Abstract
We develop a new consistent conditional moment test of functional form based on nuisance parameter indexed sample moments first presented in Bierens. We reduce the nuisance parameter space to known countable sets, which leads to a weighted average conditional moment test in the spirit of Bierens and Ploberger’s integrated conditional moment test (ICM). The weights are possibly stochastic in an arbitrary way, integer-indexed and flexible enough to cover a range of tests from average to higher quantile to maximum tests. The limit distribution under the null and local alternative belong to the same class as the ICM statistic, hence our test is admissible if the errors are Gaussian, and a flat weight leads to the greatest weighted average local power.
6 Appendix A: Assumption C
Assumption C1: 
∈ ℝ × ℝkis an iid process with non-degenerate absolutely continuous marginal distributions and finite variances. The parameter space Φ is a compact subset of ℝk+1. ϕ0=
∈ interior {Φ} f(.,ϕ) is for each ∈ Borel measurable, and twice continuously differentiable on Φ.
Assumption C2: Let 
then An(ϕ) → A(ϕ) uniformly on Φ, where A(ϕ) is a non-stochastic positive definite matrix. The NLLS estimator 
satisfies
Assumption C3: Write 
where 
Then 
uniformly on ℕk ×Φ where b(m,ϕ) is a non-stochastic function satisfying 
Assumption C4:
a finite non-stochastic matrix.ut is governed by a non-generate distribution, and
exists and is constant and finite. There exists a mapping η: ℕk→ ℝ such that 
uniformly on ℕk. if
for some q : ℕk→ ℝ then η(m) = O(q(m)).There exists a matrix functional Γ(m1,m2) on ℕk×k such that
and 
pointwise on ℕk×k. If 
for some q : ℕk→ ℝ then 
Let
. For each ζ ∈ ℝk, 
is uniformly integrable and 
for some t > 0. Moreover, 
for each u∈[0, ∞).
7 Appendix B: proofs of main results
Proof of Lemma 3.1. Recall from (6) 
and define
for arbitrary Π ∈ ℝl, π′π=1, 
and l ≥ 1. Assumption C, Cramér’s Theorem, and the Lindeberg central limit theorem guarantee
The claim follows by invoking the Cramér-Wold Theorem. The bounds 
and 
follow from Assumptions A and C. 
.
Proof of Lemma 3.2. It suffices to show 
is uniformly tight on 
for each 
by straightforward extensions of results in Bickel and Wichura (1971) and Neuhaus (1971). We will apply Lemma A.1 of Bierens and Ploberger (1997). The functional g(xt,ξ) must satisfy a Lipschitz continuity condition on 
for arbitrary 
:
for every 
and for some Kt measurable with respect to 
that satisfies 
Finally, we need
for one arbitrary 
All requirements are met under Assumption C by choosing 
Since 
is bounded 
by construction of Gm(‧) under Assumption A and 
it follows
Proof of theorem 3.3 Using the non-stochastic limiting sequence {ωm}, apply expansion (6), Lemmas 3.1 and 3.2, and the continuous mapping theorem to verify under Assumption C
By (6) it therefore suffices to prove 
By the triangle inequality and ωm≥0
Assumptions A-C and Nn→∞ imply
hence by Chebyshev’s inequality 
For A1,n observe
Assumption B states 
By Lemmas 3.1 and 3.2 
where 
Therefore 
which completes the proof.3
.
Proof of theorem 3.4 Denote by 
the inner product space of sequences 
of continuous functions y(ξ)∈C[0,∞)k with bound y(ξ)=O(ρξ), metrized with 
Let 
be the supporting inner product. Then H is separable and complete3, hence a separable Hilbert space. Separable inner product spaces have countably infinite orthonormal basis, say 
, 
(e.g. Giles, 2000: Theorem 3.27).
Now define Fourier coefficients
then z∈H admits a coordinate-wise expansion
Because 
is a symmetric positive-semi-definite 
bounded function under Lemma 3.1, it follows that 
is a linear compact self-adjoint operator (Giles, 2000: section 15). By the spectral theorem for compact self-adjoint operators on a Hilbert space there exists an orthonormal basis of H consisting of eigenfunctions of Γ, where each eigenvalue λi is real and non-negative (Giles, 2000: Theorem 20.4.1). It is immediate that 
denotes the eigenfunctions of Γ,
and Γ(m1,m2) obtains the series representation 
Use Parseval’s identity, (11) and (12) and orthonormality to get
Each z(m) under 
is mean zero Gaussian by Lemma 3.1, therefore each Fourier coefficient 
is Gaussian, completely characterized by means 
and pair-wise covariances
Thus 
which completes the proof.
Proof of Corollary 3.6 Since the argument simply mimics the proof of Theorem 3.4, we only present a sketch. Write 
Exploit 
to deduce the following. First, 
if i = j and 0 otherwise; hence 
and 
hence the Fourier coefficients 
are w1:=z(m*) and wi = 0 a.s. ∀i≥2.
Second, 
hence λ1=Γ(m*, m*) and λi=03∀i≥2.
Third, use Parseval’s identity and orthonormality to obtain the trivial identity 
Fourth, the Fourier coefficients 
are Gaussian, completely characterized by means 
if i = 1 and 0 ∀i≥2, and pair-wise covariances that reduce to 
if i≠j, i≠1 and λ1 otherwise.
Therefore 
and wi = 0 a.s. ∀i≥2. Coupled with 
this completes the proof under 
. Under 
we have 
hence by (4) 
8 Appendix C
Logistic SWACM power at 5% level
- 1
Recall a real analytic function is infinitely differentiable and therefore has an infinite order Taylor expansion.
- 2
The data are iid Gaussian and the nuisance parameter space is countable, hence it is easy to show Hansen’s approximate p-value is consistent (cf. de Jong 1996, Hill 2008).
- 3
It is straightforward to show Davidson’s (1994: Theorem 5:15) argument carries over to H due to boundedness y(ξ) = O(ρ–ξ) of every y∈H.
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