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 ?