AIC, BIC, and Copula Model Selection

Why Use Information Criteria?

Copula candidates can describe dependence differently while fitting the same observed data. A more flexible candidate may attain a higher maximized log-likelihood, but it also uses more parameters. Information criteria summarize this fit-versus-complexity trade-off in a single score. They are useful for comparing fitted candidates on a consistent basis; they do not replace diagnostics or establish that a model captures the dependence relevant to a particular risk decision.

For a copula fitted to \(n\) observations, let \(\ell\) be its maximized log-likelihood and let \(k\) be the number of free parameters estimated under the likelihood being scored. The score is calculated from the log-likelihood, not the likelihood itself. A smaller score is preferred among the candidates being compared.

Definition: The Akaike information criterion (AIC) and Bayesian information criterion (BIC) are

\operatorname{AIC}=-2\ell+2k, \qquad \operatorname{BIC}=-2\ell+k\ln(n).

Here \(n\) is the number of observations used in the fit, \(\ell\) is the maximized log-likelihood, \(k\) is the number of estimated free parameters, and \(\ln\) is the natural logarithm. For either criterion, the lower score is preferred within a valid comparison.

In a copula-only fit with fixed or empirically constructed marginal inputs, \(k\) usually counts the estimated copula parameters. For example, a bivariate Gaussian copula with an estimated correlation has one copula parameter; a bivariate Student-\(t\) copula with estimated correlation and degrees of freedom has two. If the score is based on a full likelihood that estimates marginal parameters as well, those estimated parameters must also be included in \(k\). Count free parameters actually estimated, not parameters fixed in advance.

The criteria balance two quantities. The term \(-2\ell\) rewards better likelihood: for the usual case in which \(\ell\) is negative, a larger log-likelihood (less negative) gives a smaller fit term. AIC adds \(2k\); BIC adds \(k\ln(n)\). BIC’s penalty per parameter exceeds AIC’s when \(n>e^2\), so it will often favor a simpler candidate more strongly in ordinary sample sizes.

Making a Fair Comparison

Before comparing scores, check that the candidates were fitted to the same observations and response data, and that their likelihoods use the same basis. For instance, full likelihoods calculated using the same marginal treatment are a more direct comparison than a full likelihood for one candidate and a copula pseudo-likelihood for another. Do not compare values calculated from different samples or different likelihood conventions as if the scores were interchangeable.

For pseudo-likelihood fits based on pseudo-observations, candidates can still be ranked using the same stated scoring convention, sample, and data. However, the resulting criteria should be interpreted with care: standard information-criterion theory is tied to likelihood assumptions, and a pseudo-likelihood score is not automatically equivalent to a full-likelihood AIC or BIC. In every case, report what likelihood was used and which parameters were counted.

Worked Example: Gaussian and Student-t Copulas

A hypothetical analyst fits two copulas to the same \(n=100\) paired observations using the same likelihood convention and marginal treatment. The Gaussian copula has maximized log-likelihood \(\ell=-50.0\) and \(k=1\). The Student-\(t\) copula has \(\ell=-47.8\) and \(k=2\). Calculate both criteria and compare the rankings. The method is to calculate each model’s fit term first, then add its criterion-specific penalty. This avoids confusing a larger log-likelihood with a larger information-criterion score.

Gaussian copula. Its fit term is \(-2(-50.0)=100.0\). Thus

\operatorname{AIC}_{G}=100.0+2(1)=102.000.

For BIC, \(\ln(100)=4.605170\), so

\operatorname{BIC}_{G}=100.0+1(4.605170)=104.605.

Student-\(t\) copula. Its fit term is \(-2(-47.8)=95.6\). Thus

\operatorname{AIC}_{t}=95.6+2(2)=99.600.

Its BIC is

\operatorname{BIC}_{t}=95.6+2(4.605170) =95.6+9.210340 =104.810.

AIC prefers the Student-\(t\) copula because \(99.600<102.000\). BIC prefers the Gaussian copula because \(104.605<104.810\). The Student-\(t\) fit improves the log-likelihood, but its extra parameter more than offsets that gain under BIC. The scores identify different fit-complexity trade-offs; neither result by itself confirms which model has adequate tail behavior.

Why the Penalties Change the Ranking

The comparison can be made directly when one candidate adds parameters to another. Suppose model 2 has \(q\) more estimated parameters than model 1, and its log-likelihood improves by \(g=\ell_2-\ell_1\). The difference between the models’ AIC scores is

\operatorname{AIC}_2-\operatorname{AIC}_1=-2g+2q.

Therefore, AIC favors model 2 exactly when \(g>q\): the log-likelihood gain must exceed the number of added parameters. For BIC, the difference is

\operatorname{BIC}_2-\operatorname{BIC}_1=-2g+q\ln(n),

so BIC favors model 2 only when \(g>q\ln(n)/2\). This comparison follows simply by subtracting the two score formulas; it makes clear why the same improved fit can win under AIC but lose under BIC. As \(n\) grows, BIC requires a larger log-likelihood gain for every additional parameter.

The factor \(-2\ell\) places the log-likelihood on the deviance scale: a log-likelihood improvement of \(g\) reduces that part of the score by \(2g\). The penalties prevent the maximized likelihood alone from automatically favoring the more flexible model. In broad terms, AIC is motivated by estimating predictive information loss, while BIC has a stronger large-sample penalty associated with model selection under its assumptions. Neither interpretation makes the selected candidate a guaranteed description of the true dependence.

Worked Example: AIC and BIC Select Different Candidates

Two hypothetical copula candidates are fitted to the same \(n=200\) observations under the same likelihood convention. Candidate 1 has \(\ell_1=-90.2\) and \(k_1=2\); candidate 2 has \(\ell_2=-88.9\) and \(k_2=3\). Calculate both scores. The method is to keep the fit term and penalty separate through the arithmetic; this prevents a parameter penalty from being accidentally omitted or added to the wrong candidate.

For candidate 1, the fit term is \(-2(-90.2)=180.4\). Therefore,

\operatorname{AIC}_1=180.4+2(2)=184.400.

Using \(\ln(200)=5.298317\),

\operatorname{BIC}_1=180.4+2(5.298317) =180.4+10.596634 =190.997.

For candidate 2, the fit term is \(-2(-88.9)=177.8\). Therefore,

\operatorname{AIC}_2=177.8+2(3)=183.800,

and

\operatorname{BIC}_2=177.8+3(5.298317) =177.8+15.894951 =193.695.

AIC prefers candidate 2 because \(183.800<184.400\). BIC prefers candidate 1 because \(190.997<193.695\). Candidate 2 improves the log-likelihood by \(1.3\) while adding one parameter. The gain exceeds the one-parameter AIC threshold of \(1\), but not the BIC threshold of \(\ln(200)/2=2.6492\). The rankings agree with those thresholds.

Worked Comparison With a Larger Sample

Worked Example: Interpreting the Size of the BIC Penalty

Two hypothetical candidates are fitted to the same \(n=500\) paired observations. Candidate A has \(\ell_A=-312.4\) and \(k_A=2\); candidate B has \(\ell_B=-309.9\) and \(k_B=4\). Calculate each criterion and explain what drives any disagreement. Both are scored using the same stated likelihood convention. The method is to compute each model independently, then compare the two score differences rather than relying only on which log-likelihood is largest.

Candidate A has fit term \(-2(-312.4)=624.8\). Its scores are

\operatorname{AIC}_A=624.8+2(2)=628.800,

\operatorname{BIC}_A=624.8+2\ln(500) =624.8+2(6.214608) =637.229.

Candidate B has fit term \(-2(-309.9)=619.8\). Its scores are

\operatorname{AIC}_B=619.8+2(4)=627.800,

\operatorname{BIC}_B=619.8+4\ln(500) =619.8+4(6.214608) =644.658.

AIC prefers candidate B by \(1.000\), whereas BIC prefers candidate A by \(7.429\). Candidate B gains \(g=2.5\) log-likelihood units with \(q=2\) extra parameters. Since \(2.5>2\), AIC favors B. The BIC threshold is \(q\ln(500)/2=6.214608\), and \(2.5<6.214608\), so BIC favors A. The calculations also check directly: \(644.658-637.229=7.429\), favoring A under BIC.

Information Criteria Do Not Test Tail Adequacy

AIC and BIC reduce a fitted model’s likelihood and parameter count to one number. They do not show which observations drive the likelihood, whether extreme co-movements are reproduced, or whether the candidate’s tail-dependence structure is appropriate for the intended risk question. A candidate can rank well overall while still giving an inadequate account of joint extremes. A criterion ranking is therefore evidence for a relative fit-complexity comparison, not a tail diagnostic or a validation result.

This distinction matters in risk management because the dependence in the center of the data and the dependence in joint extremes need not be alike. AIC or BIC cannot establish that simulated joint losses, stress outcomes, or tail probabilities are reliable. Use the relevant goodness-of-fit diagnostics and examine tail behavior separately; the earlier material on goodness-of-fit diagnostics for copula models is useful for that purpose. A model-selection score can help narrow candidates, but it does not settle the risk-modeling question.

Common Exam Traps

Reversing the sign of the fit term. If \(\ell=-20\), then \(-2\ell=-2(-20)=40\), not \(-40\). The negative log-likelihood contribution is positive in this case.
Using \(2k\) for both criteria. AIC uses \(2k\); BIC uses \(k\ln(n)\). Substituting one penalty for the other can change the preferred model.
Forgetting the sample size in BIC. BIC uses the natural logarithm of \(n\), not \(n\) itself and not \(\log_{10}(n)\).
Counting parameters inconsistently. Count all free parameters estimated in the likelihood being scored. Do not count a fixed parameter, and do not omit estimated marginal parameters from a full likelihood.
Choosing the largest log-likelihood as the automatic winner. The information criterion also charges for complexity; the lower criterion, not the larger \(\ell\), is preferred.
Calling the winner “the correct tail model.” A lower information criterion is not proof of tail adequacy, correct specification, or suitability for a particular capital calculation.

Key takeaway: Calculate AIC as \(-2\ell+2k\) and BIC as \(-2\ell+k\ln(n)\), using consistent data, likelihood conventions, parameter counts, and sample size. Prefer the lower score for that criterion, then assess tail adequacy separately.

Check Your Understanding

For each calculation, state the preferred candidate under the requested criterion. Scores are rounded to three decimals where needed.

Two copulas are fitted to the same \(n=80\) observations. Candidate A has \(\ell=-36.2,\ k=1\); candidate B has \(\ell=-34.9,\ k=2\). Calculate both AIC and BIC scores and identify each preference. (Answer: AIC A \(74.400\), B \(73.800\), prefers B; BIC A \(76.782\), B \(78.564\), prefers A.)
For the same \(n=250\) observations, candidate 1 has \(\ell=-120.0,\ k=2\), and candidate 2 has \(\ell=-118.7,\ k=3\). Calculate the criteria and identify their preferred candidates. (Answer: AIC 1 \(244.000\), 2 \(243.400\), prefers 2; BIC 1 \(251.043\), 2 \(253.964\), prefers 1.)
Two candidates use the same likelihood convention and \(n=64\). Model X has \(\ell=-28.0,\ k=1\), and model Y has \(\ell=-26.9,\ k=3\). Calculate both scores and state whether the rankings agree. (Answer: AIC X \(58.000\), Y \(59.800\); BIC X \(60.159\), Y \(66.277\); both prefer X.)
Two models are fitted to the same \(n=120\) pseudo-observations under the same pseudo-likelihood convention. Model P has \(\ell=-42.5,\ k=2\); model Q has \(\ell=-40.8,\ k=3\). Calculate both rankings and state one limitation of interpreting them. (Answer: AIC P \(89.000\), Q \(87.600\), prefers Q; BIC P \(94.575\), Q \(95.962\), prefers P; pseudo-likelihood criteria require care and do not establish tail adequacy.)
A model with one additional parameter improves the maximized log-likelihood by \(g=2.1\), and the sample size is \(n=400\). Determine whether the more complex model is favored by AIC and BIC using the comparison thresholds. (Answer: AIC favors it because \(2.1>1\); BIC does not because \(2.1<\ln(400)/2\approx2.996\).)

What You'll Learn