lets say now we have a set of S&P stock return in % and we would like to model a ma(1).

yes we checked the ACF and PACF and assumed it is a good fit.

the model itself is:

Observed Y = mean(u) + coefficient * Previous Error + Current Error

example:

Actual Y | Predicted Y | Error Term |

1.5% | ||

2% |

how do we get the error term?

Do we start from assuming the first predicted Y is mean(u) and 0 for previous error ? thus the error term will be = Actual - Predicted = 1.5% - ( mean(u) + 0 * coefficient )