HEC Lausanne - Advanced Econometrics
Mid-term December 2013. Christophe Hurlin
Exercise 1 (10 points): Log-normal distribution
Consider a sample fX1 ; ::; XN g of i:i:d: continuous random variables, where Xi has a lognormal distribution with parameters and 2 :
Xi
i:i:d:
ln N
2
;
(1)
or equivalently
ln (Xi )
i:i:d:
N
;
2
(2)
The pdf of the log-normal distribution is de…ned as to be:
fX (x; ) =
where
=
:
2 >
denotes the 2
x
1
p
2
exp
(ln (x)
2 2
2
)
!
(3)
1 vector of parameters.
Question 1 (2 points). Write the log-likelihood of the sample fx1 ; ::; xN g.
Question 2 (2 points). Show that the ML estimators of the parameters
b=
N
1 X
ln (Xi )
N i=1
b2 =
N
1 X
(ln (Xi )
N i=1
2
and
2
b)
are:
(4)
Question 3 (1 point). Show that the ML estimator b is weakly consistent.
Question 4 (1 point). Determine the …nite sample distribution of b and b2 (or a transformation of this estimator).
Question 5 (2 points). Determine the asymptotic distribution of b = b : b2
>
Question 6 (2 points). Propose a consistent estimator of the asymptotic variance covariance
matrix of b:
Mid-term 2013. Advanced Econometrics. HEC Lausanne. C. Hurlin
2
Exercise 2 (12 points): Probit model
Consider a binary random variable Yi that takes two values, 0 or 1:
Yi =
1
0
with Pr ( Yi = 1j Xi = xi )
with 1
Pr ( Yi = 1j Xi = xi )
(5)
The conditional probability of the event Yi = 1 given Xi = xi is given by a probit model:
Pr ( Yi = 1j Xi = xi ) =
( + xi )
(6)
where (:) denotes the cdf of the standard normal distribution. The explicative variable Xi is
a binary random variable that takes two values, 1 or 1. The parameters and are unknown.
N
In order to estimate these parameters, you have access to a sample (realisations) fyi ; xi gi=1 of
N = 190 observations. These observations are organised as follows:
Number of observations
50
30
50
60
observations
observations
observations
observations
yi
yi
yi
yi
yi
=1
=1
=0
=0
xi
xi = 1
xi = 1
xi = 1
xi = 1
The objective of this exercise is to …nd the following ML estimation results (Eviews).
Figure 1: Estimation results
Mid-term 2013. Advanced Econometrics. HEC Lausanne. C. Hurlin
3
Question 1 (1 point). Write the conditional probability to observe the event Yi = 1 as
a function of the parameters
and
for the two types of individuals in the sample
(individuals with xi = 1 and individual with xi = 1).
N
Question 2 (2 points). Assume that the random variables fYi ; Xi gi=1 are i:i:d: Write the
N
(conditional) log-likelihood of the sample fyi ; xi gi=1 as a function of the vector of para>
meters = ( : )
Question 3 (1 point). Consider a new vector of parameters
a
b
=
>
= (a : b)
de…ned as to be:
(
)
( + )
=
N
Rewrite the (conditional) log-likelihood of the sample fyi ; xi gi=1 as a function of the new
vector of parameters .
Question 4 (2 points). Calculate the maximum likelihood estimate of the vector of parameters
>
= (a : b)
Question 5 (2 points). Use the equivariance (or invariance) principle to show that the maximum likelihood estimates of the parameters and are equal to:
1
2
b=
1
2
b=
1
3
1
1
'
1
3
0:215364
(7)
' 0:215364
(8)
Question 6 (2 points). Denote Zi the 1 2 vector of explicative variables (including the
constant term) with Zi = (1 : Xi ). The conditional log-likelihood function of the sample
N
fyi ; xi gi=1 is de…ned by:
`N ( ; yj z) =
N
X
yi ln
(z i ) + (1
yi ) ln (1
(z i ))
(9)
i=1
Calculate the score vector SN ( ; Y j z) and compute its expectation. Remark 1: @ (x) =@x =
(x) ; where (:) denotes the pdf of the standard normal distribution. Remark 2 : for a
binary random variable U that takes the value 0 (with a probability 1 p) or 1 (with a
probability p), we have E (U ) = p:
Question 7 (2 points). In this probit model, the Fisher information matrix associated to the
N
sample fyi ; xi gi=1 is equal to:
IN ( ) =
N
X
i=1
2
(z i )
(z i ) [1
(z i )]
z>
i zi
(10)
An estimator of the Fisher information matrix is given by:1
b
IN b =
117:2049 10:1190
10:1190 117:2049
(11)
Propose an estimate of the asymptotic variance covariance matrix of the ML estimator
>
b= b:b
and an estimate for the standard error of b and b :
1 Here
we use the …rst estimator (of the three possible estimators) given by b
IN b = IN b :