How do AIC and BIC work for model selection, and when would they disagree?

AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are model selection tools that balance goodness-of-fit against model complexity. Both prevent overfitting by penalizing extra parameters.

The Formulas:

AIC = -2 x ln(L) + 2k

BIC = -2 x ln(L) + k x ln(n)

Where:

• L = maximized likelihood of the model
• k = number of estimated parameters
• n = number of observations
• Lower values are better for both

Key Difference — Penalty Strength:

BIC's penalty is k x ln(n), while AIC's is 2k. When n > 7 (since ln(7) = 1.95 ~ 2), BIC penalizes complexity MORE heavily. For typical financial datasets (n = 250+ daily observations), BIC is far stricter:

Example — Ridgeport Quant Research:

Compare two GARCH models for S&P 500 volatility (n = 1,000 daily returns):

• AIC prefers GARCH(2,2) (2,856.2 < 2,856.6) — the small fit improvement justifies extra params
• BIC prefers GARCH(1,1) (2,877.4 < 2,891.7) — the heavier penalty rejects the extra complexity

When They Disagree:

This happens precisely when a more complex model offers a modest improvement in fit:

• AIC (lighter penalty) accepts the extra parameters
• BIC (heavier penalty) rejects them

In practice, BIC is consistent (it selects the true model if one exists in the candidate set), while AIC tends to select models with better prediction accuracy. For risk management, BIC is often preferred because overfitting is dangerous — an overfit VaR model will fail precisely when you need it most.

FRM Exam Tips:

• Both criteria: lower = better
• BIC penalizes more harshly for large samples
• If asked which favors parsimony: BIC
• Adjusted R-squared only works for nested linear models; AIC/BIC work for any MLE-based model

Test your model selection skills in our FRM Part I question bank.