Back to course

AR, MA, and ARIMA Models

The two components

Autoregressive: today depends on recent values.

The AR(p) model
Xt=ϕ1Xt−1+⋯+ϕpXt−p+εtX_t = \phi_1X_{t-1} + \cdots + \phi_pX_{t-p} + \varepsilon_t

Today from its own recent past plus a shock. The coefficients decide whether it reverts or wanders.

Moving average: today depends on recent shocks.

Xt=εt+θ1εt−1+⋯+θqεt−qX_t = \varepsilon_t + \theta_1\varepsilon_{t-1} + \cdots + \theta_q\varepsilon_{t-q}

The distinction is subtle and matters for interpretation. AR shocks persist indefinitely, decaying geometrically; MA shocks vanish completely after qq periods.

ARIMA(p,d,qp,d,q) adds dd rounds of differencing to handle non-stationarity, and d=1d=1 (modelling changes rather than levels) covers most financial cases.

Choosing pp and qq

The classical procedure uses two diagnostics:

ACF cuts off sharply after lag qq for an MA(qq) process.

PACF cuts off sharply after lag pp for an AR(pp) process.

In practice, fit several candidates and compare by AIC or BIC, then validate out of sample.

The rest of this lesson is for subscribers

Unlock every lesson in Advanced Topics in Probability and Statistics, and every other premium course.

Subscribe to continue

Test your knowledge

Questions are only available to subscribers.
Continue
NextStationarity and Autocorrelation

Keep reading Advanced Topics in Probability and Statistics

35 lessons in this course, and every other premium course, on one subscription.

Subscribe now
  • Every lesson in every course, with the worked examples and interactive simulators
  • Graded questions on every lesson, with explanations for the wrong answers as well as the right one
  • The trainers, timed assessments and brainteaser library that go with them