Questions about cointegration and error correction model

Question

I am confused about cointegration and error correction model. I gather that, if two variables are cointegrated, they are related.

1. Is the condition for two variables $X_t\sim~I(a)$,$Y_t\sim I(b)$ to be cointegrated $a=b\geqslant 1$?

2. Is there any possibility that $X_t\sim I(0)$ and $Y_t\sim I(0)$, and the regression of them is spurious regression?

As I understand it, before developing ECM, we should have unit root test and cointegration test. Consider the following 3 situations:

The first situation is that there are 2 variables conintegrated of same order, $Y_t\sim I(2)$ and $X_t\sim I(2)$, and the long run equation is $Y_t=a+bX_t+e_t$, $e_t\sim I(0)$.

3. How can we develop short run ECM?

The second situation is that there are there 3 variables conintegrated of same order, $Z_t\sim I(1)$, $X_t\sim (1)$ and $Y_t\sim I(1)$, and the long run equation is $Z_t=a+bX_t+cY_t+e_t$, $e_t\sim I(0)$.

4. How can we develop short run ECM?
5. What if the dependent variables are $I(2)$, how can we develop short run ECM?

Answer 1

1. Yes, this is a necessary prerequisite for cointegration.

2. If we define spurious regression as a regression between two independent integrated random variables (which seems to be a common definition, as far as I know), then (by definition) the answer is "No". However, Granger & Newbold "Spurious regression in econometrics" (1974) discussed spurious regression not only among integrated but also among stationary, although highly autocorrelated, variables. So even though the answer to the question depends on the precise definition of spurious regression, you may find features of spurious regression among stationary variables, too.

3. Given a pair of integrated variables and their stationary combination, the remaining task for formulating a vector error correction model (VECM) is determining the relevant lag order. This can be done by selecting the lag order that minimizes AIC or BIC in a VAR model in levels of the original variables minus $d$ (where $d$ is the order of integration); that is, if $p$ is the AIC- or BIC-minimizing lag order in VAR in levels, use $p-d$ for VECM. Given the lag order, simply formulate the VECM as a VAR model in second differences plus the error correction term. E.g.

$$ \Delta^2 y_t = \alpha_1 (y_{t-1}-a-bx_{t-1}) + \gamma_{11} \Delta^2 y_{t-1} + \gamma_{12} \Delta^2 x_{t-1} + \varepsilon_{1,t} $$ $$ \Delta^2 x_t = \alpha_2 (y_{t-1}-a-bx_{t-1}) + \gamma_{21} \Delta^2 y_{t-1} + \gamma_{22} \Delta^2 x_{t-1} + \varepsilon_{2,t} $$

4. Analogous to point 3. E.g. $$ \Delta z_t = \alpha_1 (z_{t-1}-a-bx_{t-1}-cy_{t-1}) + \gamma_{11} \Delta z_{t-1} + \gamma_{12} \Delta y_{t-1} + \gamma_{13} \Delta x_{t-1} + \varepsilon_{1,t} $$ $$ \Delta y_t = \alpha_2 (z_{t-1}-a-bx_{t-1}-cy_{t-1}) + \gamma_{21} \Delta z_{t-1} + \gamma_{22} \Delta y_{t-1} + \gamma_{23} \Delta x_{t-1} + \varepsilon_{2,t} $$ $$ \Delta x_t = \alpha_3 (z_{t-1}-a-bx_{t-1}-cy_{t-1}) + \gamma_{31} \Delta z_{t-1} + \gamma_{32} \Delta y_{t-1} + \gamma_{33} \Delta x_{t-1} + \varepsilon_{3,t} $$

5. Analogous to points 3., 4.