# What is the virtue of a causal ARMA?

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.

• What do you think of my answer? Is it satisfactory? I am asking since you have neither accepted it (which can be done by clicking on the tick mark to the left) nor commented on it. Nov 19 '18 at 16:04