[Sebastian Kripfganz]

See for example:

• Pesaran, M. H., Y. Shin, and R. J. Smith (2001). Bounds Testing Approaches to the Analysis of Level Relationships, Journal of Applied Econometrics 16 (3), 289-326.

In Section 2 of this paper, page 291, the authors state the assumptions for the underlying data-generating process. They define a vector \( \mathbf{z}_t = (y_t, \mathbf{x}_t')' \), where \( y_t \) is the dependent variable and the vector \( \mathbf{x}_t \) contains the (weakly) exogenous regressors. They then assume

[..] the elements of \( \mathbf{z}_t \) to be purely \( I(1) \), purely \( I(0) \) or cointegrated [..]

which includes the case where \( y_t \) is \( I(0) \) and the variables in \( \mathbf{x}_t \) are a mix of \( I(0) \) and \( I(1) \) variables.

Further general references on ARDL / EC models include:

• Engle, R. F., and C. W. J. Granger (1987). Co-Integration and Error Correction: Representation, Estimation, and Testing. Econometrica 55 (2), 251-276.
• Hassler, U. and J. Wolters (2006). Autoregressive distributed lag models and cointegration. Allgemeines Statistisches Archiv 90 (1). 59-74.
• Pesaran, M. H., and Y. Shin (1999). An Autoregressive Distributed Lag Modelling Approach to Cointegration Analysis. In: Strom, S. (Ed.): Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium. Cambridge, UK: Cambridge University Press.

[Mobeen Ur Rehman]

Dear Sebastian,

Can I have any journal reference for that. My reviewer has commented on that and I need any journal reference to prove my point.

[Lajos Konok]

Hello all, if I understood properly, one may use ARDL with I(0) and I(1) variables but just one co-integrating relationship. If I have I(1) dependent variable and a mixture of three I(0) and five I(1) independent variables and I have more than one co-integrating relationship among them, may I estimate a vector error-correction model? How should I treat those I(0) variables in my model? Thanks for any help!

[Sebastian Kripfganz]

Mobeen Ur Rehman
The ARDL model is suitable for your situation, yes.

[Mobeen Ur Rehman]

I want to know that if we can apply ARDL model in case our dependent variable is stationary and the independent variables are a mix of stationary and non-stationary(integrated at order 1) variables

[Mobeen Ur Rehman]

Hi,
I would like to ask from anyone that if I have a mix of stationary and non stationary (integrated at level I) independent variables but my dependent variable is stationary, so can I still apply ARDL model?

[Faizal Amir Muhammad]

Is there a way to get impulse response functions from error correction model (not from VECM)?

[Sebastian Kripfganz]

If all roots of the AR polynomial fall outside the unit circle such that the integration properties of the dependent variable are solely driven by the integration properties of the (weakly) exogenous regressors, the short-run coefficients (and therefore the ARDL coefficients of the exogenous regressors) are \( \sqrt(T) \) consistent and asymptotically normally distributed. In the case of I(1) regressors, the corresponding long-run coefficients are \( T \) consistent and asymptotically normally distributed. See Pesaran and Shin (1999). The problem of a non-standard distribution concerns the coefficients of the autoregressive terms (the lags of the dependent variable) and the adjustment coefficient in the error-correction representation.

If you want to test for an existence of a long-run relationship, you can use the bounds testing procedure of Pesaran, Shin, and Smith (2001) and the critical values provided by them. This is implemented in the ardl command.

Bootstrap standard errors are not available.

Literature:
1) Pesaran, M.H. and Y. Shin (1999): An Autoregressive Distributed Lag Modelling Approach to Cointegration Analysis. In: Strom, S. (Ed.): Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium. Cambridge, UK: Cambridge University Press;
2) Pesaran, M.H., Shin, Y. and R.J. Smith (2001): Bounds Testing Approaches to the Analysis of Level Relationships. Journal of Applied Econometrics 16 (3), 289-326.

[super fnjfj]

I do not want to let the last question stand unanswered here. As a short answer: To do impulse response analysis, estimation of a vector error correction model may be the proper starting point. See the manual entry vec intro for further details.

Hi Sebastian,

ardl involves non-stationary variables, so the estimated coefficient can be still consistent but with non-standard distribution. My question is how to get the correct s.e. and confidence interval of the non-standard distribution to do inference? Appears bootstrap cannot work with ardl.

[Sebastian Kripfganz]

I do not want to let the last question stand unanswered here. As a short answer: To do impulse response analysis, estimation of a vector error correction model may be the proper starting point. See the manual entry vec intro for further details.

[Jaya J]

Sebastian Kripfganz Is there a way to get impulse response functions after estimating the ARDL form of the function?

[niaz_ghumro]

Thanks alot prof for laudable help.

[Daniel Schneider]

Hi dinscorpie,

the coefficients of the error-correction equation are linear or nonlinear combinations of the coefficients of the levels equation. You can find the relationship of the two sets of parameters laid out for example in a paper by Hassler/Wolters (2005). As you can see there (topmost equation on page 3), the coefficient of the dependent variable in the long-run relationship is equal to one by construction. The short-run adjustment coefficients of the dependent variable (first differences at various lags) are included in the estimation output according to your specification of options lags() and maxlags().

Best,
Daniel

[dinscorpie]

Hi Sebastian,

Firstly thanks for the code. I find it really great and helpful as I don't have Microfit software to conduct ARDL and using Eviews for ARDL is a bit of a hassle.

However, whenever I run ARDL using the 'EC' code, it seems that no matter how many lags I used for the depvar, it doesn't appear in the Long-run section of the result. From my understanding, an ARDL model should also include the lagged of depvar in estimating the Long Run relationship.

Please help, tqvm

[Sebastian Kripfganz]

Interpretation of individual short-run effects is a bit tricky here. Consider a permanent (!) shock to GDP. That means, the change in GDP is positive (or negative) in the current period but zero in the next periods (because GDP remains at its higher/lower level). The coefficient of D.GDP simply tells us how depvar is contemporaneously affected by this permanent shock conditional on being initially in the long-run equilibrium.

The one-period delayed effect due to this shock consists of multiple components. One component is the error-correction term, the short-run adjustment due to the deviation from the long-run equilibrium. Another component is the lagged difference of depvar, and a third component is LD.GDP (maybe call it the delayed direct effect, because the other two components are indirect effects). The coefficient of the latter thus gives the one-period delayed effect to this permanent shock that is not due to the long-run equilibrium adjustment and the short-run autoregressive response.

Giving a correct quantitative interpretation to this parameter therefore becomes a bit cumbersome. It might be more appropriate here to compute impulse response functions to describe the short-run dynamics.