arima : why MA polynomial needs be invertible ?

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.

Answer 1

[Wayne King]

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.

[Nath]

Crystal clear. Thank you very much Wayne.