# ARDL/Error Correction Model: long vs. short run, restricted vs. unrestricted

I have a few questions about unrestricted error correction models.

The UECM for a model where $Y$ is the dependent variable and $x$ is the sole independent variable is given by $$\Delta Y_{t}=\alpha_{0}+\sum_{n=1}^{N}\beta_1\Delta y_{t-i}+\sum_{n=1}^{N}\beta_2\Delta x_{t-i}+\gamma_{1}y_{t-1}+\gamma_{2}x_{t-1}.$$ Here are my questions:

1. Why are the $\gamma$s considered long run coefficients and the $\beta$s considered the short run coefficients? What is the idea behind interpreting coefficients on level variables as long run and those on differenced variables as short run?

2. How would I interpret the short run and log run coefficients? For instance, the $\beta$ coefficients on a cross sectional OLS provide information on how a unit change in the independent variable changes the dependent variable. Would I be interpreting the long run and short run coefficients of the UECM in the same way? If so, what does the long and short term have anything to do with this interpretation? And what span of time is considered long vs. short term in the context of these model?

3. Why is this model called an unrestricted model? What about it makes it different from a regular error correction model?

$\gamma_1y+\gamma_2x$=0 which is the long run relationship between the variables. The $\gamma$'s define this long run relationship. The $\beta$'determine the short run adjustment to this equilibrium.
2.See 1. The $\beta$'s are a measure of the persistence of the variable. I don't think that you need to pay particular attention to individual values. In the context of the model the long run relationship can be interpreted as your panel equation. There is no set rule determining the short and long run. One can estimate the half life of a disturbance to equilibrium from the estimated coefficients. This will be different for every model