[Sebastian Kripfganz]

In your case, the trace statistic and the maximum-eigenvalue statistic yield conflicting results. It is up to you to make a judgement which of the two statistics to use. If you conclude that there are two cointegrating relationships, this test is still not informative enough about the applicability of the ARDL model. The latter requires that there is at most 1 cointegrating relationship involving the dependent variable. This does not preclude the case of 2 or more overall cointegrating relationships if all of these other relationships are among the independent variables only. You might have to make an assumption that these additional cointegrating relationships indeed do not involve the dependent variable of the ARDL model.

[Anil Raj]

I have used the vecrank command in stat to check for cointegration. And the results are

vecrank bnchmrk	repo cof lgnpa, trend(constant) max
	Johansen tests for cointegration
Trend: constant	Number	of obs	=	34
Sample: 3 - 36	Lags	=	2
5%
maximum	trace critical
rank parms	LL eigenvalue statistic value
0 20	76.161843 . 57.4386 47.21
1 27	89.835272 0.55261 30.0917 29.68
2 32	97.807025 0.37433 14.1482* 15.41
3 35	104.19286 0.31315 1.3765 3.76
4 36	104.88112 0.03968
5%
maximum	max critical
rank parms	LL eigenvalue statistic value
0 20	76.161843 . 27.3469 27.07
1 27	89.835272 0.55261 15.9435 20.97
2 32	97.807025 0.37433 12.7717 14.07
3 35	104.19286 0.31315 1.3765 3.76
4 36	104.88112 0.03968

Does this means that the model has 2 cointegrating relationship or 1 cointegrating relationship, which value should I select from first table or second table?
If I have more than one cointegrating relationship as mentioned in table 1, can I use ARDL model to fit the data?

[Sebastian Kripfganz]

Cointegration does not necessarily occur pairwise but can, and often does, involve several variables.

With the ardl command, if your variables are I(1), you can test for cointegration with the bounds test that is implemented in the postestimation command estat ectest. The existence of a level relationship is equivalent to cointegration if all the variables are I(1); see slide 2 of my my presentation at last year's London Stata Conference.

The estat ectest postestimation command is part of the new version of the ARDL command that is discussed in the following Statalist topic:
ARDL: updated Stata command for the estimation of autoregressive distributed lag and error correction models

[Anil Raj]

As you replied to the query of Mr.Sebastian Li as "An underlying assumption of the ARDL model is that there exists at most one cointegrating relationship that involves the dependent variable. (There might be additional cointegrating relationships among the independent variables themselves.)". ....

1)Say I have one dependent variable and 3 independent variable. I can test the cointegration by running a regression involving one dependent variable and another independent variable and then test for stationarity of the error terms? If the dependent variable and independent variable are non-stationary and the errors are stationary we can conclude that cointegration exists. Is it right? Or as above should we run the regression for one dependent variable and all other independent variable and then test for error term's stationarity? Does conintegration happens only for a pair of variables?
2) Is there any other method in built in ardl model to test cointegration?

Here my doubt is how we know that there is atleast one cointegrating relationship?

Thanks in advance.

[Sebastian Kripfganz]

Anil Raj
When you use the maxlag() option, the underlying estimation sample (i.e. the number of observations) changes. The test results are then not directly comparable any more. Moreover, it is always possible that a test rejects the null hypothesis when it is true (type-I error) or does not reject it when it is not true (type-II error).

[Sebastian Kripfganz]

Sebastian Li
If I have counted correctly, an ARDL(3,4,4,3,2) model would estimate 21 coefficients. Given a sample size of 35 observations, this would usually be considered as too many parameters. You would either need to restrict the number of variables or the number of lags to have a chance to obtain reliable parameter estimates.

An underlying assumption of the ARDL model is that there exists at most one cointegrating relationship that involves the dependent variable. (There might be additional cointegrating relationships among the independent variables themselves.)

[Anil Raj]

Thank you Dr.Sebastian.
By default ardl option for the above data set runs ARDL(1,1,0,0) regression and estat ectest shows higher F value and t value which confirms a long term relation. But when I try to use maxlags option and give maxlag(1 2 0 2) for bounds test the F and t values is reduced and implies no long term relation. Does this means that the lagged values of variables has no long term relation?

[Sebastian Li]

Thank you Sebastian.

Regarding the maxlags you mentioned above, when I didnt specify the maxlags, an optimal selection of (3,4,4,3,2) was chosen although I only had a sample of 35 observations - I believe this ardl model cannot work in this scenario - is that correct? And why exactly doesn't it work?

(Provided I identified more than one cointegrating vector with this lag selection - I believe ardl also doesnt apply to more than one cointegrating vector.)

[Sebastian Kripfganz]

1) The number of observations is determined by the maxlags() option. By default, there are 4 observations reserved for the initial observations. The sample is left unchanged even if the optimal lag order is smaller than 4. Otherwise, the comparison of different lag orders would not be valid.

2) There is no perfect relationship between the F-statistic and the t-statistic. See my slide number 18 of my presentation at last year's London Stata Conference on how to do inference based on the test results. (Note that there is a typo on that slide. Step 3 should read as "If H₀^t is rejected, ...".) If the test based on the F-statistic rejects the null hypothesis but the test based on the t-statistic does not reject the null hypothesis, then you did not find enough statistical evidence in favor of a long-run relationship.

[Anil Raj]

My depvar is bank benchmark rate and independent variables are monetary policy rate(repo),cost of funds and non performing loans (log of gnpa) with 36 observations.
I ran the ardl command "ardl bnchmrk repo lgnpa cof, aic" ( ARDL(1,1,0,0) regression), then " ardl bnchmrk repo lgnpa cof, aic ec" (ARDL(1,1,0,0) regression) and then "estat ectest". I have the following queries
1)Why it omits 4 observations and showing number of observations as 32, is not the lags 1 1 0 0 then?
2) estat ectest returned results of F = 6.085 t = -3.806. While F is higher than I(I) at 5% value of 4.988 but t is lesss than 5% value of -3.871. How I conclude about long term relationship?

Variable list is attached. Request to please clarify.

[Sebastian Kripfganz]

1. To be a bit more precise, a 1 unit increase (1% if X is in logs) is causing an 0.5 unit increase (0.5% if Y is in logs) in the long-run equilibrium. You can look at it from the perspective of the error-correction term: EC = Y - theta * X. In the long run, EC=0. Thus, with theta = 0.5, if X increases by 1 unit, then Y must increase by 0.5 units to retain the balance.

2. The short-run coefficients are interpreted as usually in linear regression models. The coefficient of D.X measures the immediate effect of a change in D.X on D.Y, holding everything else constant.

3. The interpretation of the short-run coefficients is the same in both the ec and ec1 versions of the model, as long as the respective variable in the ARDL model enters with at least one lag. If a variable has zero lags in the ARDL form, then the ec1 representation creates an overparameterization. There will be an exact relationship between the respective short-run coefficient and the long-run and speed-of-adjustment coefficients. (If there is no lag in the ARDL form but we introduce a lagged variable in the long-run relationship, the short-run terms need to counteract this.) It does not make much sense to separately interpret the short-run coefficient in this case, which is why I usually recommend to use the ec version instead of ec1.

[Sebastian Li]

Hi Sebastian,

I have been reading through the forum and followed your ARDL EC approach with my data, which might be very stupid questions but I still can't figure it out.

My main question is about the interpretation of the results generated:

1. As far as I understand, we can directly interpret the LR coefficients (i.e. +0.5 with a significant p-value): a 1% increase in X can cause a 0.5% increase in Y (dependent) - is that correct? Even the dependent variable is in the form of d.lnY??

2. For the short run results from ec1, can we directly interpret those as coefficients just as above?? If not, what further calculations do we need to do?

3. Again with the short run results, but from ec, only one model, my model ARDL (1,0,1) only generates one variable under short run, why is this different from the results from ec1? Do we interpret it as coefficients directly?

Thank you very much for all your kind input to this command - it has been helping me a lot!

[Sebastian Kripfganz]

If you are using an older version than Stata 15, estat sbcusum is not available. You can use the community-contributed command cusum6 (available from SSC) instead.

[Anil Raj]

But estat sbcusum command says its not valid, all other post estimation are working?

[Anil Raj]

Thank you so much. I got it right.