[Esti Widyastuti]

Thank you Sebastian for your advise!

[Esti Widyastuti]

Hi Sebastian,

I have another question about the F-stat and t-stat in the cointegrating bound test.
If F-test value is in between the critical values of I(0) and I(1), it is inconclusive.
If the F-test is less than critical value of I(0) then we cannot reject the null of no cointegration, there is no evidence of cointegration.
But, how can we conclude if the t-statistic is in between the critical values of I(0) and I(1) with the F-test value is less than critical value of I(0)? Can I still conclude there is no cointegration and how can I interpret that inconclusive t-statistic?
Also for the t-statistic is in between the critical values of I(0) and I(1) with the F-test value is greater than the critical value of I(1), can I conclude there is cointegration?

Thank you very much for your help,
Regards,
Esti

[Sebastian Kripfganz]

If the F-test is less than the lower-bound critical value, then there is no evidence of a long-run relationship / cointegration, no matter what the t-statistic is. For this question, the t-statistic is only relevant when the F-test rejects the null hypothesis. In the latter case, an inconclusive t-statistic means, strictly interpreted, that your whole cointegration test is inconclusive because you need both tests to be rejected.

[Esti Widyastuti]

Thank you very much Sebastian!

[Sher Reshef]

Hi Sebastian, thank you for making this program available. I have some questions on the ARDL model and lags.

1) I'm looking over at your Stata presentation and see that ARDL can be re-parameterized to separate the rate of adjustment, long-run, and short-run coefficients.

Before parameterization, you set p>=1, q>=0, and the sum for the dependent variable ranges from i= [1,p] and from i=[0,q] for the dependent variables. When it is parameterized, the bounds become i=[1,p-1] for y and i=[1, q-1] for x. If the optimal lag p is 1, we have i bounded between [1,0]. Am I interpreting the equations wrong?

2) The stata help file says it's possible to estimate a model with zero lag regressors like ARDL(1,0) in ec1 form. I've tested the ec1 form with ARDL(1,1) and ARDL(1,0). They provide the same number of estimates, with slightly different coefficients. What's the difference in interpretation? What's the meaning of an ARDL model with no regressor lags?

Thank you!

[Sebastian Kripfganz]

1) When p=1, the sum from i=1 to p-1=0 is implicitly defined to be zero, i.e. no short-run terms exist.

2) It only appears as if there are the same number of coefficients. Effectively, in the ARDL(1,0) case, the short-run coefficient is restricted to equal the negative product of the speed-of-adjustment coefficient with the long-run coefficient. It is therefore not really a separate coefficient unlike in the ARDL(1,1) case where it is unrestricted. Here is an illustration:

.  webuse lutkepohl2

. ardl ln_inv ln_consump, ec1 lags(1 1)

ARDL(1,1) regression

Sample: 1960q2 - 1982q4                         Number of obs     =         91
                                                R-squared         =     0.1665
                                                Adj R-squared     =     0.1378
Log likelihood =  162.29382                     Root MSE          =     0.0416

------------------------------------------------------------------------------
    D.ln_inv |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
      ln_inv |
         L1. |  -.1532653   .0550371    -2.78   0.007    -.2626575   -.0438731
-------------+----------------------------------------------------------------
LR           |
  ln_consump |
         L1. |   .8582533   .0557489    15.39   0.000     .7474464    .9690602
-------------+----------------------------------------------------------------
SR           |
  ln_consump |
         D1. |   1.321705   .4083266     3.24   0.002     .5101119    2.133298
             |
       _cons |   .0084231   .0606778     0.14   0.890    -.1121805    .1290267
------------------------------------------------------------------------------

. ardl ln_inv ln_consump, ec1 lags(1 0)

ARDL(1,0) regression

Sample: 1960q2 - 1982q4                         Number of obs     =         91
                                                R-squared         =     0.0838
                                                Adj R-squared     =     0.0630
Log likelihood =  157.98774                     Root MSE          =     0.0434

------------------------------------------------------------------------------
    D.ln_inv |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
      ln_inv |
         L1. |  -.1564379   .0573642    -2.73   0.008    -.2704372   -.0424386
-------------+----------------------------------------------------------------
LR           |
  ln_consump |
         L1. |   .8338113   .0580846    14.36   0.000     .7183803    .9492422
-------------+----------------------------------------------------------------
SR           |
  ln_consump |
         D1. |   .1304397   .0510905     2.55   0.012      .028908    .2319713
             |
       _cons |   .0574683   .0608157     0.94   0.347    -.0633902    .1783267
------------------------------------------------------------------------------

. nlcom - _b[ADJ:L.ln_inv] * _b[LR:L.ln_consump]

       _nl_1:  - _b[ADJ:L.ln_inv] * _b[LR:L.ln_consump]

------------------------------------------------------------------------------
    D.ln_inv |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _nl_1 |   .1304397   .0510905     2.55   0.011     .0303041    .2305752
------------------------------------------------------------------------------

As you can see in the last output, the mentioned negative product indeed equals the short-run coefficient. The meaning is that you cannot separate long-run and short-run effects in this case because they are two sides of the same coin.

[Sher Reshef]

Hi Sebastian, I see, thank you for the clear explanation. What I take from this, is that if the optimal lag structure is ARDL(1,0) the dependent variable reacts to shocks to equilibrium in the long-term but not to short-term disruptions. Is there a useful interpretation for the "SR" product in this case?

[Sebastian Kripfganz]

Any shock to the long-term equilibrium is effectively also a short-term disruption. My interpretation would rather be that there is a smooth transition back to the long-run equilibrium that is not disturbed by additional short-term fluctuations.

The additional SR term is hard to interpret, which is why I personally prefer the ec option over the ec1 option.

[Enock Bulime]

Hello,

Under what circumstances should I use estat sbcusum or estat sbcusum, ols when testing for parameter stability?

[Sebastian Kripfganz]

I am not an expert on these CUSUM tests. Both are alternative versions of the test for parameter stability. You might want to look at both of them. If either of them rejects the null hypothesis, you might have to worry about the presence of a structural break. Some more details are provided in Stata's pdf documentation of these commands.

[Esti Widyastuti]

Hi Sebastian,
I would like to seek your advice on interpreting the cointegration result of ARDL in EC representation through bound test.
We have 3 variables: y, x1, and x2 and basically, I would like to see the cointegration among them, so that I estimated the 3 equations with y as the dependant variable in the first equation, x1 as the dependant variable in the second equation, and x2 as the dependant variable in the third equation. Y is I(1), x1 is I(0), and x2 is I(1).
I got the cointegration result based on bound test for y as dependant variable and x2 as dependant variable and inconclusive for x1 as dependant variable. Then, what is the most appropriate interpretation to conclude whether there is cointegration exist among those variables as we got one is inconclusive and two are cointegrated. I attached the estimation result.
Thank you very much for your advice.
Best Regards,
Esti
ardl_boundtest_estimationresult.docx

[Sebastian Kripfganz]

First of all, if x1 is I(0) then it cannot be cointegrated with y and/or x2.

The three equations are not interchangeable. They are based on different assumptions. In the model with y as the dependent variable, it is assumed that (x1 and) x2 are long-run forcing for y. In the model with x2 as the dependent variable, it is assumed that y (and x1) are long-run forcing for x2. You cannot have both x2 long-run forcing for y and y long-run forcing for x2. (Long-run forcing is essentially a weak exogeneity assumption.)

Therefore, you need to decide first which assumption is more reasonable. If you believe that x2 is long-run forcing (weakly exogenous) in the model explaining y, then you should use this (and only this) model to test for the existence of a long-run (cointegrating) relationship with the bounds test. The bounds test is not valid if the weak exogeneity assumption is violated, i.e. it cannot be used if there are more than 1 cointegrating relationships involving y and x2.

[Ken Mulligan]

A few months ago there was published a new ARDL package for R. I estimated the same ARDL/EC model in Stata and this new ARDL R package and the results came out differently (at first blush) between them. It seems that the coefficients of the EC paramaterization reported by the initial output is different between the two packages. In case anyone else runs into a similar issue, here is an exchange I had about this with the author of the R package.

[Sebastian Kripfganz]

Many thanks for pointing me towards the R package.

The R and the Stata package follow a different logic when reporting results. Neither of them are incorrect. The R package reports "level coefficients" of the explanatory variables in the EC form. The Stata package reports "long-run coefficients" in the EC form. The EC results in Stata should not be read as the coefficients in a linear regression model which is made clear by separating the different parts with horizontal lines. There is a clear 1:1 mapping between the coefficients in R and those in Stata:

The coefficient L(LRM, 1) in R, where LRM is the dependent variable, equals the coefficient ADJ:L.LRM in Stata. This is the so-called "speed-of-adjustment coefficient" (with opposite sign). I do not know why the author of the R package claims that the standard errors in Stata were wrong. They are identical to those in the R package. Of course, we should not use the conventional p-values to do inference under the null hypothesis that this coefficient equals zero, but Stata has the postestimation command estat ectest for this purpose. Also, the coefficients and standard errors should not depend on whether we use a restricted or unrestricted model. This distinction only matters for the reporting of the intercept or trend parameter as part of the long-run relationship and for the postestimation bounds test. There is definitely nothing "wrong" with the Stata output.

The coefficient L(LRY, 1) in R, where LRY is an explanatory variable, equals the product of the Stata coefficients LR:LRY and ADJ:L.LRM (with opposite sign), and likewise for all other explanatory variables in the long-run level relationship.

We have deliberate chosen to report the long-run coefficients instead of the level coefficients in the EC output in our Stata package because this is what applied researchers usually care about. The model parameterization for which we report the results can be found on slide 12 of my 2018 London Stata Conference presentation, and the interpretation of the reported coefficients on slide 16:

• Kripfganz, S. and D. C. Schneider (2018). ardl: Estimating autoregressive distributed lag and equilibrium correction models. Proceedings of the 2018 London Stata Conference.

[Sebastian Kripfganz]

Daniel Schneider and I are happy to announce that our paper on the computation of critical values and approximate p-values for the bounds test has been accepted for publication in the Oxford Bulletin of Economics and Statistics. The article is available open access on the publisher's website:

• Kripfganz, S. and D. C. Schneider (2020). Response Surface Regressions for Critical Value Bounds and Approximate p‐values in Equilibrium Correction Models. Oxford Bulletin of Economics and Statistics, forthcoming.

The critical values and corresponding approximate p-values have been updated in the latest version 1.0.3 of the ardl command which is now available on my personal website and on SSC, thanks to Kit Baum:

net install ardl, from(http://www.kripfganz.de/stata/) replace all

or

ssc install ardl, replace all

Compared to the previous version, there are very slight changes in the critical values and p-values due to methodological refinements during the revision of the above article.