Download Nonparametric Triangular Simultaneous Equations Models with

Assumption B u = (z; v) is continuously distributed and the density of u is bounded and
bounded away from zero on Z
V, where the support Z of z is a Cartesian product of
compact, connected intervals and the support V of v is compact.
The assumption is useful to bound below and above the eigenvalues of the transformed
second moment matrix of approximating functions (i.e., Q ). This condition is worth a discussion in the context of identi…cation and weak instruments. An identi…cation condition
like Assumption ID20 in Section 3.2 is embodied in this assumption. To see this, note that
fu being bounded away from zero means that there is no functional relationship between z
and v, which in turn implies Assumption ID20 (a)(iii).17 On the other hand, an assumption
written in terms of fw like Assumption 2 in NPV (p.574) cannot be imposed here. Observe
that w = ( (z) + v; v) depends on the behavior of
( ), and hence fw is not bounded away
from zero uniformly over n under Assumption L; in fact, it approaches a singular density.
Making use of the transformation matrix, we make an assumption in terms of fu and the
e¤ect of the weak instruments is handled separately.
The assumption for the Cartesian products of supports, namely Z
V and X
V (below),
and the compactness of V in Assumption B can be replaced by introducing a trimming function
as in NPV which ensures bounded rectangular supports. Lastly, similar to NPV, Assumption
B can be weakened to hold only for some component of the distribution of z. And, one could
allow some components of z to be discrete, as long as they have …nite supports.
Next, Assumption C is a smoothness assumption on the structural and reduced-form
functions. Let W = X
V be the support of w = (x; v).
Assumption C g0 (x) and
0 (v)
are in L2 (Fx ) and L2 (Fv ), respectively, and are Lipschitz
and continuously di¤ erentiable of order s on W.
0 (z)
continuously di¤ erentiable of order s1 on Z.
is in C(Z) and is Lipschitz and
This assumption ensures that the series approximation error shrinks as the number of
approximating functions increases. It also motivates the introduction of penalization as well.
Note that the same smoothness for g0 ( ) and
0(
) is assumed because there is no systematic
reason that one is particularly smoother than the other.
The next regularity condition restricts the rate of the growth of the number, K and L, of
approximating functions.
p
Assumption D When the approximating functions are power series, n2 K 7=2 [ L=n+L
s1 =dz ]
K 11=2 ! 0, and L3 =n ! 0. When the approximating functions are regression splines,
p
n2 K 2 [ L=n + L s1 =dz ] ! 0, n K 3 ! 0, and L2 =n ! 0.
! 0, n
This condition is stronger than the corresponding assumption in NPV (Assumption 4, p.
575) where weak instruments are not considered.
17
For the de…nition of a functional relationship, see NPV (p.568).
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