Assignment 1 Submission
Aaryaman Aggarwal, 230020
August 28, 2025
Question 1
a) A descriptive summary of the mean, median, standard deviation, maximum, and minimum
are as follows:
Variable Metrics
Mean
Median
Standard Deviation
Maximum
Minimum
IgG
5.29
5.00
2.28
Age
2.77
2.58
1.66
14.40
0.90
6.00
0.50
The histogram is as follows:
Figure 1: Histogram of IgG(Left) and Age(Right)
The IgG histogram appears to be left skewed, whereas the histogram for age doesn’t appear
1
to be skewed. This should imply that Mean < Median for IgG but from the values we can
see that isn’t the case as few high values of IgG push the mean to be higher.
The histogram for age seems to be uniformly distributed and there appears to be no skewness.
b) The M odel 1:
y i = β1 + β2 x + u i
(1)
The following are the estimates for the model using Ordinary Least Squares which was given
as follows:
Coefficients
β1 (Intercept)
β2
Estimates
3.3640
0.6951
Standard Errors
0.2214
0.068469
t-statistic
15.194
10.152
R2 = 0.258
From the coefficient estimates we can see that age has a positive effect of IgG, implying that
as age increases, the IgG value(g/L) also increases. The low value of R2 however indicates
that the model doesn’t fit very well and explains a very little amount of the variability of IgG.
c) The M odel 2:
yi = β1 + β2 x + β3 x2 + εi
(2)
The following are the estimates of the model using Ordinary Least Squares:
Coefficients
β1 (Intercept)
β2
β3
Estimates
3.0839
0.9674
-0.0454
Standard Errors
0.3833
0.3118
0.05077
t-statistic
8.0458
3.1029
-0.8954
R2 = 0.26
In the fit for Model 2 as well, we can see it is evident that there is a positive correlation between
age and IgG but the coefficient for age2 is negative with a very low t-statistic. Additionally, the
R2 is only slightly greater than Model 1 and explains around the same amount of variability
as model 1.
d) In this it is said we can assume that
ε ∼ N (0, 1) =⇒ yi − (β1 + β2 xi + β3 x2i ) ∼ N (0, 1)
(3)
From this assumption we can write the conditional distribution of yi |xi as:
yi |xi ∼ N (β1 + β2 xi + β3 x2i , 1)
Therefore the PDF of yi can be written as:
2
(4)
f (yi |xi , β) = √12π exp(
−(yi −(β1 +β2 xi +β3 x2i ))
)
2
Therefore the log likelihood is:
ln L(β) =
n
X
ln f (yi |xi , β)
(5)
i=1
ln L(β) =
n
X
1
(yi − β1 − β2 xi − β3 x2i )2
[− ln(2π) −
]
2
2
i=1
n
n
1 X (yi − β1 − β2 xi − β3 x2i )2
ln L(β) = − ln(2π) −
]
2
2 i=1
2
(6)
(7)
The model estimates using the Maximum Likelihood Method (Using Newton Rhapson) is as
follows:
Maximum Likelihood Estimates
Coefficients
β1 (Intercept)
β2
β3
Estimates
3.0839
0.9674
-0.0454
Standard Errors
0.3833
0.3118
0.05077
3
t-statistic
8.0458
3.1029
-0.8954
Question 2
a) For the response variable q85 coded as yi
Variable
yi = 1 (Yes, legal)
yi = 0 (No, illegal)
Frequency
Count
659
581
Percentages
53.1452%
46.8548%
b) For the variable ’age’ coded as x2 and ’hh1’ (Number of members in the household) as x3 .
Variable
x2 (Age)
x3 (hh1)
Mean
50.5008
2.7194
Std. Deviation
17.7579%
1.4403%
c) For the variable ’past use’ encoded as x4 which is 1 if ’Yes’ or else ’0’ if ’No’
Variable
x4 = 1 (Yes)
x4 = 0 (No)
Frequency
Count
587
653
Percentages
47.3387%
52.6613%
d) For the variable ’sex’ as x5 = 1 if Male and x5 = 0 if female. Also for the variable ’Parent’ as
x6 = 1 if ’Yes’ or x6 = 0 if ’No’
Variable
x5 = 1 (Male)
x5 = 0 (Female)
x6 = 1 (Yes for parents)
x6 = 0 (No for parents)
Count
603
637
358
882
Percentage
48.6290%
51.3710%
28.8710%
71.1290 %
e) For the variable ’Marital Status’
Category 1(’Single’) = ’x7 = 1 if x ∈’Never Been Married’
Category 2(’Post Married’) = ’x8 = 1 if x ∈ ’Divorced’, ’Separated’, ’Widowed’
Category 3(’Couple’) = ’Base Category = if ∈’Married’, ’Living with a partner’
Variable
x7 = 1 (Single)
x7 = 0 (Not Single)
x8 = 1 (Post Married)
x8 = 0 (Not Post Married)
Couple Category
Count
235
1005
285
955
720
Percentage
18.9516%
81.0484%
22.9839%
77.0161 %
58.0645
f) For the variable ’Income’. The categories are as follows
1. ’Poor’ = People ∈ Less than 10000, 10 to under 20000, 20 to under 30000, 30 to under
40000, 40 to under 50000. Encoded as x9 = 1 if individual ∈ ’Poor’ and x9 = 0 otherwise.
4
2. ’Middle’ = People ∈ 50 to under 75000, 75 to under 100000. Encoded as x10 = 1 if
individual ∈ ’Poor’ and x10 = 0 otherwise.
3. ’Rich’ = People ∈ 100 to under 150000, 150000 or more.This is considered as the Base
Category and not be included in the regression
Variable
x9 = 1 (Poor)
x9 = 0 (Not Poor)
x10 = 1 (Middle)
x10 = 0 (Not Middle)
Rich Category
Count
519
721
366
874
232
Percentage
41.8548%
58.1452%
29.5161%
70.4839 %
18.7097%
g) For the variable ’educ’. The categories are as follows:
1. ’HSandBelow’ = People ∈ Less than HS, HS Incomplete, HS. Encoded as x11 = 1 if
individual ∈ ’HSandBelow’ and x11 = 0 otherwise.
2. ’lessThanBachelors’ = People ∈ Some College, Associate Degree. Encoded as x12 = 1 if
individual ∈ ’lessThenaBachelors’ and x12 = 0 otherwise.
3. ’BachelorsandAbove’ = People ∈ Bachelors, Postgraduate Degree, Some Postgraduate.
This is considered as the Base Category and not included in the regression.
Variable
x11 = 1 (HSandBelow)
x11 = 0 (Not HSandBelow)
x12 = 1 (lessThanBachelors)
x12 = 0 (not lessThanBachelors)
BachelorsandAbove
Category
Count
414
826
381
Percentage
33.3871%
66.6129%
30.7258%
859
69.2742 %
445
35.8871%
h) For the variable ’race’. The categories are as follows:
1. ’white’ = Encoded as x13 = 1 if individual ∈ ’white’ and x13 = 0 otherwise.
2. ’black’ = Encoded as x14 = 1 if individual ∈ ’black’ and x14 = 0 otherwise.
3. ’allOther’ = Considered as Base Category which will not be included in the regression.
Variable
x13 = 1 (White)
x13 = 0 (Not White)
x14 = 1 (Black)
x14 = 0 (not Black)
allOthers Category
Count
954
286
146
1094
140
i) For the variable ’party’. The categories are as follows:
5
Percentage
76.9355%
23.0645%
11.7742%
88.2258%
11.2903%
1. ’democrat’ = Encoded as x15 = 1 if individual ∈ ’Democrat’ and x15 = 0 otherwise.
2. ’republican’ = Encoded as x16 = 1 if individual ∈ ’Republican’ and x16 = 0 otherwise.
3. ’independentOthers’ = Base Category which consists of individuals who affiliate with
all other parties excluding Democratic and Republican. This will not be included in the
regression.
Variable
x15 = 1 (Democrat)
x15 = 0 (Not Democrat)
x16 = 1 (Republican)
x16 = 0 (not Republican)
independentOthers
Category
Count
422
818
357
883
461
6
Percentage
34.0323%
65.9677%
28.7903%
71.2097%
37.1774%
Question 3
a) To derive the probability of yi = 1 and yi = 0 using a binary probit model we can use an
underlying ’Latent Variable’ zi such that:
(
0 if zi ≤ 0
f (x) =
(8)
1 if zi > 0
Where zi is defined as:
zi = x′i β + εi
(9)
β and xi are vectors kx1 and by the definition of the binary probit model:
εi ∼ N (0, 1)
(10)
Pr(y = 1) = Pr(zi > 0)
(11)
We can derive it as follows:
Pr(zi > 0) = Pr(x′i β + εi > 0)
Pr(ε >
−x′i β)
= Pr(ε <
x′i β)
(12)
(13)
=⇒ P r(ε < x′i β) = Φ(x′i β)
(14)
=⇒ Pr(yi = 1) = Φ(x′i β)
(15)
We can do this because of Equation (10). (Φ(x) is the CDF of the Standard Normal Distribution).
Similarily, we can derive the same for Pr(y = 0)
Pr(y = 0) = Pr(zi < 0)
(16)
Pr(zi < 0) = Pr(x′i β + εi < 0)
(17)
−x′i β)
Φ(−x′i β)
(18)
=⇒ P r(ε < −x′i β) = 1 − Φ(x′i β)
(19)
=⇒ Pr(yi = 0) = 1 − Φ(x′i β)
(20)
=⇒ P r(ε <
=
Due to the symmetry of the Normal Distribution
The likelihood function can be constructed as:
n
Y
f (yi |xi , β)
(21)
[f (yi = 1|xi , β)]yi [f (yi = 0|xi , β)]1−yi
(22)
L(β) =
i=1
L(β) =
n
Y
i=1
L(β) =
n
Y
[ Φ(x′i β) ]yi [ 1 − Φ(xi β)]1−yi
i=1
7
(23)
b) The following table is of the coefficient estimates:
Variable
Coefficients
Intercept
β1
Age
β2
hh1
β3
Past Use
β4
Sex
β5
Parents
β6
Single
β7
Post Married
β8
Poor
β9
Middle
β10
HSandBelow
β11
lessThanBachelors β12
White
β13
Black
β14
Democrat
β15
Republican
β16
Binary Probit Model
Estimates
Standard Errors
0.4496
0.2598
-0.0084
0.0029
-0.0552
0.0332
0.8035
0.07837
0.1771
0.0786
0.0526
0.1051
0.1242
0.1199
0.05555
0.1049
-0.2148
0.0961
-0.1711
0.1008
-0.1915
0.0949
0.0798
0.0973
-0.0346
0.1252
-0.3382
0.1598
0.2324
0.0939
-0.4638
0.0950
t-statistic
1.7302
-2.8755
-1.6625
10.2531
2.2527
0.5007
1.0363
0.5294
-2.2341
-1.6983
-2.0179
0.8205
-0.2761
-2.1159
2.4738
-4.8786
c) The covariate effect on Pr(yi = 1|x) if age is increased by 5 years is calculated as:
Covariate EffectAge = Pr(yi = 1|xi,−2 , xi,2 = xi,2 + 5, β) − Pr(yi = 1|xi , β)
(24)
All values of xi are kept constant apart from x2 which is the age variable. For each individual,
the value of age is increased by 5, while everything else is kept constant. The reported value
of the covariate effect of increasing age by 5 years is.
Covariate EffectAge = −0.0144
(25)
This is the mean of the covariate effect over all of datapoints of x.
d) The covariate effect of parents on Pr(yi = 1|β, xi )
Covariate EffectParents = Pr(yi = 1|β, xi,−6 , xi,6 = 1) − Pr(yi = 1|β, xi,−6 , xi,6 = 0)
(26)
All of xi is kept constant apart from xi,6 which is fixed as xi,6 = 1 in the first term and xi,6 = 0
for the second term. The reported value of the covariate effect of being a Parent is:
Covariate EffectParents = 0.0179
(27)
This is the mean of all the covariate effects of the individual being a parent or not in the
dataset.
8
Question 4
a) The probability Pr(yi = 1) and Pr(yi = 0) can be derived in a similar manner (i.e using a
latent variable zi )
(
0 if zi ≤ 0
f (x) =
(28)
1 if zi > 0
Where zi is defined as:
zi = x′i β + εi
(29)
β and xi are vectors of dimension kx1 and by the definition of the binary logit model:
εi ∼ Logistic(0, 1)
(30)
This is where the logit model differs from the probit model, as we can see in Equations (28)
and (10). The errors in the logit model follow a standard normal distribution.
For a random variable X, if X ∼ Logisitic(0,1) then:
E(X) = 0
π2
3
The probability density function for the standard logistic distribution is:
V ar(X) =
f (x) =
e−x
[1 + e−x ]2
(31)
(32)
(33)
The cummulative density function for the standard logisitc distribution is donated as:
Pr(X < α) = Λ(α)
(34)
It is evident that due to the symmetricity of the logistic distribution, the following will hold
true:
Λ(α) = 1 − Λ(−α)
(35)
As we can observe, since the variance of the Standard Logistic Distribution is greater than
that of the Standard Normal Distribution we can infer that the logistic distribution has comparatively fatter tails.
Using these properties of ε and Logistic distribution, we can derive the probability Pr(yi = 1)
as follows:
Pr(y = 1) = Pr(zi > 0)
(36)
Pr(zi > 0) = Pr(x′i β + εi > 0)
(37)
Pr(ε > −x′i β) = Pr(ε < x′i β)
(38)
=⇒ Pr(ε < x′i β) = Λ(x′i β)
(39)
Pr(yi = 1) = Λ(x′i β)
(40)
9
Similarily, we can also derive the probability for Pr(yi = 0) as:
Pr(y = 0) = Pr(zi < 0)
(41)
Pr(zi < 0) = Pr(x′i β + εi < 0)
(42)
=⇒ Pr(ε < −x′i β) = Λ(−x′i β)
(43)
=⇒ Pr(ε < −x′i β) = 1 − Λ(x′i β)
(44)
Pr(yi = 0) = 1 − Λ(x′i β)
(45)
Using these values of Pr(yi = 1) and Pr(yi = 0) we can construct the likelihood function:
n
Y
f (yi |xi , β)
(46)
[f (yi = 1|xi , β)]yi [f (yi = 0|xi , β)]1−yi
(47)
L(β) =
i=1
L(β) =
n
Y
i=1
L(β) =
n
Y
[Λ(x′i β) ]yi [1 − Λ(xi β)]1−yi
(48)
i=1
b) The following are the estimates, standard error and t-values of the coefficients using a binary
logit model.
Variable
Intercept
Age
hh1
Past Use
Sex
Parents
Single
Post Married
Poor
Middle
HSandBelow
lessThanBachelors
White
Black
Democrat
Republican
Coefficients
β1
β2
β3
β4
β5
β6
β7
β8
β9
β10
β11
β12
β13
β14
β15
β16
Binary Logit Model
Estimates
0.838
-0.015
-0.096
1.308
0.267
0.074
0.183
0.08
-0.364
-0.291
-0.327
0.119
-0.052
-0.523
0.36
-0.776
Standard Errors
0.43
0.005
0.055
0.13
0.13
0.173
0.198
0.174
0.16
0.167
0.158
0.161
0.206
0.264
0.155
0.158
t-statistic
1.948
-2.999
-1.737
10.052
2.058
0.426
0.921
0.46
-2.28
-1.743
-2.074
0.738
-0.25
-1.978
2.316
-4.905
c) To calculate the covariate effect of increasing age by 5 years on Pr(yi = 1|β)
Covariate EffectAge = Pr(yi = 1|β, xi,−2 , xi,2 = xi,2 + 5) − Pr(yi = 1|β, xi )
10
(49)
All values of xi are kept constant apart from x2 which is the age variable. For each individual,
the value of age is increased by 5, while everything else is kept constant. The reported value
of the covariate effect of increasing age by 5 years is.
Covariate EffectAge = −0.0151
(50)
This is the mean of the covariate effect over all of datapoints of x.
d) To calculate the covariate effect of being a parent
Covariate EffectP arent = Pr(yi = 1|β, xi,−6 , xi,6 = 1) − Pr(yi = 1|β, xi,−6 , xi,6 = 0)
(51)
While keeping all other values of xi the same, and fixing the ”Parents” variable. The value of
the covariate effect is:
Covariate Effectparents = 0.0152
(52)
This is the mean of the covariate effect of all the datapoints.
e) Yes there is a difference in the Covariate effect calculated in the probit and logit model. This
is due to the difference in their variances.
π2
V arlogit =
3
(53)
V arprobit = 1
(54)
Due to this difference in variance, the estimates can be compared as:
βlogit ≈ 1.81βprobit
(55)
So hence, the values of the covariate effects may be slightly different, as the β-estimates for
the logit model are greater. Even due to the different values, the meaning of the values can
still be interpreted as the same. For example, from both models we can conclude that if age
increases, the probability of the individual supporting the legalization of marijuana decreases.
11
Question 5
a) The ordinary probit model generalizes the binary model as yi can have more than 2 outcomes
which have some ordinal significance, i.e they can be ranked.
For this model, we also use the latent variable zi .
zi = x′i β + εi
(56)
β and xi are vectors kx1 and by the definition of the ordinal probit model:
εi ∼ N (0, 1)
(57)
γj−1 < zi ≤ γj =⇒ yi = j
(58)
j = Categories into which we have to classify the datapoint to
(59)
j = 1, 2 . . . J
(60)
γj = Threshold values for category j
(61)
And we use it in the following manner:
Where
Therefore, we can derive Pr(yi = j) as follows:
=⇒
=⇒
Pr(yi = j) = Pr(zi ≤ γj , γj−1 < zi )
(62)
=⇒ Pr(yi = j) = Pr(zi ≤ γj ) − Pr(zi ≤ γj−1 )
(63)
Pr(yi = j) = Pr(x′i β + εi ≤ γj ) − Pr(x′i β + εi ≤ γj−1 )
Pr(yi = j) = Pr(εi ≤ γj − x′i β) − Pr(εi ≤ γj−1 − x′i β)
=⇒ Pr(yi = j) = Φ(γj − x′i β) − Φ(γj−1 − x′i β)
(64)
(65)
(66)
We can come to the above equation due to Equation(57)
Now to construct the likelihood function, We define an indicator variable Zij , which equals 1
if yi = j and 0 otherwise. Then the likelihood function can be defined as follows :
L(β) =
N X
m
X
Zij ln[ Φ(γj − x′i β) − Φ(γj−1 − x′i β)]
(67)
i=1 j=0
b) A descriptive summary of categorical variables with mean and standard deviation is as follows
Variables
educlevel
motherWorked
urban
south
educFather
educMother
famIncome
female
black
age15
age16
age17
Mean
5.29
5.00
2.28
14.40
0.90
Standard Deviation
2.77
2.58
1.66
6.00
0.50
12
c)
d) For a one-unit change in covariate xjk , the marginal effect on category j is given by :
∂P r(yi = j|xi )
= βk [ Φ(γj − x′i β) − Φ(γj−1 − x′i β)]
∂xik
(68)
Now for a finite change of ∆ xik we will modify the equation a little bit and say:
∆P r(yi = j) = ( Φ(γj − x′T i β) − Φ(γj−1 − x′T i β)) − [ Φ(γj − x′i β) − Φ(γj−1 − x′i β)] (69)
where xT i = xi + ∆ xik eik and we average over i to obtain the ACE
Following are the average changes in Pr(educLevel=j) across the sample, rounded to four
decimals, and they sum to approximately zero as expected.
Categories
P r(educLevel=1)
P r(educLevel=2)
P r(educLevel=3)
P r(educLevel=4)
Reported ACE
-0.0289 (decrease)
+0.0153 (increase)
+0.0031 (increase)
+0.0105 (increase
13