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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?

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Tail dependence coefficient is by definition non-negative. You need to formulate what do you mean by "negative dependence in the tails" as it's not obvious. –  Alexey Kalmykov Jan 22 '13 at 13:31

3 Answers 3

up vote 5 down vote accepted

Here is a working paper that you may be interested in.

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Yes very nice!! –  Jase Jan 22 '13 at 15:12

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.

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Yes, you have my question right, and very nice addition! Could you please define in more detail what $(k+1)$ is? Also, is there any research on using state space modelling to get a time-varying estimate $\hat{\beta}(t)$? –  Jase Aug 14 '13 at 9:54
During the estimation, $k$ is just some fixed number. For example 50 in case of 1000 daily returns or 100 in case of 2500 daily returns. Basically, it defines which part of the empirical distribution is considered to be 'in the tail of the distribution'. Theoretically (according to Extreme Value Theory), the parameter is chosen such that $k\rightarrow \infty$ and $k/n \rightarrow 0$ as $n \rightarrow \infty$. –  Q.F. Aug 14 '13 at 10:02

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.

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