where Φ(.) is some bounded one-to-one transformation. Notice the expression above
entails infinite many weight functions, when M is a continuum of real numbers. The
key result for the test consistency is the following. Bierens (1990) shows that under
certain regularity conditions, E [(yt — g(xt, a*)) exp (mΦ(τt))] equals zero, only when
m belongs to a set of Lebesgue measure zero. Therefore, test consistency can be
achieved with a suitable choice of m. For instance, if m is chosen from some con-
tinuous distribution, the moment shown above will be non-zero a.s. (e.g. Bierens,
1987). Alternatively, a consistent test can be obtained by some appropriate func-
tional of the test statistic. Bierens (1982, 1984) and Bierens and Ploberger (1997)
consider the Cramér-Von Misses functional, while the Bierens (1990) test is based on
the Kolmogorov-Smirnov functional. The latter approach is followed here. It should
be also mentioned, that the choice of exponential function is not of crucial impor-
tance. There several families of weighting functions that can deliver consistent tests.
Stinchcombe and White (1998) show that any function that admits an infinite series
approximation on compact sets, with non-zero series coefficients, could be employed
in the place of the exponential function shown above (see also Escanciano (2006) for
further examples of families).
In the stationary framework, by virtue of the Law of Large Numbers, the sample
moment of the Bierens test statistic converges to some integral with respect to the
probability measure generated by the variables of the model. The null∕alternative
hypothesis of the Bierens (1990) test is also defined in terms of the same probability
measure as shown below:
H0 : P [E [(yt — g(xt, a)) ∣ xt] = 0] = 1 for some ao ∈ A.
H1 : P [E [(yt — g(xt,a)) ∣ xt] = 0] < 1 for all a ∈ A. (3)
Under stationarity, there is one-to-one correspondence between the truth of the
null∕alternative hypothesis and the asymptotic behaviour of the sample moment of
the Bierens test statistic.
The limit behaviour of I-regular transformations however, is characterised by
integrals with respect to the Lebesgue measure. For instance suppose that J =
1. Then under misspecification and Assumption A, a residual based test statistic
asymptotically behaves as:
where L(1, 0) is the (chronological) local time the Brownian motion V up to time 1
at the origin. The local time at the origin is L(1,0) > 0 a.s., therefore the test is
consistent as long as the Lebesgue integral in (5) is non-zero. We show that under
Assumption A, when the null hypothesis of the Bierens test is defined in terms of
the Lebesgue measure, there is one-to-one correspondence between the truth of the
null∕alternative hypothesis and the asymptotic behaviour of the sample moments of
the Bierens test statistic.
z∞
-∞
[(f (s) — g(s,a*)) Wi(s)] dsL(1,0),
(4)