My research questions require me to run 8 bivariate equations for two country cases. When I checked for unit root, some of the equations had [A] I(0) and I(1) variables, [B] both I(1) variables, and [C] both I(0) variables.
I know that I can use ARDL for cases [A] and [B]. But what about case [C]? If I cannot run ARDL, I am thinking if I should just use VAR for [C], but then, would that allow me to have a valid comparison for the bivariate equations that I used ARDL, and for some others where I used VAR?
Albertine June Din for the case of C, the results will indicates long run relation. If you want to retrieve short run relations, take difference of it.
Dear Albertine June Din
For more details on the choice of cointegration models, I recommend you consult my article on the integration and cointegration variables available on my page, in which I develop precisely the answer to your question.
To go faster, stationary variables in level or following a 0-order process are not likely to be co-integrated. Because they are already stationary. There is therefore no risk of fallacious or illusory regression.
In reality, when the models Engle and Granger (1987) and Hendry (1986); johansen (1980, 1981) do not fulfill the necessary criteria for cointegration, which is called the ARDL model (Pesaran et al (2001)).
As a result, you have the choice to model according to the characteristics in seasonality, in cycle with the following models:
- Box and Jenkins (AR, ARMA, ARIMA, SARIMA, ARMAX ...)
- Engle (ARCH; GARCH ...)
- VAR, VARMAX, VARMA, ...
... etc.
For summary, the stationarity or lack of unit root (with stationarity tests) in the variables allows us to choose the classical models.
Error correction models are possible only when the variables are not integrated in (non-stationary) level.
The same order of variable integration is required for all error-correction models except the ARDL model.
I agree with Abdi-Basid Adan .
If the number of variables N (i.e., two in your case) in the system, it means that the number of cointegrating vectors (let's say s) will be in the range 0
I am not surely if I understood your question correctly, but in general you can use ARDL when either all variables are I(0), or some of the variables are I(0) and others are I(1), or all variables are I(1). Otherwise you cannot use ARDL.
Dear Akram Shavkatovich Hasanov , for all the respect I owe you, I would only like to support a nuance in your answer
On the theoretical level, deducing stationarity from cointegrating vectors is not indicated in the literature (if I do not get wet). Moreover, it has absolutely no interest, since it is the problem of stationarity which calls for the adoption of the cointegration to finally proceed, in the valid case, to the correction of the model.
In the classical sense, cointegration is the prerequisite solution of two or more no-stationary variables taken together in a model.
It is that you propose through your answer, it is indeed, starting from the solution and affirming that there was no problem of stationarity.
However, the absence of a unit root is tested with a variety of stationarity tests, there is no reason to check the cointegration vectors to return to the stationarity assertion.
It is true that on the mathematical level, a property or a theorem gives rise to the intention of verifying its reciprocal. This is the same thing you are doing. As you know, in econometrics, the procedure is from one stage to another. To pass a later stage to return on a step of preamble is not obvious in my opinion.
I believe your question should be modified to whether the bounds test is valid in your cases. Express ARDL model in levels, it would make a statistical sense if all variables are I(0), of course. Whether it can make an economic interpretation is another issue. When I(1) variable(s) is (are) involved in an ARDL model, however, it is only statistically meaningful when the bounds test can be passed. If yes, the model can be expressed in terms of ARDL-ECM, where information about long-run equilibrium and short-run adjustments can be obtained.
Dear Xiang Lin . Thanks for your input. Your point shed light on the discussion. This is exactly what I understand.
Dear Albertine,
I will start by advising you to go and read a post by Dave Giles on this issue. A summary from there is that if,
- For case [A] with I(0) and I(1) variables, you will use the ARDL Model, but first, you have to perform the bound test (cointegration test for a combination of I(0) and I(1) variables). If positive, it shows that though the variables are integrated of different orders, there is still a long-run equilibrium relationship between them. However, the presence of the I(1) variable(s) show(s) that there is no short-run equilibrium relationship between them. You therefore need to estimate the model in level form and then estimate a separate restricted ECM. The parameter of the estimation in the level form shows the long-run relationship between the variables while the rate of adjustment from the short-run disequilibrium to the long-run equilibrium is capture by the ECM.
- For case [B] both I(1) variables. Yes you still can use the ARDL but the best approach is to test for Cointegration which is what the ARDL will do of course, but why use the complicated approach if it conforms to a simple approach? Just test for cointegration using an appropriate method and run the regression at level to capture the long-run relationship and the ECM to capture the rate of adjustment to the long-run.
- For case [C] both I(0) variables. What this is showing is that there is a short-run equilibrium relationship, which of course leads to long-run equilibrium relationship between the series. Hence, there is no need going through all the hassles of cointegration test or ARDL as they are not applicable here. Just run the regression at level and interpret it as the relationship (both short and long-run) between your variables.
- On how to compare the results, I think the answer is implicit in what I have written. The ARDL (case A) and the cointegration regression (case B) show the long-run relationship as the presence of non stationary variable(s) in the models remove short-run equilibrium relationships. The ECM in both of them shows how the short-run disequilibrium adjusts to long-run equilibrium. On the other hand, all the variables in the model being I(0) shows that they have both short and long -run equilibrium relationship and the result should be interpreted as such.
- Hope these will be of help.
I am always concerned when someone proposes to apply some particular econometric technique to a data set without specifying any underlying economic theory or giving any justification for the particular technique chosen. As I read the question you have 8 countries and 2 variables for each country. Your both were root tests imply that in some countries both variables are I(1) (Case B), in others both were I(0) Case C) and in the others one was I(1) and the other I(0). I do find it difficult to understand how such results could be compatible with any underlying economic theory. Are these different results caused by (government) policies. Are there innovations or other series breaks that might give rise to your different results. If there are such problems then you may have to take account of them in your modelling.
Case B: If both your variables are I(1) and not cointegrated then unless one is exogenous you should use a VAR in first differences to model the two variables. If you can justify the required exogeneity assumption you may use a single equation method.
If the variables are cointegrated you should use a VECM (e.g Johansen ML) if neither variable is weakly exogenous. If one variable is weakly exogenous then you can use single equation methods (e.g. ARDL, EG). Weak exogeneity can be tested within the Johansen methodology.
Case A: In this case there are two options. In the first I would first difference the I(1) variable to make it I(0) and then proceed as in case C. Alternatively I would integrate the I(0) variable and make it I(1). For example an asset return is usually I(0) and can be integrated to an asset price index which is then !(1). In this case you can proceed as in case B.
Case C: I presume that integrating both variables does not make sense (for example if you were looking at two asset returns and theory suggested that the asset prices were cointegrated you should integrate the asset returns to asset prices and look at a VECM in the asset prices). If you are satisfied that one variable is exogenous you can estimate an ARDL which looks like the cointegration ARDL but inference is different. Chapter 6 of Peseran (2015), Time Series and Panel Data Econometrics, OUP, deals with the use of ARDL methods with I(0) variables
It is possible to use ARDL to estimate 2 or more I(0) variables. You should base your Ftest in the bound test on I(0) which is the lower critical value.
Adenuga, If all variables are I(0) your significance tests have standard distributions. The bounds test only applies when you have I(1) variables. I would recommend that you look at the Pesaran reference that I have quoted in my earlier reply
My earlier contribution is inconclusive since the Ftest is only above I(0).
While I agreed with John but where endogeneity is involved, ARDL may be used for a single equation.
Adenuga
Perhaps my answer was not clear. I agree that ARDL methods may be applied when all right hand side variables are stationary and exogenous. My point is that the Pesaran bounds test is not valid in such circumstances. You should then use standard OLS inference.
The dependent variable must be I(1).
You cannot use the ARDL model when dependent variable I(0).
The independent variables can be I (0) or I (1).
Ugar
Can you have a look at Pesaran's description of the use of ARDL when all variables, including the dependent variable, are I(1). This is covered in the Pesaran reference that I quoted in my earlier reply to this question. In this case the bounds tests are not applicable but standard OLS inference is valid ( if the necessary specification tests have been passed)
John C Frain ,
You should read this article in detail:
Bootstrapping the autoregressive distributed lag test for cointegration
Answer is available on the first page.
The dependent variable must be absolutely I(1).
If all variables are I (0), yes OLS can be used.
I think we said the same thing.
Link:
Article Bootstrapping the autoregressive distributed lag test for co...
@ Uğur Korkut Pata
Sorry, I did misunderstand your comment. Long before the bounds ARDL cointegration analysis was introduced (2001) we estimated various dynamic economic models using an ARDL (Autoregressive Distributed Lag) methodology. Andrew Harvey's 1981 book contains a section on this methodology. When the variables are stationary asymptotic results similar to standard OLS hold. As many of my colleagues and I made extensive use of the method I perhaps think of that as the original ARDL methodology. I found the article you quoted interesting. I agree that the ARDL bounds test is more often misused. Very often one sees the weak exogeneity conditions or the requirement that there only be one cointegrating relationship being ignored.
I also think that we are saying the same thing.
I think the ARDL can only be applicable if the dept variable is I(1). If all variables are I(0), why not use OLS bcoz, the purpose is to establish an estimate of the relationship. If you are interested in using a dynamic specification, it depends on what you want to achieve, distributed lag can cause multicollinearity issue but AR model can do like the Koyck transformation if the interest is for dynamic model to establish short run or long run. The brain behind ARDL is combination I(1)& I(0) variables as ecm or Engle Granger/Johansen is applicable only with all variables in I(1).
Perekunah Bright Eregha
The Johansen methodology can include both I(0) and I(2) variables in addition to I(1) variables.
You will also find these matters dealt with in Juselius (2006) or Lutkepohl (2010). The problem is that these procedures are not covered in all econometric packages. As regards Autoregressive Distributed Lag models these were used in estimation of dynamic econometric models before cointegration was well known and practised. The presentation by Pesaran shows that the bounds test is an extension of this earlier analysis. These ARDL models were often the starting point of a general to specific analysis following the LSE approach to estimation. They do not impose the lag structure of Koyck and similar infinite lag structures which are also appropriate in certain citcumstances.
Johansen (1995), Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press. Juselius(2006), The Cointegrated VAR Model, Oxford University Press. Lütkepohl (2010), New Introduction to Multiple Time Series Analysis, Springer
Yes, it is possible. You can also use OLS since the variables are stationary at levels.
Yes, you can use ARDL if all the variables are I(0) but since they are all I(0), you should go ahead with your VAR as it will give a better result. As per whether the results can be compared, you need to know what the two models are testing before comparing the two results.
When all the variables are I(0), it shows that there is a short run equilibrium relationship between/among the variables and there will be no need to run the ECM. On the other hand, when some or all the variables are not I(0), i.e. not stationary, it shows that there is no short run equilibrium/stable relationship between/among the variables, and that is why we test for cointegration to see if there is a long run stable relationship. If there is, we run the model at level but interpret the parameters as long-run parameters, and then the ECM to see how quickly the short run disequilibrium adjust to long-run equilibrium relationship (speed of adjustment). Hence, one should be cautious in comparing the two results as one capture short run relationships and the other captures long run relationships.
Boubekeur Med
An ARDL model is also known as an Autoregressive Dynamic Lag model. Autoregressive means that the model includes lags of the dependent variable as explanatory variables. The Dynamic Lag implies that the model also includes lags of the explanatory variables. Prior to the introduction of cointegration, ARDL models were widely used to estimate dynamic (mainly macroeconomic) relationships. Even Pesaran's 2015 book, Time Series and Panel Data Econometrics, OUP, gives details of the estimation of ARDL models for stationary variables. Standard inference implies in such cases. Nobody is suggesting that one is testing that I(0) variables are cointegrated.
Peseran Shin and Smith (2001) extends this I(0) analysis to include I(0) and I(1) variables and they introduce what is known as the bounds test for cointegration. This is valid only in certain circumstances (weak exogeneity and only one cointegrating relationship). It is interesting to note that Pesaran in the book referenced above gives fewer pages to the case with I(1) variables than the dynamic variables involving stationary variables.
Hi
With ARDL, model, it can be used if all the variables are i(0) or i(1) or mix of the two, i(0) and i(1). But note that it cant be used when there is i(2) or higher.
Thank you
John C Frain
Thanks for the clarification, I got confused between bound testing and ARDL
Pat Obi Without an understanding of the economics of your problem it is difficult to make a recommendation, There are several possibilities.
Which method you choose depends on your understanding of the economics of the problem. If you go ahead and include the I(1) variable as an explanatory variable for the I(0) variable I would expect that the lag structure that you derive for the I(1) variable will be very similar to that for 2 above if you transform it back to levels. I would also expect a non-significant long-run multiplier. The stochastic component of an I(0) variable has an equilibrium value, to which it tends to return. The stochastic component of an I(1) variable has no equilibrium value. This there can be no relationship between a variable that returns to an equilibrium value and one that wanders.
Thus I would recommend 1 or 2 above.
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