What is the difference between cointegration and the VECM? What is the difference between cointegration and the vector error correction model (VECM)? I applied cointegration test and found long run association between variables, so should I apply VECM?
 A: Cointegration is a phenomenon that may be exhibited by a group of integrated time series; being cointegrated is a feature that may be posessed by a group of integrated time series. Let us consider a simple example. If series $x_{1,t},\dotsc,x_{m,t}$ are individually I(1) (integrated of order 1) and there exists a linear combination $y_t=\beta_1 x_{1,t}+\dotsc+\beta_m x_{m,t}$ that is I(0) (stationary), then we face the phenomenon of cointegration, and the group of series $x_{1,t},\dotsc,x_{m,t}$ posess the feature of being cointegrated. If no linear combination is I(0), then there is no cointegration and the series taken together are not cointegrated.
Vector error correction model (VECM) is a model that can be used for modelling cointegrated time series. A very simple example is a bivariate VECM with no lags for two integrated-and-cointegrated time series $x_{1,t}$ and $x_{2,t}$,
$$ \Delta x_{1,t} = \alpha_1 (x_{1,t-1}-\beta x_{2,t-1}) + \varepsilon_{1,t}, $$
$$ \Delta x_{2,t} = \alpha_2 (x_{1,t-1}-\beta x_{2,t-1}) + \varepsilon_{2,t}. $$
It shows that the series $x_{1,t}$ reacts to the most recent (as of time $t-1$) disequilibrium between itself and the other series and "corrects" (given a suitable value of $\alpha_1$) to reduce the disequilibrium (moves towards equilibrium). The same could be said about $x_{2,t}$.
