The name is absent

The reader is referred to Park and Phillips (2000, 2001) for further discussion about
the local time process and its relevance to econometrics.

We assume that the fitted model is given by:

Vt = ɛ + g(%t,a) + u_t.

additively separable of the form g(x,a) =     gj (xj ,a,j ),

J=I

with g_j∙ (x_j∙, α_j∙) being an I-regular function on a compact set Aj C R^fej. Finally, (c, a)
is the NLS estimator defined as

(c,a) = arg min Q_n(c,a), A = A₁ × ... × A_j,
(c,α)∈I×A

where

n

Qn(c, a) = ∑(yt - c - g(xt, a))² .

t=1

We assume that the fitted regression components g_j∙ (xj ,aj) are possibly “different”
than their true counterparts i.e. ∕_j∙ (xj ). This is explained precisely later.

Before we present our test, we provide a concise review of the Bierens tests pro-
posed for strictly stationary data. Typically, specification tests (e.g. Newey, 1985),
under the null hypothesis (correct specification) impose a finite number of moment
conditions of the form

E [(y_t — c — g(x_t, a)) w_i(x_t)] = 0, for some (c_o, a_o) ∈ R × A,

where w_i(.) i = 1, ..,I are weighting functions. In the stationary case, under misspeci-
fication, a test statistic based on weighted residuals, T_i say, in large samples typically
behaves as

Ti ~ √nE [(yt - c - g(xt, a )) w_i(xt)^] ,

where c* and a* are pseudo-true values (i.e. the limit of the NLS estimator under
misspecification). The test is consistent as long as the expectation shown above is
non-zero for some i = 1,...,l. Clearly, the more weighting functions are used, the more
likely is that the condition (3) will be violated under misspecification. Nonetheless, as
pointed out by Bierens (1990), a test that utilises only finite many moment conditions
cannot be consistent against all possible alternatives as it will always be possible to
find data generating mechanisms such that misspecification cannot be detected. If
infinite many moment conditions were tested, the test could be consistent against all
possible alternatives.

Bierens (1990) essentially imposes an infinite many moment conditions, by con-
sidering the following weighting function:

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