the Gaussian Kernel and that allows in most cases to neglect the tails-effect terms. Finally, the last two
assumptions A4 and A5 are standard to ensure the consistency and asymptotic normality of QMLE (White
(1982)).
4.1 Consistency and Asymptotic Normality of the NPQMLE estimator
For the NPQMLE parameter estimator bθMj we have the following results:
THEOREM 1 (Consistency): Given Assumptions A1-A4, as n ∞ ∞, θnMj —→ θMj with probability 1.
Proof: See the Appendix
The main idea is that if KInj is a contrast13 relative to the contrast function KI(g, fj (θ, x)), that is it
converges at least in probability to KI(g, fj(θ, x)), and θMj is the unique minimizer in Θ of KI(g, fj(θ, x)),
then the sequence θnMj in Θ that minimizes KInj will converge to θ*Mj.
This implies the following: given that each candidate parametric model is potentially misspecified, since
we do not know the true model and we do not even know if it belongs to the set of candidate models,
the NPQMLE estimation procedure, as QMLE, will provide an estimator that converges to the best ap-
proximation θ*Mj. In other words, it converges to the best we can attain given that we are minimizing the
Kullback-Leibler information over a space FΘMj ⊆ G rather than G. From now on, for simplicity θnMj = θn
and θMj = θ*.
Next I establish that NPQMLE has a limiting normal distribution with mean zero and variance-covariance
matrix C(θ*), and that it is root-n consistent, that is it has the same rate of convergence of parametric
method as QMLE. In particular, similarly to Powell-Stock and Stoker(1989) this estimator converges faster
than the nonparametric density fn exploited in the estimation technique, therefore avoiding the necessity for
extremely large dataset. It is much easier to understand the rationale for this convergence rate by observing
the U-statistic representation of the first order condition to derive the optimal value of the parameters:
μn^ X X hK(xj - xi )[s(θ,χi) - s(θ,χj)], where s(θ,χ)=dlog∂θ(θ, x).
i=1 j=i+1
As in Powell-Stock and Stoker(1989), it follows by the averaging of the nonparametric density estimate fbn,
which appears in the previous formula in the particular form:
n
Mxi) = n-1 X h K ( h ).
n-
Thus, I have the next result
THEOREM 2: (Asymptotic Normality): Given Assumptions A1-A5, and given that E kHn(xi,xj)k2 =
o(n), where
Hn(xi,xj ) = h K ( xj - xi ) [s(θ*,Xi) - s(θ*,xj )]
then
√n(bn - θ* ) ~a n(0,C(θ*))
13See Definition 3 and 4 Dhrymes (1998).