[wanhaiyou]

The difficulty here is that the optimal lag order for ln_inc is zero, but you are forcing it to appear in the first lag in the EC representation (option ec1). This is achieved by effectively imposing a constraint as follows:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1
constraint 1 L.ln_inc = D.ln_inc
cnsreg D.ln_consump L.ln_consump L.ln_inc L.ln_inv D.ln_inc L(0/3)D.ln_inv if e(sample), constraints(1)

The long-run coefficients in the ardl output are a nonlinear coefficient combination:

nlcom (- _b[L.ln_inc] / _b[L.ln_consump]) (- _b[L.ln_inv] / _b[L.ln_consump])

If you use the ec instead of the ec1 option, this complication does not arise and you can replicate the results directly with regress:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec
reg D.ln_consump L.ln_consump ln_inc ln_inv L(0/3)D.ln_inv if e(sample)
nlcom (- _b[ln_inc] / _b[L.ln_consump]) (- _b[ln_inv] / _b[L.ln_consump])

Dear professor, I have another question. As you mentioned in the previous post, when the optimal lag order for x is zero, and we still estimate the model in the EC representation (option ec1).
At this time, this is achieved by effectively imposing a constraint. However, some routines, such as sqreg (quantile regression), the option constraints(1) is not allowed. That is, we cannot
estimate this model with constraint. Do you have any suggestions?

Greats thank!

wanhai

[wanhaiyou]

The optimal lag orders are found based on the sample which sets aside the first 10 observations. If you set aside fewer initial observations, it is possible that you get different optimal lag orders. For comparison of the models with selection criteria (AIC, BIC), the same sample must be used.

Once you have chosen the optimal lag order, you could then use all observations as in your second code for the subsequent analysis.

I see, great thanks for your timely help!

Bests,
wanhai

[Sebastian Kripfganz]

The optimal lag orders are found based on the sample which sets aside the first 10 observations. If you set aside fewer initial observations, it is possible that you get different optimal lag orders. For comparison of the models with selection criteria (AIC, BIC), the same sample must be used.

Once you have chosen the optimal lag order, you could then use all observations as in your second code for the subsequent analysis.

[wanhaiyou]

In your first code, you have set

local maxlags = 10

This sets aside the first 10 observations in the estimation sample for computing the lags, even if the optimal lag order is smaller than 10.

In your second code, you pre-specify the lag order to be (1, 1). Here, only 1 observation is set aside.

Great thanks! Yes, I understand this, but I am question about which one is "right".

In fact, although the max lag is set to 10 for choosing the optimal lag, the sample used to estimate should based on the final optimal lag step. Is it right?

Thanks for your time!

Bests,
wanhai

[Sebastian Kripfganz]

In your first code, you have set

local maxlags = 10

This sets aside the first 10 observations in the estimation sample for computing the lags, even if the optimal lag order is smaller than 10.

In your second code, you pre-specify the lag order to be (1, 1). Here, only 1 observation is set aside.

[wanhaiyou]

The equations on page 12 are derived from the equation on page 5. With ec1, when the lag order of ln_inc is zero, its first lag is included in the error-correction form. However, in terms of the coefficients of the equation on page 5, this first lag has a coefficient equal to zero. Because this coefficient equals zero, we need the restriction for the coefficients on page 12, which I mentioned in my previous post. Put differently, for your model there is a total of 8 coefficients in the level equation on page 5:

ardl ln_consump ln_inc ln_inv, lags(1 0 4)

In the ec1 representation, page 12, you have 9 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1

For the two models to coincide, there must be 1 restriction on the coefficients in the latter version of the model.

If you do not want to have this restriction, either estimate it with option ec, which again gives you 8 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec

or allow for 1 unrestricted lag of ln_inc in the model:

ardl ln_consump ln_inc ln_inv, lags(1 1 4)
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec1
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec

In the latter case, no restriction is needed. There are 9 coefficients in each version of the model. But obviously, because we allow for a nonzero coefficient of the first lag in the level version, the estimates differ.

Hi dear Prof Kripfganz,
I am running ARDL model with the following two commands. I don't understand why different results are produced.

clear
set seed 1234
set obs 1000
gen y = uniform()
gen x1 =  rt(5)
gen time = _n
tsset time
local ylist y
local xlist x1 
local quantile "0.1 0.25 0.5"

    
local nq: word count `quantile'
di `nq'

local xnum: word count `xlist'  
di `xnum'

tempname opt

local maxlags = 10
if ("`maxlags'" != "") {
  ardl `ylist' `xlist', maxlag(`maxlags') aic ec1
  mat `opt' = e(lags)
}

mat list `opt'

The results are as follows

ARDL(1,1) regression

Sample:        11 -      1000                   Number of obs     =        990
                                                R-squared         =     0.5314
                                                Adj R-squared     =     0.5300
Log likelihood = -180.88505                     Root MSE          =     0.2911

------------------------------------------------------------------------------
         D.y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
           y |
         L1. |  -1.062711   .0318044   -33.41   0.000    -1.125124   -1.000299
-------------+----------------------------------------------------------------
LR           |
          x1 |
         L1. |   .0191763   .0099109     1.93   0.053    -.0002725    .0386252
-------------+----------------------------------------------------------------
SR           |
          x1 |
         D1. |   .0070096    .007535     0.93   0.352    -.0077768     .021796
             |
       _cons |   .5206504   .0181471    28.69   0.000      .485039    .5562619
------------------------------------------------------------------------------
.   mat `opt' = e(lags)
. }

. 
. mat list `opt'

__000000[1,2]
     y  x1
r1   1   1

As shown above, the optimal lags are 1 and 1. Then I run

ardl y x1, lags(1 1) ec1  // optimal lag equals to `opt'
ARDL(1,1) regression

Sample:         2 -      1000                   Number of obs     =        999
                                                R-squared         =     0.5334
                                                Adj R-squared     =     0.5320
Log likelihood =  -186.9914                     Root MSE          =     0.2924

------------------------------------------------------------------------------
         D.y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
           y |
         L1. |  -1.066707   .0316454   -33.71   0.000    -1.128807   -1.004608
-------------+----------------------------------------------------------------
LR           |
          x1 |
         L1. |   .0186711   .0098246     1.90   0.058    -.0006082    .0379504
-------------+----------------------------------------------------------------
SR           |
          x1 |
         D1. |   .0084251   .0074896     1.12   0.261    -.0062722    .0231223
             |
       _cons |   .5239664   .0181209    28.92   0.000     .4884069     .559526
------------------------------------------------------------------------------

I don't know why the sample obs used are different. One is 990, and the other is 999.
Which answer is correct?

Bests,
wanhai

[wanhaiyou]

The equations on page 12 are derived from the equation on page 5. With ec1, when the lag order of ln_inc is zero, its first lag is included in the error-correction form. However, in terms of the coefficients of the equation on page 5, this first lag has a coefficient equal to zero. Because this coefficient equals zero, we need the restriction for the coefficients on page 12, which I mentioned in my previous post. Put differently, for your model there is a total of 8 coefficients in the level equation on page 5:

ardl ln_consump ln_inc ln_inv, lags(1 0 4)

In the ec1 representation, page 12, you have 9 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1

For the two models to coincide, there must be 1 restriction on the coefficients in the latter version of the model.

If you do not want to have this restriction, either estimate it with option ec, which again gives you 8 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec

or allow for 1 unrestricted lag of ln_inc in the model:

ardl ln_consump ln_inc ln_inv, lags(1 1 4)
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec1
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec

In the latter case, no restriction is needed. There are 9 coefficients in each version of the model. But obviously, because we allow for a nonzero coefficient of the first lag in the level version, the estimates differ.

Dear prof, I see. Many thanks for your kindly help. Thanks again for your excellent rutine "ardl".

Bests,
wanhai you

[Sebastian Kripfganz]

The equations on page 12 are derived from the equation on page 5. With ec1, when the lag order of ln_inc is zero, its first lag is included in the error-correction form. However, in terms of the coefficients of the equation on page 5, this first lag has a coefficient equal to zero. Because this coefficient equals zero, we need the restriction for the coefficients on page 12, which I mentioned in my previous post. Put differently, for your model there is a total of 8 coefficients in the level equation on page 5:

ardl ln_consump ln_inc ln_inv, lags(1 0 4)

In the ec1 representation, page 12, you have 9 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1

For the two models to coincide, there must be 1 restriction on the coefficients in the latter version of the model.

If you do not want to have this restriction, either estimate it with option ec, which again gives you 8 coefficients:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec

or allow for 1 unrestricted lag of ln_inc in the model:

ardl ln_consump ln_inc ln_inv, lags(1 1 4)
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec1
ardl ln_consump ln_inc ln_inv, lags(1 1 4) ec

In the latter case, no restriction is needed. There are 9 coefficients in each version of the model. But obviously, because we allow for a nonzero coefficient of the first lag in the level version, the estimates differ.

[wanhaiyou]

The difficulty here is that the optimal lag order for ln_inc is zero, but you are forcing it to appear in the first lag in the EC representation (option ec1). This is achieved by effectively imposing a constraint as follows:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1
constraint 1 L.ln_inc = D.ln_inc
cnsreg D.ln_consump L.ln_consump L.ln_inc L.ln_inv D.ln_inc L(0/3)D.ln_inv if e(sample), constraints(1)

The long-run coefficients in the ardl output are a nonlinear coefficient combination:

nlcom (- _b[L.ln_inc] / _b[L.ln_consump]) (- _b[L.ln_inv] / _b[L.ln_consump])

If you use the ec instead of the ec1 option, this complication does not arise and you can replicate the results directly with regress:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec
reg D.ln_consump L.ln_consump ln_inc ln_inv L(0/3)D.ln_inv if e(sample)
nlcom (- _b[ln_inc] / _b[L.ln_consump]) (- _b[ln_inv] / _b[L.ln_consump])

Thanks very much for your reply, dear prof. Yes, I have read your slide at page 12. I think the formula at page 5 is equivalent to the formula at page 12. Whether the lag order of ln_inc is zero or not,
the x_t-1 in EC representation is always included. Therefore, the first lag of ln_inc is included. Actually, I don't follow why we need the constraint 1 L.ln_inc = D.ln_inc?

Additionally, I think the EC1 representation is more commonly in the literature.

Thanks again for your help.

Bests,
Wanhai You


Reference:

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

[Sebastian Kripfganz]

The difficulty here is that the optimal lag order for ln_inc is zero, but you are forcing it to appear in the first lag in the EC representation (option ec1). This is achieved by effectively imposing a constraint as follows:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1
constraint 1 L.ln_inc = D.ln_inc
cnsreg D.ln_consump L.ln_consump L.ln_inc L.ln_inv D.ln_inc L(0/3)D.ln_inv if e(sample), constraints(1)

The long-run coefficients in the ardl output are a nonlinear coefficient combination:

nlcom (- _b[L.ln_inc] / _b[L.ln_consump]) (- _b[L.ln_inv] / _b[L.ln_consump])

If you use the ec instead of the ec1 option, this complication does not arise and you can replicate the results directly with regress:

ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec
reg D.ln_consump L.ln_consump ln_inc ln_inv L(0/3)D.ln_inv if e(sample)
nlcom (- _b[ln_inc] / _b[L.ln_consump]) (- _b[ln_inv] / _b[L.ln_consump])

[wanhaiyou]

I do not really know enough about the Westerlund test, so cannot give you a definite answer to this question.

I(0) variables can potentially affect the long-run level relationship if the I(1) dependent variable is cointegrated with other I(1) regressors. In that case, there exists a linear combination between the dependent variable and those I(1) regressors which is I(0). The other I(0) regressors can then have a long-run effect on this I(0) linear relationship. In that regard, significant long-run coefficients of I(0) variables could still make sense.

Dear Prof Kripfganz,
I want to replicate the ardl model with regression routine, however, the results are inconsistent. The corresponding results are as follows

. webuse lutkepohl2,clear
(Quarterly SA West German macro data, Bil DM, from Lutkepohl 1993 Table E.1)

. ardl ln_consump ln_inc ln_inv, lags(1 0 4) ec1    // (1, 0, 4)

ARDL(1,0,4) regression

Sample: 1961q1 - 1982q4                         Number of obs     =         88
                                                R-squared         =     0.5385
                                                Adj R-squared     =     0.4981
Log likelihood =  306.72241                     Root MSE          =     0.0078

------------------------------------------------------------------------------
D.ln_consump |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
  ln_consump |
         L1. |  -.3335022    .043545    -7.66   0.000    -.4201595    -.246845
-------------+----------------------------------------------------------------
LR           |
      ln_inc |
         L1. |    1.00357    .032044    31.32   0.000     .9398006     1.06734
             |
      ln_inv |
         L1. |  -.0417399   .0373449    -1.12   0.267    -.1160586    .0325788
-------------+----------------------------------------------------------------
SR           |
      ln_inc |
         D1. |   .3346929    .044616     7.50   0.000     .2459042    .4234815
             |
      ln_inv |
         D1. |   .0556284   .0196031     2.84   0.006     .0166169      .09464
         LD. |   .0167529   .0200739     0.83   0.406    -.0231955    .0567012
        L2D. |   .0620601   .0198688     3.12   0.002     .0225198    .1016003
        L3D. |     .03124   .0194042     1.61   0.111    -.0073756    .0698555
             |
       _cons |    .037445   .0118549     3.16   0.002     .0138529     .061037
------------------------------------------------------------------------------

. 
. // OLS estimator
. reg d.ln_consump l.ln_consump l.ln_inc l.ln_inv d.ln_inc d.ln_inv l(1/3).d.ln_inv

      Source |       SS           df       MS      Number of obs   =        88
-------------+----------------------------------   F(8, 79)        =     11.68
       Model |  .005677968         8  .000709746   Prob > F        =    0.0000
    Residual |  .004801841        79  .000060783   R-squared       =    0.5418
-------------+----------------------------------   Adj R-squared   =    0.4954
       Total |  .010479809        87  .000120458   Root MSE        =     .0078

------------------------------------------------------------------------------
D.ln_consump |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  ln_consump |
         L1. |  -.3064987   .0563429    -5.44   0.000    -.4186463    -.194351
             |
      ln_inc |
         L1. |   .3078729   .0570286     5.40   0.000     .1943604    .4213855
             |
      ln_inv |
         L1. |  -.0130074   .0125833    -1.03   0.304    -.0380537     .012039
             |
      ln_inc |
         D1. |    .382558   .0773686     4.94   0.000     .2285596    .5365565
             |
      ln_inv |
         D1. |   .0542619   .0197379     2.75   0.007     .0149746    .0935491
         LD. |   .0116764   .0212116     0.55   0.584    -.0305443    .0538972
        L2D. |   .0586942   .0204104     2.88   0.005     .0180683      .09932
        L3D. |   .0295406   .0195846     1.51   0.135    -.0094415    .0685227
             |
       _cons |   .0337566   .0128433     2.63   0.010     .0081927    .0593205
------------------------------------------------------------------------------

. 
end of do-file

Thanks for your kindly help!


Best regards,
wanhai you

[Simra Temmy]

Your interpretation is really very helpful. Many thanks!

[Sebastian Kripfganz]

I do not really know enough about the Westerlund test, so cannot give you a definite answer to this question.

I(0) variables can potentially affect the long-run level relationship if the I(1) dependent variable is cointegrated with other I(1) regressors. In that case, there exists a linear combination between the dependent variable and those I(1) regressors which is I(0). The other I(0) regressors can then have a long-run effect on this I(0) linear relationship. In that regard, significant long-run coefficients of I(0) variables could still make sense.

[Simra Temmy]

1. You do not have to carry out separate cointegration tests. If the ARDL bounds test provides evidence in favor of a long-run level relationship, this would correspond to a cointegrating relationship if all involved variables are I(1).
2. Cointegration can only exist between I(1) variables. This does not rule out that other I(0) variables affect the short-run dynamics. In fact, including I(0) variables can help to purge the error term from serial correlation. Absence of residual serial correlation is usually an assumption underlying those tests.
3. The PMG estimator does not require that the time series are I(1). If they are I(1) but not cointegrated, then no long-run level relationship can exist among them.
4. Based on the bounds test, we may conclude that there is evidence for a long-run level relationship. If the variables are I(1), we simply call this a cointegrating relationship. If we do not know that the variables are I(1), we should not call it cointegration.
5. If you find no evidence of a long-run relationship (whether we call it cointegration or not), you could estimate a simpler ARDL model in first differences only; that is, the model would contain only short-run terms. You do not necessarily have to re-estimate the model but could also just stick to the original ARDL model (which might be less efficient as it would contain unneccessary long-run terms).

Dear Professor,
For #3, say dependent var is I(I), regressors are mix, two of them are I(0), the rest is I(I).
You say that it will be better if I add all the varibles to the Westerlung coint test; not only I(I) ones as I seen in some PMG studies? (I tried both, no cointegration)
And if we get significant long term results for I(0) regressors from PMG estimation, are these long run coefficients make any sense?

Thank you so much for replies...

[Sebastian Kripfganz]

1. You do not have to carry out separate cointegration tests. If the ARDL bounds test provides evidence in favor of a long-run level relationship, this would correspond to a cointegrating relationship if all involved variables are I(1).
2. Cointegration can only exist between I(1) variables. This does not rule out that other I(0) variables affect the short-run dynamics. In fact, including I(0) variables can help to purge the error term from serial correlation. Absence of residual serial correlation is usually an assumption underlying those tests.
3. The PMG estimator does not require that the time series are I(1). If they are I(1) but not cointegrated, then no long-run level relationship can exist among them.
4. Based on the bounds test, we may conclude that there is evidence for a long-run level relationship. If the variables are I(1), we simply call this a cointegrating relationship. If we do not know that the variables are I(1), we should not call it cointegration.
5. If you find no evidence of a long-run relationship (whether we call it cointegration or not), you could estimate a simpler ARDL model in first differences only; that is, the model would contain only short-run terms. You do not necessarily have to re-estimate the model but could also just stick to the original ARDL model (which might be less efficient as it would contain unneccessary long-run terms).