[Sebastian Kripfganz]

In the error-correction representation, depvar always appears one period lagged (ADJ). indepvars can be forced to appear in period t-1 for the long-run effects (LR) by specifying the option minlag1.

I agree that this is a bit unfortunate. The reason for this coding was that the optimal lag order could in general be zero for some indepvars. In this case, reformulating the long-run effects for period t-1 requires special treatment that we did not want to implement (so far).

[Karl Duffy]

Yeah I just figured that out with the minlag1 option. Thanks for the prompt response.

The error correction representation does not include an AR term for depvar in first difference, do you know am I doing something wrong here or is it possible?

Apologies for the queries, I think the code is very good, especially how it gives one the option to examine the regression underlying the ARDL model, very handy feature.

Karl

[Sebastian Kripfganz]

That is possible and will happen when the optimal lag order for depvar is just one in the level ARDL formulation. For additional lags in first differences, at least two lags of depvar in levels are required. For indepvars, one lag in levels is sufficient because indepvars also appear contemporaneously on the right-hand side (as opposed to depvar).

[Karl Duffy]

Apologies Sebastian I do not quite follow you in relation to my above query. So is it only possible to include an AR term in the level model and not in the first-difference model?

[Sebastian Kripfganz]

The lag order in first differences equals the lag order in levels minus one. Consider the following two examples (ignoring deterministic components), starting with the level representation:
1) y = c1 * L.y + d0 * x + d1 * L.x + u
2) y = c1 * L.y + c2 * L2.y + d0 * x + d1 * L.x + d2 * L2.x + u

The corresponding error-correction representation is as follows:
D.y = a * (L.y - b * L.x) + e1 * LD.y + f0 * D.x + f1 * LD.x + u

By analytical reformulations of models 1) and 2) above you can compute the following parameter restrictions for both cases:
1) a = c1 - 1 ; b = (d0 + d1) / (1 - c1) ; e1 = 0 ; f0 = d0 ; f1 = 0
2) a = c2 + c1 - 1 ; b = (d0 + d1 + d2) / (1 - c1 - c2) ; e1 = - c2 ; f0 = d0 ; f1 = - d2

As you can see, in model 1) the coefficient e1 of LD.y in the error-correction representation equals zero. It is only non-zero, when there are at least two lags of y in the level representation, as in model 2). The reason is simply that LD.y = L.y - L2.y but there is no L2.y in the level representation of model 1) that can be used to form LD.y in the corresponding error-correction representation.

[Karl Duffy]

I have a question in relation to interpreting one's short run dynamics in the ARDL approach to cointegration and would be very grateful if anyone had any advice on the matter!

Short run dynamics are given by the coefficients of variables in difference form in one's error correction model given a cointegrating relationship is present. My confusion lies in how does one interpret these effects if there is more than one term for a given variable included in the ECM? I.e. if in my ECM I have D.gdp and LD.gdp included how do I interpret the short run effect of GDP on my depvar?

Any help would be much appreciated! Karl.

[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.

[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

[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

[niaz_ghumro]

Thanks alot prof for laudable help.

[Jaya J]

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

[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.

[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]

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.

[Faizal Amir Muhammad]

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