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):

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