[Francisco Neto]

I really appreciate the very detailed and clear response. I just had another quick question if you do not mind.

As previously described, given that a few of my variables have q* = 0, I then use the "ec" command for the ardl; meaning that the long-run coefficients would enter at time t and not t-1. So, I assume that I could then use the ECT*(-1) notation as proposed above (e*_t-1 = y_t-1 - θ x_t). This would then mean that the short-run coefficients in the model would be the effects conditional on a deviation of lagged y from the equilibrium value prescribed by X in the current period, correct?

Thank you.

[Sebastian Kripfganz]

Yes.

[Francisco Neto]

Thanks so much. Based in the information criteria, I have an ARDL (1, 4, 0, 0 , 2, 0). In this case the optimal lag of the dependent variable (y) is 1. Meaning it enters the error correction representation of the ARDL at lag 0. Bounds test confirms cointegration among the variables and other diagnosis tests show stability. Was just wondering what that exactly implies regarding the role of the dependent variable itself (and also the error correction term) and how the mathematical representation of the model would change. Thank you again.

[Sebastian Kripfganz]

The dependent variable is always lagged in the error correction term: y_t-1, irrespective of whether you use option ec or ec1. I am not sure I understand the rest of your query.

[Francisco Neto]

I appreciate the quick response. As some of my x variables have optimal lag of zero, equation 7 attached to this post is the one estimated (using ec option). My question was regarding the differenced term of y (marked in yellow). In my case, does it still make sense to show this term in my mathematical representation, even though in the error correction representation of my ARDL my dependent variable enters at lag zero? Because no short-run results for the differenced terms of y would exist. Hope that clarifies it.

[Sebastian Kripfganz]

If the dependent variable has optimal lag order 1, then there are no differenced lags of the dependent variable among the short-run terms; you therefore do not need to list them.

[Hamid muili]

Dear Prof Sebastian Kripfganz , I set up an ardl estimate using a Lin-Log specification, on the RHS I have one of my independent variables as lpop( log of population). When I run this it shows l.pop has been omitted due to collinearity.. however if I ran the same specification in eviews, I get output but I couldn't get any output from Stata..

[ahmed mohamed ak]

I indicated that If you have pre-tested that the dependent variable is I(0), then strictly speaking the ARDL bounds test no longer makes sense. Under the null hypothesis of the bounds test, the dependent variable is I(1). Under the alternative hypothesis, it can be I(0).
Does this mean that it cannot be applied the ARDL bounds test?

[Sebastian Kripfganz]

The bounds test is valid independent of whether the dependent variable is I(0) or I(1). Pretesting is not necessary. If you are certain that the dependent variable is I(0), then - strictly speaking - the bounds test would indeed be redundant. You would then just test whether the long-run coefficients of the explanatory variables are statistically significant with conventional hypothesis tests. However, pretesting does not provide certainty. You might still be committing a type-I error - incorrectly rejecting the null hypothesis of a unit root when in fact it is true. To avoid pretesting problems, it is therefore generally recommended to directly apply the bounds test.

[ahmed mohamed ak]

Thank you very much

[ahmed mohamed ak]

What do you think, Professor, in an article “Bootstrapping the autoregressive distributed lag, Robert McNown et all (2017)
where they wrote" PSS mentions that the bounds test approach is applicable regardless the integration order either I(0) or I(1) for regressors only, which does not include the dependent variable itself." P2

[Sebastian Kripfganz]

I am aware of this statement in the mentioned article, but I am afraid this statement is incorrect.

[ahmed mohamed ak]

Pesaean points out in his research paper that it is unnecessary that the order of integration of the underlying regressors be ascertained prior to testing the existence of a level relationship between yt and xt. while he assumed that the variables must be integrated of the first order or level; That is, it should not I(2).
What is your comment, Professor?

[Sebastian Kripfganz]

It is correct that pretesting the order of integration is not necessary in the ARDL framework, as long as it is safe to assume that non of the variables is I(2). All variables must be either I(0) or I(1), at least after controling for a deterministic trend.

You might find our Stata Journal article helpful:

• Kripfganz, S., and D. C. Schneider (2023). ardl: Estimating autoregressive distributed lag and equilibrium correction models. Stata Journal 23 (4), 983-1019.

[ahmed mohamed ak]

Pesaran also points out in his article that: “ Assumption 1 permits the elements of zt to be purely I(1), purely I(0) or cointegrated but excludes the possibility of seasonal unit roots and explosive roots." P 291
How can we verify this assumption without conducting unit root tests?