# Time Series - Autocovariance and Autocorrelation of a Moving Average Process

So an MA(3) process is: $Z^*_t =\Sigma^3_0 \psi_j a_{t-j}$ where $Z^*_t = Z_t - \mu$
The autocovariance function: $\gamma_j = E(Z^*_t Z^*_{t+k})= E(\Sigma^3_{i=0} \Sigma^3_{j=0} \psi_i \psi_j a_{t-i} a_{t+k-j}) = \sigma^2_a \Sigma^3_0 \psi_j \psi_{i+k}$
The autocorrelation function: $\rho_k = \frac{\Sigma^3_0 \psi_j \psi_{i+k}}{\Sigma^3_0 \psi^2_i}$.