Long run and short run effects in ADL models

Summary

In the ADL(0, $q$ ) model

$$y_t=β_1+β_2x_t+γ_1x_{t-1}+…+γ_qx_{t-q}+ε_t , t=1,…,T$$

• $β_2$ is the short-run effect of $x$ on $y$
• $β_2+γ_1+…+γ_q$ is the long-run effect of $x$ on $y$

In the ADL(1, $q$ ) model

$$y_t=β_1+β_2x_t+ρy_{t-1}+γ_1x_{t-1}+…+γ_qx_{t-q}+ε_t , t=1,…,T$$

• $β_2$ the short-run effect of $x$ on $y$
• The long-run effect of $x$ on $y$ is given by

$$\frac{β_2+γ_1+…+γ_q}{1-ρ}$$

It is possible to find the long-run effect in a general ADL( $p,q$ ) but the formula is more complicated.