Re: [Gretl-devel] ADF p-value glitch?
Am 05.11.18 um 23:34 schrieb Allin Cottrell:
Summary, from my point of view: We're now clear that gretl is not
ignoring the sign of the test statistic, and is doing what R, Stata
and Eviews also do, namely reporting an estimate of the CDF of the
relevant distribution evaluated at the observed test statistic.
Although I think this convention is too strong to be bucked, I do
think that Sven has a point. It's not specific to ADF tests, but
pertains more generally to p-values for one-sided tests. It's not
clear to me that p-values are well defined for values of the test
statistic that are not at all adverse to the null --
Thank you for the summary, Allin, that's exactly what I meant.
(@ all others: thank you for your comments, but it seemed to me that you
were stating or proving things that I had never doubted in the first place)
For those who are still interested in the philosophy of statistics,
after some more reading (browsing, really) I think some of that
differing interpretation of p-values actually dates back to discussions
between Fisher vs. Neyman-Pearson. Fisher apparently advocated the
p-value as a continuous (or monotonous) summary statistic of how well
the data are compatible with the hypothesis. In that sense my doubts
about the one-sided test situation were Fisherian when I said, how can
data that exactly coincides with the H0 value be classified by the
software as *less* compatible (lower p-value, p<1) than data further
away from it.
However, I now actually think there is another layer of complexity added
by the special ADF situation.
In a standard one-sided test setup (non-ADF) the null hypothesis is what
is called "composite", it covers a continuum of values like in H0: b
\geq b_0. So the case bhat > b is no deviation from H0 at all. My old
argument was that P-values in that area are dubious, given that they are
supposed to refer to something "at least as extreme as the observation",
when the observation is not extreme at all.
In the ADF situation instead it really is just the unit-root point
hypothesis. Not composite. So rhohat > 1 is not part of H0, but at the
same time we would never reject H0 because of it. Very strange beast.
P-values in that region also dubious, but perhaps for different reasons.
My conjecture is that many people in R, Stata, Eviews have asked
themselves the same questions, but there is no satisfactory solution to
be communicated in a single number. Fortunately the numerical difference
is small (as someone already mentioned) and so for standard levels of
significance and decision rules it's just not relevant.
The current urcpval results in the upper tail may still be wrong and/or
meaningless when you take the meaning of p-values seriously, but I
concede it's hard to come up with something else which is also concise.
thanks,
sven