An ARMA with no roots in the unit circle has a unique stationary solution, and it is of the form $sum_{j=-infty}^{infty} psi_j Z_{t-j}$, where the $Z_i$'s are white noise, and where $sum |psi_j|<infty$.

A stationary ARMA process with no roots in the unit circle is called *causal* if it can be further written as $sum_{j=0}^{infty} psi_j Z_{t-j}$, where the $Z_i$'s are white noise, and where $sum |psi_j|<infty$.

What is the virtue of a causal ARMA? In what way is it more helpful than merely having an ARMA with no roots in the unit circle?

For example, some people *define* an ARIMA as a time series that is a causal ARMA after $d$ many instances of differencing. Why causal? Why not define an ARIMA as being a time series that is an ARMA with no roots in the unit circle after $d$ many instances of differencing?

Bonus question: same question but for invertibility.

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#### Best Answer

What is the virtue of a causal ARMA?

Causal ARMA allows for real-time forecasting as it only relies on past values but not future values. Noncausal ARMA would rely on future values which are not available when forecasting the future. Since one of the main uses of ARMA is in forecasting, the causal property is a desirable one.