ARDL Bounds Test and ECM

[Rose Banda]

Hi Everyone.

I am looking for clarification on some of the commands I have performed. I have annual data

1. Is it correct to include structural break dummies in bound test? I have 3 structural dummy variables for my model. I have run the bounds test cointegration inclusive of dummies using the command

ardl depvar indvars, exog(dum1 dum2 dum3) lags(1 1 1 3) ec

 estat ectest

2. I am confused about when to use ec1 or ec only. I run my ardl and ECM models with dummies using

ardl depvar indvars, exog(dum1 dum2 dum3) lags(1 1 1 3)

and

ardl depvar indvars, exog(dum1 dum2 dum3) lags(1 1 1 3) ec

Thanks

[Rose Banda]

Also, should I treat the structural dummies as endogenous variables since I create the dummy by interacting it with the independent var?

[Sebastian Kripfganz]

The critical values of the bounds test are not applicable if there is a structural break in the long-run relationship. If you assume that the structural breaks only affect the short-run dynamics, you can add these break dummies with the exog() option.

The ec and ec1 options do not fundamentally change the model. The long-run coefficients are still the same. It is just a reparameterization of some short-run coefficients. See slides 12 to 14 of my 2018 London Stata conference presentation:

• Kripfganz, S. and D. C. Schneider (2018). ardl: Estimating autoregressive distributed lag and equilibrium correction models. Proceedings of the 2018 London Stata Conference.
• ARDL: updated Stata command for the estimation of autoregressive distributed lag and error correction models

Btw: You do not need to start a new topics for all your related questions. This might lead to confusion because some readers may not be aware of some questions that have been answered already.

[Rose Banda]

Thank you Sebastian.
On the ec and ec1 thank you - this part is clear

I am still unclear. how do I determine whether or not a structural break should be entered as an endogenous vs exogenous variable in the bounds test, ardl and ardl ec model.

[Sebastian Kripfganz]

The only "endogenous" variable in the ARDL/EC model is the dependent variable. All other variables must at least be weakly exogenous. If you specify a variable in the list of independent variables in the command line, it will enter the long-run relationship. If you specify it with the exog() option, it will only enter the short-run part and no automatic lag selection is performed on them. As said before, I would not advise to add structural break dummies to the long-run relationship if you want to use the bounds test because the critical values might become invalid.

[Rose Banda]

Thank you very much Sebastian. I would like to reference what you have just said above regarding endogenous and exogenous variables in the ARDL/ECM. Could you Kindly suggest an article/test I can read so as to back up my decision in my thesis

[Sebastian Kripfganz]

Thank you very much Sebastian. I would like to reference what you have just said above regarding endogenous and exogenous variables in the ARDL/ECM. Could you Kindly suggest an article/test I can read so as to back up my decision in my thesis

Some literature is referenced here:
ARDL: updated Stata command for the estimation of autoregressive distributed lag and error correction models
Further references can be found within the articles mentioned there.

[Rose Banda]

Thank you

[John Costopoulos]

Would it be problematic if we used quarter dummies with the exog() option? PSS (2001, p. 307, footnote 17) note that "both the asymptotic theory and associated critical values must be modified if the fraction of periods in which
the dummy variables are non-zero does not tend to zero with the sample size T". Does their note refer only to dummies that cover a period comprising a couple of consecutive years or quarters?

[Sebastian Kripfganz]

The (asymptotic) critical values of the bounds test depend on the specification of the deterministic model components (intercept, linear time trend, quadratic time trend, ...). Seasonal dummy variables (e.g. quarter dummies) are such deterministic components as well. The same issue would apply to other unit-root or cointegration tests. In that regard, yes, it can be problematic to include quarter dummies. Critical values for such a model would have to be obtained by simulation.

The problem does not bite if you have dummies that cover only a fixed number of time periods (not necessarily consecutive). Asymptotically, they do not matter because the influence of those few time periods on the properties of the process vanishes.

[John Costopoulos]

Dear Sebastian, thank you very much for your prompt and your absolutely clear reply.

[John Costopoulos]

Please find below two questions on the ARDL approach to cointegration.

1. Let’s assume that we have a dependent variable y, which is I(1), and two independent variables x1 and x2, respectively.If y, x1 and x2 are cointegrated, a formal definition of cointegration can be easily found in any econometric textbook, while we can intuitively think of them as “moving together”. If x1 and x2 are both I(0), how a “long-run relationship” between y, x1 and x2 could be formally defined? Could we perhaps say that they are in a long-run relationship if at least one of x1 and x2 has a statistically significant long-run coefficient in the unrestricted ECM?

2. In the ADJ section of the ardl output, the coefficient of the lagged dependent variable should be statistically significant and negative, so ADJ isn’t a good place to check for a possible persistency of the dependent variable (as in the case of a GMM approach). Could the L.D.y term be used to check for the persistency of the dependent variable? If yes, which of the two representations (ec/ec1) would be the most appreciate?

[Sebastian Kripfganz]

1. If y is I(1) and x1, x2 are both I(0), then there cannot be a meaningful long-run relationship between y and the regressors. In an ECM, this should be reflected by an insignificant speed-of-adjustment coefficient. If all three variables are I(0), then there is evidence for a long-run relationship if at least one long-run coefficient of x1, x2 is statistically significant.

2. No, the ADJ coefficient is exactly what you want to look at. If the ADJ coefficient is statistically insignificant (based on the bounds test t-statistic), this implies a strong persistence of the dependent variable (essentially a unit root). If the ADJ coefficient approaches -1, then there is not much persistence in the dependent variable (conditional on x1, x2).

[John Costopoulos]

Dear Sebastian,
Thank you for your enlightening reply. Your answers are always of great value for me.

[John Costopoulos]

Dear Sebastian,

I apologize for coming back again, but as I was elaborating further on your answers I happened to read two older posts (1 Oct 2016, #139 and #141), which made me wonder if I have understood well the role of cointegrated variables in the context of the ARDL-ECM framework. More specifically, would it play any role if none of the I(1) independent variables were cointegrated with the dependent variable?
I would also like to ask you if a long-run relationship could be meaningful in the case where the dependent variable is I(1), the long-run coefficients of the I(1) independent variables are not significant, and at least one of the long-run coefficients of the I(0) variables is significant.