Marmer (2005) studies the relationship between NYSE stock returns and the
dividend yield. Marmer proposes the following non-linear model:
rt = co + f (xt-i) + ut,
with f being some integrable transformation and xt a unit root process. An inte-
grable transformation applied to an integrated process attenuates the intensity of the
process. Integrable transformations of unit root processes tend to be non-zero, when
the process is in the vicinity of some spatial point and zero, when the unit root is away
from that point. Therefore, f (xt) takes non-zero values rarely, due to the transient
behaviour of xt. An integrable transformation added to some stationary sequence
produces a seemingly stationary process. In view of this, Marmer (2005) points out
that the above model provides a small deviation from the martingale difference hy-
pothesis. It should be mentioned that locally integrable transformations (see Park
and Phillips (2001)) may also reduce the intensity of some integrated process, but
to a lesser extend. Therefore, integrable transformations produce smaller deviations
from the martingale difference hypothesis. Marmer tests for predictability of returns
by regressing returns on an intercept term and then applies his RESET test to check
for some neglected integrable function of the DY variable. The RESET test provides
evidence for a neglected non-linear component relating to the DY.
In this section we employ the Bierens tests to investigate whether returns are
predictable. Apart from DY, we also consider BM and EP as possible predictors.
Our data are the same as those used by Levellen (2004). A series of stationarity
tests suggest that DY, BM and EP are integrated series (see Table 3). In addition,
the stationarity tests suggest that the returns series a stationary one. This is not
inconsistent with the I-regular formulation of returns. The DF test performed on
an I-regural series favours the stationary alternative (see Park and Phillips, 1998).
Partial sum tests like the KPSS and the CUSUM, would also favour stationarity (see
Kasparis, 2005).
We apply the Bierens test on the residuals of a model that involves an intercept
term only. Table 4 provides results for the modified Bierens statistic (B(rn)). We also
provide results for Marmer’s RESET test5. The modified statistic has been estimated
for M = [—15,15]. Instead of using a randomised mo, we have chosen mo = 1,
15, in order to enable the reader to verify our results. We consider two weighting
functions: W,(s) = (1 + .⅛2) 1 exp(rnΦ⅛(s)), i = 1, 2, with Φι(s) = tan l(.⅛∕10) and
Φ2(s) = tan l(.⅛∕2). In addition, we consider two penalty terms, for the modified
Bierens statistic.
Both the Bierens and the RESET tests indicate that the three financial ratios
have predictive power. The Bierens tests reject then null hypothesis at 5%, when W2
is employed for all three variables. In addition the null is rejected at 1% level in most
cases for the particular weighting function. The null hypothesis is also rejected at
5We have used a single basis function (φ) for the RESET statistic.
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