Lets assume that I am planning to check for co integration between two I(1) series , X and Y and I have chosen X as the dependent and Y as explanatory variable. In such a case , do i need to generate two residual series ( X as dependent and Y as explanatory ) and ( Y as dependent and X as explanatory) and check both for stationarity to conclude co integration , or should I check only one residual series ( X as dependent and Y as explanatory) and check for its stationarity to conclude co integration. I am using the ADF Test to check for stationarity. Kindly advise.
Engle and Granger (1987) introduce a cointegration test, which consists of estimating the cointegration regression by OLS, obtaining the residual series ût, and applying unit root test for the ût series. To test an equilibrium assertion, they propose testing the null that ût series has unit root against the alternative that it has a root less than unity. Given that, ût are themselves estimates, new critical values need to be tabulated. Thus one has to use the corrected MacKinnon critical values. For more details, please, navigate to the following links.
http://www.eco.uc3m.es/~jgonzalo/teaching/phdtimeseries/cointeeviews.pdf
https://www.youtube.com/watch?v=uXG11rlWaSM
https://www.empiwifo.uni-freiburg.de/lehre-teaching-1/winter-term/makrookonometrie/methodology.pdf
Your choice of dependent and explanatory variable should ideally be determined by the theory underlying your regression. In some cases it may not be clearcut or there may be bi-directional causality. You should check using Granger causality analysis. If there is bi-directional causality maybe switch to a system framework (such as johansen's system cointegration analysis).
Assuming uni-directional causality, the Engle-Granger two step methodology (EG2SM) requires that the residual from the regression in levels (first step) be stationary - otherwise the second step is meaningless. If the residual is not stationary it can mean one of two things - either the two series are not cointegrated or alternatively there may be a third (or more) series required to achieve stationarity (the relationship between price, wage and productivity is an example).
Lastly, whilst the Engle-Granger two step method (EG2SM) is commonly used to model series which are cointegrated, one can in principle do it in one step using single equation error correction models (SEECM). This can be useful particularly in small samples where the estimates from OLS in the first step of the EG2SM, although consistent, can be biased due to serial correlation in the residuals. The dynamics allowed for in the SEECM should reduce this. An example of where this is shown to matter is Honarvar (2010, Computational Economics, Volume 36, Issue 3, pp 237-262). NB: You should still check that the residuals from the cointigrating vector are stationary.
Yes, the use of EG method requires the prior knowledge about which variable is dependent and which ones are independent. Once the researcher is aware of that according to a given theory, no need to test the stationary of residuals which are taken by running separate regressions for other variables. But the concern should be given on whether there is exactly one independent relation, and which variable is the dependent and whether all variables are integrated in the same order(I(1) or I(2)). Any errors done in the first stage can go to the second stage and then it could mislead the results. The disadvantage in this method is that, if there are more than one linearly independent (co-integration) relations among the set of variables selected, then modelling of such a relation cannot be done. Johensen's multivariate method is the usual way to resolve such a problem.
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