When do we say a process is an invertible ma process?
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heya
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Invertibility is not really a big deal because almost any Gaussian, non-invertible MA(q) model can be changed to an invertible MA(q) model representing the same process by changing the parameter values. This is mentioned in most textbooks for the MA(1) model but it is true more generally.
As an example, consider the MA(2) model
zt=(1−0.2B)(1−2B)wt,(1)
where wt is white noise with variance σ2w. This is not an invertible model because θ(B) has one root equal to 0.5 inside the unit circle. However, consider the alternative MA(2) model obtained by changing this root to its reciprocal value of 2 such that model takes the form
zt=(1−0.2B)(1−0.5B)w′t(2)
where w′t has variance σ′2w=4σ2w. You can easily verify that models (1) and (2) both have the same autocovariance functions and hence specify the same distribution for the data if the process is Gaussian.