ARDL Error Correction Model Interpretation

[Harkieran Hayer]

I am trying to estimate the relationship between real currency per capita (dependent variable) and direct tax to gdp ratio, gdp per capita, interest rate and inflation (independent variables). I have gathered quarterly data for the period 2001Q2 - 2018Q4.

After finding that my variables were stationary in the first difference and that there is presence of cointegration I estimated an ARDL error correction model as shown below. I have also shown the bounds test which shows a co integrating relationship I believe.

ardl lnrealCURPC lngdppercapita interestrate infld directtaxratiod, lag(. . . . .) m
> axlag(3 3 3 3 3) aic ec

ARDL(2,0,0,0,3) regression

Sample: 2002q3 - 2018q4                         Number of obs     =         66
                                                R-squared         =     0.3891
                                                Adj R-squared     =     0.2909
Log likelihood =  108.59474                     Root MSE          =     0.0507

---------------------------------------------------------------------------------
  D.lnrealCURPC |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
ADJ             |
    lnrealCURPC |
            L1. |  -.6820688   .1177544    -5.79   0.000    -.9179592   -.4461783
----------------+----------------------------------------------------------------
LR              |
 lngdppercapita |   1.043669   .0353481    29.53   0.000      .972858     1.11448
   interestrate |   .0020574   .0098193     0.21   0.835     -.017613    .0217279
          infld |  -.6836962   .5265204    -1.30   0.199    -1.738443    .3710509
directtaxratiod |  -33.38411    10.4536    -3.19   0.002    -54.32518   -12.44304
----------------+----------------------------------------------------------------
SR              |
    lnrealCURPC |
            LD. |   .4115564   .1239233     3.32   0.002     .1633082    .6598047
                |
directtaxratiod |
            D1. |   8.462358   13.72885     0.62   0.540    -19.03983    35.96455
            LD. |   11.06642   17.31682     0.64   0.525    -23.62333    45.75618
           L2D. |   40.31898   17.03499     2.37   0.021     6.193793    74.44416
                |
          _cons |  -.9017556   .2910219    -3.10   0.003    -1.484742   -.3187687
---------------------------------------------------------------------------------

estat ectest

Pesaran, Shin, and Smith (2001) bounds test

H0: no level relationship                                        F =     6.804
Case 3                                                           t =    -5.792

Finite sample (4 variables, 66 observations, 4 short-run coefficients)

Kripfganz and Schneider (2018) critical values and approximate p-values

   | 10%              | 5%               | 1%               | p-value         
   |    I(0)     I(1) |    I(0)     I(1) |    I(0)     I(1) |    I(0)     I(1)
---+------------------+------------------+------------------+-----------------
 F |   2.531    3.695 |   3.016    4.297 |   4.113    5.637 |   0.000    0.002
 t |  -2.539   -3.639 |  -2.865   -4.009 |  -3.512   -4.728 |   0.000    0.001

do not reject H0 if
    both F and t are closer to zero than critical values for I(0) variables
      (if p-values > desired level for I(0) variables)
reject H0 if
    both F and t are more extreme than critical values for I(1) variables
      (if p-values < desired level for I(1) variables)

I am just wondering how I interpret the ARDL output and as I am looking for the relationship between real currency per capita (dependent variable) and the independent variables what would my equation be? I do not know whether I should differentiate between a short run specification and a long run specification and if I do that, why is there only one independent variable specified in the short run.

Thanks.

[Sebastian Kripfganz]

Please see my 2018 London Stata Conference presentation, in particular slide 16, on how to interpret the coefficients:

• Kripfganz, S. and D. C. Schneider (2018). ardl: Estimating autoregressive distributed lag and equilibrium correction models. Proceedings of the 2018 London Stata Conference.
• ARDL: updated Stata command for the estimation of autoregressive distributed lag and error correction models

[Harkieran Hayer]

Please see my 2018 London Stata Conference presentation, in particular slide 16, on how to interpret the coefficients:

• Kripfganz, S. and D. C. Schneider (2018). ardl: Estimating autoregressive distributed lag and equilibrium correction models. Proceedings of the 2018 London Stata Conference.
• ARDL: updated Stata command for the estimation of autoregressive distributed lag and error correction models

Thanks. From reading through your presentation I understand how to interpret the coefficients better.

One thing I am still having trouble with is applying the general form specified on slide 12 so that I can specify my model. In particular the "−α(y_t−1 − θx_t)" part. I understand that the equation shows the negative speed of adjustment coefficient multiplied by the lag of the dependent variable minus the long run coefficient. However, what do you do when there is more than one independent variable?

[Sebastian Kripfganz]

One thing I am still having trouble with is applying the general form specified on slide 12 so that I can specify my model. In particular the "−α(y_t−1 − θx_t)" part. I understand that the equation shows the negative speed of adjustment coefficient multiplied by the lag of the dependent variable minus the long run coefficient. However, what do you do when there is more than one independent variable?

x_t can be a vector cotaining more than one independent variable. You would then get −α(y_t−1 − θ₁x_1t − θ₂x_2t − ...). The ardl command also allows you to specify variables that do not appear in the long-run relationship but only in the short-run part with option exog().

[Harkieran Hayer]

x_t can be a vector cotaining more than one independent variable. You would then get −α(y_t−1 − θ₁x_1t − θ₂x_2t − ...). The ardl command also allows you to specify variables that do not appear in the long-run relationship but only in the short-run part with option exog().

So for the output shown in my original post would my equation be;

∆lnrealCURPC = -0.90 - 0.68(Y_t-1 - 1.04*lngdppercapita - 0.0021*interestrate - (-0.68)*infld - (-33.38)directtaxratiod) + 0.41*∆lnrealCURPC_t-1 + 8.46*∆direxttaxratiod + 11.1*∆directtaxratiod_t-1 + 40.32*∆directtaxratiod<sub>t-2

Thanks for your help and sorry if these seem like simple questions.</sub>

[Sebastian Kripfganz]

That is correct.

[misso ouhibi]

Hello Sebastisan,

I want to ask you about the code of the ARDL-ECM model (estimation after the bound test) please

[Sebastian Kripfganz]

You just need to add either the ec or the ec1 option to the ardl command in order to get the EC representation.

[misso ouhibi]

ok thank you but what is the difference between ec and ec1

[misso ouhibi]

To estimate the ARDL-ECM model this code is correct?

ardl dep indep indep2 indep3, lags( ) ec
ardl dep indep indep2 indep3,lags( ) restore(ecreg)
regress

[Sebastian Kripfganz]

Please see slides 12 and following in my 2018 London Stata Conference presentation:

• Kripfganz, S., and D. C. Schneider (2018). ardl: Estimating autoregressive and distributed lag and equilibrium correction models. Proceedings of the 2018 London Stata Conference.

[misso ouhibi]

slide number 41 which there is estimation right?