What are ARMA models and when should I use AR vs MA vs ARMA for CFA Level II?

ARMA models combine autoregressive (AR) and moving average (MA) components to capture different patterns in time series data. Understanding when to use each is key for CFA Level II.

The Three Model Types:

AR(p) — Autoregressive of order p: x_t = b0 + b1x_(t-1) + b2x_(t-2) + ... + bp*x_(t-p) + e_t

Uses p lagged values of the series itself. Appropriate when the current value depends on its own recent history.

MA(q) — Moving Average of order q: x_t = mu + e_t + theta1e_(t-1) + theta2e_(t-2) + ... + thetaq*e_(t-q)

Uses q lagged error terms. Appropriate when the effect of a shock dissipates after a finite number of periods.

ARMA(p,q) — Combined: x_t = b0 + b1x_(t-1) + ... + bpx_(t-p) + e_t + theta1e_(t-1) + ... + thetaqe_(t-q)

Captures both effects. More flexible but harder to estimate.

How to Choose (Practical Guidelines):

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The ACF/PACF Identification Rules:

Pattern	ACF	PACF	Model
AR(p)	Decays exponentially	Cuts off after lag p	Use PACF to determine p
MA(q)	Cuts off after lag q	Decays exponentially	Use ACF to determine q
ARMA(p,q)	Decays	Decays	Trial and error, use AIC/BIC

Example: You're modeling monthly changes in industrial production. The ACF shows significant values at lags 1 and 2 that then cut off. The PACF decays gradually. This pattern suggests an MA(2) model.

You estimate: Delta(IP)t = 0.3% + e_t + 0.45*e(t-1) + 0.22*e_(t-2)

Interpretation: a surprise shock to industrial production affects the series for two months before fully dissipating.

Model Validation: After fitting any ARMA model, check that residuals are:

1. Not autocorrelated (Ljung-Box test)
2. Homoscedastic (no ARCH effects)
3. Approximately normally distributed

If residuals show remaining patterns, the model is misspecified — try adding AR or MA terms.

For the CFA Exam: You'll most likely need to identify the appropriate model from ACF/PACF plots rather than estimate parameters. Know the identification rules in the table above.

Explore ARMA model examples in our CFA Level II course materials.