Question
For the following MA (3) process y t  =Â
μ  + Ε t  + θ 1 Ε t -1  + θ 2 Ε t -2  + θ 3 Ε t -3  , where σ t  is a zero mean white noise process with variance σ 2Solution
MA(q) process only has memory of length q. This means that all of the autocorrelation coefficients will have a value of zero beyond lag q. This can be seen by examining the MA equation, and seeing that only the past q disturbance terms enter into the equation, so that if we iterate this equation forward through time by more than q periods, the current value of the disturbance term will no longer affect y. Finally, since the autocorrelation function at lag zero is the correlation of y at time t with y at time t (i.e. the correlation of y_t with itself), it must be one by definition.
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