are not fully consistent. The test proposed in this paper is a fully consistent test
for the I-regular class. In particular, we consider multi-covariate regressions with
additively separable I-regular components and exogenous regressors.
The test hypothesis of Bierens (1990) is determined by some probability measure.
Actually, the null hypothesis is defined in terms of a conditional moment condition.
Under the null hypothesis (correct specification) the moment condition holds with
probability one. For stationary models, 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 (1990) test statistic.
The equivalence mentioned above does not always hold when the model is I -
regular. In the context of I-regular models we show, that if the null hypothesis of
the Bierens test is defined in terms of the Lebesgue measure instead, there is one-
to-one correspondence between the truth of the null hypothesis and the asymptotic
behaviour of the sample moments of the Bierens test statistic. Our test detects
misspecification in large samples, if the functional part of the true model differs from
the fitted regression function on a set of non-zero Lebesgue measure.
The rest of this paper is organised as follows: Section 2 specifies the test hypothesis
and provides some preliminary results. In Section 3, our main results are presented.
A Monte Carlo experiment is conducted in Section 4, while Section 5 provides some
empirical application to the predictability of returns. Before proceeding to the next
section, we introduce some notation. For a matrix A = (a,ij), is ∣A∣ the matrix of
the moduli of its elements. The maximum of the moduli is denoted by Ц.Ц. For a
function g : Rp → R, define the arrays
(^gʌ ■■ = ( d2g ʌ
∖∂ai) , g ∖∂ai∂aj)
to be vectors, arranged by the lexicographic ordering of their indices. By ʃ f (s')ds,
we denote the Lebesgue integral of the function f. The Lebesgue measure of some
Borel set A on Rfe is denoted by A[A]. Finally, 1{A} is the indicator function of a set
A.
2 The model, the test hypothesis and preliminary
results
In this section we specify the model under consideration and the test hypothesis.
Some preliminary results are also provided. We assume that the series {^t}"=1 is
generated by the following additively separable regression model:
dt = co + f {χt} + ut∙, (1)