[Mobeen Ur Rehman]

Hi,
I would like to ask from anyone that if I have a mix of stationary and non stationary (integrated at level I) independent variables but my dependent variable is stationary, so can I still apply ARDL model?

[Mobeen Ur Rehman]

I want to know that if we can apply ARDL model in case our dependent variable is stationary and the independent variables are a mix of stationary and non-stationary(integrated at order 1) variables

[Sebastian Kripfganz]

Mobeen Ur Rehman
The ARDL model is suitable for your situation, yes.

[Lajos Konok]

Hello all, if I understood properly, one may use ARDL with I(0) and I(1) variables but just one co-integrating relationship. If I have I(1) dependent variable and a mixture of three I(0) and five I(1) independent variables and I have more than one co-integrating relationship among them, may I estimate a vector error-correction model? How should I treat those I(0) variables in my model? Thanks for any help!

[Mobeen Ur Rehman]

Dear Sebastian,

Can I have any journal reference for that. My reviewer has commented on that and I need any journal reference to prove my point.

[Sebastian Kripfganz]

See for example:

• Pesaran, M. H., Y. Shin, and R. J. Smith (2001). Bounds Testing Approaches to the Analysis of Level Relationships, Journal of Applied Econometrics 16 (3), 289-326.

In Section 2 of this paper, page 291, the authors state the assumptions for the underlying data-generating process. They define a vector \( \mathbf{z}_t = (y_t, \mathbf{x}_t')' \), where \( y_t \) is the dependent variable and the vector \( \mathbf{x}_t \) contains the (weakly) exogenous regressors. They then assume

[..] the elements of \( \mathbf{z}_t \) to be purely \( I(1) \), purely \( I(0) \) or cointegrated [..]

which includes the case where \( y_t \) is \( I(0) \) and the variables in \( \mathbf{x}_t \) are a mix of \( I(0) \) and \( I(1) \) variables.

Further general references on ARDL / EC models include:

• Engle, R. F., and C. W. J. Granger (1987). Co-Integration and Error Correction: Representation, Estimation, and Testing. Econometrica 55 (2), 251-276.
• Hassler, U. and J. Wolters (2006). Autoregressive distributed lag models and cointegration. Allgemeines Statistisches Archiv 90 (1). 59-74.
• Pesaran, M. H., and Y. Shin (1999). An Autoregressive Distributed Lag Modelling Approach to Cointegration Analysis. In: Strom, S. (Ed.): Econometrics and Economic Theory in the 20th Century: The Ragnar Frisch Centennial Symposium. Cambridge, UK: Cambridge University Press.

[Mobeen Ur Rehman]

Dear Sebastian, thank you very much for these references. I hope that my reviewer will be satisfied with these.

[Ken Mulligan]

Hi Sebastian. Thank for the ardl command in Stata.

I want to test for serial correlation in the residuals of my ardl model. However, ardl does not support 'estat dwatson' or 'estat durbinalt,' right? In the ardl documentation, you say that one can use the 'estimates store' command to recover "the estimation results from Stata's regress which underlies ardl ... and then use the many tools of regress postestimation to perform the desired calculations."

The short version of my question is: How does one test for serial correlation of residuals with ardl?

Thank you!

[Sebastian Kripfganz]

Ken Mulligan
The ardl command is built on Stata's regress command. The following example shows you how to restore the underlying regress estimation results that can be used for subsequent postestimation analysis.

webuse klein
tsset yr
ardl consump wagegovt, regstore(regress_res)
estimates store ardl_res
estimates restore regress_res
regress                                  // can be skipped
estat dwatson
estat durbinalt

The three main steps are:

1. Store the underlying regress estimation results with the ardl option regstore() for later use.
2. Store the ardl estimation results with estimates store.
3. Restore the underlying regress results with estimates restore.

[Ken Mulligan]

Thank you Sebastian very much for your fast, detailed, helpful response.

[Ken Mulligan]

Is there a glitch in the ardl output or am I missing something here?

I replicated the problem I'm having using the klein data. Here are two models. They have identical variables but different lag commands:

webuse klein
tsset yr
ardl invest consump govt wagepriv , minlag1 lags(1) ec
ardl invest consump govt wagepriv , minlag1 lags(1 1 . .) ec

The first model assigns one lag to all variables. The second model assigns one lag to the DV and first IV, and allows ardl to use the BIC to choose the appropriate number of lags for the last two IVs.

In the output of the two models (below), the variable labels on the left hand side are the same in the two models, but the coefficients differ across the two models. Why the different results? How are the two models different? [Also, the first model has 21 observations and the second has 18]

Thanks!

. ardl invest consump govt wagepriv , minlag1 lags(1) ec

ARDL regression
Model: ec

Sample:      1921 -      1941 
Number of obs  = 21
Log likelihood = -19.451219
R-squared      = .94273306
Adj R-squared  = .91189701
Root MSE       = .77654282

------------------------------------------------------------------------------
    D.invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
      invest |
         L1. |    -.43948   .1420406    -3.09   0.009    -.7463399     -.13262
-------------+----------------------------------------------------------------
LR           |
     consump |
         L1. |  -1.644985   .4096043    -4.02   0.001    -2.529882   -.7600889
             |
        govt |
         L1. |  -.5135741   .5555708    -0.92   0.372    -1.713812    .6866637
             |
    wagepriv |
         L1. |   2.373294   .5085385     4.67   0.000     1.274664    3.471925
-------------+----------------------------------------------------------------
SR           |
     consump |
         D1. |  -.0098738   .1633918    -0.06   0.953    -.3628603    .3431127
             |
        govt |
         D1. |  -.8931402   .1879847    -4.75   0.000    -1.299256   -.4870239
             |
    wagepriv |
         D1. |   1.033818   .2239478     4.62   0.000     .5500081    1.517628
             |
       _cons |   2.164687   2.346015     0.92   0.373     -2.90357    7.232943
------------------------------------------------------------------------------

. ardl invest consump govt wagepriv , minlag1 lags(1 1 . .) ec

ARDL regression
Model: ec

Sample:      1924 -      1941 
Number of obs  = 18
Log likelihood = -16.883436
R-squared      = .9393438
Adj R-squared  = .89688446
Root MSE       = .82938027

------------------------------------------------------------------------------
    D.invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
      invest |
         L1. |  -.4370741   .1878032    -2.33   0.042    -.8555258   -.0186224
-------------+----------------------------------------------------------------
LR           |
     consump |
         L1. |  -1.999773   .6131518    -3.26   0.009    -3.365961   -.6335859
             |
        govt |
         L1. |   -.429871   .7541322    -0.57   0.581    -2.110182     1.25044
             |
    wagepriv |
         L1. |   2.748364   .6742054     4.08   0.002     1.246141    4.250587
-------------+----------------------------------------------------------------
SR           |
     consump |
         D1. |  -.1411733   .2117196    -0.67   0.520     -.612914    .3305674
             |
        govt |
         D1. |  -1.026757   .2495717    -4.11   0.002    -1.582838   -.4706769
             |
    wagepriv |
         D1. |   1.232643   .3016226     4.09   0.002      .560586      1.9047
             |
       _cons |   4.486252   4.068839     1.10   0.296    -4.579685    13.55219
------------------------------------------------------------------------------

.

[Sebastian Kripfganz]

You have already made the right observation: The sample sizes used to compute the two results differ. As a consequence also the estimates differ. So, why do the samples differ?

The data set contains annual data from 1920 to 1941. In the first example, one lag is pre-specified for all variables which reduces the available sample by one observation (1920 drops out). In the second example, for the third and fourth variable the number of lags are no longer pre-specified. Instead, ardl is supposed to determine the optimal lag order based on information criteria (which is by default the Schwarz/Bayesian information criterion). To obtain the optimal lag order, ardl estimates the model with all possible combinations up to the maximum lag order which is 4 by default. That means that for the largest allowed lag order we lose four observations (1920 to 1923). At the end, ardl compares the computed information criteria for all combinations and chooses the model with the minimal criterion. In this case, it is the model with one lag for all variables.

The crucial point is that the comparison of the information criteria for different model specifications is only valid if all are based on the same sample! That means, when the maximum allowed lag order is 4, it uses the sample starting at 1924 for all lag combinations. Since the choice of the optimal lag order relies on the specific sample, it would not be consistent with this approach to finally estimate the model again based on the optimal lag order for a different sample even though we would have more data at hand given that we do not need all the lags up to 4. This is why the two samples differ in your example with the consequence that also the estimates differ.

[Rohan]

Is it also correct that the interpretation of the LR coefficients can be made in the same way as a "simple linear" regression. For example, in the first estimation results provided by Ken, a one unit increase in consumption results in a 1.6 unit increase in investment in the long-run (since the LR coefficient on investment is normalised to equal one)? Or is an additional calculation required? Cheers.

[Sebastian Kripfganz]

Hi Rohan,
Sorry for the late response. Your interpretation is correct.

[Sebastian Kripfganz]

Thanks to Daniel Schneider, a new update is available for the ardl package. This update fixes several bugs that led to unexpected error messages in previous versions.

Another improvement of the new version 0.6.0 is that the maximum number of lag permutations is no longer constrained by Stata's matsize settings. In previous versions, this could be very restrictive if a large number of regressors were used.

The help file has also been extended and improved.

To update an existing installation type:

adoupdate ardl, update

For a fresh installation type:

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