Interpreting Copula Spread Through Conditional Density
Summary
The document asks how to describe the apparent spread of points in copula scatter plots when comparing dependence models fitted to asset log returns. The motivating observation is that a Clayton copula fit worse by AIC than Normal and Gumbel alternatives, despite an expectation that its lower-tail dependence might suit crash co-movement. The question also notes that tail dependence alone may not describe how broadly points are distributed in other regions.
The response interprets spread as dispersion in the bivariate copula plot and connects that dispersion to the copula density: regions with density near the uniform baseline tend to have more evenly distributed points, while concentrated density produces clustered points. It suggests examining density conditional on being in the upper tail and comparing it with a constant uniform density. This is a qualitative diagnostic, not a specified scalar metric or formal test, and the document offers no calculation or empirical validation showing that it explains the cited AIC ranking.
Key ideas
- A copula's density describes how probability mass is distributed across its support.
- Scatter may appear more uniform where density is near the uniform baseline and more clustered where density concentrates.
- Conditional tail density can be compared with uniform density to inspect tail-region dispersion.
- Tail dependence coefficients alone may not describe the full shape of dependence across a copula.
- The suggested comparison is a diagnostic idea, not a validated standalone spread metric.
Tags
Full text
# Metric for measuring the "spread" of a copula # Metric for measuring the "spread" of a copula I am fitting copula to log returns data for my undergraduate thesis, and comparing the quality of the fit with AIC. One interesting thing that I found is that the Clayton copula, which has negative tail dependence, provides a worse fit then the Normal and Gumbel copulas, which have no and positive tail dependence, respectively. It seems that a copula family which has negative tail dependence should have a better fit because markets are more correlated on crashes. I looked at the support set of the Clayton copula, and it has a large degree of "spread" on the positive tail. I believe that this is the reason why the Clayton copula fails to fit the returns data well. However, I can't find a copula-based metric that explains this. Tail dependence would be the most obvious candidate, but it doesn't really explain this behavior (compare the negative tails of a Gumbel copula with the positive tail of a Clayton copula, both have 0 dependence at those tails but the spread is vastly different). I was wondering if there was any alternative metric that could be used to explain this. Thanks! ## Answer by g g (score 1) https://quant.stackexchange.com/a/45198 I assume with "spread of a copula" you mean the spread/dispersion of points in a bivariate scatter plot of the copula. This spread is related to the density of the copula. In an area where the copula is near uniform, scatter will be somewhat uniform. Obviously in areas where the density is concentrated, scatter will be concentrated. You can verify this by comparing the density of the copula conditional on being in the upper tail with the uniform density, i.e. a constant.
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