[Sebastian Kripfganz]

Hi Valérie,

In your second specification, lnx appears in the long-run relationship in period t instead of period t-1. The long-run coefficient is not affected by the timing of the regressors in the long-run relationship. However, the short-run coefficients are affected. For a detailed discussion about this issue please have a look at the Stata help file of the ardl command, in particular the remarks section "Long-run coefficients expressed in time t or t-1".

[valérie orozco]

Hello everybody,

I've just learned the ARDL model so maybe I miss something.
I estimate first an ARDL with variable of interest in level and then the ARDL in difference.

. ardl lny lnx

ARDL regression
Model: level

Sample:       1963 -       2011
Number of obs  = 49
Log likelihood = 109.9019
R-squared      = .98733233
Adj R-squared  = .98648782
Root MSE       = .02680262

------------------------------------------------------------------------------
         lny |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         lny |
         L1. |    .278142   .1371756     2.03   0.049     .0018561    .5544278
         L2. |   .3199927   .1265193     2.53   0.015     .0651699    .5748156
             |
         lnx |   .0633735   .0198776     3.19   0.003     .0233379    .1034091
       _cons |   1.806711   .5328952     3.39   0.001     .7334054    2.880017
------------------------------------------------------------------------------

So my ARDL in level can be written y_t=mu + g_1 y_{t-1} + g_2 y_{t-2} + b_1 x_t + eps_t
Manipulating a bit this equation, I am able to write the ARDL in ECM form :

D.y_t = mu + (g_1-1+g_2) * [y_{t-1} - (b_1 x_{t-1})/(-g_1+1-g_2)] + b_1 D.x_t - g_2 LD.y_t

The term in bracket being the LR term.
So I expect to find :
- for the adjustment parameter : (g_1-1+g_2) so around 0.278-1+0.320=-.402
- for the coefficient of x_{t-1} of the LR term : b_1 /(-g_1+1-g_2) so around .157
- for the coefficient of LD.y_t of the SR part : - g_2 so -0.31999
- for the coefficient of D.x_t of the SR part : b_1 so around 0.063

. ardl lny lnx, ec

ARDL regression
Model: ec

Sample:       1963 -       2011
Number of obs  = 49
Log likelihood = 109.9019
R-squared      = .38612047
Adj R-squared  = .34519517
Root MSE       = .02680262

------------------------------------------------------------------------------
       D.lny |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
         lny |
         L1. |  -.4018653   .1205984    -3.33   0.002     -.644763   -.1589675
-------------+----------------------------------------------------------------
LR           |
         lnx |   .1576984   .0070317    22.43   0.000     .1435359     .171861
-------------+----------------------------------------------------------------
SR           |
         lny |
         LD. |  -.3199927   .1265193    -2.53   0.015    -.5748156   -.0651699
             |
       _cons |   1.806711   .5328952     3.39   0.001     .7334054    2.880017
------------------------------------------------------------------------------

So I find what I expect, except for the coefficient of D.x_t of the SR part which doesn' appear in this estimation. And I don't understand why.
Could anyone help me?

thank you very much.

[Sebastian Kripfganz]

The ardl command cannot handle instrumental variables, if that is what you had in mind.

[mwai mwai]

Is there a way of solving the omitted variables bias in stata for ARDL models?

[Viktorija Mano]

Hi to all,

I am also using the ARDL in my model. So I used the varsoc variable to determine the optimum lag for my variables, than I used the ADF to test whether my variables have a unit root or not and also I used the Zivot-Andrews test for structural breaks. Now I am struggling with the ardl itself. Can someone advise how to do the ardl? I used:

regress GDPD1 L1.GDPD1 L1.OPEN INF L1.POP L1.ED FDI L1.GOV L1.OPEND1 INFD1 L1.POPD1 L1.EDD1 FDID1 L1.GOVD1

and

ardl GDPD1 OPEND1 INFD1 POPD1 EDD1 FDID1 GOVD1, lags(1)

but for some reason when I add the variables at level and first difference it says to me that the variables are correlated and ardl cannot be completed.

I also conducted the Wald -test :

test _b[L1.OPEN]= _b[L1.INF] = _b[L1.POP]= _b[L1.ED]= _b[L1.FDI]= _b[L1.GOV] = 0

Any help will be appreciated and I can provide more details of my analysis if needed.

Thanks
Viki

[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/)

[Sebastian Kripfganz]

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

[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]

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.

[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
------------------------------------------------------------------------------

.

[Ken Mulligan]

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

[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]

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!

[Mobeen Ur Rehman]

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

[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.