How does the ARIMA model work for time series forecasting in risk management?

ARIMA stands for AutoRegressive Integrated Moving Average and is one of the most important time series models in the FRM curriculum. It combines three components to model and forecast data that may not be stationary.

The Three Components

1. AR(p) — AutoRegressive of order p: The current value depends on p previous values.

Y_t = phi_1 Y_{t-1} + phi_2 Y_{t-2} + ... + epsilon_t

1. I(d) — Integrated of order d: The series is differenced d times to achieve stationarity. If the original series has a unit root (non-stationary), taking first differences (d=1) often makes it stationary.

If Y_t is non-stationary, define Z_t = Y_t − Y_{t-1} and model Z_t instead.

1. MA(q) — Moving Average of order q: The current value depends on q past error terms.

Y_t = epsilon_t + theta_1 epsilon_{t-1} + theta_2 epsilon_{t-2} + ...

Choosing (p, d, q)

Practical Example

Elmwood Asset Management wants to forecast monthly credit spreads for their corporate bond portfolio:

1. Stationarity test: The Augmented Dickey-Fuller test on raw credit spreads gives p=0.23 (non-stationary). First-differencing yields p=0.001 (stationary). So d=1.
2. PACF analysis: The PACF of differenced spreads shows significant spikes at lags 1 and 2, then cuts off. So p=2.
3. ACF analysis: The ACF shows a significant spike at lag 1 only. So q=1.
4. Model: ARIMA(2,1,1). The fitted model is:

Delta_S_t = 0.35 Delta_S_{t-1} − 0.18 Delta_S_{t-2} + epsilon_t + 0.22 * epsilon_{t-1}

1. Forecast: Plug in the last two spread changes and the last residual to project next month's spread change.

Common FRM Exam Traps:

• Forgetting to check stationarity before fitting AR/MA
• Confusing ACF (for MA order) with PACF (for AR order)
• Over-differencing (d=2 when d=1 suffices) introduces unnecessary noise

Practice more time series problems in our FRM quantitative analysis question bank.