Can you please explain what you mean by "element' here?

About the coefficients: Yes, bokhandel the long-run coefficient of the Capital share as well as the short-run (error correction term) of the Debt ratio are significant, at the 1 per cent lever.

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From: Sven Schreiber <sven.schreiber@fu-berlin.de>
Sent: Sunday, October 30, 2022 9:28:37 AM
To: gretl-users@gretlml.univpm.it <gretl-users@gretlml.univpm.it>
Subject: [Gretl-users] Re: small Johansen test - minimum sample size

Am 30.10.2022 um 08:53 schrieb Lars Ahnland:
According to Lütkepohl and Krätzig (2004, Applied Time Series Econometrics, p. 89) it is viable to relax the assumption that all variables should be I(1) in levels so that the concept of cointegration can is extended by including any stationary linear combination (providing variables are I(0) in first differences).

Exactly, and what we're saying is that in general a linear combination can also mean to just pick one element by having a unit vector as the cointegration vector in this broader sense. This then would not mean that really two variables are cointegrated in the narrower sense.

Från: Cottrell, Allin <cottrell@wfu.edu>

No, what Sven is pointing out is a common "gotcha" with the Johansen
test: one of the series in the llst proposed for testing is I(0). This
in itself does _not_imply that there's any cointegration going on.

To be clear: I don't _know_ that one series is I(0). Nor does Lars _know_ that it is I(1). The evidence is just not super clear, given the small sample size. So the test result could be a reflection of the capital share being I(0), or there could alternatively be genuine cointegration where the debt ratio enters with a small coefficient. (Check the estimates of the cointegration vector under rank 1.)

So, coming back to the original question which is only a little gretl-related in the sense of clarifying if the "johansensmall" package by Andreas Noack Jensen and myself can be used with 30 observations or so: Yes, I think it's possible and the results are not "wrong". Of course, the bootstrap approach never promises to achieve _exactly_ the nominal test size (for example, 5% rejection under the null). We just hope that it corrects the size distortion in small samples _to some extent_. At the same time you will lose test power. So it's still the case that the uncertainty surrounding the test result gets bigger when you have less data, and you definitely have to interpret the results with caution.

cheers

sven