[John Costopoulos]

If we used ardl with quarterly data, should we first have to remove any seasonal components of the series? I know of course that we can specify quarterly seasonal dummies with the exog option, but - unless I am wrong - they will affect the short-dynamics only.

[Justin Niakamal]

I recommend working with seasonally-adjusted data.

[Sebastian Kripfganz]

Specifying time dummies with the exog() option is conceptually not very different from removing any seasonal components of the series first. While you could in principle still calculate long-run effects of seasonal dummy variables, I am not sure what can be learned from that. Whether removing seasonal components is advisable in your case is something that depends on your particular application. The existing literature in your field should hopefully have an answer for you.

[John Costopoulos]

Dear Justin and Sebastian, thank you very much for your prompt response. My question was raised after I read the documentation of a package in R implementing the bounds testing. I understand now that a prior removal of the seasonal components is proposed by programs that do not offer an option similar to exog, which enables the use of dummies .

[John Costopoulos]

I have noticed that the "Number of obs", reported by the ardl program, is the difference between the number of observations contained in the sample and the maximum number of lags defined with the maxlags option (e.g. 8). Is there a specific reason for that? Someone might expect that the number of observations taking part in the regression would rather be defined by the maximum of the lags of the optimal model (e.g. 2 for an optimal model ARDL(1,2,1,1)).

[Sebastian Kripfganz]

Different models are only comparable (e.g. by means of the AIC/BIC selection criteria) if they are based on the same number of observations. This is why the ardl command estimates all models on the same number of observations determined by the maximum lag, even if some model specifications with fewer lags in principle could be estimated on more observations.

[John Costopoulos]

Dear Sebastian,
Thank you for your very enlightening answer.

[Stephen Esaku]

Dear Sebastian,
I have two questions about interpreting ARDL coefficients (both long and short run): (1): How should these coefficients be interpreted? If am examining level relationship, is it correct to say an increase in X leads to an increase of --- units in/of Y? (2): Can these coefficients be taken to infer causal effects?? Thanks

[Sebastian Kripfganz]

(1) I would interpret it as follows: "A permanent increase of 1 unit in X leads to a long-run equilibrium increase of ... units in Y." You need to be clear in your interpretation that this is not an immediate change. When X changes, the equilibrium relationship becomes distorted. It usually takes time to restore the equilibrium relationship (depending on how large the speed-of-adjustment coefficient is). In the new equilibrium, Y will have changed according to the long-run coefficient.

(2) Causality requires that the regression model can be interpreted as a structural model and not just a reduced-form model. There should not be any relevant omitted variables or feedback mechanisms. That is often difficult to justify with macroeconomic data. While the regression results tell us something about the empirical relationship between the variables, a causal interpretation would require that exogenously shifting X (say, by a policy intervention) has a certain effect on Y. In practice, X can change for many reasons. So, the observed effect may easily capture the change of another variable that leads to a change in X, and then indirectly to a change in Y.

[John Costopoulos]

Dear Sebastian,

I apologize for my long question, but I'm trying to understand how the ardl output is produced. As I think of it, a possible approach might be the one described below.

(1) A regression is performed using the "conditional" ECM.
(2) The coefficient of the first lag of the dependent variable is reported without any transformation. However, there is the problem with the misleading p-value, so - as you have mentioned in an older post - we must always check the reported t-value against the critical values provided by your program in the bounds test.
(3) The long-run coefficients are the normalized values (on the ADJ coefficient) of the coefficients of the lagged "level" regressors (a minus sign is used in the normalization).
(4) The short-run coefficients are exactly those reported by the regression.

I would be grateful if you could confirm the validity of the above description.

[Sebastian Kripfganz]

It would be possible to directly estimate the conditional ECM and then to report the coefficients according to your procedure. Instead, the ardl command estimates the unrestricted ARDL model in levels and then recovers the ECM parameters from those ARDL coefficients (see slide 12 of my 2018 London Stata Conference presentation for the speed-of-adjustment and long-run coefficients). Standard errors are calculated with the delta method.

[John Costopoulos]

Dear Sebastian,

Thank you very much for your prompt response and your clarifications.

To be sure that I am on the right track, please allow me to summarize the steps taken by your program:
1. The unrestricted ardl model in levels (Slide 5) is estimated, using the lags obtained by minimizing AIC or BIC.
2. The speed of adjustment coefficient and the long-run coefficients of the regressors x1, x2, ... are computed using the formulas described in Slide 12, while their standard errors are calculated with the delta method.
3. A conditional ECM (Slide 12) is formed, using the lags obtained in step 1 above.
4. The estimation of the conditional ECM provides the short-run coefficients.
5. The bounds test is implemented as a post-estimation command of step 4 above.

[Sebastian Kripfganz]

I need to correct myself. Actually, we do indeed estimate the EC model as you describe in your earlier post #25. Apologies for the confusion. (It is quite a while since we implemented this procedure.) Standard errors for the long-run coefficients are still obtained with the delta method.

[John Costopoulos]

Dear Sebastian,

Your clarification is highly appreciated. After all, your posts #26 and #28 provided the extremely useful for me information regarding the use of the delta method. I am really grateful for all your answers so far, which have greatly helped me to understand how your excellent program works.

[John Costopoulos]

Dear Sebastian,
I would appreciate it very much if you could find the time to answer the following questions:
1. Taking into account the two EC parameterizations in Slide 12 of your 2018 presentation, it is clear to me what does the label "ADJ" in the regression output stand for, as well as why the regression results for the first lag of the dependent variable y are reported in that section. However, these notations might be not so clear for someone who is not familiar with the ardl program. From this point of view, would it be correct if we used the label "Error correction term (ECT)" instead of ADJ, as well as the label "ECT(-1)" instead of y.L1, in a working paper ?
2. In older posts, you have described in detail the interpretation of the SR coefficient of the contemporaneous (not lagged) FD term. Based on your great experience, which representation (ec or ec1) would you give a vote as having the most clear interpretation of the short-run effects?
3. If we used the ec1 presentation, would it be acceptable if we avoided to report the "L1." prefix in the LR section? This notation is not used in the literature, while the long-term estimates are exactly the same with those of the ec representation.