The name is absent

for random variables with positive density functions (see for example Halmos, 1950
p.104). For purposes of generality and convenience, we formulate the test hypothesis
as in Definition 1 without any further reference.

Finally, we present some preliminary results. In order to derive the asymptotic
properties of our test, we need to characterise the limit of the NLS estimator both
under the null and the alternative hypothesis. Let σ~¹ Wj, 1 ≤ j ≤ J indepen-
dent standard Gaussian. In addition, W_j∙ ^,s are independent of Lj (0,1)^,s and U. The
following lemma demonstrates the limit theory of the NLS under the null hypothesis:

Lemma 2. Let

(a) Assumption A and H₀ hold.

(b) For 1 ≤ j ≤ J:

(i) gj and ]j are !-regular on Aj.

(ii) There are a_oj ∈ Aj such that j'f° (fj(s') — gj(s,aj))² ds > 0, for all aj = a_oj in

^Aj.  ∞

⁽ⁱⁱⁱ⁾ £L ^gj^(s,ao,j^)gj^(s,ao,j^)>ds > 0.

Then we have

fn (C — Co) → U(1)

as n → ∞.

To obtain the limit properties of the tests under the alternative hypothesis, we need
to establish that the NLS estimator has a well defined limit. Sufficient conditions for
this are provided by Marmer (2005) in the context of single covariate models. The
following result holds for multi-covariate models:

Lemma 3. Let

(a) Assumption A and H₁ hold.

(b) For j = {1,..., J}, there are aj ∈ Aj such that
for all aj = aj in Aj.

Then, as n → ∞, we have

^ p. „  „„J ^ p. „*

c —→ Co and a —→ a ,

where a* = (af,..., a*f)^r.

Lemmas 2 and 3 are essential for the subsequent analysis as they establish that à has
a well defined limit both under H₀ and H₁.

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