What are AR and MA models and when do you use each for financial time series?

AR and MA models are the building blocks of time series analysis in risk management. They capture different patterns in how a variable relates to its own history.

Autoregressive (AR) Model — AR(p):

The current value depends on its own past values:

Yt = c + φ₁Yt-1 + φ₂Yt-2 + ... + φpYt-p + εt

• Intuition: Tomorrow's volatility is partly determined by today's and yesterday's volatility
• Stationarity: Requires |φ| < 1 for AR(1)
• ACF pattern: Decays gradually (exponentially)
• PACF pattern: Cuts off sharply after lag p

Moving Average (MA) Model — MA(q):

The current value depends on past error terms:

Yt = c + εt + θ₁εt-1 + θ₂εt-2 + ... + θqεt-q

• Intuition: The effect of a shock dissipates over q periods
• Always stationary (no restrictions on θ)
• ACF pattern: Cuts off sharply after lag q
• PACF pattern: Decays gradually

ARMA(p,q) — Combined model:

Yt = c + φ₁Yt-1 + ... + φpYt-p + εt + θ₁εt-1 + ... + θqεt-q

Combines both past values AND past errors.

How to identify which model to use:

Financial examples:

AR(1) for volatility: Today's stock return volatility is related to yesterday's:

σt = 0.02 + 0.85σt-1 + εt

The 0.85 coefficient means volatility is highly persistent — high-vol days tend to be followed by high-vol days.

MA(1) for return shocks: A market shock affects returns for one additional period:

Rt = 0.001 + εt + 0.3εt-1

A -2% unexpected shock today contributes -0.6% to tomorrow's return.

Model selection criteria:

• AIC (Akaike Information Criterion): Balances fit vs. complexity — lower is better
• BIC (Bayesian Information Criterion): Penalizes complexity more than AIC
• Ljung-Box test: Checks if residuals are white noise (no remaining autocorrelation)

Exam tip: FRM Part I tests your ability to identify model order from ACF/PACF plots, interpret coefficients, and apply stationarity conditions. Practice reading correlograms.

Practice time series analysis on AcadiFi's FRM platform.