Unit root

Summary

• Given: a theta function, \(θ\left( L \right)=1-θ_1L-θ_2L^2- \ldots -θ_pL^p\)
• We say that \(θ\left( L \right)\) has a unit root if the equation \(θ\left( x \right)=0\) has a solution \(x=1\) .
• If \(θ\left( L \right)\) has a unit root then \(θ\left( L \right)\) is not invertible.
• If \(θ\left( L \right)\) has a unit root then we can write

\[θ\left( L \right)=\left( 1-ϕ_1L \right)\left( 1-ϕ_2L \right) \ldots \left( 1-ϕ_{p-1}L \right)\left( 1-L \right)=θ^*\left( L \right)Δ\]

• where \(θ^*\left( L \right)=\left( 1-ϕ_1L \right)\left( 1-ϕ_2L \right) \ldots \left( 1-ϕ_{p-1}L \right)\) is a lag polynomial of order \(p-1\) and \(Δ=1-L\) .
• An ARMA(p,q) process \(θ\left( L \right)y_t=α(L)ε_t\) where \(θ\left( L \right)\) has a unit root is also called an ARIMA(p-1,1,q) which we can write as

\[θ^*\left( L \right)Δy_t=α(L)ε_t\]

• If \(θ^*\left( L \right)\) is invertible then \(Δy_t\) is a stationary process (if started at \(t=-∞\) ).
• Any nonstationary process which becomes stationary after first-differencing is said to be integrated of order 1, \(I\left( 1 \right)\) .
• Similarly, a process is \(I\left( 2 \right)\) if it is nonstationary but becomes stationary after first-differencing twice.