[Abu Kom]

Hello everyone,

I have followed several comments on this forum and I have really benefited from the discussions. I have used ARDL to carry out estimations for my thesis, but I am unclear as to what step-by-step Stata codes (commands) I need to work through to get the error correction term. I tried the ardl help on Stata, but I could not still do it. Does anyone want to help me with this? Thanks.

[charles martens]

The statements in your first question are correct. The long-run estimates (LR) are the coefficients theta, and the speed-of-adjustment coefficient (ADJ) is the coefficient alpha (without subscript t). (As an aside, it should be Y_{t-1} instead of Y_t in the error-correction term.)

The answer to your second question is a bit more tricky. There should be at most one cointegrating relationship that involves Y_t; but that does not exclude the possibility that the cointegration rank exceeds one if there is an additional cointegrating relationship just between X_{1t} and X_{2t} that does not involve Y_t.

If the F-statistic falls in between the two bounds, the result is generally inconclusive. You could proceed by testing the order of integration for each variable. If all variables are individually I(1), then the upper bound is the relevant one and you would not reject the null hypothesis of no long-run relationship. If all variables are individually I(0), then the lower bound becomes relevant and you would reject the same null hypothesis. It is getting tricky, if some of the variables are I(0) and others are I(1). In principal, you would need to simulate the respective critical value for your specific case. Alternatively, you might want to be conservative and not reject the null hypothesis.

Depending on your conclusion, you would continue using the error-correction specification if you do reject the null hypothesis while it would be more efficient to estimate a model purely in first differences if you cannot reject it.

Thank you so much for your quick reply. It makes better sense now!
I will probably not reject the null hypothesis when the F-statistic falls between bounds, just to be sure. Thanks again.

[Sebastian Kripfganz]

The statements in your first question are correct. The long-run estimates (LR) are the coefficients theta, and the speed-of-adjustment coefficient (ADJ) is the coefficient alpha (without subscript t). (As an aside, it should be Y_{t-1} instead of Y_t in the error-correction term.)

The answer to your second question is a bit more tricky. There should be at most one cointegrating relationship that involves Y_t; but that does not exclude the possibility that the cointegration rank exceeds one if there is an additional cointegrating relationship just between X_{1t} and X_{2t} that does not involve Y_t.

If the F-statistic falls in between the two bounds, the result is generally inconclusive. You could proceed by testing the order of integration for each variable. If all variables are individually I(1), then the upper bound is the relevant one and you would not reject the null hypothesis of no long-run relationship. If all variables are individually I(0), then the lower bound becomes relevant and you would reject the same null hypothesis. It is getting tricky, if some of the variables are I(0) and others are I(1). In principal, you would need to simulate the respective critical value for your specific case. Alternatively, you might want to be conservative and not reject the null hypothesis.

Depending on your conclusion, you would continue using the error-correction specification if you do reject the null hypothesis while it would be more efficient to estimate a model purely in first differences if you cannot reject it.

[charles martens]

Dear Sebastian Kripfganz,

regarding the use of the ardl package I have the following two questions:

1) When co-integration is established, estimating an error correction model (using ardl depvar indepvar, ec) provides the following insight:

Consider a regression model with two independent variables X_1 and X_2. By using "ardl depvar indepvar1 indepvar2, ec
", an error correction term is generated, namely: [ Y_t - theta_1 * X_{1t} - theta_2 * X_{2t} ].This term is stationary and usable in regression analysis.

Consequently, in the output table of "ardl depvar indepvar1 indepvar2, ec", the "LR" estimates are the corresponding coefficients for the theta's in the aforementioned error correction term.
The "ADJ" estimate is alpha in the following regression equation: delta*Y_t = alpha_t*[Y_t - theta_1 * X_{1t} - theta_2 * X_{2t}].

Are these two last statements right? (The theta's corresponding to the LR estimates and the ADJ estimate being the estimate for alpha).

2) When the found F-statistic (using estat btest / ardl, noctable btest) falls within the bounds of the critical values, it is necessary to check the co-integration rank. If the rank is equal to 1, we can continue using the error correction specification by using "ardl depvar indepvar, ec". Is this right as well? Or should we not use the error correction specification when the found F-statistic falls within the bounds?

Thanks in advance

[Sebastian Kripfganz]

In the ardl package are there the critical values implemented that have been calculated by Banerjee et al. (1996), as the usual critical values do not apply.

Which paper by Banerjee et al. (1996) are you refering to? This reference is not precise enough to identify the exact piece of work.

The relevant critical values in the context of the bounds testing procedure implemented in the ardl package are the asymptotic critical values by Pesaran et al. (2001) and the finite-sample critical values by Narayan (2005):

• Pesaran, M. H., Y. Shin, and R. Smith (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics 16(3): 289–326.
• Narayan, P. K (2005). The saving and investment nexus for China: evidence from cointegration tests. Applied Economics 37(17): 1979–1990

[Tino Tinsky]

In the ardl package are there the critical values implemented that have been calculated by Banerjee et al. (1996), as the usual critical values do not apply.

[Mahana Noorma]

Dear Sebastian,

Thank you so much for your reply and your help.

Kind regards

[Sebastian Kripfganz]

There is nothing that guarantees that the coefficient of your error-correction term falls into the interval [-1, 0] even though this is indeed the only meaningful range. Moreover, your coefficient is not statistically significantly different from -1 in either model, so you should not overemphasize the observation that it slightly falls outside of this range in your second model.

That said, your second model imposes the restriction that all your variables z1 to z8 do not enter the long-run relationship but only affect the short-run dynamics. There is nothing wrong with that per se if you have a good reason for justifying this assumption given your particular application, but the inclusion of these additional variables will of course affect the estimates of the other coefficients including that of the error-correction term.

Personally, I advise against estimating the error-correction term in separate regression first and then including it in a second step as a generated regressor in the error-correction model. This is a very inefficient and error-prone way of doing it. Instead, my suggestion would be to estimate all the coefficients jointly in an ARDL / EC model. You might want to have a look at the help file for the ardl command that has been discussed extensively in this topic.

[Mahana Noorma]

Dear all,

I estimated the following ARDL model and tested whether y_t and x_1t are cointegrated by means of bounds testing approach :

Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + θ₀y_t-1 + θ₁x_1t-1 + e_t


Bounds testing indicates that y_t and x_1t are cointegrated. When I estimated the corresponding error correction model (ECM), the coefficient of the error correction term (ϕ) lies between -1 and 0 and is statistically significant.

Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + ϕECT_t-1 + e_t

where ECT_t=y_t-c₀ -a₁x_1t - v_t

However, when I introduce additional explanatory variables (z1 to z8) in the specification (ECM), the coefficient of the error correction term doesn't lie between -1 and 0 anymore. Does this mean that I should have introduced the additional explanatory variables at the beginning of the procedure (before bounds testing) already? Should I test whether y_t and x_1t are cointegrated while considering the additional explanatory variables as exogenous variables?

Thank you so much for your help.

Here are my results:

Note: the optimal lags length is 1 for y and 0 for x1.

 regress d.y l.d.y d.x1 l.ect, robust

Linear regression                                      Number of obs =      42
                                                       F(  3,    38) =   12.69
                                                       Prob > F      =  0.0000
                                                       R-squared     =  0.5385
                                                       Root MSE      =  2.1817

------------------------------------------------------------------------------
             |               Robust
D.           |
y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y |
         LD. |   .2215078   .1441625     1.54   0.133     -.070334    .5133496
x1 |
         D1. |  -.2948656   .0597863    -4.93   0.000    -.4158966   -.1738347
ect |
         L1. |  -.9550015   .1773393    -5.39   0.000    -1.314006   -.5959969
       _cons |   .4485086   .3293098     1.36   0.181    -.2181444    1.115161
------------------------------------------------------------------------------

 regress d.y l.d.y d.x1 l.ect d.z1 d.z2 d.z3 d.z4 d.z5 d.z6 d.z7 d.z8, robust

Linear regression                                      Number of obs =      38
                                                       F( 11,    26) =    8.37
                                                       Prob > F      =  0.0000
                                                       R-squared     =  0.6882
                                                       Root MSE      =  2.0607

------------------------------------------------------------------------------
             |               Robust
D.           |
y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y |
         LD. |   .1602478   .1880296     0.85   0.402    -.2262527    .5467482
x1 |
         D1. |  -.2354829    .086665    -2.72   0.012    -.4136252   -.0573405
ect |
         L1. |  -1.029924   .3138605    -3.28   0.003    -1.675074    -.384775
z1 |
         D1. |  -.0198289   .0288202    -0.69   0.498    -.0790696    .0394119
z2 |
         D1. |  -.9476361   .6397082    -1.48   0.151    -2.262575    .3673028
z3 |
         D1. |  -.1182584   .2679417    -0.44   0.663    -.6690205    .4325037
z4 |
         D1. |   1.555395   6.224749     0.25   0.805    -11.23976    14.35055
z5 |
         D1. |   1.638369   4.030418     0.41   0.688    -6.646273    9.923011
z6 |
         D1. |   .0062724   .1266548     0.05   0.961    -.2540703    .2666151
z7 |
         D1. |  -.1416688   .1269392    -1.12   0.275    -.4025961    .1192586
z8 |
         D1. |   275.7578   182.6913     1.51   0.143    -99.76966    651.2852
       _cons |   .6476965   .5997187     1.08   0.290     -.585043    1.880436
------------------------------------------------------------------------------

[Sebastian Kripfganz]

Valentin:
Please see the ardl help file for the syntax and available options of the command. Please also have a look at the remarks section of the help file. In particular, the options ec or ec1 will give you the error-correction representation (with parameterization of the error-correction term in period t or t-1, respectively).
After having estimated the ardl model with either of the ec or ec1 option, you can obtain the error-correction term as a new variable with the ec option of the postestimation command predict. Please see help ardl postestimation for details.

[Tino Tinsky]

Hello

First of all thank you for your time, it is appreciated. I wanted to ask whether you could briefly explain how it was possible to estimate an error correction model which in my case is simply the reparametrization of an ARDL. How can I obtain the error correction term from the ARDL in Stata?

Thank you for your help!
Best Regards
Valentin

[Sebastian Kripfganz]

We have just released a major update of the ardl package. Thanks to all the suggestions and bug reports, the new version has considerably improved functionality. Some of the highlights are as follows:

1. As an alternative to the option ec there is a new option ec1 which parameterizes the error-correction representation of the long-run coefficients as of period t-1. This now does work even if there are no lags of a regressor in the ARDL levels equation. (Note that the former minlag1 option that has been discussed earlier in this topic is thus no longer needed. However, the results will be different with ec1 compared to the former minlag1 option if the optimal lag order for some regressors is indeed zero. For backward compatibility, minlag1 continues to work but it is advised to now use ec1 instead.)

2. The optimal lag order selection for larger models has been improved. In particular, if the new option fast is specified then ardl will use Mata to do the computations which results in sizeable speed gains. (We kept it as an option because Stata's regress command that is otherwise used is computationally more robust.)

3. The new option dots now displays a progress bar during optimal lag order selection which can be useful in the case of large models.

4. The postestimation functionality has been improved. Predictions are now available in the familiar way with the predict command. In particular, predict can be used to obtain predictions of the error-correction term. Please see help ardl postestimation for details. (The former regstore() option now becomes redundant but also continues to work for backward compatibility reasons.)

5. The computation of the bounds test for the existence of a long-run relationship is now implemented as a postestimation feature. After the estimation of the ARDL model in error-correction form, you can now type estat btest to obtain the bounds test. See help ardl postestimation for details. (The former ardl option btest is now redundant but again continues to work.)

6. The new command ardlbounds has been added to the package that can be used to display critical value tables. In addition to the large-sample critical values by Pesaran, Shin, and Smith (2001, Journal of Applied Econometrics), also the small-sample critical values by Narayan (2005, Applied Economics) can be displayed. See help ardlbounds for details.

7. Some minor bugs have been fixed.

The above list of improvements is not exhaustive. Please consult help ardl for a full list of available options, their description, and remarks concerning the error-correction representation, the bounds test, and other related topics. Note that at least Stata version 11.2 is now required to run the command.

To update an existing version of the ardl package type

adoupdate ardl, update

into Stata's command window.

To install a fresh version of the package type

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

into Stata's command window.

[Sebastian Kripfganz]

Ngan:
We are aware that the current version of the ardl command is not well suited for the estimation of large models with "many" variables and high lag orders. It is mainly a memory problem. Stata slows down considerably the more memory it requires and eventually it may not respond any more.

For the moment, all I can suggest is to reduce the number of maximum lags or to fix the lag order for some of your variables with the lags() option. We will try to implement some improvements in future versions of the ardl package.

[Ngan Tran]

Dear Advisers,

I found the package of ARDL model in STATA very useful to my research. Just have a small question relating to running ardl with the optional of max lag length.

Particularly, with large number of max lag length and variables (for ex: I have 6 variables and choose the maximum lag length of 8), the STATA takes too long to find out the optimal lag length (the lag permutations is approx 400,000). In fact, I let it run a whole day but it has never responded. I do try many times by using different laptops and desktops, but unfortunately it's still not working.

However, when I use Eviews, it just takes few seconds to give me a result.

So I do not understand why ARDL package cannot work with large number of lag permutations? And what should I do to fix this problem? I prefer using STATA to Eviews, since STATA has more functions which I need.

Many thanks for your consideration!

Best,
Ngan

[Sebastian Kripfganz]

Maruf Ahmed:
Please understand that I do not have the time to give such detailed individual advice. Please consult the ardl help file which contains a lot of information about how to use the command and the available options.