I'm confused about multistep forecasting from VECM model for 2 cointegrated series. The model is pretty simple, in error-correction form: $$ \Delta x_{t+1} = \alpha_1 (y_t - \beta x_t -\beta_0) +\varepsilon_{1,t}, \\ \Delta y_{t+1} = \alpha_2 (y_t - \beta x_t -\beta_0) +\varepsilon_{2,t}. $$
Where $y_t - \beta x_t -\beta_0$ is the long-term cointegrating relation and $\alpha$'s are short-term error correction coefficients.
Now, what I need is multistep forecast of $y_{t+k}$, conditional on $x_{t+k},\dots,x_{t}$. So $x$'s are known (including future) and I need forecast $y$'s $$\hat y_{t+k} = \mathbf{E}(y_{t+k} | x_{t+k},\dots,x_t).$$ As usual $y_{t+k}$ doesn't depend on next values of $x$, only on past.
First thought - Ok, just use long-run cointegration relation $$\hat y_{t+k} = \beta \hat x_{t+k} + \beta_0.$$ Here $\hat y, \hat x$ are conditional expectations. But this doesn't account for short-term error-correction dynamics and residuals $(y_t - \hat y_t)$ from these forecasts have long'ish periods of deviations from 0 - i.e. residuals have autocorrelation. Would be good to improve forecasts using short-term error-correction as well.
I try to do that using VECM formulation above. I illustrate this for 1-step forecast, but of course it's easy to iterate formulas and obtain k-step forecasts.
If I use only the second equation of the model, this gives: $$ \hat y_{t+1} = (1+\alpha_2)\hat y_t - \alpha_2 (\beta \hat x_t +\beta_0) $$ and doesn't look good because forecast for $y_{t+1}$ doesn't depend on $x_{t+1}$ - we don't use all available information for forecasting.
If I use both equations of the model and substitute one to the other, we have: $$\Delta \hat y_{t+1} = {\alpha_2 \over \alpha_1} \Delta \hat x_{t+1}.$$ Here $\hat y_{t+1}$ forecast depends on $\hat x_{t+1}$, but now it is only determined as short-term relation between differences via error-correction coefficients! So now long-term cointegration relation doesn't comе into play.
So the question is - how to derive conditional forecast for $y$ that uses both cointegration and error-correction dynamics (I guess the latter will affect only several first steps predictions) and also conditions on all information available?