[Kyla Antonio]

Hello. So far, I've only been able to run the ardl command in stata with Y lags. I was wondering if there is a certain ardl command format that includes lagged X variables in the ardl model in stata.

[Sebastian Kripfganz]

The following example is taken from the ardl help file:

webuse lutkepohl2
ardl ln_inv ln_inc ln_consump, lags(. . 4) maxlag(3 3 3)

The first two arguments (dots) of the option lags(. . 4) specify that the number of lags for the dependent ("Y") and the first of the regressors ("X variables") should be chosen automatically. The last element imposes 4 lags on the second regressor. The maxlag(3 3 3) option specifies a maximum of 3 lags for each of these variables when lags are chosen automatically. Notice that the last argument of the maxlag(3 3 3) option is ignored because we have specified a specific lag order for the last variable with the lags(. . 4) option.

[Simon Suleiman]

ardl lnM lnY π lnEX, ec1 lags(2 4 1 1) regstore(ecreg)

ARDL regression
Model: ec

Sample: 2001q1 - 2015q4
Number of obs = 60
Log likelihood = 68.247426
R-squared = .61706315
Adj R-squared = .52930678
Root MSE = .08674141

------------------------------------------------------------------------------
D.lnM | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ |
lnM |
L1. | -.3756424 .1177529 -3.19 0.003 -.6124004 -.1388845
-------------+----------------------------------------------------------------
LR |
lnY |
L1. | 2.046759 .2340766 8.74 0.000 1.576116 2.517401
|
π |
L1. | -.0746791 .0324809 -2.30 0.026 -.1399863 -.0093719
|
lnEX |
L1. | .629879 .4384167 1.44 0.157 -.2516169 1.511375
-------------+----------------------------------------------------------------
SR |
lnM |
LD. | -.1253645 .1352892 -0.93 0.359 -.3973816 .1466526
|
lnY |
D1. | 1.402742 .3396997 4.13 0.000 .7197298 2.085754
LD. | .6953933 .3693675 1.88 0.066 -.0472697 1.438056
L2D. | .3779132 .3489156 1.08 0.284 -.3236286 1.079455
L3D. | -.0649744 .2872861 -0.23 0.822 -.6426018 .512653
|
π |
D1. | -.0137279 .0039108 -3.51 0.001 -.021591 -.0058648
|
lnEX |
D1. | -.5099346 .3896824 -1.31 0.197 -1.293444 .2735745
|
_cons | -8.565543 2.564517 -3.34 0.002 -13.72185 -3.409237
------------------------------------------------------------------------------

Hello everyone. These are the results of ARDL model. Bounds Test showed that there is a long run relationship between variables. I would like to know how can I generate and compute the error term (ECM t-1)?

[Sebastian Kripfganz]

I would like to know how can I generate and compute the error term (ECM t-1)?

Please see the help file for ardl, in particular the remarks section on "The error-correction term".

webuse lutkepohl2
ardl ln_consump ln_inc, ec
predict ecterm, ec

[Simon Suleiman]

Thank you Sebastian.

[Mahana Noorma]

Dear Sebastian,
Dear all,

I tested whether GDP growth and public debt ratio are cointegrated by means of the bounds testing approach (Pesaran et al. (2001)) for several countries. For some of them, the results indicate that these two variables are not cointegrated. In such cases, I don't know how the econometric specification should be in order continue my analysis of the impact of public debt ratio on GDP growth.

Thus, my question is the following: when the results indicate that there is no cointegration between the two variables of interest, should the variables still be in first difference (see Equation (1))? This would correspond to the ARDL model developed within the bounds testing procedure but without any error correction term. If yes, can the coefficients still be interpreted as the short term effects of the various explanatory variables on the first-difference of GDP growth?

Equation (1):

Δy_t = β₀ + Σ β_1iΔy_t-i + Σ β_{2 j}Δx_t-j+ Σ β_3k Δz_t-k + Σ β_4l Δm_t-l + Σ β_5n Δd_t-n + e_t

y : real GDP per capita growth
β_{0 :} constant
x : public debt ratio
z, m and d : various additional explanatory variables

Or should the variables be in first difference when there are I(1) and in levels when there are I(0) (the econometric specification would thus contain a mixture of variables (in first difference and in levels))? In that case, how the coefficents must be interpreted?

I kindly thank you for your consideration and your precious help.

Best regards

[Sebastian Kripfganz]

Strictly speaking, the bounds testing approach by Pesaran et al. (2001) is a test on the existence of a long-run relationship in the levels of the variables. That is not necessarily the same as cointegration. The latter requires the variables to be individually I(1). In your case, GDP growth would typically be a stationary variable. (If the log of GDP is an I(1) variable, then the GDP growth is I(0) because it is approximately equal to the first difference of log(GDP).)

This does not invalidate your estimation and testing procedure. It is only relevant for the interpretation of the results, i.e. do not call the existence of a long-run relaionship "cointegration" if the variables themselves are already stationary.

If you do not find evidence for a long-run relationship, you can estimate the model in first differences without the error-correction term. The error-correction term would be an irrelevant regressor and excluding it improves the efficiency of the estimates for the remaining coefficients, which would still be interpreted as short-run coefficients. (There would not exist a long-run relationship in this case.)

Estimating the model in levels (irrespective of whether the variables are I(0) or I(1)) would also be an acceptable approach. The coefficients are still interpreted as short-run coefficients. The quantitative interpretation of course differs because once you looking at the effect of a change in the levels and in the first-differenced model the coefficients give you the effect of a change in the differences. Given your example, the level coefficients might be easier to interpret if you are interested in the quantitative short-run effects.

[Mahana Noorma]

Dear Sebastian,

Thank you so much for having replied so quickly and for your explanations, which are very clear. I also thank you for having reminded me that the bounds testing approach is a test on the existence of a long-run relationship and not a cointegration test, this is important.

Kind regards

[Muhammad Azam]

Unable to install ARDL Module.

Its says:
net install ardl, from(http://www.kripfganz.de/stata/)
host not found
http://www.kripfganz.de/stata/ either
1) is not a valid URL, or
2) could not be contacted, or
3) is not a Stata download site (has no stata.toc file).

[Muhammad Azam]

Found the solution. You need Stata 14 64 bit. Will not install on 32 bit.

[Lajos Konok]

Dear all,
I would like to ask, may I use I(2) variable, if I treat it as exogenous variable usind exog option?
Shall I take the first difference of I(2) to make it I(1)?
After ARDL model, is it possible to estimate something similar to Impulse response function or Variance decomposition (normally used after VAR) ?
Thank you very much for your help!

[Sebastian Kripfganz]

The variables specified with the exog() option should in general be I(0). All other variables may be I(0) or I(1).

Unfortunately, the irf command does not work after ardl. A similar postestimation command specific to ardl is currently not available.

[Lajos Konok]

Thank you Sebastian

[Louison Cahen-Fourot]

Dear Sebastian, dear Daniel,

Thanks a lot for your ardl command, it is very helpful. I am a bit puzzled by some results I get regarding the presence of a long-run relationship. Here are my results:

. ardl lny lnx2 lnx3, ec1 lags(2 . .) bic dots regstore(regress_res1)

Optimal lag selection, % complete:
----+---20%---+---40%---+---60%---+---80%---+-100%
..................................................
BIC optimized over 25 lag combinations

ARDL regression
Model: ec

Sample:       1955 -       2013
Number of obs  = 59
Log likelihood = 112.33055
R-squared      = .50835557
Adj R-squared  = .46197402
Root MSE       = .03803596

------------------------------------------------------------------------------
       D.lny |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
         lny |
         L1. |  -.1443632   .0714981    -2.02   0.049    -.2877702   -.0009562
-------------+----------------------------------------------------------------
LR           |
        lnx2 |
         L1. |   .5013196   .1632799     3.07   0.003     .1738216    .8288177
             |
        lnx3 |
         L1. |  -.5857345   .4965066    -1.18   0.243    -1.581601    .4101316
-------------+----------------------------------------------------------------
SR           |
         lny |
         LD. |  -.1524721   .1150116    -1.33   0.191    -.3831561    .0782119
             |
        lnx2 |
         D1. |   1.632613   .2792421     5.85   0.000     1.072525    2.192702
             |
        lnx3 |
         D1. |  -.0845585   .1055645    -0.80   0.427    -.2962941    .1271771
             |
       _cons |   1.760881   .8828525     1.99   0.051    -.0098966    3.531659
------------------------------------------------------------------------------

. estat btest

Pesaran/Shin/Smith (2001) ARDL Bounds Test
H0: no levels relationship             F =  2.733
                                       t = -2.019

Critical Values (0.1-0.01), F-statistic, Case 3

      | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]
      |    L_1     L_1 |   L_05    L_05 |  L_025   L_025 |   L_01    L_01
------+----------------+----------------+----------------+---------------
  k_2 |   3.17    4.14 |   3.79    4.85 |   4.41    5.52 |   5.15    6.36
accept if F < critical value for I(0) regressors
reject if F > critical value for I(1) regressors

Critical Values (0.1-0.01), t-statistic, Case 3

      | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]
      |    L_1     L_1 |   L_05    L_05 |  L_025   L_025 |   L_01    L_01
------+----------------+----------------+----------------+---------------
  k_2 |  -2.57   -3.21 |  -2.86   -3.53 |  -3.13   -3.80 |  -3.43   -4.10
accept if t > critical value for I(0) regressors
reject if t < critical value for I(1) regressors

k: # of non-deterministic regressors in long-run relationship
Critical values from Pesaran/Shin/Smith (2001)

. estat btest, n

Pesaran/Shin/Smith (2001) ARDL Bounds Test
H0: no levels relationship             F =  2.733

Critical Values (0.1-0.01), F-statistic, Case 3

      | [I_0]   [I_1]  | [I_0]   [I_1]  | [I_0]   [I_1]
      |    L_1     L_1 |   L_05    L_05 |   L_01    L_01
------+----------------+----------------+---------------
  k_2 |   3.27    4.26 |   4.00    5.06 |   5.70    6.99
accept if F < critical value for I(0) regressors
reject if F > critical value for I(1) regressors

k: # of non-deterministic regressors in long-run relationship
Critical values from Narayan (2005), N=60

. estimates store ardl_res

. estimates restore regress_res1
(results regress_res1 are active now)

. qui regress

. testparm L.lny L.lnx2 lnx3

 ( 1)  L.lny = 0
 ( 2)  L.lnx2 = 0
 ( 3)  lnx3 = 0

       F(  3,    53) =    2.73
            Prob > F =    0.0528

(one remark: I specified two lags for lnY to solve an autocorrelation problem.)

As you can see, I get the same f-stat using estat btest after ardl and testparm after the underlying regress. This is not very surprising since the PSS test is a Wald test if I understood it correctly.

However, both commands yield differing conclusions: I fail to reject H0 following btest while I can reject H0 at 10% following testparm. I ran many regressions using the ardl command and comparing estat btest with testparm, and it is usually concordant. But for some specifications like the one above, I get puzzling results. I know I am missing something here but I don't understand what it is. Could you enlighten me ?

Many thanks for your help

[Daniel Schneider]

Dear Louison,

The distributions of the F and t statistics of the PSS 2001 bounds test are non-standard. That means for your case that you cannot simply calculate p-values off of a standard F(3,53) distribution. The bounds test output is informative, the standard F-test is not.

Best,
Daniel