How do AIC, BIC, and HQC differ in penalizing model complexity, and which should I use?

Information criteria balance goodness of fit (log-likelihood) against model complexity (number of parameters). They differ in how severely they penalize additional parameters, which leads to different model selection behavior.

Formulas:

All three have the form: IC = -2 x ln(L) + penalty

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

where k is the number of estimated parameters, n is the sample size, and L is the maximized likelihood.

Penalty Comparison:

For a model with k = 5 parameters and sample sizes:

Sample Size (n)	AIC Penalty	BIC Penalty	HQC Penalty
50	10	19.56	12.51
200	10	26.49	14.61
1000	10	34.54	16.27
10,000	10	46.05	18.53

AIC penalty is constant regardless of n. BIC grows with ln(n), becoming increasingly harsh with larger samples. HQC falls between them, growing with ln(ln(n)).

Practical Example:

Analyst Duncan at Greystone Partners evaluates three models for predicting GDP growth:

Model	Parameters (k)	-2 ln(L)	AIC	BIC (n=120)	HQC
AR(1)	2	185.4	189.4	191.0	189.9
AR(2) + Inflation	4	178.2	186.2	189.3	187.2
Kitchen Sink (8 vars)	9	170.8	188.8	198.9	192.0

AIC selects the AR(2) + Inflation model (186.2 is lowest). BIC also selects AR(2) + Inflation (189.3). But notice the Kitchen Sink model has the best fit (lowest -2 ln(L)) yet ranks last on all criteria because the penalty for 9 parameters outweighs its marginal fit improvement.

When They Disagree:

• BIC tends to select more parsimonious models, especially with large n
• AIC can overfit by selecting models with too many parameters
• BIC is consistent (selects the true model as n approaches infinity) if the true model is among the candidates
• AIC is efficient (minimizes prediction error) even if the true model is not among candidates
• HQC is consistent like BIC but penalizes less aggressively

CFA Exam Guidance: For the exam, know that BIC penalizes more heavily than AIC, leading to simpler model selection. Lower values are always preferred. Use AIC when prediction accuracy matters most; use BIC when you want the most parsimonious model.

Practice information criteria problems in our CFA Quantitative Methods question bank.