The lag operator and the ARMA process

Summary

• If \(\{ Y_t \}\) follows an ARMA(p,q) process and \(y_t=Y_t-E\left( Y_t \right)\) then

\[y_t=θ_1y_{t-1}+θ_2y_{t-2}+ \ldots +θ_py_{t-p}+ε_t+α_1ε_{t-1}+ \ldots +α_qε_{t-q}\]

• Using the lag operator,

\[y_t=θ_1Ly_t+θ_2L^2y_t+ \ldots +θ_pL^py_t+ε_t+α_1Lε_t+ \ldots +α_qL^qε_t\]

• or

\[y_t-θ_1Ly_t-θ_2L^2y_t- \ldots -θ_pL^py_t=ε_t+α_1Lε_t+ \ldots +α_qL^qε_t\]

• or, factoring

\[\left( 1-θ_1L-θ_2L^2- \ldots -θ_pL^p \right)y_t=\left( 1+α_1L+ \ldots +α_qL^q \right)ε_t\]

• We define the theta function as the following polynomial of degree \(p\) :

\[θ\left( x \right)=1-θ_1x-θ_2x^2- \ldots -θ_px^p\]

• where \(θ_1, \ldots ,θ_p\) are scalars. \(θ\left( L \right)\) will then be an operator defined by

\[θ\left( L \right)=1-θ_1L-θ_2L^2- \ldots -θ_pL^p\]

• We define the alpha function as the following polynomial of degree \(q\) :

\[α\left( x \right)=1+α_1x+α_2x^2+ \ldots +α_qx^q\]

• where \(α_1, \ldots ,α_q\) are scalars. \(α\left( L \right)\) will then be an operator defined by

\[α\left( L \right)=1+α_1L+α_2L^2+ \ldots +α_qL^q\]

• ARMA processes using the operator polynomials:
• AR(p):

\[θ\left( L \right)y_t=ε_t\]

• MA(q):

\[y_t=α(L)ε_t\]

• ARMA(p,q):

\[θ\left( L \right)y_t=α(L)ε_t\]