I'm pretty new to econometrics and I've been taking a class at university which uses the book "Econometric theory and methods" by Davidson and MacKinnon. It's a pretty good book but there's one thing I'm trying to realize completely.
My question is mainly: What is the relationship between cointegration and Error Correction Models? I have to say that the book tells me little about it, it briefly talks about estimating non-stationary data at page 629 but I'm still missing this "relationship". Of course I tried to google it and found some papers from Engle and Granger, where they talk about this "close relationship" without really mentioning it in a specific manner.
Maybe it's a silly question or a difficult relationship to estimate in few words, but any help would be greatly appreciated.
Best Answer
A cointegration relation means that two or more dependent variables are related. Assume that the equilibrium relation between two economic dependent variables is $y_{1,t}=\beta_1 y_{2,t}$.
If $y_{1,t}$ and $y_{2,t}$ deviate from this relation, the market forces them back to the equilibrium. This can be represented as follows: \begin{align} \Delta y_{1,t} &= \alpha_1 (y_{1,t-1} - \beta_1 y_{2,t-1}) + u_{1,t}\\ \Delta y_{1,t} &= \alpha_2 (y_{1,t-1} - \beta_1 y_{2,t-1}) + u_{2,t}. \end{align} That is, the change depends on the deviation form the equilibrium. In addition, $\Delta y_{i,t}$ may also depend on previous changes in both variables. That is: \begin{equation} \Delta y_{i,t} = \alpha_i (y_{1,t-1} - \beta_1 y_{2,t-1}) + \gamma_{i,1} \Delta y_{1,t-1} + \gamma_{i,2} \Delta y_{2,t-1} + u_{i,t}, \quad i = 1,2 \end{equation}
This is the vector error correction model. Now about the relationship between this model and cointegration. If you rewrite the previous equation as: \begin{equation} \alpha_i (y_{1,t-1} - \beta_1 y_{2,t-1}) = \Delta y_{i,t} - \gamma_{i,1} \Delta y_{1,t-1} - \gamma_{i,2} \Delta y_{2,t-1} - u_{i,t}, \quad i = 1,2, \end{equation} you will see that all the terms on the RHS are stationary. Hence, the term on the LHS is stationary as well, and this term coincides with a scaling coefficient times the cointegration relation.