PROBLEM SET MANNY
ECON 315: Labor Economics
Fall 2015
FAQs
Do I have to type the homework?
Yes! Homework must be typed. You will receive a zero for turning in handwritten homework. You must
type all graphs (including labels and arrows) and equations. Handwritten portions will not be graded.
Does it need to be stapled?
You must staple your homework together. Unbound homework will lose 10% credit.
Can I email it to you?
NO! Homework is collected before class. Do not slide it under my door. No late homework is accepted.
NO EMAIL SUBMISSIONS. Please don’t forget to bring an extra copy to class with you so that you can
participate during homework day.
Can I work with a friend?
You may work in groups to discuss answers, but you must write up you own unique responses. If two items
are too closely related, it will be submitted to the Academic Integrity Office.
Econ 425 – Problem Set 8
Due March 20 (Thursday) in Class
Instructions. The questions below review the identiÖcation and inference on enodgenous regression
models. Do your best to make your arguments rigorous. You may discuss this problem set with
your classmates and consult any books or notes, but write out the answers on your own using your
own words and show your derivation so that your understanding is transparent from the answers.
1. Consider the following regression model:
Yi = 0 + 1Xi + ui
;
where V ar(Xi) > 0.
(i) Find out the maximum likelihood estimator (MLE) of
2
0
, under the assumption that the
conditional distribution of ui given Xi
is N(0; 2
0
). Is the estimator unbiased for
2
0
? Explain
your answer. (Recall that given an estimator ^ of , it is unbiased if E^ = .)
(ii) Suppose that the assumption in (i) is not true, and instead, the truth is that the conditional
distribution of ui given Xi
is in fact a logistic distribution (with mean 0 and variance 1) whose
CDF is given as
Fu(t) = exp(t)
1 + exp(t)
:
Is the estimator obtained in (i) consistent as n ! 1? Explain your answer.
2. Suppose that we have i.i.d. observations Xi
; i = 1; :::; n; where each random variable Xi
follows uniform distribution on [0; ] for some > 0. Find out the MLE of . (HINT: The
density of the uniform distribution on [0; ] is given by
f(x; ) = 11 f0 x g :
The function is discontinuous in . So, taking the Örst order condition of the likelihood
function over does not work. Think about the shape of the likelihood function carefully,
and the meaning of the indicator functions.)
3. Consider the following binary choice model:
Yi = 1 f0 + 1Xi + 2XiWi uig ;
where the conditional distribution of ui given Xi
; Wi
is equal to N(; 2
).
(i) Find out the log-likelihood function whose maximizers are the MLEs, say,
0
;
1 and
2
;
for
~
0 =
0
; ~
1 =
1
, and ~
2 =
2
;
1
respectively.
(ii) Suppose that we have MLEs
0
;
1 and
2
. Then using these estimators provide a
sample analogue estimator of the average partial e§ect with respect to Wi
, i.e., provide a
sample analogue estimator of
Z Z @P fYi = 1jXi = x; Wi = wg
@w f(x; w)dxdw;
where f(x; w) is the joint density of (Xi
; Wi).
4. (i) Show that for any two random variables Yi and Xi and a set A such that PfXi 2 Ag > 0,
we can write
E[Yi
jXi 2 A] = E [m(Xi)1fXi 2 Ag]
PfXi 2 Ag
;
where m(x) = E [Yi
jXi = x]. (There are many ways to tackle this problem depending on
your knowledge about probability. You can assume that Xi and Yi are continuous or discrete
random variables if you would like, and use the deÖnition of the conditional expectations
through the conditional probability mass functions or density functions. Also, there is another
way to solve this problem without using distributional assumptions, which starts by noting
that for any random variables Z and X and any set A:
E [Z] = E [ZjX 2 A] P fX 2 Ag + E [ZjX =2 A] P fX =2 Ag ;
which is the expectation version of the law of total probability.)
(ii) Consider the following binary choice model
Yi = 1f0 + 1Xi uig;
where ui
is independent of Xi
. Assume that ui follows a standard normal distribution N(0; 1).
Here we assume that Xi
is a continuous random variable taking values in [0; 1]. Suppose that
one has consistent estimators of 0 and 1
. Using these estimators, provide a sample analogue
estimator of
0 = PfYi = 1j0 Xi 0:5g
PfYi = 1j0:5 Xi 1g:
2
