[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.

[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]

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!

[Georg Neuschaeffer]

Thank you for your answer and the useful comment.

[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]

Hello!

Regarding the bounds testing of the ARDL Model I have the following issue:

The critical values of the bounds provided by the lists of Pesaran, Shin and Smith (2001) and also by the syntax fbounds(casenum) are based on stochastic simulations using T=1000 and 40.000 replications (Pesaran et al. 2001, p.301). Narayan and Smyth (2004, p. 2082-2083) however calculate critical bound values based on a smaller T, which are different to the values of Pesaran et al. (2001).

My question is if it is possible to calculate also bounds with Stata using a smaller T OR what kind o f limitation arises using the values provided by Pesaran et al. (2001).

I am thankful for any help.


Pesaran, M. H., Shin, Y. and Smith, R. J. (2001) Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16, 289–326.

Paresh Kumar Narayan and Russell Smyth (2004) Crime rates, male youth unemployment and real income in Australia: evidence from Granger causality tests, Applied Economics, 36:18, 2079-2095.

[mahsa mesgarani]

Hi everyone. I've estimated an ARDL model and when I assessed the ecm(-1), the value I hot was more than 2 ! Does anybody knows what's the problem?

[Sebastian Kripfganz]

This is already on our list for potential future improvements but I cannot make a promise that (or when) we will eventually implement it.

[valérie orozco]

OK, thanks again. Do you plan to allow an ECM specification with x(t-1) in the LR relationship with a only zero lag of this regressor in the levels equation in a next version of ardl?

[Sebastian Kripfganz]

The probably unsatisfactory answer is: With the current version of ardl you cannot obtain an ECM specification with \(x_{t-1}\) in the long-run relationship if you have only zero lags of this regressor in the levels equation. With the option minlag1, the ardl command enforces to have at least one lag of each regressor which, however, might not be the optimal lag number.

See again the help file:

You can estimate (ECF0) using option ec and (ECF1) using the option combination ec and minlag1. Note that (ECF0) and (ECF1) feature identical coefficient expressions in the long-run relationship. Despite this fact, you may get different results for the long-run coefficients for (ECF0) and (ECF1) if you use automatic optimal lag selection. This is because option minlag1 will enforce that all regressors have at least one lag. Since ardl does not require a minimum regressor lag of 1 for (ECF0) but it does for (ECF1), the results will differ if the optimal lag for at least one regressor in (ECF0) is zero. If you want to consider models that have or that potentially have zero lag regressors, use option ec to estimate (ECF0).

Note that it is possible to estimate a model with zero lag regressors and express it in (ECF1). The resulting covariance matrix of the estimates, however, would be singular. Since this is an awkward result, this feature is not currently implemented in ardl.

The reason for your empirical differences is precisely that in your first specification the regressor lnx enters with zero lags into the levels equation (automatically chosen optimal lag order by selection criteria), while in your second specification this regressors enters with one lag into the levels equation (which is not optimal according to the selection criteria).

[valérie orozco]

Thank you Sebastian. I agree with your now and I'm ok theoretically!
Empirically, no problem to obtain your specification (with lnx_t in the LR part), it corresponds to results posted earlier.

. ardl lny lnx, ec

ARDL regression
Model: ec

Sample:       1963 -       2011
Number of obs  = 49
Log likelihood = 109.90187
R-squared      = .38612073
Adj R-squared  = .34519545
Root MSE       = .02680264

------------------------------------------------------------------------------
       D.lny |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
         lny |
         L1. |  -.4018656   .1205985    -3.33   0.002    -.6447635   -.1589677
-------------+----------------------------------------------------------------
LR           |
         lnx |   .1576984   .0070317    22.43   0.000     .1435359     .171861
-------------+----------------------------------------------------------------
SR           |
         lny |
         LD. |  -.3199929   .1265193    -2.53   0.015    -.5748158   -.0651701
             |
       _cons |   1.806713   .5328956     3.39   0.001     .7334061     2.88002
------------------------------------------------------------------------------

But I still wonder how to obtain the other specification (with ln_x(t-1) in the LR part). I tried the minlag1 option of ardl but I don't succeed to obtain all expected coefficients :

eststo ECminlag1 :  ardl lny lnx, ec minlag1
eststo ECminlag1_lag2 :  ardl lny lnx, ec minlag1 lags(2 .)
eststo ECminlag1_lags21 :  ardl lny lnx, ec minlag1 lags(2 1)

. esttab ECminlag1  ECminlag1_lag2  ECminlag1_lags21

------------------------------------------------------------
                      (1)             (2)             (3)   
                    D.lny           D.lny           D.lny   
------------------------------------------------------------
ADJ                                                         
L.lny              -0.703***       -0.562***       -0.472**
                  (-5.15)         (-3.66)         (-3.39)   
------------------------------------------------------------
LR                                                          
L.lnx               0.155***        0.154***        0.157***
                  (35.66)         (28.87)         (27.22)   
------------------------------------------------------------
SR                                                          
D.lnx              -0.254          -0.175         -0.0708   
                  (-1.77)         (-1.19)         (-0.57)   

LD.lny                             -0.245          -0.320**
                                  (-1.85)         (-2.72)   

_cons               3.181***        2.548***        2.134**
                   (5.16)          (3.69)          (3.42)   
------------------------------------------------------------
N                      49              49              51   
------------------------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001

The third estimation (ECminlag1_lags21) is the closest but with a different adjustment parameter (-0.472 instead of -0.402) and a very different D.lnx (-0.0708 instead of 0.063).

I suspect maybe automatic optimal lag selection to be source of differences but I'm still puzzled not to be able to find consistent coefficients in line with the ARDL in level.

[Sebastian Kripfganz]

There is nothing wrong with your derivation of the ECM. All I am saying is that there are two different ways how we can formulate the long-run relationship in the ECM. Take your last line and notice that
\[(\gamma_1 - 1 + \gamma_2) \left( - \frac{\beta_1}{1 - \gamma_1 - \gamma_2} \ln x_{t-1} \right) + \beta_1 \Delta \ln x_t = \beta_1 \ln x_{t-1} + \beta_1 \Delta \ln x_t = \beta_1 \ln x_t = (\gamma_1 - 1 + \gamma_2) \left( - \frac{\beta_1}{1 - \gamma_1 - \gamma_2} \ln x_t \right)\]

This shows that you can rewrite this model equivalently with having \(\ln x_t\) instead of \(\ln x_{t-1}\) in the long-run relationship. In the latter case, the corresponding short-run component disappears. This is how the ardl command estimates the ECM by default.

[valérie orozco]

Hi Sebastian, I understand your argument but writting the ECM ARDL from the basic ARDL give me a specification with dx(t) :

So I have the D.x_t term.
Where am I wrong?
Moreover I read a published paper having only x_t (no lag) in the basic ARDL and a D.x_t term in the conditional ECM ARDL.
Again, thank you for your help.

[Sebastian Kripfganz]

There would be such a term in the SR part if there was at least one lag of that regressor in the model in levels. In your case, there lnx appears only contemporaneously without any lag. Essentially, we have \(b_1 = 0\).

[valérie orozco]

Thank you for your answer Sebastian.
I carefully read the help of the ardl command but maybe didn't get everything. In the section you mentionned, your model (ECF0) is written as
dy(t) = c - (1 - a_1) * [y(t-1) - (b_0 + b_1) / (1 - a_1) * x(t) ] - b_1 * dx(t) + e(t)
and so there is a dx(t) term in th SR part, isn't it?