[Mahana Noorma]

Dear Sebastian,
Dear all,

I applied the bounds testing approach (developed by Pesaran et al. (2001)) to a specific case and I have a question regarding the long-run coefficients.
According to Giles’s blog (2013: http://davegiles.blogspot.ch/2013/06...nds-tests.html), and further papers on the bounds testing approach, there is a correspondence between the coefficients of the long-run equation and those of the "unconstrained error correction model":

Long-run equation
y_t = α₀ + α₁x_1t + α₂x_2t + v_t

Unconstrained ECM:
Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + Σδ_kΔx_2t-k + θ₀y_t-1 + θ₁x_1t-1 + θ₂ x_2t-1 + e_t

Source: http://davegiles.blogspot.ch/2013/06/ardl-models-part-ii-bounds-tests.html

As explained by Giles, the long-run coefficients can be extracted from the unconstrained ECM:
the long-run coefficients for x₁ and x₂ are -(θ₁/ θ₀)= α₁ and -(θ₂/ θ₀)= α2 respectively.

However, according to my results, this “coefficients correspondence” is not realized and I am unable to understand why.
I already posted this question a few months ago but since then, I read the above mentioned blog of Giles and a few papers in which the bounds testing procedure was applied. Most of the authors mention this “coefficients correspondence”. Thus, I am confused and would need some further help.

Here are my results:

Unconstrained ECM

ARDL(1, 0, 0, 0, 0, 0, 0, 0)
Dependent Variable: D(Y)
Sample (adjusted): 1973 2013
Number of obs = 41 after adjustments
Variable	Coef.	Std. Err.	t	P>|t|
D(Y(-1))	-0.149489	0.118913	-1.257125	0.2208
D(X1)	-0.06227	0.075834	-0.821134	0.4197
D(X2)	-1.641474	0.493317	-3.32742	0.0028
D(X3)	0.135946	0.273259	0.4975	0.6234
D(X4)	7.858689	2.3929	3.28417	0.0031
D(X5)	-0.468855	0.108366	-4.326576	0.0002
D(X6)	-0.011257	0.01183	-0.951626	0.3508
D(X7)	-101.6557	142.0518	-0.715624	0.4811
Y(-1)	-0.945555	0.261508	-3.615784	0.0014
X1(-1)	-0.036331	0.012209	-2.975747	0.0066
X2(-1)	-0.632525	0.293756	-2.15323	0.0416
X3(-1)	-0.149936	0.139885	-1.071849	0.2944
X4(-1)	-2.337494	1.939226	-1.205375	0.2398
X5(-1)	-0.291691	0.10937	-2.667013	0.0135
X6(-1)	-0.012298	0.015918	-0.772534	0.4473
X7(-1)	-66.49316	154.2351	-0.431116	0.6702
C	28.02979	7.87529	3.559208	0.0016
R-squared	0.952677	Mean dependent var		-0.130624
Adjusted R-squared	0.921128	S.D. dependent var		2.622106
S.E. of regression	0.736398	Akaike info criterion		2.519657
Sum squared resid	13.01475	Schwarz criterion		3.230162
Log likelihood	-34.65296	Hannan-Quinn criter.		2.778384
F-statistic	30.19684	Durbin-Watson stat		2.025592
Prob(F-statistic)	0

Long-run equation

Long-run equation
Dependent Variable: Y
Sample (adjusted): 1971 2013
Number of obs = 43 after adjustments
Variable	Coef.	Std. Err.	t	P>|t|
C	32.37835	8.3102	3.896217	0.0004
X1	-0.032359	0.012787	-2.530517	0.016
X2	-1.013083	0.25197	-4.020655	0.0003
X3	-0.136791	0.156776	-0.872522	0.3889
X4	2.914955	2.137927	1.363449	0.1814
X5	-0.389945	0.165139	-2.36131	0.0239
X6	-0.030134	0.010307	-2.923763	0.006
X7	-294.3322	158.4416	-1.85767	0.0716
R-squared	0.571425	Mean dependent var		1.861392
Adjusted R-squared	0.48571	S.D. dependent var		1.955403
S.E. of regression	1.402299	Akaike info criterion		3.680344
Sum squared resid	68.82546	Schwarz criterion		4.008009
Log likelihood	-71.12739	Hannan-Quinn criter.		3.801176
F-statistic	6.666562	Durbin-Watson stat		2.387055
Prob(F-statistic)	0.000049	Wald F-statistic		4.324163
Prob(Wald F-statistic)	0.001537

For example, if I want to get the long-run coefficient of the variable X1: - (-0.036331)/( -0.945555)= -0.03842 is not equal to -0.032359
I can not find what I am doing wrong…and would appreciate some help.

Thank you very much.
Kind regards

[Sebastian Kripfganz]

Dear Sebastian:
I tried run "net install ardl, from(http://www.kripfganz.de/stata/)" in my STATA-14 but i always get the error messages as followed:

net install ardl, from(http://www.kripfganz.de/stata/)
connection timed out -- see help r(2) for troubleshooting
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).


Most was due to some restrictions to Internet access. what should i do?

Much appreciated.

I have sent you a private message here on Statalist.

[John heric]

Dear Sebastian:
I tried run "net install ardl, from(http://www.kripfganz.de/stata/)" in my STATA-14 but i always get the error messages as followed:

net install ardl, from(http://www.kripfganz.de/stata/)
connection timed out -- see help r(2) for troubleshooting
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).


Most was due to some restrictions to Internet access. what should i do?

Much appreciated.

[Daniel Schneider]

Dear Louison,

-ardl- and -nardl- are separate projects and I do not see any merging of functionality happening in the future. Merging functionality / code would require a substantial work effort and it is probably better to keep them as separate, tested entities. But many thanks for pointing towards the -nardl- command and for your suggestion.

Best,
Daniel

[Louison Cahen-Fourot]

Dear Daniel,

Thanks a lot for this quick and neat reply. My bad for not having kept in mind the point on the non-standard distributions from the Pesaran et al. paper.

As a side thought, if I may make a suggestion for further development of the ardl command: It would be nice to include an option to run nonlinear asymetric ardl models such as proposed by Shin et al. (2014). I am aware of the command nardl by Marco Sunder, I have used it quite a lot and it works fairly well, but as far as I know it doesn't allow for non-zero threshold in partial sum decompositions. That would be a nice improvement, and it will would be great to have it all in the same place within the ardl command. In the meantime, your command works well with hand-made partial sum decompositions.

Many thanks again for your command and your time,

Best,

Louison

[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

[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

[Lajos Konok]

Thank you Sebastian

[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]

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!

[Muhammad Azam]

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

[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).

[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

[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,
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