[Sebastian Kripfganz]

Thanks for raising this issue. We are aware of the Narayan critical values for smaller sample sizes and we might implement them in a future version of the ardl command. These finite-sample critical values are a decreasing function of the number of observations and they tend to approach the Pesaran-Shin-Smith asymptotic critical values in the limit. Based on the latter, one would therefore too often classify the regressors to be I(1). Without having made my own investigation, it can be expected that the use of appropriate finite-sample critical values improves the size and power properties of the bounds test.

[Georg Neuschaeffer]

Thank you for your answer and the useful comment.

[Georg Neuschaeffer]

Hi, just another short question on the interpretation of the output: Am I correct that the adjustment coefficient (ADJ) in the ECM, which shows deviations from the long-run equilibrium, is the same as the equilibrium correction term. I am a bit confused because different sources refer to it differently. Thanks for any help!

[Sebastian Kripfganz]

Consider the following error-correction model:
\[ \Delta y_t = \alpha (y_{t-1} - \beta x_{t-1}) + \gamma_1 \Delta y_{t-1} + \gamma_2 \Delta x_t + \gamma_3 \Delta x_{t-1} + u_t \]

The adjustment coefficient is \(\alpha\) and the equilibrium correction term is \((y_{t-1} - \beta x_{t-1})\).

[Georg Neuschaeffer]

Thank you for the clarification Sebastian.

For the critical values the paper of Narayan (2005) is very helpful, he provides a proper lists of critical values with lower T for the different cases. I will use them instead of the ones from Pesaran et al.(2001).

Narayan, Paresh (2005). The savings and investment for China: evidence from cointegrating tests. Applied Economics 37 (17):1979-1990.

[valérie orozco]

I tried Microfit software (demo version 5.01) developped by Pesaran and tried to compare with my Stata results.I don't have the same F-stat value, neither same indicated bounds. Do you know why? Is it related to Georg' point? Or did I miss something?

For example, for the same model (and estimated parameters are the same) in Stata :

Pesaran/Shin/Smith (2001) Bounds Test
H0: no levels relationship             F =  6.395
                                       t = -3.448

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_1 |   4.04    4.78 |   4.94    5.73 |   5.77    6.68 |   6.84    7.84
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_1 |  -2.57   -2.91 |  -2.86   -3.22 |  -3.13   -3.50 |  -3.43   -3.82
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

in Microfit :

Testing for existence of a level relationship among the variables in the ARDL model
*******************************************************************************
 F-statistic  95% Lower Bound  95% Upper Bound  90% Lower Bound  90% Upper Bound
   11.8907         12.6082         12.6082         10.3543         10.3543
 
 W-statistic  95% Lower Bound  95% Upper Bound  90% Lower Bound  90% Upper Bound
   11.8907         12.6082         12.6082         10.3543         10.3543
*******************************************************************************

[Sebastian Kripfganz]

If you have the same estimated coefficients both with the ardl command and Microfit, then also the F-statistic and the corresponding asymptotic critical values should be the same. I actually do not where the critical values reported by Microfit are taken from. It would be helpful, if somebody could provide some insight about the critical values used there. The ones reported by the ardl command are taken from Pesaran, Shin, and Smith (2001). Bounds Testing Approach to the Analysis of Level Relationships. Journal of Applied Econometrics 16, 289-326.

Did you check that the critical values from Microfit also refer to case 3 (unrestricted intercept and no trend)?

What strikes me, though, is that your Microfit critical values are the same for the upper and lower bound. This implies that those critical values refer to a model with k=0 long-run forcing variables, while the ardl output provides critical values for k=1. That does not make sense if both models are the same. Without seeing the full estimation output I cannot say more at this point.

[valérie orozco]

Thank you Sebastian for your answer. I'm not a Microfit specialist at all. I'm just continuing a work of some colleagues that were using it and my goal is to use Stata, which is a reproducible software (Microfit is a point and click soft). But I have to reproduce previous results and it is not very easy. The Microfit output I put in my previous message was the results obtained by my colleagues. But when I tried it, I don't obtain the same one. Here I really put what I obtain in Stata and Microfit (again F stat are different, but not so far).

Microfit results :

 
                   Autoregressive Distributed Lag Estimates                  
           ARDL(2,0) selected based on Akaike Information Criterion          
*******************************************************************************
 Dependent variable is LNY
 50 observations used for estimation from 1963 to 2012
*******************************************************************************
 Regressor              Coefficient       Standard Error         T-Ratio[Prob]
 LNY(-1)                  .26434             .13493             1.9592[.056]
 LNY(-2)                  .31955             .12454             2.5658[.014]
 LNX                     .067817            .020605             3.2913[.002]
 C                        1.8713             .53340             3.5082[.001]
*******************************************************************************
 R-Squared                     .98785   R-Bar-Squared                   .98706
 S.E. of Regression           .026446   F-Stat.    F(3,46)      1246.4[.000]
 Mean of Dependent Variable    5.1312   S.D. of Dependent Variable      .23244
 Residual Sum of Squares      .032173   Equation Log-likelihood       112.7696
 Akaike Info. Criterion      108.7696   Schwarz Bayesian Criterion    104.9456
 DW-statistic                  2.0606
*******************************************************************************
 
 
    Error Correction Representation for the Selected ARDL Model        
           ARDL(2,0) selected based on Akaike Information Criterion          
*******************************************************************************
 Dependent variable is dLNY
 50 observations used for estimation from 1963 to 2012
*******************************************************************************
 Regressor              Coefficient       Standard Error         T-Ratio[Prob]
 dLNY1                   -.31955             .12454            -2.5658[.014]
 dLNX                    .067817            .020605             3.2913[.002]
 ecm(-1)                 -.41611             .12067            -3.4483[.001]
*******************************************************************************
 List of additional temporary variables created:
 dLNY = LNY-LNY(-1)
 dLNY1 = LNY(-1)-LNY(-2)
 dLNX = LNX-LNX(-1)
 ecm = LNY   -.16298*LNX   -4.4971*C
*******************************************************************************
 R-Squared                     .39903   R-Bar-Squared                   .35984
 S.E. of Regression           .026446   F-Stat.    F(3,46)     10.1810[.000]
 Mean of Dependent Variable   .014236   S.D. of Dependent Variable     .033054
 Residual Sum of Squares      .032173   Equation Log-likelihood       112.7696
 Akaike Info. Criterion      108.7696   Schwarz Bayesian Criterion    104.9456
 DW-statistic                  2.0606
*******************************************************************************
 
Estimated Long Run Coefficients using the ARDL Approach            
           ARDL(2,0) selected based on Akaike Information Criterion          
*******************************************************************************
 Dependent variable is LNY
 50 observations used for estimation from 1963 to 2012
*******************************************************************************
 Regressor              Coefficient       Standard Error         T-Ratio[Prob]
 LNX                      .16298           .0068402            23.8264[.000]
 C                        4.4971            .036804           122.1924[.000]
*******************************************************************************
 
 
 
Testing for existence of a level relationship among the variables in the ARDL model
*******************************************************************************
 F-statistic  95% Lower Bound  95% Upper Bound  90% Lower Bound  90% Upper Bound
    6.9345          5.2102          6.0627          4.2070          4.9431
 
 W-statistic  95% Lower Bound  95% Upper Bound  90% Lower Bound  90% Upper Bound
   13.8691         10.4204         12.1254          8.4140          9.8863
*******************************************************************************
 If the statistic lies between the bounds, the test is inconclusive. If it is
 above the upper bound, the null hypothesis of no level effect is rejected. If
 it is below the lower bound, the null hypothesis of no level effect can't be
 rejected. The critical value bounds are computed by stochastic simulations
 using 20000 replications.

So it is written that the critical value bounds are calculted by simulations.

It seems it is a restricted intercept case. In Stata, I estimate ARDL in level and then the ARDL ECM form both with restriction (to check the intercept coefficient) and without restriction (I read the F-test has to be those from an unrestricted case).

/*level ardl*/
ardl lnY lnX, aic
 
ARDL regression
Model: level
 
Sample:       1963 -       2012
Number of obs  = 50
Log likelihood = 112.7696
R-squared      = .98784778
Adj R-squared  = .98705524
Root MSE       = .02644628
------------------------------------------------------------------------------
         lnY |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         lnY |
         L1. |   .2643447   .1349268     1.96   0.056    -.0072488    .5359382
         L2. |   .3195464   .1245427     2.57   0.014     .0688549    .5702378
              |
         lnX |   .0678166   .0206047     3.29   0.002     .0263414    .1092918
       _cons |   1.871289   .5333987     3.51   0.001     .7976133    2.944965
------------------------------------------------------------------------------
 
/*ECM ARDL (restriction on the constant)*/
ardl lnY lnX, ec restricted    aic
 
ARDL regression
Model: ec
 
Sample:       1963 -       2012
Number of obs  = 50
Log likelihood = 112.7696
R-squared      = .39903176
Adj R-squared  = .35983818
Root MSE       = .02644628
 
------------------------------------------------------------------------------
     D.lnY |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ        |
        lnY |
        L1. |   -.416109    .120671    -3.45   0.001    -.6590071   -.1732108
-------------+----------------------------------------------------------------
LR         |
        lnX |   .1629779   .0068402    23.83   0.000     .1492092    .1767466
    _cons |   4.497114   .0368035   122.19   0.000     4.423032    4.571195
-------------+----------------------------------------------------------------
SR         |
        lnY |
        LD. |  -.3195464   .1245427    -2.57   0.014    -.5702378   -.0688549
------------------------------------------------------------------------------
 
. ardl, noctable btest
 
ARDL regression
Model: ec
 
Sample:       1963 -       2012
Number of obs  = 50
Log likelihood = 112.7696
R-squared      = .39903176
Adj R-squared  = .35983818
Root MSE       = .02644628
 
 
Pesaran/Shin/Smith (2001) Bounds Test
H0: no levels relationship             F =  13.422
 
Critical Values (0.1-0.01), F-statistic, Case 2
 
      | [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_1 |   3.02    3.51 |   3.62    4.16 |   4.18    4.79 |   4.94    5.58
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 ARDL regression
 
/*ECM ARDL (without any restriction)*/
ardl lnY lnX, ec     aic
 
ARDL regression
Model: ec
 
Sample:       1963 -       2012
Number of obs  = 50
Log likelihood = 112.7696
R-squared      = .39903176
Adj R-squared  = .35983818
Root MSE       = .02644628
 
------------------------------------------------------------------------------
       D.lnY |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
         lnY |
         L1. |   -.416109    .120671    -3.45   0.001    -.6590071   -.1732108
-------------+----------------------------------------------------------------
LR           |
         lnX |   .1629779   .0068402    23.83   0.000     .1492092    .1767466
-------------+----------------------------------------------------------------
SR          |
         lnY |
         LD. |  -.3195464   .1245427    -2.57   0.014    -.5702378   -.06885
     _cons |   1.871289   .5333987     3.51   0.001     .7976133    2.944965
------------------------------------------------------------------------------
  
. ardl, noctable btest
 
ARDL regression
Model: ec
 
Sample:       1963 -       2012
Number of obs  = 50
Log likelihood = 112.7696
R-squared      = .39903176
Adj R-squared  = .35983818
Root MSE       = .02644628
 
 
Pesaran/Shin/Smith (2001) Bounds Test
H0: no levels relationship             F =  6.395
                                       t = -3.448
 
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_1 |   4.04    4.78 |   4.94    5.73 |   5.77    6.68 |   6.84    7.84
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_1 |  -2.57   -2.91 |  -2.86   -3.22 |  -3.13   -3.50 |  -3.43   -3.82
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

So F-test in Stata is 6.395 whereas it was 6.9345 in Microfit.

[Sebastian Kripfganz]

Dear Valérie, the full output helps to shed some light on this matter. Indeed, it seems that your initial guess is correct that the differences in the critical values are related to Georg's comment. The critical values reported by Microfit come very close to those reported by Narayan (2005) for case 3 with 50 observations and 1 long-run forcing variable. In that respect, these small-sample critical values are probably better suited for inference than the asymptotic critical values reported by the ardl command. As mentioned earlier, we are planning to implement those finite-sample critical values as well, as time permits.

If at some time you come across a source for the critical values displayed by Microfit, I would be happy if you could let us know.

[valérie orozco]

Thank you Sebastian.
I don't find the critical values of Microfit so close from Narayan (2005).
At 10% the Microfit critical bounds are 4.2070 and 4.9431. In Narayan(2005) for k=1 and n=50 they are 3.177 and 3.653. In Pesaran and al. (2001), like in the ardl command, they are 4.04 and 4.78.
At 5% the Microfit critical bounds are 5.2102 and 6.0627. In Narayan(2005) for k=1 and n=50 they are 3.860 and 4.440. In Pesaran and al. (2001), like in the ardl command, they are 4.94 and 5.73.
So Microfit critical bounds are closer from Pesaran and al. (2001) than from Narayan(2005).
And how can we explain the differences of the F-test? (here 6.395 in Stata, 6.9345 in Microfit).

[Sebastian Kripfganz]

The Microfit critical values are close to those of Narayan (2005) for case 3. You are referring to case 2 instead.

Admittedly, I did not look at the F-statistic itself. I do not know why it differs between Microfit and Stata. I just had a brief look into the ardl code but could not find a mistake in the computation of the F-statistic. Again, I do not know how the F-statistic is computed by Microfit.

[Georg Neuschaeffer]

Dear Valerie, I think you have mistakenly looked up the wrong critical bounds from the Narayan-tables. The critical bounds you referring to are the ones for Case II (Narayan 2005, p.1987). The correct ones should be from Case III (Narayan 2005, p. 1988). At 10% and for k=1 and T=50 they are 4.190 and 4.940. For 5% the bounds are 5.220 and 6.070 for the K01 T=50 case. Both bounds are quite similar to the ones from the Microfit codes.

[valérie orozco]

Sebastian, Georg, thank you for your answers and help. OK for the Narayan case III. For the F test, I'm still surprised they are different (I also had a look inside the ardl command and didn't find anything that strikes me). If someone has an idea...

[Georg Neuschaeffer]

Sebastian, regarding my post about the Error Correction Term I have two questions:
1) In Valéries output the "adjustment coefficient" of the Stata output is equal to the ecm(-1) from the Microfit output with a rounding difference. Are both having the same interpretation? I am a bit confused on this matter with regard to your earlier answer.
2) Is it possible to save the terms of the ecm(-1)/adjustment coefficient in Stata as a variable, because I need it for a Granger Causality test. See for example Narayan and Smyth (2004).

[Sebastian Kripfganz]

1) What Microfit refers to as ecm(-1) is the coefficient of the 1-period-lagged error-correction term. That is the same as the "adjustment coefficient" in the Stata output.

2) You do not want to save the coefficient as a variable because this coefficient is just a constant. Instead you want to save the error-correction term itself as a new variable (see my earlier post about the distinction). Based on Valérie's Stata output you could do this as follows:

/* ECM ARDL (restriction on the constant) */
ardl lnY lnX, ec restricted aic
gen ect = lnY - _b[LR:lnX] * lnX - _b[LR:_cons]

/* ECM ARDL (without any restriction) */
ardl lnY lnX, ec aic
gen ect = lnY - _b[LR:lnX] * lnX

Notice that this is the error-correction term for period t. To obtain the error-correction term for period t-1, that is \(ECT_{t-1}\) used by Narayan and Smyth (2004), you simply have to take the first lag: L.ect

We will put the computation of the error-correction term as a new variable on our to-do list for future improvements of the ardl package. Thanks for this suggestion.