Page 451 - Special Topic Session (STS) - Volume 3

STS555 Mohamed Salem Ahmed et al.
{ , = 1, … , } and { , = 1, … , } and { , = 1, … , } is independent

of { , = 1, … , } We give asymptotic results according to Increasing

domain asymptotic.
2.1 Estimation procedure
We propose an estimation procedure based on a combination of a weighted
likelihood method and a generalized method of moments. We first fix the parametric
components and of the model and estimate the nonparametric component using
a weighted likelihood. The obtained estimate is then used to construct generalized
residuals where the latter are combined to instrumentals variables to propose GMM
parametric estimates. This approach will be described as follow. By equation (2) we
have

where denotes the expectation under the true parameters (i.e , and
0
0
0
(∙)), Φ(∙) is the cumulative distribution function of a standard normal
0
distribution, and ( ( )) = ( ), 1 ≤ ≤ , = 1,2, … are the diagonal
2
0

0

elements of ( ).
0

For each ∈ Θ , ∈ Θ , ∈ and ∈ ℝ , we define the conditional

expectation on of the log-Likelihood of given ( , ) for 1 ≤ ≤

, = 1,2, …,

with ℒ(; ) = log( (1 − ) 1− ). Note that (; , , ) is assumed to be
constant over (and ). For each fixed ∈ Θ , ∈ Θ and ∈ , ,()

denotes the solution in of
(5)

Then, we have () = () for all z ∈ Z.
0 , 0 0
Now using , (∙), we construct GMM estimates of and as Pinkse and
0
0
Slade (1998). For that, we define the generalized residuals, replacing ( )
0

in (1) by , ( );

(6)

where ∅(∙) is the density of the standard normal distribution and

(, , , )=( ()) −1 ( + , ( )).

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