Causality is a subtle issue. You can justify it from long run economic  theor. The johansen approach is often criticized because the cointegrated relationship is identified from a statistical and not economic point of view. Pesaran make one step ahead, he imposes long run relationship (also for coefficients) and test them statistically (based on the data).

Causality is a subtle issue. You can justify it from long run economic  theor. The johansen approach is often criticized because the cointegrated relationship is identified from a statistical and not economic point of view. Pesaran make one step ahead, he imposes long run relationship (also for coefficients) and test them statistically (based on the data).

Cointegration is indicator for the existence of long-run relation between two variables. the exsistence of causality need further test

The situation you describe is certainly largely possible, although the existence of cointegration puts some restrictions on the system. Helmut Lütkepohl has an example in his book New Introduction to Multiple Time Series Analysis (sect. 6.6, pp. 261-262). The problem of testing Granger causality in cointegrated systems appears to be complicated (see sect. 7.6). I once attempted a test like this as a by-product for a paper but decided not because there it would not have been worth the efforts. This should not deflect you from pursuing it. 

It may be considered, with the precautions, that causality is a short-term relationship while cointegration is a long-term relationship.

It was one of the main results of the paper by Engle and Granger (1987, Econometrica) that they showed the equivalence of cointegration and error correction. This means that if there is cointegration then there must be error correction, and at least one of the variables (X1,say) in the (here, three-dimensional) system must react to the "long-run disequilibrium" (described by X2 and X3, say) and thus there must be some Granger causality, running from X2 and X3 to X1. This, of course, does not rule out that you may find statistical evidence that is intrinsically contradictory, such as evidence on cointegration but not on causality in any direction, the situation described here.

There are several problems with inferring causality from cointegrated models. While one of the objectives of a cointegrated VAR is to reduce the probability of a spurious regression, unless your variables have weak exogeneity and structural invariance leading to robust parameter constancy, there is no complete assurance  that hidden endogeneity is not the real cause of what might appear to be a long-run causal relationship.  If you can't prove a closed system and inclusion of all the relevant variables, it is possible that that one or more unknown variables is the source of what you appear to be  a situation of long-term causality.  

Moreover, there may be a simultaneous relationship between the first difference of x_t and the first  difference of y_t, which a test for Granger non-causality among lags will miss, owing to the focus on predictive precedence rather than Geweke's focus on instantaneous simultaneity.  After all, Granger causality will not accommodate non-linear models, contemporaneous simultaneity,  omitted variable bias, highly collinear time series without an adequate set of lags or sample size.

Yes, it is possible. Cointegration is characterizing a long-run relationship between these variables without any commitment on causality. You mention that there is no causlaity - but: how did you test it? The test is usually applied to the differences which, if significant, will produce more convincing results. It is important to find the right number of lags.

You have to understand that equilibrium is not the natural state of things.  Theoretically, we would like to assume that some sort of dynamic equilibrium exists.  Perhaps it does within confined systems for short periods of time.  However, nature is evolving. Nature is more like entropy in long run.  The Bayesian framework accommodates this adaptation nicely, whereas in the frequentist state of mind, we often make unrealistic assumptions to support our conclusions--such that we have conducted monte carlo resampling of our dataset, which over the long-haul gives us confidence intervals that we really got from only one sample. You can if you're not careful about your sampling wind up with an unappy randomization.

The Granger causality test, is not a test on ausality, he called it causality in the sense of GRanger, but in fact you can't measure causality you are measuring other things.

When using the Johansen cointegration model, it is more appropriate to test for Granger causality within the model and not without (see, Lütkepohl, 1991, p. 35; Toda and Phillips, 1993). This amounts to testing for weak exogeneity among the variables concerned in the model.

Granger representation theorem asserts that cointegration implies causality so there should be error correction in at least one of the equation. The significance of error correction confirms long run causality from all right hand side variables to the left side variable.  But software often report the results of short run causality (based on the restricted least square F or Chi Sq test) this is in differences of variablles. So i think it is quite possible that one can get results that may look contradictory as found by Mounir. Also i suggest to perform impulse response analysis since by definition of impulse responses (dyi,t+s/dej,t) there is possibility of contemporaneous causality (s=0) as pointed out by Robert Yafee.

Three years later, i am facing the same problem.Gratitude to the respondents here, your responses to Mounir really helped!