Page 449 - Special Topic Session (STS) - Volume 3

STS555 Mohamed Salem Ahmed et al.
of the model and estimate nonparametrically the nonlinear component by
weighted likelihood. The obtained estimator depending on the values at which
the parametric components were fixed is used to construct a GMM estimator
(Pinkse and Slade, 1998) of these components.

2. Model

We consider that at n spatial locations { , ,… , } satisfying ‖ − ‖ >

1
2

with > 0, observations of a random vector (, , ) are available. Assume
that these observations are considered as triangular arrays (Robinson, 2011)
and follow the partially linear model of a latent dependent variable :
∗

(1)
with

(2)

where and are explanatory random variables taking values in two
compacts subsets ⊂ ℝ ( ≥ 1) and ⊂ ℝ ( ≥ 1) respectively. The

parameter is an unknown × 1 vector that belongs to a compact subset
0
Θ ⊂ ℝ , (∙) is an unknown smooth function valued in the space of functions

0
2
2
= { ∈ (): ‖‖ = ∈ |()| < } with () the space of twice
differentiable functions from to ℝ, a positive constant. In model (1),
0
and (. ) are constant over (and ). Assume that the term of disturbance
0

in (1) is modeled by the following spatial autoregressive process (SAR):

(3)

where is the autoregressive parameter, valued in the compact subset Θ ⊂

0
ℝ, , = 1, … , are the elements in the –th row of a non-stochastic × spatial
weights matrix , that contains the information on the spatial relationship between

observations. This spatial weight matrix is usually constructed as a function of the
distances (with respect to some metric) between locations, see Pinkse and Slade
(1998) for more of details. The × matrix ( − ) is assumed to be
0

nonsingular for all where denotes the × identity matrix, and { , 1 ≤ ≤ }

are assumed to be independent random Gaussian variables; ( ) = 0 and

( ) = 1 for = 1, … , = 1,2, … . Note that one can rewrite (3) as:
2

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