.e.
E ( ϵi | X ) = 0 (1)
simplifies to
E ( ϵi | xi 2 ) = 0_._ (2)
As such, for the OLS estimator to be unbiased we only need to worry about the explana-
tory variable not being correlated with the error term within each draw. It will also have
implications for the structure of the variance of our estimator.
(b) random variable
Answer: A random variable is a variable whose outcome is uncertain/unknown.
Given that yi and xik ( i = 1 ,... , n and k = 1 ,... , K ) are random variables, the OLS
estimator, βk ( k = 1 ,... , K ) is also a random variable. It has a mean and a variance,
and as such, has statistical properties and we can carry-out hypothesis testing.
b. Discuss how the addition of a control variable has implications for
(a) the variation used to estimate a parameter of interest
Answer: By the FWL theorem, the addition of a control variable means that we are not
using some of the variation in x 2 when estimating its parameter (i.e. β 2 ). More precisely,
we are not using the variation in x 2 that is correlated with the new control.
(b) the validity of a test
Answer: Adding a control may result in assumption A3 holding, if the additional variable
belongs in the model and is correlated with the explanatory variable. Recall that if A
does not hold the t-test and F-test are not valid.
c. Briefly evaluate the following statement
“ The R^2 measures the proportion of the variation in the dependent variable that is explained by
the regression line, and as such, a high R^2 implies the estimate of the parameter of interest is
probably close to the true parameter.”
Answer: The first part of the statement is true. The R^2 measures how much of the variation in
the dependent variable is explained by the OLS regression line. Having said this, the second
part of the statement is not true. Having a high R^2 has no bearing on whether assumption
A3 holds, and as such no bearing on whether the estimator is unbiased or consistent. As such,
it has no bearing on whether the estimate is close (or probably close) to the true.
d. You are interested in exploring whether men, on average, have a higher hourly wage than
women in the Canadian labour market. Your co-author suggests the following estimator
i ∈ male (^) wage i — nm
i ∈ female wagei nf
where wagei is the hourly wage of individual i , and nm and nf are the number of males and
females in the sample, respectively.
Would this estimator, when applied to data, generate a guess that close to the true parameter
of interest? Justify your answer.
Answer: This is a method of moment estimator (i.e. the sample analogs of the population
moments). It is made up of sample means, and we know that sample means converge in
probability to population means if we have a random sample. Now, if the estimator is con-
sistent (i.e. we have a random sample) and we have a large sample, one can conclude that the
estimator is probably close to the true population parameter. We cannot, however, say it
with certainty.
Question 2
Assume the true population model takes the form
yi = β 1 + β 2 xi 2 + ϵi
Σ x Σ Σ Σ x
Σ Σ Σ x Σ Σ x Σ x
i = n i =
2 i 2 i =
i =1 i 2 n i =
i =1 i 2 i =
n i =
2 i 2
i =
i = 2 i 2 i =
n i = i = n i = i =
Answer: Σ n yixi 2
Σ n ( β 2 xi 2 + ϵi ) xi 2 = (^) n i = n (^) [ β 2 x 2
2 i 2
β 2
Σ n x^2
Σ n ϵixi 2
Σ n ϵixi 2
c. Will this slope parameter estimator be unbiased? Show your work.
Answer: Σ n ϵixi 2
Σ n ϵixi 2
Σ n xi 2 E [ ϵi | X ]
Σ n xi 2 · 0 = β 2 +
= β 2
n (^) 2 i =1 i 2
It will be unbiased if A3 holds.
Question 3
a. Show the following
(a)
PM = 0
= E [ β 2 | X ] + E
x
x
x
x
b 2 =
2 i 2 =
2 i 2
= β 2 +
E [ b 2 | X ] = E β 2 +^ n i =
| X
2 i 2
| X
= β 2 + (^2) i 2
n i =1 x
2 i 2
Answer:
PM = X ( X ′ X )−^1 X ′[ I − X ( X ′ X )−^1 X ′]
= [ X ( X ′ X )−^1 X ′ I ] − [ X ( X ′ X )−^1 X ′ X ( X ′ X )−^1 X ′]
= [ X ( X ′ X )−^1 X ′ I ] − [ X ( X ′ X )−^1 IX ′]
= [ X ( X ′ X )−^1 X ′] − [ X ( X ′ X )−^1 X ′]
(b)
M is idempotent
Answer:
MM = [ I − X ( X ′ X )−^1 X ′][ I − X ( X ′ X )−^1 X ′]
= [ II ] − [ IX ( X ′ X )−^1 X ′] − [ X ( X ′ X )−^1 X ′ I ] + [ X ( X ′ X )−^1 X ′ X ( X ′ X )−^1 X ′]
= I − X ( X ′ X )−^1 X ′^ − X ( X ′ X )−^1 X ′^ + X ( X ′ X )−^1 IX ′
= I − X ( X ′ X )−^1 X ′^ − X ( X ′ X )−^1 X ′^ + X ( X ′ X )−^1 X ′
= I − X ( X ′ X )−^1 X ′
= M
(c)
[( M 1 X 2 )′( M 1 X 2 )]−^1 ( M 1 X 2 )′( M 1 y ) = [ X ′^ X 2 − X ′^ X 1 ( X ′^ X 1 )−^1 X ′^ X 2 ]−^1 [ X ′^ y − X ′^ X 1 ( X ′^ X 1 )−^1 X ′^ y ] 2 2 1 1 2 2 1 1
where the underlying population model is
y = X 1 β 1 + X 2 β 2 + ε
Answer:
[( M 1 X 2 )′( M 1 X 2 )]−^1 ( M 1 X 2 )′( M 1 y ) = [ X ′^ M 1 X 2 ]−^1 [ X ′^ M 1 y ] 2 2 = [ X ′^ ( I − X 1 ( X ′^ X 1 )−^1 X ′^ ) X 2 ]−^1 [ X ′^ ( I − X 1 ( X ′^ X 1 )−^1 X ′^ ) y ] 2 1 1 2 1 1 = [( X ′^ − X ′^ X 1 ( X ′^ X 1 )−^1 X ′^ ) X 2 ]−^1 [( X ′^ − X ′^ X 1 ( X ′^ X 1 )−^1 X ′^ ) y ] 2 2 1 1 2 2 1 1 = [ X ′^ X 2 − X ′^ X 1 ( X ′^ X 1 )−^1 X ′^ X 2 ]−^1 [ X ′^ y − X ′^ X 1 ( X ′^ X 1 )−^1 X ′^ y ] 2 2 1 1 2 2 1 1
(c) Another co-author suggests the following estimator
b = ( X ′ PZX )−^1 X ′ PZy
where P Z = Z ( Z ′ Z )−^1 Z ′, with Z being the X matrix where the information for the xK -
th variable is replaced with information on z 1 and z 2. Therefore Z is n × K + 1. P Z is
both symmetric and idempotent. Show that, under certain identifying assumptions, this estimator will be consistent. (5 marks)
Answer:
b = ( X ′ PZX )−^1 X ′ PZy
= ( X ′ PZX )−^1 X ′ PZ ( X β + ε )
= β + ( X ′ PZX )−^1 X ′ PZ ε (prop. of matrices)
= β + ( X ′ Z ( Z ′ Z )−^1 Z ′ X )−^1 X ′ Z ( Z ′ Z )−^1 Z ′ ε
X ′ Z Z ′ Z
− 1 Z ′ X
X ′ Z Z ′ Z
− 1 Z ′ ε
X ′ Z Z ′ Z
− 1 Z ′ X
X ′ Z Z ′ Z
− 1 Z ′ ε
plim ( b ) = plim β + n n n n^ n^ n
= plim β + plim
X ′ Z
n
plim
Z ′ Z
− 1
n
plim
Z ′ X
− 1
n
plim
X ′ Z
n
plim
Z ′ Z
− 1
n
plim
Z ′ ε
n
= β + ( Q XZ Q −^1 Q ZX )−^1 Q XZ Q −^1 0 ZZ ZZ = β
Question 5
The STATA output needed to answer this question can be found in the following pages.
Assume the following population model
wagei = β 1 + β 2 unioni + β 3 indigenousi + β 4 westi + β 5 easti
- β 6 indigenousi · westi + β 7 indigenousi · easti + εi
where wagei is the hourly wage (in dollars) of individual i. union is a binary variable equal to
one if the individual is unionized or covered by a union (and zero otherwise), and indigenous is
a binary variable that equals one if the person identifies themselves as being indigenous (and zero
n n n n n n = β +
2
2
otherwise). Finally, west is a binary variable that equals one if the person lives in Western Canada
(and zero otherwise) and east is a binary variable that equals one if the person lives in Eastern Canada
(and zero otherwise).^1
a. Interpret the coefficient estimate of β 2.
Answer: Holding all other factors constant, a worker that is unionized (or covered by a union)
makes $ 5.04 more per hour than a worker that is not unionized (nor covered by a union).
b. Test whether being unionized or covered by a union has an impact on the wage. Show your
work. (5 marks)
Answer:
Step 1:
H 0 : β 2 = 0
Ha : β 2 ̸= 0
Step 2: Under A 1, A 2 a , A 3, A 4, A 5, and A 6
tstat^ =
bols^ − β 2 se ( bols )
~ tn − K ( t 53779 )
Step 3:
tstat^ =
Furthermore, the critical values are is 1_._ 962 and - 1_._ 962
Step 4 (conclusion): Since
tstat^ = 35_._ 652 > 1_._ 962
one rejects H 0 in favour of Ha at the 5% level of significance. Said less formally, being unionized
(or covered by a union) impacts the wage.
c. Do you believe that the test carried above is valid? Justify your answer. (5 marks)
Answer: Gender probably belongs in the wage equation and women tend to be more unionized
than men. As such, assumption A3 probably does not hold which means that the test is not
valid. (^1) Central Canada is the reference group.
i
sw 2 = i = n − 1
sw,z = i = n − 1
Equations
Properties of sums
n n n Σ ( xi 2 + yi ) =
xi 2 +
yi i =1 (^) n i =1 n i =
ayi = a
yi i = n
i =
a = n · a i =
other equations
n n n Σ ( yi − y ¯) = 0
( xi 2 − x ¯ 2 )( yi − y ¯) =
( xi 2 − x ¯ 2 ) yi
Expectation properties (with a being a constant)
E ( wi + zi ) = E ( wi ) + E ( zi )
E ( awi ) = aE ( wi )
E ( a ) = a
E ( wi ) = Ez { E [ wi | zi ] }
Variance and covariance
V ar ( wi ) = E ( wi − E ( wi ))^2
= E ( w^2 ) − E ( wi ) E ( wi )
Cov ( wi, zi ) = E ( wi − E ( wi ))( zi − E ( zi ))
= E ( wizi ) − E ( wi ) E ( zi )
Sample variance and sample covariance
Σ n ( wi − w ¯ )^2
Σ n ( wi − w ¯)( zi − z ¯)
i =1 i =1 i =
i =
s^2 =
1 Σ^
e^2
Simple linear regression model
yi = β 1 + β 2 xi 2 + εi
Σ n ( xi 2 − x ¯ 2 )( yi − y ¯)
var ( bols | X ) = (^) n i =
σ^2 ( xi 2 — x ¯ )^2
est. var ( bols | X ) = (^) n i =
s^2 ( xi 2 — x ¯ )^2
where
se ( bols ) =
n − 2
s
Σ n
i =
s^2 ( x
i
— x ¯)^2
i =1 i
yi = y ˆ i + ei where y ˆ i = b 1 + b 2 xi 2
Simple linear regression model assumptions
A 1: Linearity (in the parameters) of the regression model
A 2 a : Variation in x 2 (in the sample)
A 3 (Regarding the error term ( ε )): Exogeneity of X , E ( εi | X ) = 0
A 4: Homoskedasticity and absence of autocorrelation var ( εi ) = σ^2 and cov ( εi, εj ) = 0 for i ̸= j
A 5: Data Generation (stochastic process for the explanatory variable(s))
n i =1 ( x^ i^2 —^ x ¯^2 ) 2
ols ols ols b 1 = y ¯ − b 2 x ¯^2 b 2 =
n
e
X
R-squared
R^2 =
SSR
SST
where SST (total sum of squares) is
n ( yi − y ¯)^2 i =
SSR (regression sum of squares) is
n ( y ˆ i − y ¯)^2 i =
and SSE (sum of squared error) is
n 2 i i =
Properties of Matrices
a. Given two matrices ( A and B ) of equal dimensions
A + B = B + A
b. Given three matrices ( A , B and C ) of equal dimensions
( A + B ) + C = A + ( B + C )
c. Given two matrices ( A and B ) of equal dimensions
( A + B )′^ = A ′^ + B ′
d.
( A ′)′^ = A
e. In general,
AB ̸= BA
f.
( AB ) C = A ( BC )
g.
A ( B + C ) = AB + AC and ( A + B ) C = AC + BC
h.
( AB )′^ = B ′ A ′
i.
AI = IA = A
where I is the identity matrix
j. If A is square and of full rank, A −^1 exists, and
A −^1 A = AA −^1 = I
k.
( A −^1 )−^1 = A
l.
( A −^1 )′^ = ( A ′)−^1
m.
( AB )−^1 = B −^1 A −^1
Multiple linear regression model assumptions
A 1: Linearity (in the parameters) of the regression model
A 2: X is full rank (i.e. rank ( X ) = K ).
A 3 (Regarding the Error Term ( ε )): Exogeneity of X , E ( ε | X ) = 0
A 4: Homoskedasticity and absence of autocorrelation
E ( εε ′| X ) = σ^2 I n
A 5: Data Generation (stochastic process for the explanatory variable(s))
A 6: Normality of the error terms - The disturbance terms are normally distributed, conditional on
x
Fstat^
( R^2 − R ∗^2 ) /J
(1 − R^2 ) /n − K
