What are the differences between long and short term impacts in ARDL model? Here I am using ARDL as suggested by Pesaran and Shin, 1999 to deal with variables that are integrated of a different order. They derive, assuming you reject the null of no cointegration, long and short term effects (the later through an error correction model). But I don't know how to interpret these, are they similar to the interpretation of slopes in say OLS? And what is the practical difference between short term and long term effects (short and long term relationships)? I am not really sure what these terms mean in practice.
 A: Hi: Suppose you had a simple ARDL(1,1). ( they might refer to this as a (0,1). I forget tthe convention )
$y_{t} = \rho y_{t-1} + \beta x_{t} + \epsilon_t$.
This is probably the simplest ARDL that there is and it's also called a Koyck distributed lag model. But the concept carries over to the more general case.
The short term effect of $x_{t}$ in this model is $\beta$ because a 1 unit impulse in $x_{t}$ increases the response by $\beta$ immediately.
To calculate the long term effect, one can re-write the model as follows:
$y_{t} = \frac{x_{t}}{(1 - \rho L)} +  \frac{\epsilon_{t}}{(1-\rho L)} = $
$ = \beta \sum_{i=0}^{\infty} \rho^{i} x_{t-i} +  \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i}  $
The infinite sum of the first term on the RHS can be written as $\frac{1}{(1-\rho)}$. Therefore, the long term effect of a one unit impulse in $x_{t}$ is $\frac{\beta}{(1-\rho)}$.
The pdf at this link contains more of the details on this.
https://www.reed.edu/economics/parker/312/tschapters/S13_Ch_3.pdf
