Effect of Logarithm of Annual Family Income Questions

DescriptionProblem Set II
Data Driven Economic Analysis (DSE)
Microeconometrics Module
Prof. Michele De Nadai
Trimester II 2023
1. The following table reports the value of potential outcomes Y0 and Y1 , along with the observed outcome
Y , the treatment value D and a control X (gender) for a population with 10 statistical units:
unit
1
2
3
4
5
6
7
8
9
10
Y0
10
4
10
1
11
20
10
3
2
5
Y1
12
8
10
3
9
21
12
5
15
12
Y
12
4
10
3
9
20
12
5
2
12
D
1
0
0
1
1
0
1
1
0
1
X
male
male
male
male
male
male
female
female
female
female
a) Compute the value of the individual treatment effect of D on the outcome Y for the first unit in this
population.
b) Compute the value of the individual treatment effect of D on the outcome Y for the fifth unit in this
population.
c) Compute the value of the Average Treatment Effect (ATE): E(Y1 − Y0 ) in this population.
d) Compute the value of the Average Treatment Effect on the Treated (ATT): E(Y1 − Y0 |D = 1).
e) Compute the value of the Average Treatment Effect conditional on X: τ (X) = E(Y1 − Y0 |X).
f) Compute the quantity ∆ = E(Y |D = 1) − E(Y |D = 0). Under which condition can you expect ∆ =
ATE?
1
2. Suppose we have availability of a binary instrument Z. Below you find an augmented version of the table
above where you also find the values of the instrument and potential treatments for each unit in the
population.
unit
1
2
3
4
5
6
7
8
9
10
Y0
10
4
10
1
11
20
10
3
2
5
Y1
12
8
10
3
9
21
12
5
15
12
Y
12
4
10
3
9
20
12
5
2
12
D0
0
0
0
1
0
0
1
1
0
1
D1
1
0
0
1
1
1
1
1
0
1
Z
1
0
1
0
1
0
1
0
1
0
D
1
0
0
1
1
0
1
1
0
1
X
male
male
male
male
male
male
female
female
female
female
a) Is the monotonicity assumption satisfied for this population?
b) Compute the Local Average Treatment Effect (LATE): E(Y1 − Y0 |D1 > D0 ) in this population.
c) What is the proportion of compliers in this population?
3. Using data from question 4 in the previous problem set we are interested in estimating the parameters of
the following linear model:
read = α0 + α1 female + α2 cathhs + u,
(1)

where read is the (standardized) score on reading; female
is a binary variable equal to one for females
and zero otherwise and cathhs is a dummy equal to one if the student attended a catholic High School.
a) Is there any reason to believe that the OLS estimator of α2 might be biased?
b) Some researchers are suggesting the use of parcath, a binary variable equal to one if parents are
catholic and zero otherwise, as an instrumental variable for cathhs in equation (1). Under which
conditions is this justified?
c) Describe the OLS regressions you would run to obtain the Two Stage Least Squares (2SLS) estimate
of α2 using parcath as an instrument for cathhs.
d) Suppose you have availability of two separate binary variables:
• mothcath: equal to one if the mother is catholic and zero otherwise
• fathcath: equal to one if the father is catholic and zero otherwise
Consider using both mothcath and fathcath as instruments for cathhs. Could you test for the
validity of the instruments in this scenario? How would you run the test and what are the assumptions
behind such a test?
e) Under which conditions is the 2SLS estimator for α2 using parcath as an instrument for cathhs
estimating the Average Treatment Effect of attending catholic high school on reading scores?
f) Describe in your own words who are the compliers when using parcath as an instrument for cathhs?
Also, how would you describe the no-defiers assumption (monotonicity) in this context?
2
Problem Sets
Data Driven Economic Analysis (DSE)
Microeconometrics Module
Prof. Michele De Nadai
Trimester II 2023
1. Consider a linear regression of y on X:
y = Xβ + u
The OLS estimator for β is obtained as (X′ X)−1 X′ y. It follows that ŷ = X(X′ X)−1 X′ y Obtain the OLS
estimator for a regression of ŷ on X.
2. The orthogonal matrix in a linear regression of y on X has the following expression:
M = I − X(X′ X)−1 X′
Show that M is idempotent: M = MM.
3. Consider a random variable y and denote θ = E(y 3 ).
a) Construct a moment estimator (θ̂) for θ.
b) Show that θ̂ is unbiased.
c) Derive an expression for V ar(θ̂)
d) Propose an estimator for V ar(θ̂)
e) Derive the asymptotic distribution for θ̂
4. You are interested in the relationship between attending catholic schools and students’ standardized scores
on reading. We have information from a cross-section of 7, 430 US students and we consider the following
econometric model:
read = β0 + β1 female + β2 cathhs + β3 lfaminc + β4 cathhs × female + u,
(1)

where read is the (standardized) score on reading on a scale from 0 to 70; female
is a binary variable
equal to one for females and zero otherwise; cathhs is a dummy equal to one if the student attended a
catholic High School; lfaminc is the logarithm of annual family income.
Call:
lm(formula = read12 ~ cathhs + female + lfaminc + I(cathhs *
female), data = catholic)
Residuals:
Min
1Q
-27.697 -6.745
Median
1.192
3Q
7.190
Max
23.129
Coefficients:
1
Estimate Std. Error t value Pr(>|t|)
(Intercept)
19.0234
1.3846
cathhs
1.5247
0.6001
female
1.9117
0.2165
lfaminc
3.0570
0.1328
I(cathhs * female)
0.6010
0.8795
–Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 9.029 on 7425 degrees of freedom
Multiple R-squared: 0.07947,
Adjusted R-squared: 0.07898
F-statistic: 160.3 on 4 and 7425 DF, p-value: < 2.2e-16 a) How can we interpret the coefficient β3 in equation (1)? b) Is there any evidence of an effect of attending catholic high schools on reading scores? c) Somebody is suggesting that females are benefiting more from attending catholic high schools than men. How would you respond to this based on the available estimates? d) Under which conditions can you interpret these effects of attending catholic schools as causal? Is this a credible assumption? e) Derive the expression for the effect of being a female, as opposed to being a male, using the above specification. 5. With reference to the previously considered dataset we estimate a Probit model to analyze the relationship between attending a catholic high school and actually graduating from high school (hsgrad). In this sample 93% of students actually graduated. The model we consider is hsgrad = Φ (α0 + α1 female + α2 cathhs + α3 lfaminc + α4 cathhs × lfaminc) + e Call: glm(formula = hsgrad ~ cathhs + female + lfaminc + I(cathhs * lfaminc), family = binomial(link = "probit"), data = catholic) Deviance Residuals: Min 1Q Median -3.0434 0.2671 0.3148 3Q 0.4033 Max 1.2110 Coefficients: Estimate Std. Error z value Pr(>|z|)
(Intercept)
-2.442099
0.293242
cathhs
1.895165
1.632514
female
0.001609
0.051604
lfaminc
0.385023
0.028878
I(cathhs * lfaminc) -0.123952
0.156729
–Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 3018.6 on 5969 degrees of freedom
Residual deviance: 2808.4 on 5965 degrees of freedom
(1460 observations deleted due to missingness)
AIC: 2818.4
Number of Fisher Scoring iterations: 7
a) How can we interpret the coefficient α2 in the above equation?
b) Provide an estimate for the probability of graduating from high school for a male student attending
a catholic highschool whose parents earn 30000 dollars in annual income.
c) What about an estimate of the same probability for the same student, but with parents earning
10000 dollars in annual income?
2
d) Provide an expression for the marginal effect of attending catholic schools on the probabilty of
graduating from highschool. What is the value of such marginal effect for a male student whose
parents earn 30000 dollars per year?
3

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