Distinguish between short run and long run effects

Question

I read in a paper the following sentence:

The fact that there is a difference between short-term and long-term coefficients is a result of our specification which includes lagged endogenous variables.

They run a regression in first differences and include a lag of the dependent variable.
Now they argue, that if you look at an estimate (e.g. lets call this estimate $p$) from the output, that this is the short run effect of $p$ on the dependent variable.
Further they argue that looking at $p$ / (1 - estimate for the lag) gives the long run effect of p on the dependent variable.

The paper can be found: https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp1328.pdf and their discussion about short/long run effect on page 20 in the footnote 23.

I don't exactly understand why you can differentiate between the short and the long run effect of $p$ on the dependent variable. If someone could explain their idea more detailed it would be very helpful.

Answer 1

Suppose you have a model $$y_t=\alpha+\beta y_{t-1}+\gamma x_t+\varepsilon_t.$$ $\gamma$ measures the instantaneous effect (or the short-term effect) of $x_t$ onto $y$.

Note that $y_{t-1}$ is included in the model. Since $x_t$ has an effect on $y_t$, $x_t$ will also have an effect on $y_{t+1}$ through the lagged dependent variable, and the size of this effect will be $\beta \gamma x_t$.

The story does not end here. The effect of $x_t$ on $y_{t+2}$ will be $\beta^2 \gamma x_t$. The effect of $x_t$ on $y_{t+3}$ will be $\beta^3 \gamma x_t$. And so on, and so forth. If you sum up the instantaneous effect and all the delayed effects all the way to the infinite future, you will obtain the cumulative effect of $x_t$ onto $y$ which will be $\frac{1}{1-\beta}\gamma x_t$ (where you use a formula for the infinite sum of a decaying geometric series, see Wikipedia). That is what is called the long-term effect.

The model above can be generalized to more complex lag structures, but the idea remains the same; lagged dependent variables perpetuate an effect into infinite future.