Why are we using ARMA model that mixes AR and MA model. Isn't AR or MA sufficient?

Question

Why are we using ARMA model that mixes AR and MA model. Isn't AR or MA sufficient?

I know that AR model is a function of previous readings and MA model is a function of previous errors, also know that identifying AR model is best done with PACF and know that identifying MA model is best done with ACF (and identifing AR model is best done with PACF)... But what is the intiuition behind using mixed model between the two

Answer 1

Short answer: The AR shows how the variable evolved with time, the MA makes the forecast error diminish with time.

It assumes :

• AR: x_t depends linearly of x_t-1, x_t-2,...x_t-p, where x_t is the variable of interest.

• AM: ε_t depends linearly of x_t-1, ε_t-1, ε_t-2,...ε_t-q, where ε_t is the error.

AR is a characteristic of the model, AM a characteristic of the error.