AR model: An autoregression mode is a regression of the variable against itself (past values of the forecast variable). An autoregressive model of order p, AR(p) can be written as yt=c+ϕ1yt−1+ϕ2yt−2+⋯+ϕpyt−p+et, where c is a constant and et is white noise.
MA model: In contrast to the AR model, a moving average model uses past forecast errors in a regression-like model. A moving average model of order q, MA(q) can be written as yt=c+et+θ1et−1+θ2et−2+⋯+θqet−q, where et is white noise.
In both cases, the error term is white noise. And from the formula above, we can clearly see how error terms are modeled differently in the two models.
In an AR model, the lagged values of yt are predictors. And the error term et in the model is just like the error term in a multiple linear regression.
In an MA model, the past forecast errors are predictors.
One thing to notice is that It is possible to write any stationary AR(p) model as an infinite MA model, and an (invertible) MA(p) can be written as an infinite AR.
