ARDL Model Interpretation

[Albertus van Niekerk]

Dear forum,

I'm new on here, so hi to you all!

I have three basic conceptual questions regarding the use of the ARDL model. Just by way of background, I am attempting to construct an ARDL model that includes a dummy variable in order to estimate the overcharge inflicted by a cartel in a certain industry. We have data on prices as well as demand side and supply side variables. We are essentially modelling, in an ARDL setup, a reduced form pricing equation with the dummy variable as an exogenous regressor. The dummy variable is set up such that it equals 1 during the cartel period and 0 otherwise. The idea is that the coefficient on the dummy variable will give as an indication of the percentage overcharge inflicted by the cartel. My questions are as follows:

1. Is it correct to simply add the dummy variable as an exogenous regressor (using the exog() option of the ARDL model) and interpret its coefficient as one would normally?
2. How does on interpret the results of the bounds test if the null can be rejected based on the F-test, but not based on the t-test?
3. If the model does not pass the bounds test, in other words there is not a long run relationship between the variables, would the interpretation of the dummy variable still be the same?

Sorry for the basic questions - I am trying to get the foundations right for the ARDL modelling, I will be doing quite a lot of it in the future.

(@Sebastian Kripfganz, this may be a question you could answer, but I don't wont to bother you with such basic questions!! Nevertheless, thanks for all your useful information on the topic, I applaud the effort!)

Thanks in advance for your time and all the useful comments on this forum, it has helped me a lot!!

Albertus

[Sebastian Kripfganz]

1. This is what I would recommend, yes.
2. Taken at face value, a rejection of the t-test would imply that there is evidence that the dependent variable follows a unit-root process without being cointegrated with the exogenous variables. The F-test in this situation is then not very informative any more.
3. The interpretation of the dummy variable is not affected. It is supposed to capture a particular "short-term" deviation during the cartel period but in the long-run it would be assumed that there is no cartel.

[Albertus van Niekerk]

1. This is what I would recommend, yes.
2. Taken at face value, a rejection of the t-test would imply that there is evidence that the dependent variable follows a unit-root process without being cointegrated with the exogenous variables. The F-test in this situation is then not very informative any more.
3. The interpretation of the dummy variable is not affected. It is supposed to capture a particular "short-term" deviation during the cartel period but in the long-run it would be assumed that there is no cartel.

@Sebastian Kripfganz Thank you for your useful input. Much appreciated!

[Albertus van Niekerk]

@Sebastian Kripfganz I have one further question regarding the output of the ARDL models. We have run various models, and in all of them the ECM term is significantly negative. However, the bounds test is not passed in all cases, even though the ECM term suggests that the model is converging to a steady state. What I do notice, is that for the model where the bounds test is not significant while the ECM term is, the rate at which the system adjusts (i.e. the coefficient on the ECM term) is smaller than in the models where the bounds test is passed. Why would this be the case? In a nutshell, my question is basically how one interprets the output if the bounds test is not passed, but the ECM term is still significantly negative?

[Sebastian Kripfganz]

You cannot directly infer from the regression output whether the coefficient of the speed-of-adjustment term is statistically significant because it has a nonstandard asymptotic distribution. You need to consider the t-statistic from the bounds test instead. It would be slightly surprising to reject the bounds test for the t-statistic when you do not reject the bounds test for the F-statistic. In such a case, you could possibly ignore the bounds test for the t-statistic given that the F-test already provides conclusive evidence against the existence of a long-run relationship.

[Albertus van Niekerk]

@Sebastian Kripfganz Thank you for your reply - this makes sense. Indeed this anomaly has not surfaced in my modelling so far.

One last thing I want to check with you, is the way in which I run my model tests. I have read your slide deck, and perform all the normal tests (normality, Ramsey RESET, etc.). I have run these tests after I have estimated the ARDL model, specifying option ec as well as regstore(ardlreg) and obtaining the estimates with estimate restore ardlreg. Would it make a difference if I do the same tests, but not specifying option ec in the model? In other words, the dependent variable would be ln_depvar instead of D.ln_depvar.

Just out of interest sake, what is the conceptual difference between option ec and option ec1?

Also, thank you for all your feedback and the immensely useful ARDL code you developed. It is so refreshing to see someone so friendly and willing to help!

Regards,
Albertus

[Sebastian Kripfganz]

The important thing to keep in mind is that the EC (or EC1) representation is just a reformulation of the ARDL model that leaves the error term unchanged. In other word, even though the dependent variable changes from ln_depvar to D.ln_depvar, the residuals are the same. Hence, any residual-based tests should be the same.

You can see the formal difference of the ec and ec1 option on slide 12 of my 2018 London Stata Conference presentation. The key difference is the term \(\boldsymbol{\omega}' \Delta \mathbf{x}_t\) in the latter representation. If you have variables that enter the ARDL model with 0 lags, then the ec1 representation is overparameterised due to this term. You would have a 1-to-1 mapping between the short-run coefficients \(\boldsymbol{\omega}\) and the corresponding long-run coefficients. The variance-covariance matrix would not be of full rank. This is not a problem in the ec representation because in that case no short-run coefficients will be reported when a variable has 0 lags in the ARDL representation. The additional short-run coefficient in the ec1 representation becomes necessary because the long-run variables are modelled with a one-period lag even though there was no lag in the ARDL representation.

[Themis Hernandez]

ARDL model interpretation

24 Oct 2019, 13:16
Professor Sebastian

Sorry for bothering you but I have some questions Regarding too the interpretation of the ARDL model:

1. According to your presentation in slide 15, I understand the ADJ value corresponds to the error correction model coefficient, right? And that one of the conditions is that it should be negative?

2. In the same example, how should I interpret LR value? given that, in slide 16 you explain the normalization procedure? How should this value be should normalized?

3. In a paper can I use directly the LR value to interpretate the impacts or should I normalize the value?

In my work I use first the command

ardl Lpproprice Lnatwprice Ladretailprice, aic ec

D.Lpproprice	Coef.	Std. Err.	t	P>t	[95% Conf.	Interval]
ADJ
Lpproprice
L1.	-.5090201	.0937077	-5.43	0.000	-.6961673	-.3218728
LR
Lnatwprice
L1.	1.23731	.1817724	6.81	0.000	.8742852	1.600334
Ladretailprice
L1.	.5088632	.3380518	1.51	0.137	-.1662728	1.183999
SR
Lnatwprice
D1.	.9868068	.0923886	10.68	0.000	.8022939	1.17132
Ladretailprice
D1.	-.1706345	.3345306	-0.51	0.612	-.8387383	.4974693
LD.	-.454341	.2597332	-1.75	0.085	-.973064	.064382
L2D.	-.5489642	.2585797	-2.12	0.038	-1.065384	-.0325448
L3D.	-.6358164	.2740161	-2.32	0.023	-1.183064	-.0885685
_cons	-1.313742	.5133485	-2.56	0.013	-2.33897	-.2885147

from what I understand to normalize the value I run

ardl Lpproprice Lnatwprice Ladretailprice, aic ec1 regstore(ardl)
estimates restore ardl
estimates replay ardl

D.Lpproprice Coef.	Std. Err.	t	P>t	[95% Conf.	Interval]
Lpproprice
L1.	-.5090201	.0937077	-5.43	0.000	-.6961673	-.3218728
Lnatwprice
L1.	.6298155	.1254681	5.02	0.000	.3792383	.8803926
Ladretailprice
L1.	.2590216	.1879451	1.38	0.173	-.1163307	.6343739
Lnatwprice
D1.	.9868068	.0923886	10.68	0.000	.8022939	1.17132
Ladretailprice
D1.	-.1706345	.3345306	-0.51	0.612	-.8387383	.4974693
LD.	-.454341	.2597332	-1.75	0.085	-.973064	.064382
L2D.	-.5489642	.2585797	-2.12	0.038	-1.065384	-.0325448
L3D.	-.6358164	.2740161	-2.32	0.023	-1.183064	-.0885685
_cons	-1.313742	.5133485	-2.56	0.013	-2.33897	-.2885147

Which long-run values should I use to interpret?

Which is the difference between the two outcomes

I really appreciate your explanations

[Themis Hernandez]

The previous was a question I posted respecting ARDL model to professor Sebastian Kripfganz; I'm sharing his kind reply:

Dear Themis,
Please understand that writing questions as private messages to members of this forum is strongly discouraged. The aim of the forum is that everyone can benefit from publicly posted answers.

Nevertheless, a brief response:
1. Yes, that is the coefficient of the error-correction term and it should be negative.
2./3. The coefficient of the lagged dependent variable in the long-run relationship is normalized to 1. As a consequence, you can interpret the reported long-run coefficient as the long-run effects of the respective variable on the dependent variable, as in a standard linear regression. These are the long-run coefficients that should usually be reported in a paper. Your second output does not show any long-run coefficients.

[Themis Hernandez]

Dear Members

I have a question regarding on code for wald test with ARDL model, should I group long and short-run coefficients or the should be checked independently.

So far I' coding test (LD.var1) (L2D.var1) (L3D.var1) (LD.var2) (L2D.var2) (L3D.var2) (_cons), mtest(noadjust)

[Sebastian Kripfganz]

There is usually no reason to group long-run and short-run coefficients as they have a very different interpretation.

[Themis Hernandez]

Dear professor Sebastian Kripfganz, thank you so much for your help, I have a question. When I use the ARDL model with data transform to natural logarithm, the ADJ, LR and SR value how they should be interpreted? Sorry for the basic question.

[Sebastian Kripfganz]

As with standard linear regression models, the coefficients are interpreted as elasticities once the natural logarithm is used as a transformation for all variables.