Estimating Negative Tail Dependence with Extreme-Value Indicators
Summary
The document asks whether copulas can capture negative tail dependence, meaning that one variable is extremely high while another is extremely low. It contrasts this cross-tail relationship with familiar lower-tail and upper-tail dependence measures, which describe joint extremes in the same direction. One response proposes estimating the cross-tail probability nonparametrically: select upper and lower sample thresholds, form indicators for observations beyond them, and regress the lower-tail indicator on the upper-tail indicator without an intercept. The slope estimates the tail association at those thresholds.
The discussion also points to parametric copula approaches and a paper on tail negative dependence, but provides no model specification or comparative evidence. The indicator method is a practical alternative to fitting a copula, though the document does not discuss threshold selection, uncertainty estimates, or finite-sample performance. Its usefulness therefore lies in defining the target and outlining an estimator, not in establishing which method is best.
Key ideas
- Negative tail dependence describes opposite-direction extremes, such as one variable being high while the other is low.
- An indicator regression without an intercept can estimate cross-tail association at chosen sample thresholds.
- The document also notes that parametric copula methods for negative tail dependence exist.
- It does not compare estimators or explain threshold choice and statistical uncertainty.
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Full text
# Is there a copula that can estimate negative tail dependence?
# Is there a copula that can estimate negative tail dependence?
I have encountered numerous copula estimators that can estimate time-invariant and time-varying linear and non-linear correlations on the interval $[-1,1]$, and these estimators are fully consistent with arbitrary univariate marginals and different forms of the bivariate joint distribution.
I have also encountered copulas (Gumbel, Clayton, and others) that can estimate time-varying lower and upper tail dependence on the interval $[0,1]$.
However, I believe that these tail dependence measures can only detect positive dependence.
Does there exist a time-invariant OR time-varying copula estimator that can detect negative dependence in the tails?
## Answer by GAM (score 5, accepted)
https://quant.stackexchange.com/a/7070
Here is a working paper that you may be interested in.
## Answer by Q.F. (score 3)
https://quant.stackexchange.com/a/8750
Do you refer with 'negative tail dependence' to the case that one variable has a extremely low value and the other random variable has an extremely large value, i.e.,
$$\tau=\lim_{p \rightarrow 0} \frac{Pr[x>Q_x(1-p),y<Q_y(p)]}{p},$$
where $Q_x(1-p)$ and $Q_y(p)$ refer to the $(1-p)$-th quantile of the random variable $x$ and the $p$-th quantile of the random variable $y$, respectively?
In this case, 'negative tail dependence' can easily be estimated non-parametrically by performing an ols regression.
With $n$ observations $x_1,\cdots,x_n$ and $y_1,\cdots,y_n$, the non-parametric estimate of $\tau$ can be obtained as $\hat{\beta}$ after performing an ols regression on the model $$\bf{1}_{y_t<Y_{k+1}}=\beta \bf{1}_{x_t>X_{n-k-1}},$$ where $\bf{1}$ denotes the indicator function for the condition in the subscript, and where $Y_{k+1}$ and $X_{n-k-1}$ denote the respectively the $(k+1)$-th lowest observation of $y_t$ and the $(k+1)$-th highest observation of x, respectively. Make sure not to include a constant in the regression.
For more information, see the article "The simple econometrics of tail dependence", Economics Letters 116(3), 371-373, http://dx.doi.org/10.1016/j.econlet.2012.04.016.
## Answer by Lei Hua (score 0)
https://quant.stackexchange.com/a/17801
There are different methods to get parametric copulas that have tail negative dependence.
You might be interested in the following paper: Tail negative dependence and its applications for aggregate loss modeling and the reference therein.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.