Getting understand error correction model

Question

I have several intuitive problems with error correction model. I will write below how I understand derivation of ECM model with my queries.

Let $I(y_t)=I(X_t)=1$ and consider model : $$y_t=\alpha_0+\alpha_1y_{t-1}+\beta_0x_t+\beta_1x_{t-1}+u_t$$ Now substitute $y_{t-1}$.

$$\Delta y_t= \alpha_0+p_1y_{t-1}+\beta_0x_t+\beta_1x_{t-1}+u_t$$

After that in the right sight of the equation we add $\beta_0x_{t-1}-\beta_0x_{t-1}$ to get

$$\Delta y_t= \alpha_0+p_1y_{t-1}+\beta_0\Delta x_t+\theta_1x_{t-1}+u_t$$ Now we have to think for a while. Because $I(y_t)=1$ we have stationarity of variable $\Delta y_t$. The same justification we can apply to variable $\Delta x_t$. In terms of that way of thinking we have the equivalence following : $$y_t \; \text{and} \; x_t \; \text{cointegrated} \Leftrightarrow u_t \text{stationary}$$

For testing stationarity of $u_t$ we can use certain tests. The most popular one is ADF test. After claiming cointegration we are following the algorythm :

(1) Estimate our linear regression model $y_t=c+\beta x_t+u_t$

(2) Extract $u_{t-1}=(y_{t-1}-c-\beta x_{t-1})$ from the model (1)

(3) Create new model using equivalent form of $u_t$ derived in step (2) $\Delta y_t=\beta_0+\beta_1 \Delta x_t + \beta_2 u_{t-1}+\epsilon_t=\beta_0+\beta_1 \Delta x_t +\beta_2(y_{t-1}-c-\beta x_{t-1})$

And the model derived in the point (3) is the final error correction model that we should use. I have several queries about that

(1) What is exactly variable $y_{t-1}$ ? Let's say that y is a vector created by numbers from 1 up to 100. What exactly is $y_{t-1}$ ? It can't be just numbers from 1 to 99 because then $y_t$ and $y_{t-1}$ have different lengths and model cannot be created (the very first one)

(2) Why we are deriving the ecm model after claiming stationarity of $u_t$ ? If $x_t$ and $y_t$ is cointegrated why we do not just use standard model $y_t=\alpha_0+\alpha_1 x_t +\epsilon_t$ ? I understand that ECM is better than simply $\Delta y_t= \alpha_0 + \alpha_1 \Delta x_t + \epsilon_t$ but I couldn't find any explanation why it's better than standard model.

(3) What exactly is the interpretation of $\beta_2$ in the ECM (value next to $u_{t-1})$? What does it mean when $\beta_2$ is big or small ?

(4) Does ECM can be applied only to univariate regression ?

Answer 1

I'll answer very briefly. Most of your questions are answered in the texts of Lutkephol or Banerjee & Dolado.

1. The reason for building an ecm is that the original error term, $\epsilon_t$, in the original regression is not stationary. In your derivations, you never changed the error term from $\epsilon_t$ to $u_t$ and wrote the same thing each time. It's only when you make the equation I(0) on both sides that the TRANSFORMED error term becomes stationary. Also, one needs to test for non-stationarity of $\epsilon_t$ using the ADF or some other test.

2. Yes, one will lose a data point in the ECM because $y_t$ is one side and $y_{t-1}$ is on the other. There's not much that can be done about this. So, if you have 100 data points, the ECM will consist of 99 equations.

3. $\beta_2$ measures the amount of level reversion at each step. So, how much does the difference between $y_{t-1}$ and ( c+ $x_{t-1}$) cause $y_t$ to kick back in the other direction.

4. The VECM is the multivariate analogue of the ECM. I would check out Lutkepohl for that.

This is obviously a brief answer. The gory details are in the texts mentioned. I hope it helped some but I recommend reading those texts or atleast parts of them.