Page 450 - Special Topic Session (STS) - Volume 3

STS555 Mohamed Salem Ahmed et al.

where = ( , … , ) and = ( , … , ) .Therefore, the variance-

1

1
covariance matrix of is

This matrix allows to describe the cross-sectional spatial dependencies between the
n observations. Furthermore, the fact that the diagonal elements of ( ) depend

0
on and particularly on and allows some spatial heteroscedasticity. These
0
spatial dependence and heretoscedasticity depend on the neighborhood structure
established by the spatial weights matrix . Before going further, let us give some

particular cases of the model.
If one consider i.i.d observations, that is ( ) = , with σ depending on , the
2

0

0
obtained model may be seen as a particularly case of the classical generalized
partially linear models (e.g Severini and Staniswalis, 1994) or the classical generalized
additive model (Hastie and Tibshi-rani, 1990). Several approaches of estimating this
particular model have been developed, among others, we cite that of Severini and
Staniswalis (1994), based on the concept of generalized profile likelihood (e.g Severini
and Wong, 1992). This approach consists to first fix the parametric parameter and
estimate nonparametrically (∙) = 0 by using the weighted likelihood method. This
0
last estimate is then used to construct a profile likelihood to estimate .
0
When (∙) = 0 (or is an affine function), that is without a nonparametric
0
component, several approaches have been developed to estimate the parameters
0
and . The basic difficulty encountered is that the likelihood function of this model
0
involve a n dimensional normal integral, thus when n is high, the computation or
asymptotic properties of the estimates may be difficult (e.g Poirier and Ruud, 1988).
Various approaches have been proposed to address this difficulty, among these we
cite:
• Feasible Maximum Likelihood approach: it consists of replacing the true
likelihood function by a pseudo likelihood function constructed via marginal
likelihood functions. Smirnov (2010) proposes a pseudo likelihood function
obtained by replacing ( ) by some diagonal matrix by the diagonal

0
elements of ( ). Alternatively, Wang et al. (2013) proposed to divide the
0

observations by pairwise groups where the latter are assumed to be
independent with bivariate normal distribution in each group and estimate
0
and by maximizing the likelihood of these groups.
0
• GMM approach used by Pinkse and Slade (1998). These authors used
̃
the generalized residuals defined by (, ) = ( | , , ), =

1, … , , = 1,2, … , with some instrumentals variables to construct
moments equations to define GMM estimators of and .
0
0
In what follows, using the observations ( , , ) we propose

parametric estimators of , and a nonparametric estimator of the
0
0
smooth function (∙).
0
To this aim, assume that, for all = 1,2, … , { , = 1, … , } is independent of

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