well. It is not our objective to make close power comparisons between the Bierens and
the RESET tests3. It seems however, that the Bierens modified test has better power
than the RESET test, for the particular choice of weighting and basis functions4.
The RESET test performs better, when f = f⅛. Notice that in this case, the basis
function used resembles the neglected component and as a result it provides a good
approximation. It should be stressed again that the RESET test can be converted
into a fully consistent test. We expect that the utilising a Bierens weighting function
in the RESET test statistic, could improve power.
It follows from our simulation experiment, that all the tests under consideration
have reasonable power in the presence of a neglected locally integrable component.
Hong and Phillips (2005) and Kasparis (2005) develop specification tests that have
power against locally integrable alternatives but have no power, when there is some
neglected integrable term. Contrary to the tests proposed in the two aforementioned
papers, our Bierens tests (as well as the Marmer’s test) provide valid testing proce-
dures for both families of transformations. It is not difficult to see why the Bierens
tests have power in the presence of neglected locally integrable components. For in-
stance, suppose that the true model is given by (1), with f locally integrable, and
the fitted model by (6). Then Theorem 3(ii) still holds.
5 Application
The question whether certain financial ratios (i.e. dividend yield (DY), book-to-
market (BM) and equity-to-price ratio (EP)) can predict stock returns has received
much attention over the years. A substantial body of applied work in this area focuses
on the following linear model (see for example Levellen (2004) and the references
therein):
rt = co + α,jx∕ ∣ + ut,
where rt is stock returns and xt is some financial ratio. Stambaugh (1999), Levellen
(2004), Goyal and Welch (2003) among others, explore the predictability of NYSE
returns using a t-test for the regression slope parameter.
A simple inspection of the NYSE data, reveals that the returns and the finan-
cial ratios series have very different properties. Namely, the NYSE returns series
exhibits mean reversion, constant variance and little autocorrelation. On the other
hand financial ratios, exhibit no mean reversion, strong persistence and time varying
variance. Actually, the financial ratio series are reminiscent of integrated processes.
The fact that two sets of series exhibit very different characteristics give support to
a possible non-linear relationship between returns and financial ratios. A non-linear
transformation applied to some trending series may attenuate its intensity and bound
its variance.
3Clearly, for a close comparison, we should consider a variaty of basis and weighting functions.
4The basis functions utilised for the RESET test are ψk(x) = xkφ(x), к = {1, 2, 3}.
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