ADL(p,q) model

Summary

Setup:

• Given: time series data on a dependent variable \(y\) and one explanatory variable \(x\) .
• \(y_1,…,y_T\) and \(x_1,…,x_T\) are stationary processes.
• The static LRM can be formulated

\[y_t=β_1+β_2x_t+ε_t , t=1,…,T\]

Lagged variables

• If we believe that \(y_t\) is directly affected by the value of the explanatory variable in the previous period, \(x_{t-1}\) , we can specify a dynamic LRM

\[y_t=β_1+β_2x_t+θx_{t-1}+ε_t , t=1,…,T\]

• \(x_{t-1}\) is called a lagged explanatory variable.
• If we believe that \(y_t\) is directly affected by the value of the dependent variable in the previous period, \(y_{t-1}\) , we can specify our LRM

\[y_t=β_1+β_2x_t+ρy_{t-1}+ε_t , t=1,…,T\]

• \(y_{t-1}\) is called a lagged dependent variable which is then an explanatory variable in our LRM.

ADL(p,q)

• If we believe that \(y_t\) is directly affected by the values of the dependent and explanatory variable even further back in time we can assume the LRM

\[y_t=β_1+β_2x_t+θ_1x_{t-1}+…+θ_qx_{t-q}+ρ_1y_{t-1}+…+ρ_py_{t-p}+ε_t , t=1,…,T\]

• \(q\) is called the lag length of the explanatory varriable while \(p\) is the lag length of the dependent variable.
• The model can easily be extended to include several explanatory variables and they can all have different lag lengths.
• We say that \(y\) follows an ADL( \(p,q)\) (Autoregressive Dynamic Lag) model if it can be modeled as a linear regression model where \(p\) is the lag length of the dependent variable and \(q\) is the maximum lag length of the explanatory variables.

Most common ADL models

ADL(0,0), static model:

\[y_t=β_1+β_2x_t+ε_t t=1,…,T\]

ADL(1,0):

\[y_t=β_1+β_2x_t+ρy_{t-1}+ε_t t=1,…,T\]

ADL(0,1):

\[y_t=β_1+β_2x_t+θx_{t-1}+ε_t t=1,…,T\]

ADL(1,1):

\[y_t=β_1+β_2x_t+θx_{t-1}+ρy_{t-1}+ε_t t=1,…,T\]