Autoregressive distributed lag models (ARDL) do not require nonstationary data at all.  Note that AR(p) models are often stationary.

You can do ARDL on stationary or non stationary variables (endo and exogenous) as long as the data does not exceeding integrated 2, or I(2) after differencing if the data is non stationary. You shall check the stationarity of every variable with the root test. For details on procedure to perform ARDL please refer to my paper; Wan Ahmad Wan Omar, et. al., (2015). The empirical effects of Islam on Economic Development in Malaysia. Research in World Economy. Vol. 6, No. 1. pp:99-111.

Yes , You can do it. No need for all variables to be non stationary . Variables can have different stationarity/ non stationarity levels . This is said to be the advantage for ARDL models

The single equation approach by Pesaran et al. (2001) does not need to identify whether the regressors are I(0), I(1), or mutually cointegrated when finding a long-run level relationship among the dependent variable and the regressors. As discussed clearly by Pesaran et al.(2001), this is an advantage of their ARDL approach over

conventional single equation cointegration tests such as in Engel and Granger (1987) because the latter requires to pretest the I(1) property of all underlying variables. However, none of the time series should be I (2) as the calculated F-statistics cannot determine long-run relationship in this case.

Note: Pesaran et al.(2001) method is fundamentally a single equation approach. As discussed in their conclusion as a caveat, a limitation of the ARDL approach is that it restricts only one level-relationship among the variables under consideration and does not allow for a greater number of long-run relationships. In fact they say "The analysis of this paper is based on a single-equation approach. Consequently, it is inappropriate in situations where there may be more than one level relationship." Applying their single equation ARDL approach to a multivariate stochastic environment is not such an easy and trivial exercise as done by many these days.

If Autoregressive distributed lag models (ARDL) are meant (not easy to find out) I would like to adress some restrictions:

So called autoregression is not in most cases a functional relationship but a variable reacting to similar events in a similar manner - statistically a "nonsense correlation".

If volatility the variable this is definitely true. And volatility is not stationary regarding stock, indexs and forex.  

It  is precisely the interest of using ARDL. The degree of integration is not important ...

I think the questions is particularly focused on "if, our dependent variable is stationary". I haven't seen an empirical study using stationary variable as DV in ARDL and intuitively, e.g. if you are using stock returns (log differenced prices);  in ARDL, DV enters as the first difference; so  difference of a differenced variable would certainly have some implications, may be not on the estimates but on the comparison (comparison of F or Wald test statistics with critical bounds by Narayan or Pesaran). Are critical bounds meant to deal with this as well, i am not sure.   

I agree with Jawad that the question is particularly focused on "if, our dependent variable is stationary".  I also did not see the paper which uses ARDL in this case. The advantage of ARDL that order of integration does not matter is refereed in the case of independent variables.

Thank you all 

 I com[letrly agree with all the above comments.In short  Hrishikesh D. Vinod and Jamal, Bouoiyour gave the shortest answer and straight fo the point.

To use or not to use an estimation method (like ARDL) should not depend on the data, but mainly on the model (specification) one thinks is best for ananlysing (economic) relations. I have the impression that ARDL is often chosen because it is modern and one has a comfortable econometric package.

Hello everyone, When I see some papers where the authors have estimated the unconditional ARDL first, and then conditional ARDL (p,q1,q2..qn) to obtain long-run and short-run coefficients, for that they have used two separate models one for long-run another for short-run. Therefore, they produce separate results so that each result has constant (one constant for each model)

But when I use STATA (I use 13 version), it gives results, together LR for long-run and SR for short-run coefficients with one constant.

Could you please give an idea of this difference?

Dear Bhavan, I think that STATA uses an ECM approach, which estimates the short and long term coefficients in one regression. The 2-step approach estimates first the long-term effect, in a second regression the short-term relation is estimated by using the first differences of the variables and the residuals of the first regression. Therefore, there are two constants, but - I think - the constant of the second equation should be near zero and insignificant, such that one can estimate with suppressing the constant (in a third step). Otherwise, I would suspect that something is wrong with the model. I would never accept an estimate for differences of the variables with a high and significant constant, unless there is a good, convincing explanation for that.

In my knowledge, the dependent variable should only be I(1) in order for the ARDL-bound test to be vaild. My conclusion is primarily based on original paper ot Pesaran et al (2001) p.301 where it clearly shows that the hypothesis under which the test statistics are based on it clearly indicated that the Data-Generating-Process of the independent variable is I(1) (particularly a Random-Walk).

Can anyone who agrees or disagrees with this provide any evidence to suppots his/her statemets?

Thank you

Dear Anton Rainer, thank you for prompt suggestion. I got it now

Hi, I have a question. If I found that some of my endogeneous variable are stationary at level also, can I still use it to estimate ARDL??

I would not try to estimate ARDL for stationary endogenous variables, because, in general, autoregressive processes are not compatible with stationary exonenous variables. If you take the simplest AR-process y=ay(-1)+bx, one can show that y(t)=a^t*y(0)+b*sum(a^i*x(t-i)), and for a constant x that the difference between y(t+1) and Y(t) is something like

(a-1)*a*y(0)+a^t*b*x. For y to be stationary this difference should be zero, but this expression can, if ever, only be zero for one point of time t. As a has to be lower than 1, the first part of the expression will be negative and, to get zero, the second part has to be positive and constant. That means that the x's would have to shrink according to the increase of a^(-t) and b must have a special value. For simplicity, this calculation is done without random error, but I think including such a "well-behaved" error would not principally change the result and one can certainly prove it for more general processes.

If the dependent variable is stationary, ARDL model fails to apply. There is greater need to apply other models.

Yes as per pesaran paper...in order to apply Ardl....your dependent variable must be I(1)...independent variables can be mixture of I(0) or I(1)...However...it shld b checked that no variable should be I(2)...

it i my question too

I have the similar case Rasmhi, my endogenous variable is stationary at level and one exogenous, stationary stationary I(2). Which should be the appropriate model?

It does not make sense to apply ARDL without using simpler statstic methods before. In many cases, ARDL may be unnessesary, at all.

absolutely you can proceed, ARDL gives you the flexibility of using variables of different integration order{i(0) & i(1)} into single equation but avoid ARDL in case of integration order 2

If integrated of order 2, you might as well try Dynamic Ordinary Least Squares (DOLS) for cointegration. ARDL is applicable for variables with a mixture of integration of order 1 and 0. Whether DV with integration of order 0 (stationary) will affect the results will need to relook at the whole idea of doing a cointegration using ARDL.

I also have a question. Can ARDL be used if all Independent variables are stationary at first difference

If there are good (convincing) theoretical reasons, I think. one can apply ARDL even for variables which are (partly) non-stationary. Often the application of ARDL (of other methods) is chosen to save the trouble of carefully thinking about the problem to analyse, i.e. to delegate "brain effort" to a statistical procedure. It may also be that professors would not accept papers of their students, if they do not use "modern" methods. In this case, one should try to change to another professor.

Mr. Chung Tin Fah, you said that the DOLS method can be used if the variables are stationary in i [2]. For DOLS methods, the dependent variable must be I [1], right?

In addition, since NARDL methods are ARDLbased estimators, do dependent variables have to be stationary at level I [1]?

if the dependent variable is stationary at levels, that is I(0), you cannot use ARDL. Although ARDL allows for the mix of I(0) and I(1) variables, according to Peasaran et al. the dependent variable has to be I(1). You cannot regress I(1) variables (of independent variables) on I(0) (dependent) variable.

Yes I agree that independent variables can be mix of I(1) and I(0), dependent variable has to be I(1) to apply ARDL .However people have used ARDL in this case also as question arises as what else may be done may not be a perfect solution.

@Hikmet Akyol

In addition, since NARDL methods are ARDLbased estimators, do dependent variables have to be stationary at level I [1]?

I am a bit confused here... When variables are stationary at levels, they are I(0). When they are I(1), they are stationary after first difference.

Yes. but not OLS

does anyone have a summary which model to use in each case ?

You are supposed to understand that, Ardl is applied when the dependent variable is either stationary at level 1(0) or stationary at first difference 1(1)! Thus, stationarity is a condition for all time series analysis to avoid spurious regression.

The only advantage of ardl is to control endogeneity of variables which are so common in macroeconomics.

Yes you can

ARDL could be used for IV or DV stationary at level (I(0)) or at the first difference (I(1)) or both, but not at I(2) or more. You may use Eviews or Microfit to conduct ARDL tests.

Yes the above advice is right and appropriate. Thank you Dr Wan Omar 🙏

If it is I(0) you can not use the ARDL Model

The dependent variable has to I(1) or else you run into degenerate cases of non-cointegration where your cointegration is incomplete.

The dependent variable has to be I(1). If not you cannot regress I(1) variables (of independent variables) on I(0)