ARMA(p,q) process

Summary

• A stochastic process \(\{ Y_t \}\) is called an autoregressive process of order p or an AR(p) process if

\[Y_t=δ+θ_1Y_{t-1}+ \ldots +θ_pY_{t-p}+ε_t\]

• where \(δ\) and \(θ_1, \ldots ,θ_p\) are parameters and \(\{ ε_t \}\) is white noise.
• If \(Y_t\) follows an AR(p) process then \(y_t=Y_t-E\left( Y_t \right)\) will follow an AR(p) process with intercept 0

\[y_t=θ_1y_{t-1}+ \ldots +θ_py_{t-p}+ε_t\]

• A stochastic process \(\{ Y_t \}\) is called an moving average process of order q or an MA(q) process if

\[Y_t=μ+ε_t+α_1ε_{t-1}+ \ldots +α_qε_{t-q}\]

• where \(μ\) and \(α_1, \ldots ,α_q\) are parameters and \(\{ ε_t \}\) is white noise.
• If \(Y_t\) follows an MA(q) process, then \(y_t=Y_t-E\left( Y_t \right)\) will follow an MA(q) process with intercept 0

\[y_t=ε_t+α_1ε_{t-1}+ \ldots +α_qε_{t-q}\]

• A stochastic process \(\{ Y_t \}\) is called an autoregressive moving average process of order p and q or an ARMA(p,q) process if

\[Y_t=δ+θ_1Y_{t-1}+ \ldots +θ_pY_{t-p}+ε_t+α_1ε_{t-1}+ \ldots +α_qε_{t-q}\]

• If \(Y_t\) follows an ARMA(p,q) process then \(y_t=Y_t-E\left( Y_t \right)\) will follow an ARMA(p,1) process with intercept 0,

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