Master Econométrie et Statistique Appliquée.
Statistique et Econométrie
Janvier 2015. Christophe Hurlin
Exercise 1: MLE, parametric tests and the trilogy (28 points)
Consider a sample fX1 ; ::; XN g of i:i:d: continuous random variables, where Xi 2 [0; c] has
a pdf given by
1
1
fX (x; ) = 1= x
8x 0
(1)
c
where the boundary c is known and is a positive parameter (unknown).
Part I: Maximum Likelihood Estimation (8 points)
Question 1 (2 points). Write the log-likelihood of the sample fx1 ; ::; xN g as a function of the
unknown parameter :
Question 2 (2 points). Show that the ML estimator b of the parameters
b = ln (c)
is:
N
1 X
Xi
N i=1
Question 3 (2 points). Admit that E (ln (Xi )) = ln (c)
weakly consistent.
(2)
: Show that the ML estimator b is
Question 4 (2 points). Determine the asymptotic distribution of b:
Part II: Parametric tests (10 points)
Question 6 (2 points). Consider the test :
H0 :
=
0
against H1 :
=
1
with 1 < 0 : Show that the rejection region of the uniformly most powerful (UMP) test
of size has the general form:
n
o
W = x : b (x) < A
(3)
and b (x) denotes the realisation of the
where A is a constant determined by the size
estimator b for the sample x:
Question 7 (2 points). Consider that N is su¢ ciently large to assume that:
where
size :
b asy N
2
;
N
(4)
denotes the true value of : Determine the critical value A of the UMP test of
Mid-term 2013. Advanced Econometrics. HEC Lausanne. C. Hurlin
2
Question 8 (2 points). Consider the one-sided test:
H0 :
=
0
against H1 :
<
0
Determine the rejection region of the UMP test of size .
Question 9 (2 points). Consider the two-sided test:
H0 :
=
0
against H1 :
6=
0
Determine the rejection region of the test of size .
Question 10 (2 points). Show that the two-sided test of question 9 is consistent.
Part III: The trilogy (10 points)
Question 11 (3 points). Assume that ln (c) = 4: Consider a sample of size N = 100 for which
the realisation of the ML estimator b (x) is equal to 2. We want to test:
H0 :
= 1:5 against H1 :
6= 1:5
Compute the LR test statistic and the corresponding critical region for a signi…cance
level of 5%. What is your conclusion for this sample? Remark: the 95% critical value of
chi-squared distribution with one degree of freedom is 20:95 (1) = 3:8415:
Question 12 (3 points). Consider the same test and the same sample as in question 11.
Compute the Wald test statistic and the corresponding critical region for a signi…cance
level of 5%. What is your conclusion for this sample? Remark: a consistent estimator of
the asymptotic variance of the ML estimator b is given by:
b asy b =
V
b2
N
(5)
Question 13 (4 points). Consider the same test and the same sample as in questions 11 and
12. Compute the LM (or score) test statistic and the corresponding critical region for a
signi…cance level of 5%. What is your conclusion for this sample? Remark: in this case,
the LM test statistic is de…ned as to be:
LM =
s2N ( c ; xi )
b
I N ( c)
(6)
where c denotes the constrained value of the parameter under the null, sN ( ; x) is the
score of the unconstrained model and b
I N ( c ) is an estimator of the Fisher information
number of the unconstrained model (for the sample) evaluated at c .
Mid-term 2013. Advanced Econometrics. HEC Lausanne. C. Hurlin
3
Exercise 2: Logit model and MLE (6 points)
Consider a binary random variable yi that takes two values, 0 or 1:
yi =
1
0
with Pr ( yi = 1j xi )
with 1
(7)
Pr ( yi = 1j xi )
The conditional probability of the event yi = 1 given xi is given by a logit model:
Pr ( yi = 1j xi ) =
(x|i )
(8)
|
|
where
= ( 1 :: K ) is a K 1 vector of parameters, xi = (x1i ::xKi ) is a K 1 vector
of individual characteristics (stochastic) including the constant term (x1i = 1; 8i) and (:)
denotes the cdf of the logistic distribution given by:
(x) =
exp (x)
1 + exp (x)
(9)
N
In order to estimate these parameters, you have access to an i.i.d. sample fyi ; xi gi=1 . Remark :
in these notations, I don’t distinguish a random variable (or vector) from its realisation.
N
Question 1 (2 points). Show that the log-likelihood of the sample fyi ; xi gi=1 is:
`N ( ; yj x) =
N
X
yi ln
(x|i ) +
i=1
N
X
(1
yi ) ln (1
(x|i ))
(10)
i=1
Question 2 (2 points). Denote by (x) = (x) (1
(x)) the pdf of the logistic distribution.
Show that the score vector of the logit model can be expressed as:
sN ( ; yj x) =
N
X
xi (yi
i=1
Question 3 (2 points). Determine the Hessian matrix.
(x|i ))
(11)