[Viktorija Mano]

Difficult to say. My answer would be that 22 observations is probably too small to meaningfully fit such a model. There is not much hope to obtain precise estimates of a third-order polynomial (and corresponding standard errors for the long-run coefficients) in your FDI variable.

Hi Sebastian,

Thanks for your response, I agree that the data is limited, but I am puzzled why does the specific variable (FDI^2) shows me r498 error. I have ran a number of combinations of a similar model investigating GDP and even thoug including 5-6 variables it never showed me this error.

This is a PhD research that I am doing and I am aware that the data is limited, but one of my examiners wanted me to include polynomial.

Thanks again.

[Sebastian Kripfganz]

I am affraid that I cannot say much more other than guessing that there might be a collinearity problem between FDI and FDI^2. It might help to rescale the variable FDI, i.e. dividing it by 1.000 or one million, depending on its unit of measurement, before squaring it.

[Viktorija Mano]

Thanks, I will try to rescale it. In terms of collinearity that didn't show a high correlation.

[Mahana Noorma]

Dear all,

I have a question regarding the interpretation of the coefficients in an ARDL model.

I estimated the following ARDL model

Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + e_t

 
 regress d.y  c  l(1/2).d.y  l(0/2).d.x1

I would like to interpret the impact of x₁ on y. As several lags of the variable x₁ is included in the econometric specification, I do not know how to interpret the impact of x₁ on y. Should I consider the sum of the coefficients of the different lagges values of x₁ or should I interpret each of them separately? What if the coefficients are not all statistically significant?

I kindly thank you for your help.

Kind regards

[Mustapha Immurana]

If your variables are I(1) and you have more than one co-integrating relationship among them, the single-equation ARDL model would be misspecified as it can accommodate only one co-integrating relationship. In that case you would prefer to estimate a vector error-correction model (VECM).

If your variables are I(1) and you have exactly one co-integrating relationship, you can rewrite the ARDL model analytically in error-correction representation with first-differences of depvar on the left-hand side, the co-integrating relationship of the level variables as well as additional lags of first-differenced depvar and indepvars on the right-hand side. All those components are then I(0) which shows that you can safely estimate this ARDL model in levels.

If your variables are I(1) but you do not have any co-integrating relationship among them, estimation is still fine because there exist values for the population parameters such that the error term can be I(0) due to the inclusion of lags of the dependent variable (the sum of the coefficients for the lags of depvar would equal unity in the underlying data generating process such that the level term drops out in the error-correction representation of the model; similarly for indepvars that are I(1)). However, in this case it would be more efficient to estimate an ARDL model directly in first differences.

If all of your variables are I(0) then you obviously do not have any problem with the ARDL model.

The point that I want to make is the following: Testing for non-stationarity and co-integration of your variables is still useful as it guides you towards the optimal model choice (VECM, ARDL in levels, ARDL in first differences).

Dear Sebastian,

1. Please based on the above, can I confidently say that if all vars are I(1), establishing long-run relationship or cointegration among variables does not matter anymore?

2. Also from what you have said above, does it mean using ARDL directly in first differences imply applying the ec1?

3. Further, I would want to know the reason behind getting a result that shows only the constant term and not other vars in a short run ARDL model using stata when ec is used and not ec1.

4. Last and not the least, please is it appropriate to use the ec1 even when some explanatory variables are dummies?

[Sebastian Kripfganz]

Mahana:

It really depends on what you are interested in. The first coefficient of your x-variable gives you the contemporaneous short-run effect of a change in that variable on the change in y. The second coefficient gives you the one period lagged effect, and so on. Of course, if the respective coefficient is statistically insignificant, than you would conclude that the corresponding effect is zero.

Adding all the coefficients would yield a cumulative effect. You can compute it with the lincom command. For this exercise, the significance of individual coefficients is not crucial but the significance of the sum of all coefficients.

Since you estimated the model in first differences without an error-correction term, you are imposing that there exists no long-run effect of the level of x on the level of y.


Mustapha:

1. No, if all variables are individually I(1) you can still have a cointegrating / long-run relationship among them.

2. The ardl options ec or ec1 both produce estimates for an error-correction model including the long-run terms. If you want to estimate an ardl model purely in first differences, you should use the time-series first-difference operator D. to generate first-differenced variables and then run the ardl command with these differenced variables and without the option ec or ec1. (Unfortunately, ardl does not allow time-series operators right now. You would have to generate new variables in a first step.) But note that such a first-differenced model would be misspecified if there is in fact a long-run relationship.

3. I do not really understand this point. Can you show an example with the command line as you have typed it and the corresponding Stata output?

4. You should specify the dummy variables with the exog() option of the ardl command. Then there will be no problem using the ec or ec1 option.

[Mahana Noorma]

Sebastian,

Thank you so much for your quick reply and your explanations.

I just have one additional question regarding your last comment. What would the presence of an error correction term would imply?

The model would be:
Δy_t = β₀ + Σ β_iΔy_t-i + Σγ_jΔx_1t-j + ϕz_t-1+ e_t

where z_t-1=y_t-1-c-ρx_1t-1 and is the error correction term.

Thank you very much for your help.

Kind regards

[Sebastian Kripfganz]

If the speed-of-adjustment coefficient ϕ and the long-run coefficient ρ are nonzero, then there exists a long-run relationship in the levels between x and y. Otherwise there does not exist such a relationship. The long-run coefficient tells you how large the effect of a change in x on y in the long-run equilibrium is, and the speed-of-adjustment coefficient tells you how fast the convergence back to this equilibrium is.

[Mustapha Immurana]

Mahana:

It really depends on what you are interested in. The first coefficient of your x-variable gives you the contemporaneous short-run effect of a change in that variable on the change in y. The second coefficient gives you the one period lagged effect, and so on. Of course, if the respective coefficient is statistically insignificant, than you would conclude that the corresponding effect is zero.

Adding all the coefficients would yield a cumulative effect. You can compute it with the lincom command. For this exercise, the significance of individual coefficients is not crucial but the significance of the sum of all coefficients.

Since you estimated the model in first differences without an error-correction term, you are imposing that there exists no long-run effect of the level of x on the level of y.


Mustapha:

1. No, if all variables are individually I(1) you can still have a cointegrating / long-run relationship among them.

2. The ardl options ec or ec1 both produce estimates for an error-correction model including the long-run terms. If you want to estimate an ardl model purely in first differences, you should use the time-series first-difference operator D. to generate first-differenced variables and then run the ardl command with these differenced variables and without the option ec or ec1. (Unfortunately, ardl does not allow time-series operators right now. You would have to generate new variables in a first step.) But note that such a first-differenced model would be misspecified if there is in fact a long-run relationship.

3. I do not really understand this point. Can you show an example with the command line as you have typed it and the corresponding Stata output?

4. You should specify the dummy variables with the exog() option of the ardl command. Then there will be no problem using the ec or ec1 option.

Dear Sebastian,

Thanks so much.

1. Please I need clarity concerning response 1 above. Thus if all variables are individually I(1) and we do not have a cointegrating / long-run relationship is there the need to be worried?

2. In using the ardl command in stata and applying the exog option, the variables stated in the parenthesis do not appear in the long run results, so please how do you explain this especially when that exog var assuming is a dummy, is the main variable of interest as shown in the results below:

. ardl lngdp lnalindirect lnaldirect, exog(dummy) ec1 regstore(obj2)

ARDL regression
Model: ec

Sample: 1975 - 2015
Number of obs = 41
Log likelihood = 99.629501
R-squared = .48803084
Adj R-squared = .37943132
Root MSE = .02374474

------------------------------------------------------------------------------
D.lngdp | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ |
lngdp |
L1. | -.2676916 .0792778 -3.38 0.002 -.4289835 -.1063996
-------------+----------------------------------------------------------------
LR |
lnalindirect |
L1. | .69882 .0862262 8.10 0.000 .5233916 .8742484
|
lnaldirect |
L1. | .2651334 .0800871 3.31 0.002 .1021949 .4280719
-------------+----------------------------------------------------------------
SR |
lngdp |
LD. | .3157183 .1478288 2.14 0.040 .0149583 .6164782
|
lnalindirect |
D1. | .2418461 .094818 2.55 0.016 .0489374 .4347549
LD. | -.2319939 .0905123 -2.56 0.015 -.4161426 -.0478453
|
lnaldirect |
D1. | .070974 .0356959 1.99 0.055 -.00165 .1435979
|
dummy | -.0172295 .0214643 -0.80 0.428 -.0608989 .0264399
_cons | -.3642064 .1204868 -3.02 0.005 -.6093387 -.1190741
------------------------------------------------------------------------------

[Usman Bashir]

I just released a user-written command by myself and Daniel Schneider that accomplishes this task.

The command ardl fits a linear regression model of depvar on indepvars with lagged depvar and indepvars as additional regressors. Information criteria can be used to find the optimal lag lengths. Estimation output is delivered either in levels form or in error-correction form. As an option, results from the Pesaran/Shin/Smith (2001) bounds testing procedure for the existence of a levels relationship can be displayed.

You can find and install the ardl package by typing the following line in the Stata command window:
net from "http://www.kripfganz.de/stata/"

Please see the Stata help file for additonal information about the command. Comments, suggestions, and bug reports are highly welcome.

Dear Sebastian i am currently using STATA 14 and i am unable to download your program this is the error when i run "net install ardl, from(http://www.kripfganz.de/stata/)"

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).
r(2);

.
your help would be appreciated

[Sebastian Kripfganz]

Mustapha:

1. Worry about what? You can still consistently estimate the ARDL model but estimation purely in first differences would be more efficient.

2. If you want your dummy to appear in the long-run relationship, you could alternatively use the following specification:

ardl lngdp lnalindirect lnaldirect dummy, lags(. . . 0) ec1

which forces the dummy to enter the ARDL model with zero lags while the lag order for all other variables is still chosen optimally.


Usman:

This can happen for example if you do not have full administrator rights on a company PC. I have sent you a private message here on Statalist.

[Mahana Noorma]

Again, thank you very much for your precious help
Kind regards

[Mustapha Immurana]

Mustapha:

1. Worry about what? You can still consistently estimate the ARDL model but estimation purely in first differences would be more efficient.

2. If you want your dummy to appear in the long-run relationship, you could alternatively use the following specification:

ardl lngdp lnalindirect lnaldirect dummy, lags(. . . 0) ec1

which forces the dummy to enter the ARDL model with zero lags while the lag order for all other variables is still chosen optimally.


Usman:

This can happen for example if you do not have full administrator rights on a company PC. I have sent you a private message here on Statalist.

Dear Sebastian,

Thanks so much for the knowledge you have been sharing.

Could you Please kindly help with the following questions.

You have explained that if all vars are I(1) and there is no cointegration, then it is better to use first differences of the vars and run the ardl directly without the ec or ec1.

1. So please what I want to be sure about is whether vars are all I(1) or Mixed I(0) and I(1) , provided there is no cointegration, the ardl should be directly run WITHOUT the ec or ec1 options but lags options can be stated?

2. Based on your response above, can I say that If vars are mixed I(0) and I(1) and there is no cointegration, running the ardl drectly using the vars in LEVEL forms and WITHOUT the ec or ec1 options is appropriate?

3. Please what could be the reason for such a result below where the lag or first difference of the dependent var was not seen in the Short run estimates?

. ardl lndvlp lntexpnd lntrev dumy, lags(. . . 0) ec1 regstore(reg1)

ARDL regression
Model: ec

Sample: 1977 - 2015
Number of obs = 39
Log likelihood = 72.370326
R-squared = .48755266
Adj R-squared = .42726473
Root MSE = .04051888

------------------------------------------------------------------------------
D.lndvlp | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ |
lndvlp |
L1. | -.2975397 .0727746 -4.09 0.000 -.4454355 -.149644
-------------+----------------------------------------------------------------
LR |
lntexpnd |
L1. | .2827662 .2020133 1.40 0.171 -.1277741 .6933065
|
lntrev |
L1. | .7414265 .2378828 3.12 0.004 .2579904 1.224863
|
dumy |
L1. | -.1744303 .0955046 -1.83 0.077 -.3685191 .0196585
-------------+----------------------------------------------------------------
SR |
lntexpnd |
D1. | .0841342 .0674223 1.25 0.221 -.0528845 .2211529
|
lntrev |
D1. | .2206038 .0818039 2.70 0.011 .0543582 .3868494
|
dumy |
D1. | -.0519 .0282168 -1.84 0.075 -.1092434 .0054435
|
_cons | -.1322512 .1203211 -1.10 0.279 -.3767731 .1122708
------------------------------------------------------------------------------

[Sebastian Kripfganz]

1./2. You can always run the ARDL model in levels no matter whether your variables are I(0), I(1), or a mixture of both. The ec and ec1 options are just reparameterizations of the ARDL model, another way of displaying the results, but the underlying estimation is the same as without these options.

3. If there is only 1 lag of the dependent variable in the ARDL specification of the model, then there will not be any additional short-run term (lagged differences of the dependent variable). The one lag of the dependent variable determines the speed-of-adjustment coefficient in the error-correction representation.

[Mustapha Immurana]

1./2. You can always run the ARDL model in levels no matter whether your variables are I(0), I(1), or a mixture of both. The ec and ec1 options are just reparameterizations of the ARDL model, another way of displaying the results, but the underlying estimation is the same as without these options.

3. If there is only 1 lag of the dependent variable in the ARDL specification of the model, then there will not be any additional short-run term (lagged differences of the dependent variable). The one lag of the dependent variable determines the speed-of-adjustment coefficient in the error-correction representation.

Dear Sebastian,
Thanks so much

So please does response1./2. mean if I use the ec1 option and the short run and long run results are displayed BUT the estat btest command showed no cointegration I can still maintain the results?