AR, MA, and ARIMA

Linear models for time-dependent data.

Three linear time-series models cover most of the territory.

An AR( $p$ ) model regresses the current value on the past $p$ values:

X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + \epsilon_t

An MA( $q$ ) model expresses the current value as a moving average of the last $q$ shocks:

X_t = \epsilon_t + \theta_1 \epsilon_{t-1} + \cdots + \theta_q \epsilon_{t-q}

ARMA( $p, q$ ) combines both. ARIMA( $p, d, q$ ) first differences the series $d$ times to make it stationary before fitting ARMA.

Box-Jenkins methodology: identify orders via ACF/PACF, fit by maximum likelihood, then diagnose residuals for any leftover structure. In quant trading, AR models often appear in mean-reversion strategies on residuals from a hedge — even short half-lives translate to real edge if execution is cheap enough.