ASSIGNMENT 2
For each part of each question below, state whether or not the answer given by ChatGPT is
correct or incorrect. To be correct, both the final answer and all workings/reasoning used
to obtain it must be correct. Otherwise the answer is incorrect. You should not base your
decision on the notation ChatGPT uses, which may be slightly different to what we have used
on this course.
If you decide that ChatGPT’s answer is correct, simply write ”Correct”. For example, if you
think question 1 part (a) is correct, write
1.(a) Correct.
If you decide that ChatGPT’s answer is incorrect, write 1-2 sentences to explain why.
Then provide the correct answer, including all workings/reasoning. For example, if you
think question 1 part (a) is incorrect, write
1.(a) Incorrect. ChatGPT’s answer confuses the probability density function with the dis-
tribution function. The correct answer is
E[X] =
∫ ∞
−∞
xfX(x)dx =
∫ 1
0
x× 1dx = [x2/2]10 = 1/2− 0 = 1/2.
To submit your assignment you should scan/photograph your answers and then upload to
Blackboard using the link provided. Answers may be handwritten or typed. Please make sure
that your answers are legible prior to submission.
Questions
1. Are the following statements true or false? Explain your answers.
(a) Suppose that we have a random sample X1, X2, ..., Xn from a population with mean
µ and variance σ2 and let n > 10,m = n− 10. Given the estimators
µ̂n =
1
n
n∑
i=1
Xi, µ̂m =
1
m
m∑
i=1
Xi,
we have
n(µ̂n − µ)
σ2
+
m(µ̂m − µ)
σ2
d−→ χ22.
(5 marks)
(b) An unbiased estimator must be consistent. (5 marks)
(c) Let θ̂ be an estimator such that limn→∞E[θˆ] = 0 and limn→∞ V [θ̂] = θ. Then θˆ
converges in probability to θ. (5 marks)
(d) Given a random sample X1, X2, ..., Xn from a normal distribution with mean µ and
variance σ2, and
X =
1
n
n∑
i=1
Xi, s
2 =
1
n− 1
n∑
i=1
(Xi −X)2,
we have
P
[
(n− 1)s2
Cn0.9
< σ2 <∞
]
= 0.9,
where Cn0.9 is the 0.9 percentile of the chi-squared distribution with n degrees of
freedom. (5 marks)
2. Let X1, X2, ..., Xn be a random sample from
fX(x) =
{
1
2σ
√
3
µ−√3σ ≤ x ≤ µ+√3σ
0 otherwise
with parameters µ and σ2 > 0.
(a) Show that fX(x) is a proper density function. (2 marks)
(b) Find the method of moments estimator of θ = (µ, σ2). (5 marks)
(c) Derive the bias of the methods of moments estimator of σ2. Is it unbiased? Is it
consistent? Justify your answers. (5 marks)
(d) Find the maximum likelihood estimator of θ. (8 marks)
3. Consider the probability function for the discrete random variable Y given by
P [Y = y] =
(
1− θ
3
)y−1
θ
3
, y = 1, 2, 3, ...
for 0 < θ < 1/3 and a random sample Y1, Y2, ..., Yn. Note also that the random variable
Y verifies E[Y ] = 3/θ.
(a) Derive the maximum likelihood estimator of θ. (7 marks)
(b) Derive the asymptotic distribution of the maximum likelihood estimator of θ. (7
marks)
(c) Propose a 0.95 confidence interval for eθ. (6 marks)
4. Consider the regression model
X = µ+ U
where X,U are random variables, E[U ] = δ, E[U2] = 1 + δ2 <∞, and the two unknown
parameters are µ, δ.
(a) Derive the ordinary least squares estimator of µ. (4 marks)
(b) Find the probability limit of the ordinary least squares estimator of µ. (3 marks)
(c) Find the Mean-Squared Error of the ordinary least squares estimator of µ. (3 marks)
(d) For which value(s) of δ is ordinary least squares consistent? (2 mark)
(e) Find the asymptotic distribution of the ordinary least squares estimator. (8 marks)
5. Consider the regression model
E[Y |X] = p(β0 + β1X)
where Y is binary, X is continuous, and p(·) is a known function.
(a) Explain why the choice of
p(a) =
exp(a)
1 + exp(a)
is a good one. (3 marks)
(b) Using the choice of p(·) from part (a), derive the marginal effect of X on Prob[Y =
1|X] and evaluate it at β0 = 0, β1 = 1 and X = 2. (6 marks)
(c) Consider instead the linear model with
p(a) = a.
Explain one potential issue with this model. (2 marks)
(d) Consider the choice of p(·) from part (c) and the corresponding regression equation
Y = β0 + β1X + U, i = 1, ..., n.
Does E[U |X] satisfy the exogeneity assumption? Does E[U2|X] depend on X? (i.e.,
is homoskedasticity violated?). Justify your answers. (9 marks)