19.6 MA(q)

In these models the stationary series \((Y_t)\) is the result of adding a fixed value \((Y_t)\) plus a linear combination of present and past shocks of zero mean and constant variance:

\[ Y_t=\mu+\omega_t+\theta_1\omega_{t-1} +\theta_2\omega_{t-2} + \ldots + \theta_q\omega_{t-q} \]

  • The number of lags of the $ _t$ determine the order \(q\) of the process.
  • It will be found that \(Y_t\) fluctuates around its mean \(mu\).
  • The connection with the past depends on the order \(q\).

\[ \begin{aligned} MA(1) &:& Y_t &=\mu+\omega_t + \theta_1\omega_{t-1}; \quad \omega_t \sim WN(0,\sigma^2_\omega) \\ MA(2) &:& Y_t &=\mu+\omega_t + \theta_1\omega_{t-1} + \theta_2\omega_{t-2}\\ MA(q) &:& Y_t &=\mu+\omega_t + \theta_1\omega_{t-1} + \ldots + \theta_q\omega_{t-q} \end{aligned} \]