Bivariate Copula Fitting with MLE

The Log-Likelihood of a Copula Model

Let's begin by constructing the objective function we want to maximize. Recall from Sklar's theorem that the joint probability density function (PDF) $f(x_1,x_2)$ can be expressed using the copula density $c(u_1,u_2)$ and the marginal PDFs $f_1(x_1)$ and $f_2(x_2)$:

where $u_1=F_1(x_1)$ and $u_2=F_2(x_2)$ are the marginal cumulative distribution functions (CDFs).

For a set of $n$ independent bivariate observations $\{(x_{1,i},x_{2,i}){\}}_{i=1}^n$, the total log-likelihood function $\ell$ is the sum of the log of the joint PDF for each observation. This naturally separates into three components:

Here, ${\theta }_{copula}$ represents the parameter(s) of the copula (e.g., $\theta$ for a Gumbel copula), and ${\theta }_{marg1},{\theta }_{marg2}$ represent the parameters of the marginal distributions (e.g., shape and scale for a Gamma distribution).

The following resource provides a formal statement of this log-likelihood function and introduces the R tools used to work with it.

Estimation Strategies

As the reading suggests, there are three primary strategies for maximizing the log-likelihood function.

1. Full Maximum Likelihood (One-Stage): This involves maximizing the entire log-likelihood function simultaneously with respect to all parameters (${\theta }_{copula}$, ${\theta }_{marg1}$, ${\theta }_{marg2}$). While theoretically the most efficient, this can be a complex, high-dimensional optimization problem that is computationally intensive and may struggle to converge.

2. Inference Functions for Margins (IFM) (Two-Stage): This is a more practical, widely-used approach that breaks the problem down:

   • Step 1: Fit each marginal distribution separately via MLE to get parameter estimates ${\hat{\theta }}_{marg1}$ and ${\hat{\theta }}_{marg2}$.
   • Step 2: Transform the original data into "pseudo-observations" on the unit square using the fitted marginal CDFs: ${\hat{u}}_{1,i}=F_1(x_{1,i};{\hat{\theta }}_{marg1})$ and ${\hat{u}}_{2,i}=F_2(x_{2,i};{\hat{\theta }}_{marg2})$.
   • Step 3: Maximize only the copula component of the log-likelihood with respect to ${\theta }_{copula}$, using the pseudo-observations from Step 2.

3. Canonical Maximum Likelihood (CML) (Semi-Parametric): This approach avoids specifying parametric forms for the marginal distributions altogether.

   • Step 1: Transform the data into pseudo-observations using their scaled ranks (i.e., the empirical CDF): ${\hat{u}}_{1,i}=\frac{\text{rank}(x_{1,i})}{n+1}$ and ${\hat{u}}_{2,i}=\frac{\text{rank}(x_{2,i})}{n+1}$.
   • Step 2: Maximize the copula log-likelihood using these rank-based pseudo-observations.

The CML method is robust and very common in practice because it isolates the estimation of the dependence structure from the choice of marginals. Most dedicated copula fitting software uses this as the default.

Choosing the Right Copula Family

Before we can fit the parameters, we need to select a suitable copula family. How do we decide whether a Gumbel, Clayton, Frank, or other copula is the best fit for our data? This is a model selection problem.

Visually, we're trying to find a copula family that generates a dependence pattern similar to our observed data. The image below illustrates this. The top-left plot shows the empirical pseudo-observations from a real dataset. The other plots show random samples from different copula families that have been fitted to this data. The goal of fitting is to find the family and parameter that best replicate the empirical pattern.

While visual inspection is helpful, we need a quantitative method. This is where information criteria like the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) come in. The AIC is defined as:

where $\ell$ is the maximized log-likelihood and $m$ is the number of parameters in the model. When comparing several fitted copula models, the one with the lowest AIC or BIC is preferred, as it represents the best trade-off between goodness-of-fit and model complexity.

Fitting Copulas in R: A Practical Walkthrough

The VineCopula package in R provides a powerful and user-friendly toolkit for fitting bivariate copulas, primarily using the semi-parametric CML approach. Let's walk through the typical workflow.

Let's summarize the key steps from the case study:

1. Prepare Data: The raw data is first transformed into pseudo-observations on $[0,1{]}^2$ using the pobs() function. This function computes the scaled ranks, implementing the first step of the CML method.

2. Select a Model: The BiCopSelect() function is the workhorse. You provide the pseudo-observations, and it automatically fits a wide range of bivariate copula families. It then returns the best-fitting model based on a specified criterion (e.g., familyset = "BIC"). The summary of the output shows the chosen family (e.g., BB8), the estimated parameters, the corresponding Kendall's tau, and the fit statistics (logLik, AIC, BIC).

3. Estimate Parameters (for a specific family): If you have a prior reason to use a particular copula family (e.g., you expect tail dependence in a certain corner), you can use BiCopEst() to fit only that one. The case study does this for the Survival Clayton copula for comparison. By comparing the AIC/BIC from this fit to the one chosen by BiCopSelect, you can confirm that the automatically selected model is indeed a better choice.

4. Goodness-of-Fit: Finally, functions like BiCopGofTest() can be used to formally test whether the chosen copula is a good fit for the data. These tests often check if the empirical dependence structure is statistically distinguishable from the one implied by the fitted copula.

A Note on Censored Data

In many actuarial and survival analysis contexts, the data is subject to censoring. The MLE framework is flexible enough to handle this. The contribution to the likelihood function for a censored observation is no longer a density but a probability, which can often be expressed using the copula function $C$ or the survival copula $\overline{C}$. The ISO data case study in the resource LINK provides a full example of this more advanced scenario.

Conclusion

You have now moved beyond the simple method of moments to the more robust and widely-used maximum likelihood estimation framework for fitting copulas. This approach leverages the full information in the data to provide efficient parameter estimates and a formal basis for model selection.

Key Takeaways:

• The log-likelihood for a copula model can be separated into a component for the dependence structure (copula) and components for each of the marginal distributions.
• Three main estimation strategies exist: Full MLE (one-stage), Inference for Margins (two-stage), and Canonical MLE (semi-parametric, based on ranks). The latter is the most common in practice for its robustness.
• Model selection between different copula families is typically done by comparing AIC or BIC values, with the lowest value indicating the preferred model.
• The VineCopula package in R provides a comprehensive suite of tools (pobs, BiCopSelect, BiCopEst) to efficiently fit and select the best copula model for a given dataset.

Preview of the Next Lesson

Once we have fitted a copula model and estimated its parameters, what can we do with it? A primary application in risk modeling is simulation. In our next lesson, we will learn how to simulate pairs of dependent random variables from a chosen copula model. This is a crucial skill for assessing the distribution of aggregate losses from dependent risks, calculating risk measures, and pricing complex insurance products.