MATLAB: Arima : why MA polynomial needs be invertible

arima polynomial-inverse

When providing myself an initial point to arima.estimate(), I got the error: Error using arima/validateModel (line 1298) The non-seasonal moving average polynomial is non-invertible.
As far as I know, only the AR polynomial is required to be invertible. The MA part (with no common roots) needs not. For instance X(t)= u(t) + 1.5 u(t-1) is a stationnary MA(2) with non invertible polynomial for which classical estimation procedure work well.

Best Answer

MA models are required to be invertible because of non-uniqueness. For example, consider an MA(1) model:
X_t = W_t+\theta W_{t-1} where W_t \sim N(0,\sigma^2)
The autocorrelation at lag 1 is \dfrac{\theta}{1+\theta^2}
Given \theta = 5 or \theta = 1/5, you cannot tell the difference in the autocorrelation.
So we restrict \theta to a value that yields an infinite AR representation, which is \theta = 1/5.
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