[Alvaro Gornes Jimenez]

Hi @Babacar Mbengue, have you found out how to apply the Cusum after the ARDL estimation? I'm in need of such a thing now.

Thanks,

[Sebastian Kripfganz]

In Stata 15 or newer, you can use the official postestimation command estat sbcusum. You can find an example in my 2018 London Stata Conference presentation from slide 27 onwards:

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

In Stata 14 or earlier, you need to use the community-contributed cusum6 command instead.

[Monir Ahmed]

Dear Professor,
Thank you for your reply through email. You are so much helpful in the STATA blog and I have learned a lot about ARDL estimation. I used ARDL estimation using Microfit. Your's package seems better than the Microfit because of controlling the lag and exogenous variables.

I have a simple question.

1. I have a specification in which year is a regressor ( following the literature, I found that year could be a good proxy for technological change and its related concepts). Now if I add the year in the main function of ARDL, I cannot estimate the ARDL This is logical. But if I add trend(year) then it gives the result ( no year effect of SR after ec1 or ec command, although it is logical. I did it after both (2,1) and (2,0) lag order). My question is: To check the year effect, is it correct to add trend(year) if I assume that my cointegration has no trend part?

Thanks in advance.

[Sebastian Kripfganz]

Yes, you would always specify the trend variable with the trend() option. If you want to include the trend in your long-run (cointegrating) relationship, you would additionally specify the option restricted. If you do not want to include the trend in the long-run relationship, simply do not specify the option restricted.

[Monir Ahmed]

Thank you Professor for your prompt answer.

[Simra Temmy]

Hi Professor,
Your comments really help us a lot. With the ARDL, something still not clear. If you can answer my questions, you will once again be a great help to both me and those who search for answers to these questions on google.
1- If ARDL is the best option for mix integration series, why we still apply cointegration tests? As you already mentioned, cointegration can only be exists among I(I) series.
2- Based on question 1, some panel cointegration tests allow using mix order series such as Westerlund (2008), right? How is it possible to find cointegration using those tests even if the series are mix of II) and I(0)?
3- More precisely, for panel series, if we have mix order integration series, and if we applied Westerlung and found no cointegration, Can we still go on for PMG for both short-and long run? If yes, what is the rationale for this process?
4- For time series, if we apply ARDL-bound for mix order series, how is it possible to conclude cointegration as in some studies? Or, whay they found is in fact, a relationship, not cointegration, right?
5- For time series, if no cointegration is found for mix order series, should we keep going on ARDL for only short-run coefficients, or is it still OK to continue with ARDL even without cointegration?


I have no doubt that your answers will help everyone. Thank you very much in advance

[Sebastian Kripfganz]

1. You do not have to carry out separate cointegration tests. If the ARDL bounds test provides evidence in favor of a long-run level relationship, this would correspond to a cointegrating relationship if all involved variables are I(1).
2. Cointegration can only exist between I(1) variables. This does not rule out that other I(0) variables affect the short-run dynamics. In fact, including I(0) variables can help to purge the error term from serial correlation. Absence of residual serial correlation is usually an assumption underlying those tests.
3. The PMG estimator does not require that the time series are I(1). If they are I(1) but not cointegrated, then no long-run level relationship can exist among them.
4. Based on the bounds test, we may conclude that there is evidence for a long-run level relationship. If the variables are I(1), we simply call this a cointegrating relationship. If we do not know that the variables are I(1), we should not call it cointegration.
5. If you find no evidence of a long-run relationship (whether we call it cointegration or not), you could estimate a simpler ARDL model in first differences only; that is, the model would contain only short-run terms. You do not necessarily have to re-estimate the model but could also just stick to the original ARDL model (which might be less efficient as it would contain unneccessary long-run terms).

[Simra Temmy]

1. You do not have to carry out separate cointegration tests. If the ARDL bounds test provides evidence in favor of a long-run level relationship, this would correspond to a cointegrating relationship if all involved variables are I(1).
2. Cointegration can only exist between I(1) variables. This does not rule out that other I(0) variables affect the short-run dynamics. In fact, including I(0) variables can help to purge the error term from serial correlation. Absence of residual serial correlation is usually an assumption underlying those tests.
3. The PMG estimator does not require that the time series are I(1). If they are I(1) but not cointegrated, then no long-run level relationship can exist among them.
4. Based on the bounds test, we may conclude that there is evidence for a long-run level relationship. If the variables are I(1), we simply call this a cointegrating relationship. If we do not know that the variables are I(1), we should not call it cointegration.
5. If you find no evidence of a long-run relationship (whether we call it cointegration or not), you could estimate a simpler ARDL model in first differences only; that is, the model would contain only short-run terms. You do not necessarily have to re-estimate the model but could also just stick to the original ARDL model (which might be less efficient as it would contain unneccessary long-run terms).

Dear Professor,
For #3, say dependent var is I(I), regressors are mix, two of them are I(0), the rest is I(I).
You say that it will be better if I add all the varibles to the Westerlung coint test; not only I(I) ones as I seen in some PMG studies? (I tried both, no cointegration)
And if we get significant long term results for I(0) regressors from PMG estimation, are these long run coefficients make any sense?

Thank you so much for replies...

[Sebastian Kripfganz]

I do not really know enough about the Westerlund test, so cannot give you a definite answer to this question.

I(0) variables can potentially affect the long-run level relationship if the I(1) dependent variable is cointegrated with other I(1) regressors. In that case, there exists a linear combination between the dependent variable and those I(1) regressors which is I(0). The other I(0) regressors can then have a long-run effect on this I(0) linear relationship. In that regard, significant long-run coefficients of I(0) variables could still make sense.

[Simra Temmy]

Your interpretation is really very helpful. Many thanks!

[wanhaiyou]

I do not really know enough about the Westerlund test, so cannot give you a definite answer to this question.

I(0) variables can potentially affect the long-run level relationship if the I(1) dependent variable is cointegrated with other I(1) regressors. In that case, there exists a linear combination between the dependent variable and those I(1) regressors which is I(0). The other I(0) regressors can then have a long-run effect on this I(0) linear relationship. In that regard, significant long-run coefficients of I(0) variables could still make sense.

Dear Prof Kripfganz,
I want to replicate the ardl model with regression routine, however, the results are inconsistent. The corresponding results are as follows

. webuse lutkepohl2,clear
(Quarterly SA West German macro data, Bil DM, from Lutkepohl 1993 Table E.1)

. ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1    // (1, 0, 4)

ARDL(1,0,4) regression

Sample: 1961q1 - 1982q4                         Number of obs     =         88
                                                R-squared         =     0.5385
                                                Adj R-squared     =     0.4981
Log likelihood =  306.72241                     Root MSE          =     0.0078

------------------------------------------------------------------------------
D.ln_consump |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
  ln_consump |
         L1. |  -.3335022    .043545    -7.66   0.000    -.4201595    -.246845
-------------+----------------------------------------------------------------
LR           |
      ln_inc |
         L1. |    1.00357    .032044    31.32   0.000     .9398006     1.06734
             |
      ln_inv |
         L1. |  -.0417399   .0373449    -1.12   0.267    -.1160586    .0325788
-------------+----------------------------------------------------------------
SR           |
      ln_inc |
         D1. |   .3346929    .044616     7.50   0.000     .2459042    .4234815
             |
      ln_inv |
         D1. |   .0556284   .0196031     2.84   0.006     .0166169      .09464
         LD. |   .0167529   .0200739     0.83   0.406    -.0231955    .0567012
        L2D. |   .0620601   .0198688     3.12   0.002     .0225198    .1016003
        L3D. |     .03124   .0194042     1.61   0.111    -.0073756    .0698555
             |
       _cons |    .037445   .0118549     3.16   0.002     .0138529     .061037
------------------------------------------------------------------------------

. 
. // OLS estimator
. reg d.ln_consump l.ln_consump l.ln_inc l.ln_inv d.ln_inc d.ln_inv l(1/3).d.ln_inv

      Source |       SS           df       MS      Number of obs   =        88
-------------+----------------------------------   F(8, 79)        =     11.68
       Model |  .005677968         8  .000709746   Prob > F        =    0.0000
    Residual |  .004801841        79  .000060783   R-squared       =    0.5418
-------------+----------------------------------   Adj R-squared   =    0.4954
       Total |  .010479809        87  .000120458   Root MSE        =     .0078

------------------------------------------------------------------------------
D.ln_consump |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  ln_consump |
         L1. |  -.3064987   .0563429    -5.44   0.000    -.4186463    -.194351
             |
      ln_inc |
         L1. |   .3078729   .0570286     5.40   0.000     .1943604    .4213855
             |
      ln_inv |
         L1. |  -.0130074   .0125833    -1.03   0.304    -.0380537     .012039
             |
      ln_inc |
         D1. |    .382558   .0773686     4.94   0.000     .2285596    .5365565
             |
      ln_inv |
         D1. |   .0542619   .0197379     2.75   0.007     .0149746    .0935491
         LD. |   .0116764   .0212116     0.55   0.584    -.0305443    .0538972
        L2D. |   .0586942   .0204104     2.88   0.005     .0180683      .09932
        L3D. |   .0295406   .0195846     1.51   0.135    -.0094415    .0685227
             |
       _cons |   .0337566   .0128433     2.63   0.010     .0081927    .0593205
------------------------------------------------------------------------------

. 
end of do-file

Thanks for your kindly help!


Best regards,
wanhai you

[Sebastian Kripfganz]

The difficulty here is that the optimal lag order for ln_inc is zero, but you are forcing it to appear in the first lag in the EC representation (option ec1). This is achieved by effectively imposing a constraint as follows:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1
constraint 1 L.ln_inc = D.ln_inc
cnsreg D.ln_consump L.ln_consump L.ln_inc L.ln_inv D.ln_inc L(0/3)D.ln_inv if e(sample), constraints(1)

The long-run coefficients in the ardl output are a nonlinear coefficient combination:

nlcom (- _b[L.ln_inc] / _b[L.ln_consump]) (- _b[L.ln_inv] / _b[L.ln_consump])

If you use the ec instead of the ec1 option, this complication does not arise and you can replicate the results directly with regress:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec
reg D.ln_consump L.ln_consump ln_inc ln_inv L(0/3)D.ln_inv if e(sample)
nlcom (- _b[ln_inc] / _b[L.ln_consump]) (- _b[ln_inv] / _b[L.ln_consump])

[wanhaiyou]

The difficulty here is that the optimal lag order for ln_inc is zero, but you are forcing it to appear in the first lag in the EC representation (option ec1). This is achieved by effectively imposing a constraint as follows:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1
constraint 1 L.ln_inc = D.ln_inc
cnsreg D.ln_consump L.ln_consump L.ln_inc L.ln_inv D.ln_inc L(0/3)D.ln_inv if e(sample), constraints(1)

The long-run coefficients in the ardl output are a nonlinear coefficient combination:

nlcom (- _b[L.ln_inc] / _b[L.ln_consump]) (- _b[L.ln_inv] / _b[L.ln_consump])

If you use the ec instead of the ec1 option, this complication does not arise and you can replicate the results directly with regress:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec
reg D.ln_consump L.ln_consump ln_inc ln_inv L(0/3)D.ln_inv if e(sample)
nlcom (- _b[ln_inc] / _b[L.ln_consump]) (- _b[ln_inv] / _b[L.ln_consump])

Thanks very much for your reply, dear prof. Yes, I have read your slide at page 12. I think the formula at page 5 is equivalent to the formula at page 12. Whether the lag order of ln_inc is zero or not,
the x_t-1 in EC representation is always included. Therefore, the first lag of ln_inc is included. Actually, I don't follow why we need the constraint 1 L.ln_inc = D.ln_inc?

Additionally, I think the EC1 representation is more commonly in the literature.

Thanks again for your help.

Bests,
Wanhai You


Reference:

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

[Sebastian Kripfganz]

The equations on page 12 are derived from the equation on page 5. With ec1, when the lag order of ln_inc is zero, its first lag is included in the error-correction form. However, in terms of the coefficients of the equation on page 5, this first lag has a coefficient equal to zero. Because this coefficient equals zero, we need the restriction for the coefficients on page 12, which I mentioned in my previous post. Put differently, for your model there is a total of 8 coefficients in the level equation on page 5:

ardl ln_consump ln_inc ln_inv, lags(1 0 4)

In the ec1 representation, page 12, you have 9 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1

For the two models to coincide, there must be 1 restriction on the coefficients in the latter version of the model.

If you do not want to have this restriction, either estimate it with option ec, which again gives you 8 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec

or allow for 1 unrestricted lag of ln_inc in the model:

ardl ln_consump ln_inc ln_inv, lags(1 1 4)
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec1
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec

In the latter case, no restriction is needed. There are 9 coefficients in each version of the model. But obviously, because we allow for a nonzero coefficient of the first lag in the level version, the estimates differ.

[wanhaiyou]

The equations on page 12 are derived from the equation on page 5. With ec1, when the lag order of ln_inc is zero, its first lag is included in the error-correction form. However, in terms of the coefficients of the equation on page 5, this first lag has a coefficient equal to zero. Because this coefficient equals zero, we need the restriction for the coefficients on page 12, which I mentioned in my previous post. Put differently, for your model there is a total of 8 coefficients in the level equation on page 5:

ardl ln_consump ln_inc ln_inv, lags(1 0 4)

In the ec1 representation, page 12, you have 9 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1

For the two models to coincide, there must be 1 restriction on the coefficients in the latter version of the model.

If you do not want to have this restriction, either estimate it with option ec, which again gives you 8 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec

or allow for 1 unrestricted lag of ln_inc in the model:

ardl ln_consump ln_inc ln_inv, lags(1 1 4)
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec1
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec

In the latter case, no restriction is needed. There are 9 coefficients in each version of the model. But obviously, because we allow for a nonzero coefficient of the first lag in the level version, the estimates differ.

Dear prof, I see. Many thanks for your kindly help. Thanks again for your excellent rutine "ardl".

Bests,
wanhai you