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.