[Sebastian Kripfganz]

If none of the I(1) independent variables are cointegrated with the dependent variables, their long-run coefficients should all be zero. They could still have significant short-run effects, such that keeping them in the model might be meaningful nevertheless.

An I(1) dependent variable with nonzero ADJ coefficient and a nonzero long-run coefficient of an I(0) variable constitutes a contradiction. Possibly one or more of the unit-root tests gave a wrong indication, or the deterministic components of the model are misspecified (e.g. a missing linear time trend).

[John Costopoulos]

Dear Sebastian,

Thank you for your reply and (mainly) for your patience. I see now that I had not interpreted correctly the condition "there can be at most one cointegrating relationship involving the dependent variable". At most one may mean zero or one, but as I understand now the former refers only to the case of a I(0) dependent variable. The problem is that I never expected that a I(0) dependent variable could in practice participate in a long-run relationship in the context of the ARDL-ECM approach to cointegration, since the dependent variable had to be I(1) in order to rule out one of the two degenerate cases that are presented in PSS (2001).

Based on the above, as well on your first answer in post #16, it is clear that if the dependent variable were I(1), then our model should always have to include one (but only one) I(1) independent variable which should have to be integrated with the dependent variable y. If x1 is I(1) and cointegrated with y, while x2 and x3 are the rest I(1) independent variables, should we expect that the coefficient of x1 would be always significant, while the coefficients of x2 and x3 would be always non-significant?

Regarding the role of the I(0) variables in a long-run relationship, should they always have a non-significant long-run coefficient if the dependent variable is I(1), otherwise it would be a sign of model mispecification or a wrong indication from a unit-root test? Until today I believed that a I(0) variable might theoretically have a significant long-run coefficient in order to help the system "retain" its long-run equilibrium.

[Sebastian Kripfganz]

Based on the above, as well on your first answer in post #16, it is clear that if the dependent variable were I(1), then our model should always have to include one (but only one) I(1) independent variable which should have to be integrated with the dependent variable y.

I think you are still misinterpreting the condition that "there can be at most one cointegrating relationship involving the dependent variable". It is perfectly fine to have one cointegrating relationship that involves multiple variables, e.g.
y = x1 theta1 + x2 theta2
but there should not be two separate cointegrating relationships:
y = x1 theta1
y = x2 theta2

If x1 is I(1) and cointegrated with y, while x2 and x3 are the rest I(1) independent variables, should we expect that the coefficient of x1 would be always significant, while the coefficients of x2 and x3 would be always non-significant?

"always" isn't the right word here. If you choose a 5% significance level, then even the best test should reject the null hypothesis that the coefficients of x2 and x3 are zero in 5% of the cases.

Regarding the role of the I(0) variables in a long-run relationship, should they always have a non-significant long-run coefficient if the dependent variable is I(1), otherwise it would be a sign of model mispecification or a wrong indication from a unit-root test? Until today I believed that a I(0) variable might theoretically have a significant long-run coefficient in order to help the system "retain" its long-run equilibrium.

I guess you could come up with an argument that there can be long-run relationship between an I(0) variable and the "gap" between two cointegrated variables. A permanent change in an I(0) variable could widen the gap between the cointegrated variables but the latter variables continue moving together. This argument could also go the other way round, the deviation from a cointegrated relationship could affect an I(0) variable. Thus, when the dependent variable is I(0), nonzero long-run coefficients of two I(1) variables are possible if they are cointegrated with each other.

[John Costopoulos]

Dear Sebastian,

To understand it better, if a Johansen test showed "There is no cointegration relationship between y, x1, x2 as a whole (I mean your first equation)", should we expect to get zero long-run coefficients for both x1 and x2?

Regarding your second answer, I still miss the case "should we expect that the coefficient of x1 would be always significant".

I wish I could really come up with so insightful arguments as those presented in your third answer. I'm just trying to compile a short list of cases that may lead to erroneous results. If I understood correctly your third answer, then: (1) If the dependent variable is I(1), a I(0) variable may possibly have a significant long-run coefficient through its contribution in widening the gap between two cointegrated variables, but never as a result of a direct permanent impact of it on the dependent variable. (2) When the dependent variable is I(0), nonzero long-run coefficients of two I(1) variables are possible if they are cointegrated with each other. As I understand it, if there were only one I(1) variable in the model, it would be impossible to have a non-zero long-run coefficient. If, in addition, it had a zero short-run coefficient should we exclude it from the model or leave it in the model as a control variable?

[Sebastian Kripfganz]

To understand it better, if a Johansen test showed "There is no cointegration relationship between y, x1, x2 as a whole (I mean your first equation)", should we expect to get zero long-run coefficients for both x1 and x2?

In principle, yes.

Regarding your second answer, I still miss the case "should we expect that the coefficient of x1 would be always significant".

"If x1 is I(1) and cointegrated with y, while x2 and x3 are the rest I(1) independent variables", the true long-run coefficient of x1 should be nonzero and the true long-run coefficients of x2, x3 should be zero.

I wish I could really come up with so insightful arguments as those presented in your third answer. I'm just trying to compile a short list of cases that may lead to erroneous results. If I understood correctly your third answer, then: (1) If the dependent variable is I(1), a I(0) variable may possibly have a significant long-run coefficient through its contribution in widening the gap between two cointegrated variables, but never as a result of a direct permanent impact of it on the dependent variable. (2) When the dependent variable is I(0), nonzero long-run coefficients of two I(1) variables are possible if they are cointegrated with each other. As I understand it, if there were only one I(1) variable in the model, it would be impossible to have a non-zero long-run coefficient. If, in addition, it had a zero short-run coefficient should we exclude it from the model or leave it in the model as a control variable?

Yes.
If neither the long-run nor the short-run coefficients are (statistically significantly) nonzero, then you could obtain more efficient estimates by removing this variable from the model.

[John Costopoulos]

Dear Sebastian,

I am grateful to your for your detailed and enlightening answers, as well for the time you devoted to me even in the weekend.

After all, I am unfortunately fully disappointed with the ARDL approach to cointegration. It seems that I had too many and unrealistic expectations from that method (the blame only to me). If the dependent variable is I(1), then the only case where I should expect to find a significant long-run coefficient is when there is an independent variable which is I(1) and cointegrated with y. All the other independent variables, either I(1) or I(0), should be expected to have a zero long-run coefficient (except for a few cases you have so far mentioned in your posts). Therefore, the term "long-run relationship" resembles (to my eyes, at least) as a cointegration between the dependent variable and an independent variable only, while controlling for the other independent variables which however cannot normally have a significant long-run impact on the dependent variable.

The only positive thing in this journey was that I had the opportunity to use your ardl program, which really impressed me for the following reasons: (1) it is very stable (never crashed), (2) you do not have to wait for a long time to calculate the optimal lags (your Mata code calculates them in seconds), (3) it automatically checks the F- and t-statistic against the advanced critical values provided by you and Daniel Schneider, and (4) it automatically accommodates to problematic cases, such as that of a zero optimal lag in a ec1 representation. My warm congratulations to both of you!

[Sebastian Kripfganz]

Yes, with I(1) variables, a long-run relationship in an ECM essentially corresponds to a cointegrating relationship.

Many thanks for your warm words. Good luck with your research.

[Eric de Souza]

If the dependent variable is I(1), then the only case where I should expect to find a significant long-run coefficient is when there is an independent variable which is I(1) and cointegrated with y. All the other independent variables, either I(1) or I(0), should be expected to have a zero long-run coefficient

Your two sentences taken together are ambiguous. Cointegration requires I(I) variables but there can be several such variables in the relationship, not only one independent variable.
The whole idea of cointegration was developed because if you arbitrarily select several I(1) variables and run a regression between them you have a high probability of getting a very significant relation, the so-called spurious regression. It was to distinguish spurious relationship from non-spurious relationship that the concept of cointegration was developed.

[John Costopoulos]

Dear Eric,

Thank you very much for your interest and your willingness to help.

Sebastian and I did not refer to the well-known concept of cointegration between a I(1) dependent variable y and and some I(1) independent variables, where all variables (e.g. y, x1, x2, and x3) "move together".

As you can see in the second answer in post #20 above, we referred to the case where only x1 is cointegrated with y, while x2 and x3 are not. Just by eyeballing, I think that such a case is presented on pages 309-310 in Pesaran et al.(2001) where the dependent variable "Real wages" seems to co-trend with "Productivity", while the rest independent variables in Figures 2 and 3 do not seem to participate in the cointegration.

Although I always thought, probably due to a misinterpretation of the PPS' example, that the long-run coefficients of x2 and x3 might have equally good chances with the long-run coefficient of x1 to be significant (as well as that even I(0) variables might have significant long-run coefficients in a long-run relationship), Sebastian's answer in post #20 is crystal clear (although very disappointing for me!) that only x1 can have a significant long-run coefficient.

[Eric de Souza]

What I am trying to say is that the word "cointegration" is applied to the specific case of I(1) variables which co-integrate to form a stationary variable. It is a question of semantics. Gouriéroux and Peaucelle called the long run relationship between stationary varibales "co-dependence".
See https://ideas.repec.org/p/cpm/cepmap/8902.html
On Edit: the purpose of providing a link to the paper was not to get you to look at it: it is quite arduous to read. It is just to say that stationary variables be involved in a "long-run" relationship: they are just not needed to form a stationary variable. It all depends on how one understands the expression "long run".

[John Costopoulos]

Dear Eric,

I' am very sorry but I had not understood that you referred to cointegration from a semantics point of view. Your statement "It all depends on how one understands the expression 'long run'" is what actually I'm trying to understand through some "technical" questions (please forgive me for my English). To be honest, I considered it as a kind of "co-dependence" (thank you for your helpful reference). However, what exactly kind of co-dependence? Which is the "glue" among so heterogeneous variables (I(0) and I(1)) mixed together!) that is able to retain their equilibrium? I have not yet answered this question, but at least Sebastian's enlightening answers helped me to come across some practical rules which provide a great help in checking critically the regression results.

[Sam Helen]

Dear Sebastian Kripfganz ,

On a slightly different note: May we discuss what the purpose of including a time trend in the error correction specification of the ARDL model is? Especially when the variables are already specified in 1st differences?

I am referring to the following equation:

Delta Y_{t} & = c_{0} + c_{1}t - alpha(Y_{t-1} - theta X_{t-1}) + Sum_{j=1}^{p-1} gamma_{j} Delta Y_{t-j} + Sum_{j=1}^{q-1} phi_{j} Delta X_{t-j} + epsilon_{t}

TREND ERROR CORRECTION TERM 1st DIFFERENCED Y VARIABLE 1st DIFFERENCED X VARIABLE

Does first differencing the variables not take care of any time-trends already, making t obsolete? The only reason I can imagine including a trend nevertheless is to control for any other unobserved time-varying variables that could influence the left and right hand side simultaneously.

Very curious to hear your thoughts, und alles Gute aus Oxford.

Sam

[Sebastian Kripfganz]

You are not estimating the model entirely in first differences. There are still the lagged levels of Y and X on the right-hand side. These variables could possibly be trend-stationary variables. To account for the possible existence of a deterministic linear time trend in these variables, we might want to include the time trend in the specification. Otherwise, if both X and Y have a time trend but we do not account for it, we might estimate a spurious relationship between the two variables simply because of the similar trend even though there might not be any fundamental relationship among them.

[Sam Helen]

I see Sebastian Kripfganz ! So what I missed is the level variables in the error correction term, of course! Thanks Sebastian, this makes a lot of sense. Do you agree with my assessment of the usefulness of time trends for capturing unobserved time-varying factors?

[Sebastian Kripfganz]

If these unobserved time-varying factors exhibit a linear trend, then to some degree yes. But there could still be an omitted variables bias if these unobserved factors are related to the observed variables not just by the time trend.