When to use AR and when to use MA model? When to use an AR model and when to use an MA model to model time-series data. What aspects of data are modelled by the AR process which can't be done by MA and vice-versa?
 A: TL;DR

*

*You would choose an AR model if you believe that previous observations have a direct effect on the time series.

*You would choose an MA model if you believe that the weighted sum of differences (errors) have a direct effect on the time series.


To see what I mean with this, suppose you have a time series of the form $\{S_t\}_{t=1}^T$. Let's say that $S_t$ is the price of a stock at time $t$. Since $S_t$ is a random variable, we assume that it is normally distributed (stock prices cannot take negative values, as it is assumed on a normal distribution, but for the sake of the example, let's assume that this is true) with constant variance $\sigma^2$. If we were to model an AR(P) process, then $S_t$ would be distributed as
$$
S_t \sim \mathcal{N}\left(\psi + \sum_{p=1}^P \phi_pS_{t-p}, \ \sigma^2\right).
$$
As you can see, under an AR(P) process, we assume that the price at time $t$ has a mean that is given by the weighted sum of past observations plus a constant. Of course, the higher $\phi_p$, the greater effect a previous time-step has on what we think will be the current value of $S_t$. You can think this one as the weighted cumulative sum of previous returns, up to $Q$ periods.
On the other hand, if we were to model an MA(Q) process, then $S_t$ would be distributed as
$$
S_t \sim \mathcal{N}\left(\psi + \sum_{q=1}^Q \phi_pr_{t-q}, \ \sigma^2\right).
$$
If we denote $r_t$ as the return of the stock at time $t$, i.,e, $r_t= S_t - S_{t-1}$, then we can see that the MA(Q) process assumes that the current price has a mean that is given by the weighted sum of past returns plus a constant.
A: Generally , if the pure AR model has p coefficients AND the pure MA model has q coefficients select/use the AR model if p = < q OTHERWISE use the MA model.
