6:38 p.m.
I'm trying to upgrade gretl's testing for cointegration rank in a
variety of cases. But I'm stuck on one point: understanding the
relationship between Johansen's famous "five cases" (lucidly
described by Jack in the Gretl User's Guide) and the five cases
discussed by Pesaran, Shin and Smith (Journal of Econometrics,
2000), or "PSS" for short.
Differences arise in relation to the "unrestricted constant" and
"unrestricted trend" cases only. Let's forget the unrestricted
trend, leaving the unrestricted constant. I think I understand
Johansen on this (thanks, Jack!). If the constant, $\mu$, is
restricted to the cointegration space its effect does not "leak
out" into a trend, since $\mu$ is expressible as a linear
combination of the columns of $\alpha$, and $\alpha_{\perp}'\mu =
0$. If the constant is not so restricted, it does leak out and
generate a trend in the levels of the (possibly) cointegrated
variables (call them Y).
Now, as I understand it, PSS have a "Case II" which is identical
to Johansen's restricted constant. Fine. But they also have a
"Case III" which in some way resembles Johansen's unrestricted
constant case except that it does _not_ generate a trend in Y.
This I'm not getting. Can anyone help? I don't understand how you
can "unrestrict" the constant and yet not have it leak out of the
cointegration space into a trend in Y.
Allin
5:29 a.m.
Am 03.04.2011 00:38, schrieb Allin Cottrell:
Now, as I understand it, PSS have a "Case II" which is identical
to Johansen's restricted constant. Fine. But they also have a
"Case III" which in some way resembles Johansen's unrestricted
constant case except that it does _not_ generate a trend in Y.
This I'm not getting. Can anyone help? I don't understand how you
can "unrestrict" the constant and yet not have it leak out of the
cointegration space into a trend in Y.
Maybe it's just a misunderstanding: when they say "Case III:
(Unrestricted intercepts; no trends.)" I interpret it as "the trend term
in the equation for the VECM disappears", not as "the trend in the
process is absent". So it's just the standard cases IMO.
If that's not what you meant, could you be more specific with page
numbers etc.?
cheers,
sven
5:01 p.m.
On Tue, 5 Apr 2011, Sven Schreiber wrote:
Am 03.04.2011 00:38, schrieb Allin Cottrell:
> Now, as I understand it, PSS have a "Case II" which is identical
> to Johansen's restricted constant. Fine. But they also have a
> "Case III" which in some way resembles Johansen's unrestricted
> constant case except that it does _not_ generate a trend in Y.
> This I'm not getting. Can anyone help? I don't understand how you
> can "unrestrict" the constant and yet not have it leak out of the
> cointegration space into a trend in Y.
Maybe it's just a misunderstanding: when they say "Case III:
(Unrestricted intercepts; no trends.)" I interpret it as "the trend term
in the equation for the VECM disappears", not as "the trend in the
process is absent". So it's just the standard cases IMO.
I think I now understand this better: the test statistic is indeed
just the standard one, but PSS refer it to a different
distribution from Johansen. I've written up a note on this, by way
of response to a 2009 piece in the Journal of Applied Econometrics
by Paul Turner, which I think gets it wrong. My note is at
http://www.wfu.edu/~cottrell/tmp/coint_model.pdf
Comments, criticisms welcome.
Allin
6:41 p.m.
Am 05.04.2011 23:01, schrieb Allin Cottrell:
On Tue, 5 Apr 2011, Sven Schreiber wrote:
> Am 03.04.2011 00:38, schrieb Allin Cottrell:
>> Now, as I understand it, PSS have a "Case II" which is identical
>> to Johansen's restricted constant. Fine. But they also have a
>> "Case III" which in some way resembles Johansen's unrestricted
>> constant case except that it does _not_ generate a trend in Y.
>> This I'm not getting. Can anyone help? I don't understand how you
>> can "unrestrict" the constant and yet not have it leak out of the
>> cointegration space into a trend in Y.
>
> Maybe it's just a misunderstanding: when they say "Case III:
> (Unrestricted intercepts; no trends.)" I interpret it as "the trend term
> in the equation for the VECM disappears", not as "the trend in the
> process is absent". So it's just the standard cases IMO.
I think I now understand this better: the test statistic is indeed
just the standard one, but PSS refer it to a different
distribution from Johansen. I've written up a note on this, by way
of response to a 2009 piece in the Journal of Applied Econometrics
by Paul Turner, which I think gets it wrong. My note is at
http://www.wfu.edu/~cottrell/tmp/coint_model.pdf
Comments, criticisms welcome.
Hm, I took a look at your note and also some further look at the PSS
paper (still without a proper reading though, so take everything with
grains of salt). I understand why they want to work with their framework
where the lag polynomial is applied to the de-trended levels (X_t - \mu
- \gamma t). IIRC the Saikkonen&Lütkepohl-type approach does it in a
similar way (including dummies for shifts). But then I don't understand
why they have their Case III at all, where the corresponding restriction
is "ignored". Your table 4 seems to show exactly the problems with that
"ignorance", if I understand correctly.
Also, the way you describe the Turner exercise, it seems that he pointed
out that the trace test is not "similar" (in the statistical jargon
sense) in the famous unrestricted constant case. But that had been well
known for a long time. Or am I missing something here (haven't looked at
the Turner paper myself)?
Anyway, in many applications the presence of deterministic trend
components in the data is pretty obvious IMHO. And at the same time we
don't really want to "explain" long-run connections between variables
with exogenous deterministic terms. I mean we want long-run forcing
variables, elasticities, and all that! So the famous unrestricted
constant case remains highly relevant, despite the statistical
drawbacks. (Not that I think that you were doubting that...)
cheers,
sven
8:05 p.m.
On Wed, 6 Apr 2011, Sven Schreiber wrote:
Am 05.04.2011 23:01, schrieb Allin Cottrell:
> I think I now understand this better: the test statistic is indeed
> just the standard one, but PSS refer it to a different
> distribution from Johansen. I've written up a note on this, by way
> of response to a 2009 piece in the Journal of Applied Econometrics
> by Paul Turner, which I think gets it wrong. My note is at
> http://www.wfu.edu/~cottrell/tmp/coint_model.pdf
> Comments, criticisms welcome.
Hm, I took a look at your note and also some further look at the PSS
paper (still without a proper reading though, so take everything with
grains of salt). I understand why they want to work with their framework
where the lag polynomial is applied to the de-trended levels (X_t - \mu
- \gamma t). IIRC the Saikkonen&Lütkepohl-type approach does it in a
similar way (including dummies for shifts). But then I don't understand
why they have their Case III at all, where the corresponding restriction
is "ignored". Your table 4 seems to show exactly the problems with that
"ignorance", if I understand correctly.
Thanks for following upon this. I agree: it seems to me there's
something faintly bogus about PSS Case III: they calculate the
statistic "ignoring" the restriction implied by their VAR
formulation, but then refer it to a distribution that is
constructed on the assumption that the restriction holds.
Also, the way you describe the Turner exercise, it seems that he
pointed
out that the trace test is not "similar" (in the statistical jargon
sense) in the famous unrestricted constant case. But that had been well
known for a long time. Or am I missing something here (haven't looked at
the Turner paper myself)?
I don't think you're missing anything here. In fact I'm surprised
that the Turner paper made it into JAE. He says, "Look, the
Johansen Case 1 test over-rejects!" and backs this up by feeding
it data that by construction belong in Johansen Case 0.
Anyway, in many applications the presence of deterministic trend
components in the data is pretty obvious IMHO. And at the same time we
don't really want to "explain" long-run connections between variables
with exogenous deterministic terms. I mean we want long-run forcing
variables, elasticities, and all that! So the famous unrestricted
constant case remains highly relevant, despite the statistical
drawbacks. (Not that I think that you were doubting that...)
Well, I don't _think_ I was doubting that ;-) But this is a pretty
subtle area.
Allin
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