I have a forthcoming paper on this but you may check the following references:

Österholm, P (2010) ‘Unemployment and labor-force participation in Sweden’, Economics Letters, ISSN 0165-1765, 2010, Volume 106, Issue 3, pp. 205 - 208

Kakinaka, Makoto, Miyamoto, Hiroaki (2012) ‘Unemployment and labor force participation in Japan’,Applied Economics Letters, ISSN 1350-4851, 07/2012, Volume 19, Issue 11, p. 1039

There you have an analysis framework that is fairly clear, first checking for unit roots, then for cointegration, then, if cointegration is found, use a VEC model.

In order to understand the issue you can read this

http://www.kva.se/documents/priser/nobel/2003/sciback_ek_en_03.pdf

Hope this helps

One need to check for i) structural breaks, ii) stationarity/non-stationarity (presence of unit-root), iii) if non-stationary, then order of integration (first difference or second difference where series turns stationary), iv) if two or more series are of same order (I(0), I(1), ...), then check for co-integration, and v) check for Granger-Casuality and auto-correlation.

Now different scenarios are:

1. two or more cointegrated series (series have unit roots, inetgareted of same order and co-integrated), go for error correction model like Vector Error Correction Model.

2. two or more series - non-stationary and integrated of same order, but not co-integrated: go for models in difference (like VAR in difference), use first difference if series are I(1), second difference if series are I(2), ...

3. Two or more series are stationary: go for models in levels like VAR in levels.

4. Now if the series are of different order (no question of co-integration, cointegration is applicable only in case of series of same integration order), use models in difference with highest order of integration.

Notes: Correct for auto-correlation if any. If structural breaks are there, consider period without any break. Use variable on RHS, which are Granger-Cause of a variable on RHS.

You can see one of my publication on Price Transmission analysis on ResearchGate, where I have applied these steps: Price Transmission in Thai Fish Markets, submitted to MRE. Other publications might interest you are Price Transmission in Bangladesh Fish Markers, and Error Corrected Almost Ideal Demand System for Crustaceans in USA.

Dear Annees. This is to add to add what others have already written. If you are faced with say a dependent variable that is I(1) and X1 and X2 are also I(1) but X3 is I(0), you can still estimate this model even if X3 is not obviously co-integrated with the rest. In this case you can either use Dynamic Ordinary Least Squares or Fully Modified OLS. These two methods allow you to estimate models with mixed orders of integration, that is, the explanatory variables of I(1) and I(0) orders.

Kiiza Barnabas

Now, Kehar, what happens when you have tested your variables for unit roots and four of them turn up to to be non-stationary and their order of integration is the same (first difference), but the other variable is stationary? Should you continue to test for cointegration for all of the variable or you should exclude the stationary one? Or must you go for VAR model in levels?

Dumisani: If two series have unit root and the order of integration is same, they might be co-integrated. In your case, you need to check for co-integration between four series. No question of checking co-integration with stationary series.

In case when series have different orders of integration, as in your case - one series I(0) and other four I(1), one need to use maximum order of integration for differencing (in your case first difference).

Regarding use of VAR or VECM, if series are co-integrated, VECM is appropriate one; but if not co-integrated, then VAR in difference.

Please also see Barnbas Kiza comment. In case, when two or more series are co-integrated and other non-stationary series are no co-integrated; and series appear in same equation. In this case we need to see the relationship with LHS variable.

If LHS variable is co-integrated with one or more RHS variables and not co-integrated with other non-stationary variables; one need to use model in difference with maximum order of integration and use error correction term simultaneously. Keep in mind, two co-integrated series are always co-integrated at higher order of integration.

If LHS variable is not co-integrated with any of the series, use of a model in difference with maximum order of integration is sensible.

Thank you Kehar. I have excluded the stationary variable in my regression and proceeded with VECM using only the cointegrated variables. Also I did a fully modified OLS to cater for the variable I excluded in the VECM estimation. Can I include both estimations and respective the results in my paper?

Dumisani: Is there theoretical or empirical basis for exclusion of stationary or non co-integrated series? Did you check whether excluded variable(s) is/are the "Granger cause" of other series? If you checked, and did not find the excluded series as the Granger cause of any other series in question, exclusion is not a problem. However, if any of the excluded series is the "Granger cause" of the series on LHS, exclusion of series is not appropriate. to me exclusion of series is not a welcome step to address an empirical problem.

I could not understand your last question: inclusion of estimations and respective the results.

Dumisani: You may see my manuscript available of researchgate.net "Price Transmission in Thai Fish Market: An Analysis along Value Chain and across Species" for presentation of results.

The views are really appreciated and rich in context. I am learning from this too much.

Hence we need further and in depth investigation (theoretical rather than empirical) to find the sensitivity of such models which include variables with different orders of integration. ARDL is one approach to such issues but still this has a limitation that is is limited to combination of I(0) and I(1) or between. So further theoretical evidence on the cointegration analysis of variable order of integration is required. If you have one such piece of research, can I request for sharing the details here. Not needed the copy but the citations is perfect. Thank you in advance.

Sorry for my late response. No there is no theoretical or empirical basis for excluding the variable. I just thought that if I include the stationary variable with non-stationary ones it may influence cointegration results, hence I thought of doing the fully modified OLS. Does your question imply that I may go on with co-integration and VECM for all the variables including the one which is stationary at levels?

Dumisani: ARDL (mention by Anees) is a single equation model. This is in a family of ARIMA models. In ARDL, apart from AR terms (lags of LHS variable), lags of variables on RHS are also there. On other hand, VAR and VECM are system of equations model. In a typical equation of VAR/VECM model, AR and DL terms are there. In addition to this, in VECM model moving average term (which is error correction term) is there. While specifying VAR/VECM, researchers generally take same number of AR, DL terms in each equation; however, one can take different AR and DL terms in different equations.

In ARDL/VECM model, the number of AR terms is order of integaration of LHS variable plus additional lags to correct for auto-correlation. Means while specifying these models, we take care of auto-correlation; and one can use OLS/MLE. For system of equations like VAR and VECM, one can use SURE/ISURE methods.

I am of the view to specify VAR/VECM including all important variables, based on the Granger casuality and theoretical grounds, with different AR and DL terms on each equation, and estimate using system technique like ISURE (iterative Seemingly Unrelated Regressions).

Thank you Kehar. You have provided me with great stuff.

@Barnabas Kiiza, can you give reliable reference in favour of using DOLS or FMOLS in case of mixed order of integration, please???

Ghulam,

See this paper below.

Seema Narayan, Paresh Kumar Narayan. 2004. “Determinants of Demand for Fiji’s Exports:An Empirical Investigation,” The Developing Economies, XLII-1, pp:95−112

Can i use ARDL in plave of VECM or can i use both

Any method used will give similar results

Thanks