# Journal of Operational Risk

**ISSN:**

1744-6740 (print)

1755-2710 (online)

**Editor-in-chief:** Marcelo Cruz

####
Need to know

- The paper compares different approaches that are either meant to measure the tail indexes or to test for tail (in)dependence.
- It shows the limitations when willing to measure or test for tail dependence in small samples
- It enables practitioners to understand what are the main parameters involved in this kind of analysis
- It provides a framework to get benefits from that notion, even when considering the bad theoretical properties of the approaches studied in the paper.

####
Abstract

Tail dependence is a probability-based concept meant to address the challenge of detecting and modeling the extreme comovements that can be observed in many real-life situations. Huge financial losses for a bank, floods and epidemics are obvious instances of such extreme comovements. Like extreme value theory in the univariate case, tail dependence depends on asymptotic theory. Therefore, the statistical assessment of tail dependence faces exactly the same problem as extreme value theory: a scarcity of extreme event observations. In the field of dependence modeling, copulas have stood out as a tool of singular importance. They are widely used to account for the various dependence structures that can be encountered in real life. In 2009, Genest *et al *provided a series of tests to achieve copula selection but showed that these tests were not greatly powerful. This is all the more true when it comes to selecting a copula where tail dependence is crucial. In this paper, we suggest the use of tail indexes in order to detect the presence of tail dependence in a given data set and thus improve the process of selecting a copula. Because tail dependence often goes with data scarcity, we focus on this specific issue through an application to operational losses in the banking industry and propose a way to apply the benefits from theory in practice, while being conscious of the boundaries of such a notion.

####
Introduction

## 1 Introduction

Tail dependence has been studied considerably in the past few years, but the conditions relying on its estimation and the performance of the tools leading to estimate it or test for it are not always obvious. Many parametric and nonparametric estimators have been proposed. In particular, the latter were introduced due to the use of empirical copulas, and multivariate extreme value theory has been the basis of many papers wanting to estimate tail dependence. For example, Frahm et al (2005) and Schmidt and Stadtmüller (2005) proposed estimators that rely on both the extreme statistics and the notion of tail copulas. Schmidt and Stadtmüller suggested that tail dependence is observable at extreme values and consequently worked under the extreme value theory to detect the multivariate extreme threshold above which a nonparametric estimator can be used to measure tail dependence. Frahm et al, in a similar way, built several estimators (parametric, semiparametric and nonparametric) depending on the available information (class of copulas, marginal distributions, etc) to measure this dependency. Both studies pointed out that the estimated tail coefficients are biased when there are less than 250 observations, and several studies showed that these methods cannot be applied without any other measurements or tests, since they generate a bias–variance trade-off that could be misleading. Malevergne and Sornette (2004) proposed an approach that relies not on the multivariate extreme value theory but on the specification of a factor model between random variables. This approach is very restrictive and requires a priori calibration of the linear factor model. Finally, Caillault and Guegan (2007) suggested using the specific properties of the tail dependence by selecting the value for which the behavior of the empirical copula trajectory changes; in addition, a bootstrap is used and confidence intervals are built in order to refine the measurement. They specify that this method cannot enable a copula to be selected, but can only be used to figure out the tail properties. Finally, all these methods try to estimate the tail coefficient, but they all conclude that the results are not fully reliable, mostly when dealing with small data sets.

Statistical testing of tail dependence is another approach that allows us to focus on its detection rather than its measure. It seems an interesting way to deal with this theory, since it can exhibit the presence of extreme dependencies and potential symmetries, giving us clues about how to select the right copula. Most of the methods dealing with such issues, which characterize asymptotic independence and provide the statistical tests to assess an asymptotic dependence, are presented in de Carvalho and Ramos (2012).

Thus, our paper focuses on two approaches to deal with tail dependence and define the best way to benefit from this notion in practice (selecting a copula, for instance). It also provides a quantitative comparison of the performance of different approaches and sets the conditions of application when seeking to use tail dependence theory in practice. Moreover, a new test is proposed, derived from the Caillault and Guegan (2007) estimation method, to help us select a copula.

This paper is organized as follows. Section 2 gives the basic roots of tail dependence. Section 3 focuses on the estimation of the tail coefficient. Section 4 introduces two ways of testing for tail dependence. Section 5 provides simulation studies to set up the limits of such tools to select an appropriate copula. Finally, Section 6 applies some of the methods to a real data set to demonstrate how these tools can be useful in practice.

## 2 Tail dependence

The tail dependence coefficient is a measurement of the dependence between extreme realizations of a multivariate distribution. Let $d$ be the number of dependent random variables, for instance. For random variables ${({X}_{i})}_{i=1,\mathrm{\dots},d}$, the coefficient is an indication of the probability that an extreme value of ${X}_{i}$ will appear, given that all ${({X}_{j})}_{j=1,\mathrm{\dots},d;j\ne i}$ take an extreme value. We formally define the coefficient of upper tail dependence as follows:

${\lambda}_{\mathrm{U}}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{d})=\underset{q\to 1}{lim}\mathbb{P}({X}_{1}>{Q}_{({X}_{1})}(q)\mid {X}_{2}>{Q}_{({X}_{2})}(q),{X}_{3}>{Q}_{({X}_{3})}(q),\mathrm{\dots},{X}_{d}>{Q}_{({X}_{d})}(q)).$ |

Then,

$${\lambda}_{\mathrm{U}}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{d})=\underset{q\to 1}{lim}\frac{\mathbb{P}({\bigcap}_{i=1,\mathrm{\dots},d}\{{X}_{i}>{Q}_{({X}_{i})}(q)\})}{\mathbb{P}({\bigcap}_{i=2,\mathrm{\dots},d}\{{X}_{i}>{Q}_{({X}_{i})}(q)\})},$$ |

where ${Q}_{({X}_{i})}(q)$ is the quantile function of the distribution of ${X}_{i}$ assessed at the $q$ level. If the limit exists and is positive, then there is a right tail dependence between all ${({X}_{i})}_{(i=1,\mathrm{\dots},d)}$.

This can also be written as

$${\lambda}_{\mathrm{U}}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{d})=\underset{x\to 1}{lim}\frac{\mathbb{P}({\bigcap}_{i=1,\mathrm{\dots},d}\{{X}_{i}>x\})}{\mathbb{P}({\bigcap}_{i=2,\mathrm{\dots},d}\{{X}_{i}>x\})}.$$ |

This measurement is independent of the marginal distributions of ${({X}_{i})}_{(i=1,\mathrm{\dots},d)}$. If ${\lambda}_{\mathrm{U}}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{d})>0$, extreme events tend to arise simultaneously. Conversely, if ${\lambda}_{\mathrm{U}}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{d})=0$, there is no right tail dependence.

In the bivariate case, letting $({x}_{1},{x}_{2})\in {[0,1]}^{2}$, and letting ${X}_{1}$ and ${X}_{2}$ be two random variables of the dependence structure (the copula $C$), we can write

$$\mathbb{P}({X}_{1}>{x}_{1}\mid {X}_{2}>{x}_{2})=\frac{\mathbb{P}({X}_{1}>{x}_{1},{X}_{2}>{x}_{2})}{\mathbb{P}({X}_{2}>{x}_{2})}.$$ |

The copula $C$ is defined by $C({u}_{1},{u}_{2})=C({F}_{1}({x}_{1}),{F}_{2}({x}_{2}))=F({x}_{1},{x}_{2})$, where ${u}_{1}$ and ${u}_{2}$ are respective realizations of ${U}_{1}$ and ${U}_{2}$, two uniform random variables. When using the copula, the tail dependence becomes

$\mathbb{P}({X}_{1}>{x}_{1}\mid {X}_{2}>{x}_{2})$ | $={\displaystyle \frac{\mathbb{P}({X}_{1}>{x}_{1})-\mathbb{P}({X}_{1}>{x}_{1},{X}_{2}\le {x}_{2})}{\mathbb{P}({X}_{2}>{x}_{2})}}$ | ||

$={\displaystyle \frac{1-{u}_{1}-{u}_{2}+C({u}_{1},{u}_{2})}{1-{u}_{2}}}.$ |

Thus, the bivariate upper tail coefficient is

$${\lambda}_{\mathrm{U}}({X}_{1},{X}_{2})=\underset{u\to 1}{lim}\frac{1-2u+C(u,u)}{1-u}.$$ |

Now we focus on the lower tail dependence; for the smallest losses of the same sample, ${({X}_{i})}_{(i=1,\mathrm{\dots},d)}$:

$$ |

Then,

$$ |

where ${Q}_{({X}_{i})}(q)$ is the quantile function of the distribution of ${X}_{i}$ assessed at the $q$ level. If the limit exists and is positive, then there is a left tail dependence between all ${({X}_{i})}_{(i=1,\mathrm{\dots},d)}$. This measurement is also independent of the marginal distributions of ${({X}_{i})}_{(i=1,\mathrm{\dots},d)}$. If ${\lambda}_{\mathrm{L}}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{d})>0$, small losses tend to arise simultaneously. Conversely, if ${\lambda}_{\mathrm{L}}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{d})=0$, there is no left tail dependence.

Letting $({x}_{1},{x}_{2},\mathrm{\dots},{x}_{d})\in {[0,1]}^{d}$, we can write

$$ |

Thus, in the bivariate case, using the same notation as above, we can write

$${\lambda}_{\mathrm{L}}({X}_{1},{X}_{2})=\underset{u\to 0}{lim}\frac{C(u,u)}{u}.$$ |

For some copulas, this coefficient is directly bounded by the parameters of the copula (Gumbel, Clayton, Student $t$), and knowing such an index may allow the parameters’ estimation. Nevertheless, this estimation can be very difficult when we do not know the margin distribution. It becomes a nonparametric estimation and could be very biased, mostly in lower dimensions.

## 3 Estimation of the tail dependence

Most nonparametric estimation methods rely on the empirical copula or the use of a tail copula. Throughout the paper we focus on the bivariate case. This concept can be extended to the multivariate case, but for small data sets the assumption that multiple margins also exceed some high threshold is much harder to verify (for instance, with fifty observations, and the suggestion that excesses are expected from the 90% quantile, we would need five observations of several margins to obtain extreme values at the same time, ie, the margins should all tend to $1$ at the same time, which is very rare). Consequently, we will consider that if the following methods provide poor results, there is no need to deepen our analysis to the multivariate case.

### 3.1 Empirical copula trajectory

Let $d$ be the dimension of copula $C$ and $n$ be the number of realizations for the $d$ random variables fixed by $C$. Let $?=({U}_{1},{U}_{2},\mathrm{\dots},{U}_{n})$ be the pseudo-observations vector of the copula realizations $(?\in {\mathcal{M}}_{(n,d)},{({U}_{s})}_{(s=1,\mathrm{\dots},n)}\in {\mathcal{M}}_{(1,d)})$. The empirical copula function $\widehat{C}$ is defined by

$$\widehat{C}=\frac{1}{n}\sum _{i=1}^{n}?\{{U}_{i1}\le {u}_{1},\mathrm{\dots},{U}_{id}\le {u}_{d}\}$$ |

with $u=({u}_{1},\mathrm{\dots},{u}_{d})\in {[0,1]}^{d}$.

If $X$ and $Y$ are two random variables, for example, $?={\{({x}_{k},{y}_{k})\}}_{k=1}^{n}$ are $n$ observations of the sample fixed to the random vector $Z=(X,Y)$ of copula $\widehat{C}$, then

$$\widehat{C}(\frac{i}{n},\frac{j}{n})=\frac{1}{n}\sum _{k=1}^{n}?\{{x}_{k}\le {x}_{k}^{i},{y}_{k}\le {y}_{k}^{j}\},$$ |

where ${x}_{k}^{i}$, ${y}_{k}^{j}$, $1\le i,j\le n$, are the ordered statistics of the sample.

From the empirical copula, we can define the empirical trajectories of the tail dependencies for $i\in \{1,\mathrm{\dots},n-1\}$ by

$${\widehat{\lambda}}_{\mathrm{U}}\left(\frac{i}{n}\right)=\left(1-2\frac{i}{n}+\widehat{C}(\frac{i}{n},\frac{i}{n})\right){\left(1-\frac{i}{n}\right)}^{-1}$$ | |||

and | |||

$${\widehat{\lambda}}_{\mathrm{L}}\left(\frac{i}{n}\right)=\widehat{C}(\frac{i}{n},\frac{i}{n})\left(\frac{n}{i}\right).$$ |

Caillault and Guegan (2007) propose using the specific properties of the tail dependence and selecting the value ${i}_{0}$ at which the trajectories stopped increasing or decreasing with $i/n$. Indeed, ${\widehat{\lambda}}_{\mathrm{U}}$ (respectively, ${\widehat{\lambda}}_{\mathrm{L}}$) is a decreasing (respectively, increasing) function of $1-i/n$ (respectively, $i/n$). They then deduce that the value of $\widehat{\lambda}$ for which we observe an inflection is the tail dependence coefficient. However, as this method can be biased, Caillault and Guegan suggest applying bootstrap resampling to obtain an average curve and confidence intervals for the trajectories in order to get a more robust tail index estimator. We will perform this method on simulated data to quantify the performance of such a technique. As Caillault and Guegan (2007) point out, convergence toward the true value is difficult when the upper tail dependence is less than 0.2. Moreover, this technique seems more efficient when the sample size increases, which could raise some issues with small data sets. Nevertheless, we will see that this method can be a useful tool when attempting to point out some dependence features, and that it can be adapted to test for tail dependence.

### 3.2 Other estimates

Many nonparametric estimators for the tail indexes have been proposed in the literature (see, for example, Fischer and Dörflinger 2006). Most rely on similar logic to that provided by the tail trajectory principle. For instance, we can compute the empirical copula and choose the previous ${i}_{0}$ to be $\sqrt{n}$ (linear regression of the copula on $i/n$), or we can use the same method on the logarithm of the empirical copula. These estimators are approximations that are very similar to the previous ones. Consequently, we will not study them here.

Other approaches propose working with tail copulas or extreme copulas. The idea is that we can always convert distributions at an extreme scale due to the empirical distribution functions and a unit Fréchet scaling. These approaches are studied in many papers and explore asymptotic properties that are not obtained with small data sets. Moreover, they prove to be inefficient, mostly in lower dimensions. Consequently, we will not study these either.

Another method, proposed in Dobrić and Schmid (2005), approximates the unknown copula $C(u,v)$ by $\stackrel{~}{C}(u,v)$, a convex combination of the maximum (comonotonicity) copula ${C}_{\mathrm{max}}(u,v)=\mathrm{min}\{u,v\}$ and the independence copula ${C}_{\u27c2}(u,v)=uv$:

$$\stackrel{~}{C}(u,v)=\alpha {C}_{\mathrm{max}}(u,v)+(1-\alpha ){C}_{\u27c2}(u,v).$$ |

Consequently, we get

$\underset{u\to 0}{lim}{\displaystyle \frac{\stackrel{~}{C}(u,u)}{u}}$ | $=\underset{u\to 0}{lim}(\alpha +(1-\alpha )u)=\alpha $ | |||

and | ||||

$\underset{u\to 1}{lim}{\displaystyle \frac{1-2u+\stackrel{~}{C}(u,u)}{1-u}}$ | $=\underset{u\to 1}{lim}(1-(1-\alpha )u)=\alpha .$ |

A nonparametric estimate of the upper tail index ${\widehat{\lambda}}_{{U}_{1}}$ would be an $\alpha $ that minimizes

$$F(\alpha )=\sum _{i=1}^{k}{\left[\widehat{C}(1-\frac{i}{n},1-\frac{i}{n})-\stackrel{~}{C}(1-\frac{i}{n},1-\frac{i}{n})\right]}^{2}.$$ |

Another estimate, derived from the same idea, takes advantage of ${C}_{\mathrm{max}}$ and ${C}_{\u27c2}$ properties:

$${\widehat{\lambda}}_{{U}_{2}}=\underset{\alpha \in [0,1]}{\mathrm{arg}\mathrm{min}}\sum _{i=1}^{k}{\left[\widehat{C}(1-\frac{i}{n},1-\frac{i}{n})-{\left(1-\frac{i}{n}\right)}^{2-\alpha}\right]}^{2}.$$ |

We get analogous estimates for ${\widehat{\lambda}}_{{L}_{1}}$ and ${\widehat{\lambda}}_{{L}_{2}}$, the lower tail indexes.

## 4 Testing instead of estimating tail dependence

### 4.1 Testing using the empirical copula trajectory

We propose to adapt the Caillault and Guegan (2007) method to approach the tail index by computing its mean trajectory on a given data set and to compare this to the confidence interval bounds of the trajectories of a theoretical dependence structure. In this way, we can perform a statistical test of the following null hypothesis: “the tail dependence is that of the theoretical structure”. This method uses the bootstrap resamplings of the mean trajectories computed on the empirical sample and is completed by another bootstrap resampling on a simulated data set stemming from the theoretical structure. We plot the 5% and 95% quantiles of these trajectories to make the statistical test. It is important to note that this test cannot be seen as a goodness-of-fit test since it only allows us to compare tail behaviors. Consequently, it must be complemented with other criteria when dealing with copula selection.

### 4.2 Testing tail independence using the quotient-correlation or Gamma test

Zhang (2008) proposes a quotient that could be compared with a nonlinear correlation coefficient, from which we can test either the independence of two random variables, or tail independence when adapting this coefficient. This quotient is presented as an alternative to Pearson’s correlation or to Spearman’s rank correlation in its rank-based version. Suppose $X$ and $Y$ are identically distributed positive random variables satisfying

$$\mathbb{P}(X\ge Y)>0,\mathbb{P}(Y\ge X)>0.$$ |

The quotients between $X$ and $Y$ are $X/Y$ and $Y/X$. Assuming that ${({X}_{i},{Y}_{i})}_{i=1,\mathrm{\dots},n}$ is a bivariate random sample of $(X,Y)$, we get $2n$ quotients.

The quotient correlation is defined as

$${q}_{n}=\frac{{\mathrm{max}}_{i\le n}\{{X}_{i}/{Y}_{i}\}+{\mathrm{max}}_{i\le n}\{{Y}_{i}/{X}_{i}\}-2}{{\mathrm{max}}_{i\le n}\{{X}_{i}/{Y}_{i}\}+{\mathrm{max}}_{i\le n}\{{Y}_{i}/{X}_{i}\}-1}$$ |

and can take values between $0$ and $1$. Moreover, the larger the ${\mathrm{max}}_{i\le n}\{{X}_{i}/{Y}_{i}\}$ and ${\mathrm{max}}_{i\le n}\{{Y}_{i}/{X}_{i}\}$, the smaller the ${q}_{n}$, ie, the smaller the change in magnitude between $X$ and $Y$. All this theory relies on these changes in magnitude to explore dependence between two random variables. For instance, if ${q}_{n}$ is near $1$, we can conclude that there are comovements between ${X}_{i}$ and ${Y}_{i}$, and consequently that $X$ and $Y$ are nearly completely dependent. We propose to convert the random variables to unit Fréchet scale, since the quotient can converge asymptotically toward a Gamma distribution. Moreover, this permits us to respect the “identically distributed” assumption. Once this quotient correlation is defined, the main issue is to determine a value for which we can conclude there is independence between the variables. Since this value is subjective, a statistical test can be performed. Nevertheless, in this paper we focus not on the dependence between two series but only on the tail dependence that could exist. We thus do not give details of this test and will present the adapted quotient correlation to test tail independence.

Assuming that $u$ is a positive threshold ($u$ large; quantile of a unit Fréchet distribution), and that ${W}_{i}$ and ${V}_{i}$ are exceedances over the threshold $u$ of positive scaled Fréchet random variables ${X}_{i}$ and ${Y}_{i}$, respectively, the tail quotient correlation coefficient is defined as

$${q}_{u,n}=\frac{{\mathrm{max}}_{i\le n}\{(u+{W}_{i})/(u+{V}_{i})\}+{\mathrm{max}}_{i\le n}\{(u+{V}_{i})/(u+{W}_{i})\}-2}{{\mathrm{max}}_{i\le n}\{(u+{W}_{i})/(u+{V}_{i})\}+{\mathrm{max}}_{i\le n}\{(u+{V}_{i})/(u+{W}_{i})\}-1},$$ |

where $u$, ${W}_{i}$ and ${V}_{i}$ are unit Fréchet scaled realizations of $X$ and $Y$. While testing tail independence between two variables $X$ and $Y$, under the null hypothesis we get

$$n{q}_{u,n}\underset{\text{law}}{\overset{}{\to}}\mathrm{\Gamma},$$ |

where $\mathrm{\Gamma}$ is $\mathrm{Gamma}(2,1-{\mathrm{e}}^{-1/u})$ distributed. This is true only if $X$ and $Y$ are Fréchet distributed. Consequently, most of the time, a conversion must be carried out; this is called the Gamma test for the null hypothesis of tail independence. When $n{q}_{u,n}>{\mathrm{\Gamma}}_{\alpha}$, where ${\mathrm{\Gamma}}_{\alpha}$ is the upper $\alpha $th percentile of the Gamma distribution ($\alpha =95$% for a 5% confidence level test), the tail independence assumption is rejected. The parameter $u$ can impact the results, as we will see in Section 5.

## 5 Simulation studies

### 5.1 Performance of the estimation methods

#### 5.1.1 Estimating thanks to the empirical copula trajectory

In order to ensure the best conditions of application for this method, we propose to model a $t$-copula, as it has symmetric tail dependence. Since this method is deemed delicate for ${\lambda}_{\mathrm{U}}\le 0.2$ (equivalent, for instance, to five degrees of freedom and correlation 50%), we choose a $t$-copula with three degrees of freedom and correlation 50% (or ${\lambda}_{\mathrm{U}}=0.3$) as a good configuration to detect tail dependence. We do not provide the estimates, as Caillault and Guegan (2007) provide statistics on the estimation. Instead, we will focus on the graphs that permit us to observe tail dependencies in order to illustrate the underlying logic of such a method. We will also compare these results to those obtained with different degrees of freedom to ensure that this graphical tool can help us to find boundaries for estimating the tail indexes accurately.

We test the sensitivity of this method first to the number of observations, then to the correlations and finally to the degree of freedom combined with the other parameters.

Correlation (%) | |||||
---|---|---|---|---|---|

df | 0 | 5 | 10 | 20 | 50 |

3 | 0.12 | 0.13 | 0.14 | 0.18 | 0.31 |

5 | 0.05 | 0.06 | 0.07 | 0.09 | 0.21 |

10 | 0.01 | 0.01 | 0.01 | 0.02 | 0.08 |

We then simulate 1000 copulas in each configuration with a given number of observations and different correlations. On each copula, we apply 1000 bootstrap resamplings to compute a mean trajectory. We then get 1000 mean trajectories for each copula, from which we compute the median and the 90% confidence interval of the trajectories stemming from all the bootstrapped mean trajectories. In Table 1, we give different values of the tail coefficient depending on the correlation and the degrees of freedom (df) of a $t$-copula in order to illustrate the low levels of tail dependency that can be expected with correlations of less than 50%.

We first test the sensitivity to the number of observations, in the case of “high” tail dependence (correlation 0.5 and three degrees of freedom). The results are shown in Figure 1.

We plot the true tail index with the solid horizontal line and the bounds of the confidence intervals with dotted curves. We can see that the behavior of the trajectories changes for $n\ge 100$. The coordinates corresponding to these breakdowns have similar values, which points out the symmetry between the two tail dependence indexes of a $t$-copula. The estimated tail indexes are less observable, though, when the number of observations is low ($n\le 50$).

We will now test the sensitivity of this method to the correlation parameter. The results are given in Figure 2.

It seems that, whatever the correlation, the tail coefficients remain observable. We set $n=1000$, as it enables us to focus only on the sensitivity to the correlation level. However, since the tail indexes of the $t$-copula rely also on the degree of freedom, we also study the sensitivity to this degree of freedom. As we have shown that for $n=1000$ the correlation parameter does not impact the estimation, we next try this configuration to isolate the marginal effect of the degree of freedom. However, we will still focus on two different levels of correlation in order to ensure that there is no cross-effect, since both parameters play a part in the level of tail dependence. Finally, we also study the sensitivity to the degree of freedom in different cases of sample size. Indeed, if $n=1000$ seems to allow ideal properties, in practice we deal with sample sizes closer to $n=50$ or $n=100$. Thus, we want to study the sensitivity to the degree of freedom in such cases. The results are given in Figures 3 and 4.

First, we provide results with a 10% correlation parameter. With $n=1000$, a sample size deemed sufficient to ensure good properties, we note a slight bias for $\text{df}=5$ and a high one when $\text{df}=10$. This result depicts the sensitivity to the degree of freedom, ie, the difficulties in properly estimating tail dependence when it is less than 0.1. When $n$ is smaller, a small bias exists when $\text{df}=3$, but the results are much better than for higher degrees of freedom. We might then rely on this method for tail dependence greater than 0.1. We also note that this method does not perform well for more than three degrees of freedom. This can be due to the low level of tail dependence. In order to validate this assumption, we will now reproduce the same study with a higher correlation parameter. Indeed, we observe the sensitivity to the degree of freedom for a low level of correlation and show that results get poorer when $\text{df}>3$. However, the correlation parameter can lead to greater tail dependence even with $\text{df}>3$. The results are shown in Figure 4.

The results improve for a higher correlation parameter when $\text{df}>3$. Indeed, relative biases for $\text{df}=5$ or $\text{df}=10$ are smaller than with a 10% correlation parameter, whatever the sample size. However, results are still very biased for $\text{df}=10$, ie, for a tail dependence of less than 0.1. For $\text{df}=5$ (or a tail dependence of 0.21), results are also biased (overestimation of the tail dependence). Thus, we can conclude there is a cross-effect of correlation and degree of freedom.

Finally, trajectories seem an efficient way to measure tail dependence, mostly when it is high. Nevertheless, they are a very interesting tool for detecting potential symmetry and obtaining clues on the underlying dependence structure in any configuration. Yet, the conclusions drawn from using such trajectories need to be challenged by statistical goodness-of-fit tests and a proper calibration. This is especially true when the size of the data set is small and the tail dependence very low (0.05).

##### Is the size of the data set the main issue?

In order to point out issues raised by the use of the empirical copula, we propose here to model a $t$-copula for which tail dependence indexes are directly estimated from the empirical copula. For different ${i}_{0}$, we will test the ability of this method to provide a good estimation of the tail indexes. To do this we adapt the method of Caillault and Guegan (2007): instead of finding ${i}_{0}$, we will try several quantile levels and deduce each tail index dependence corresponding to these levels. This is also an adaptation of Schmidt and Stadtmüller (2005) that consists in finding the optimal threshold $k$, above which we consider values to be in the tail of the distribution, and then deduce the tail indexes by applying estimators to the tail copulas. Here, we do not find the optimal threshold but try several thresholds $k$ (or different ${i}_{0}$, in Caillault and Guegan (2007)), and estimate tail indexes for all these, assuming that if we get a good estimate for one, it can be set as our ${i}_{0}$ or $k$. We simulated 1000 bivariate $t$-copulas for different degrees of freedom and with correlation parameter $\rho =0.5$. We reproduce in Figure 5 the mean trajectories for quantiles of the margins from 0.5 to 1 for robustness.

Finally, whatever the size of the data set, it seems difficult to obtain the true value of the tail index using the empirical copula. We assume that the empirical copula is a fine proxy of the true copula in terms of convergence to the theoretical distribution, but it is not close enough to allow computations like these. Moreover, even when finding the true value, the quantile level is so high that very few observations for modeling extreme dependencies remain.

European Banking Authority (2015) states that “competent authorities shall verify … that the institution carefully considers dependence between tail events”, which seems tricky given these results. Indeed, empirical copulas are required in order to assess the dependence structure, either by using goodness-of-fit tests or by looking for clues with methods such as those proposed in this paper. The European Banking Authority (EBA) and US Federal Reserve System also asked banks to consider Student-like copulas, which seems quite arbitrary when looking at the performances of the possible selection processes for small dimensions.

#### 5.1.2 Other estimates

(a) Correlation 0%, ${\lambda}_{\text{th}}=\text{0.12}$ | ||||

$?$ | Mean | SD | R-bias (%) | R-RMSE (%) |

50 | 0.07 | 0.10 | $-$41 | 99 |

100 | 0.06 | 0.08 | $-$51 | 85 |

150 | 0.05 | 0.07 | $-$57 | 80 |

200 | 0.04 | 0.06 | $-$62 | 79 |

300 | 0.04 | 0.05 | $-$66 | 78 |

500 | 0.03 | 0.04 | $-$73 | 80 |

750 | 0.03 | 0.03 | $-$74 | 79 |

900 | 0.03 | 0.03 | $-$75 | 79 |

1000 | 0.03 | 0.03 | $-$76 | 80 |

(b) Correlation 20%, ${\lambda}_{\text{th}}=\text{0.18}$ | ||||

$?$ | Mean | SD | R-bias (%) | R-RMSE (%) |

50 | 0.14 | 0.14 | $-$20 | 79 |

100 | 0.14 | 0.11 | $-$22 | 65 |

150 | 0.14 | 0.09 | $-$20 | 56 |

200 | 0.14 | 0.08 | $-$23 | 52 |

300 | 0.14 | 0.07 | $-$21 | 45 |

500 | 0.14 | 0.06 | $-$21 | 38 |

750 | 0.14 | 0.04 | $-$23 | 34 |

900 | 0.14 | 0.04 | $-$22 | 30 |

1000 | 0.14 | 0.04 | $-$22 | 31 |

(c) Correlation 50%, ${\lambda}_{\text{th}}=\text{0.31}$ | ||||

$?$ | Mean | SD | R-bias (%) | R-RMSE (%) |

50 | 0.30 | 0.14 | $-$3 | 45 |

100 | 0.32 | 0.11 | 2 | 35 |

150 | 0.32 | 0.09 | 4 | 29 |

200 | 0.33 | 0.08 | 5 | 25 |

300 | 0.33 | 0.07 | 5 | 22 |

500 | 0.33 | 0.05 | 6 | 18 |

750 | 0.33 | 0.04 | 7 | 15 |

900 | 0.33 | 0.04 | 7 | 14 |

1000 | 0.33 | 0.03 | 7 | 13 |

(a) Correlation 0%, ${\lambda}_{\text{th}}=\text{0.12}$ | ||||

$?$ | Mean | SD | R-bias (%) | R-RMSE (%) |

50 | 0.10 | 0.14 | $-$13 | 124 |

100 | 0.09 | 0.12 | $-$25 | 105 |

150 | 0.08 | 0.10 | $-$33 | 92 |

200 | 0.07 | 0.09 | $-$41 | 87 |

300 | 0.06 | 0.08 | $-$48 | 81 |

500 | 0.05 | 0.06 | $-$58 | 78 |

750 | 0.05 | 0.05 | $-$60 | 75 |

900 | 0.05 | 0.05 | $-$61 | 74 |

1000 | 0.04 | 0.05 | $-$63 | 75 |

(b) Correlation 20%, ${\lambda}_{\text{th}}=\text{0.18}$ | ||||

$?$ | Mean | SD | R-bias (%) | R-RMSE (%) |

50 | 0.2 | 0.18 | 12 | 100 |

100 | 0.2 | 0.15 | 15 | 87 |

150 | 0.21 | 0.13 | 19 | 76 |

200 | 0.21 | 0.12 | 16 | 69 |

300 | 0.21 | 0.10 | 20 | 61 |

500 | 0.22 | 0.08 | 22 | 51 |

750 | 0.21 | 0.07 | 18 | 41 |

900 | 0.22 | 0.06 | 22 | 38 |

1000 | 0.22 | 0.06 | 21 | 39 |

(c) Correlation 50%, ${\lambda}_{\text{th}}=\text{0.31}$ | ||||

$?$ | Mean | SD | R-bias (%) | R-RMSE (%) |

50 | 0.38 | 0.14 | 22 | 50 |

100 | 0.41 | 0.1 | 32 | 46 |

150 | 0.43 | 0.08 | 37 | 46 |

200 | 0.44 | 0.07 | 40 | 46 |

300 | 0.44 | 0.06 | 41 | 46 |

500 | 0.45 | 0.05 | 45 | 48 |

750 | 0.46 | 0.04 | 47 | 48 |

900 | 0.46 | 0.04 | 47 | 49 |

1000 | 0.46 | 0.03 | 48 | 49 |

We simulate 1000 $t$-copulas with specific figures. We set $k=n$, as that is the best framework for this estimation. Dobrić and Schmid (2005) argue that $k=\sqrt{n}$ is enough to ensure good properties. For each figure, we estimate the tail indexes using the two estimators defined in Section 3.2 and compute the mean, standard deviations, relative biases and relative root mean square error (R-RMSE) for both estimates. We recall that the R-RMSE is defined as

$$\text{R-RMSE}(\widehat{\theta})=\frac{1}{\theta}\sqrt{\frac{1}{S}\sum _{s=1}^{S}{({\widehat{\theta}}_{s}-\theta )}^{2}},$$ |

where $\widehat{\theta}$ is the estimate of $\theta $, and ${\widehat{\theta}}_{s}$ the $s$th bootstrapped estimation ($S$ is the number of bootstraps, 1000 here). This indicator enables us to study the accuracy of the estimation. We show the results in Tables 2 and 3 for a $t$-copula of $\text{df}=3$. As the copula chosen is symmetric, the upper and lower tail indexes are supposed to be the same. Consequently, we only estimate the lower tail index.

First, we can note that ${\widehat{\lambda}}_{{L}_{1}}$ seems more accurate and less biased regardless of the correlation or the number of observations. Standard deviations of the estimations decrease with $n$, but we could use the estimates to assess the existence of a positive tail dependence in any configuration. Moreover, when the tail index is larger, results seem more reliable. We also estimated tail indexes of a Gaussian copula to focus on the sensitivity to the correlation parameter and noted that the estimates were more biased when the correlation increased. This is mainly due to the approximation, which is not ensured for any copula. Indeed, we expect to get better results when the underlying copula belongs to the class of copulas derived from the convex combination used for approximation, as is the case in Dobrić and Schmid (2005), in which different settings are defined, raising the issue of the class of copulas. This approximation is not ensured for the Gaussian copula, for instance, where other terms should complete this decomposition. This estimator is, then, not fully reliable.

### 5.2 Conditions required to test tail dependence

As already pointed out in Section 5.1.1, the tail index trajectory seems to demonstrate tail dependence, even if the estimation can be biased for small data sets and low correlation levels, when the tail dependence is expected to be greater than 0.2. When testing for tail dependence, the bias should not be considered, and the condition retained is $n\ge 50$ whatever the correlation, when tail dependence is assumed to be important. However, in practice, other copulas can be appraised and the degree of freedom is not available. As we observed, when this index is less than 0.2, there are many configurations where the tail trajectories do not change much with small data sets. In these cases, we found that correlations can have an impact, and for those of less than 10% we need at least 150 observations to observe moves in the trajectories; 100 observations seems sufficient for correlations above 10%. In Figure 6, we show different cases leading to these boundaries. Note that we do not perform a statistical test for now, since the confidence intervals are still those of the underlying $t$-copula. Further, we will test whether or not the median trajectory is contained by the trajectories of a tail-dependent copula.

On the first row, we see that, for a 5% correlation, as soon as the degree of freedom increases, even slightly, $n=50$ will not allow us to observe changes in the trajectory. On the second row, we see that for a $\text{df}>3$ and 5% correlation, $n=200$ seems enough even if the move is not very explicit. Finally, on the last row, we see that, when the correlation increases, $n=100$ seems reasonable even if it is not very obvious when tail dependence is almost null ($\text{df}=10$).

### 5.3 Conditions required to test tail independence: the Gamma test

#### 5.3.1 Asymptotic property: number of observations and correlation parameters

We simulated 1000 Gaussian copulas 1000 times with different numbers of observations and different correlations. The margins have been converted to the unit Fréchet scale. We provide the statistical adjustment of $n{q}_{u,n}$ to a Gamma distribution of parameters $(2,1-{\mathrm{e}}^{-1/u})$ for a given $u$ (95% quantile of a unit Fréchet distribution since the initial sample is converted). The statistical test is the Anderson–Darling goodness-of-fit. Each test is based on the realizations of 1000 quotients derived from 1000 Gaussian copulas for each correlation parameter. We then perform 1000 goodness-of-fit tests. In Table 4, we provide the median $p$-values of the 1000 goodness-of-fit tests.

Correlation (%) | |||||
---|---|---|---|---|---|

$?$ | 0 | 10 | 20 | 30 | 50 |

50 | 0.01 | 0.03 | 0.03 | 0.03 | 0.03 |

100 | 0.03 | 0.19 | 0.26 | 0.29 | 0.27 |

150 | 0.06 | 0.27 | 0.28 | 0.36 | 0.21 |

200 | 0.10 | 0.24 | 0.33 | 0.33 | 0.17 |

250 | 0.13 | 0.31 | 0.35 | 0.32 | 0.13 |

300 | 0.15 | 0.37 | 0.33 | 0.27 | 0.11 |

500 | 0.29 | 0.31 | 0.33 | 0.26 | 0.07 |

750 | 0.33 | 0.32 | 0.32 | 0.25 | 0.05 |

900 | 0.34 | 0.29 | 0.25 | 0.19 | 0.04 |

1000 | 0.34 | 0.34 | 0.28 | 0.21 | 0.03 |

Whatever the correlation is, at least 100 observations are required to ensure the goodness-of-fit test to the Gamma distribution, the parameter of which depends on the 95% quantile of a unit Fréchet distribution (the threshold above which observations are assumed to be extreme). Actually, with less data, the quantity $n{q}_{u,n}$ is constrained by small values of $n$ and the asymptotic property seems to be compromised, especially for high thresholds that lead to setting almost all components of the quotient to the threshold value. The results improve with larger numbers of observations. Consequently, the value of the threshold might hamper the asymptotic distribution assumption and must be chosen carefully.

Moreover, it seems that when the correlation is more than 30% the asymptotic property is not always ensured. The level of correlation in operational risk is around 10–15%, leading to good results when the number of observations is greater than 100.

#### 5.3.2 Does the asymptotic property depend on the extremal threshold?

Percentile level of the threshold | |||||||

$?$ | 0.50 | 0.60 | 0.70 | 0.75 | 0.80 | 0.90 | 0.95 |

50 | 0.27 | 0.22 | 0.16 | 0.13 | 0.1 | 0.04 | 0.01 |

100 | 0.34 | 0.31 | 0.28 | 0.25 | 0.21 | 0.1 | 0.03 |

150 | 0.37 | 0.35 | 0.33 | 0.3 | 0.27 | 0.16 | 0.06 |

200 | 0.35 | 0.35 | 0.34 | 0.35 | 0.32 | 0.21 | 0.1 |

250 | 0.36 | 0.34 | 0.34 | 0.34 | 0.34 | 0.26 | 0.13 |

300 | 0.37 | 0.35 | 0.35 | 0.35 | 0.34 | 0.29 | 0.15 |

500 | 0.35 | 0.35 | 0.33 | 0.33 | 0.37 | 0.34 | 0.29 |

750 | 0.33 | 0.35 | 0.35 | 0.33 | 0.34 | 0.34 | 0.33 |

900 | 0.34 | 0.33 | 0.35 | 0.32 | 0.35 | 0.35 | 0.34 |

1000 | 0.32 | 0.33 | 0.38 | 0.33 | 0.33 | 0.34 | 0.34 |

Percentile level of the threshold | |||||||

$?$ | 0.50 | 0.60 | 0.70 | 0.75 | 0.80 | 0.90 | 0.95 |

50 | 0.12 | 0.17 | 0.24 | 0.31 | 0.32 | 0.24 | 0.03 |

100 | 0.07 | 0.11 | 0.18 | 0.2 | 0.31 | 0.32 | 0.26 |

150 | 0.05 | 0.08 | 0.14 | 0.16 | 0.18 | 0.29 | 0.28 |

200 | 0.04 | 0.07 | 0.11 | 0.16 | 0.18 | 0.32 | 0.33 |

250 | 0.04 | 0.08 | 0.11 | 0.15 | 0.16 | 0.3 | 0.35 |

300 | 0.04 | 0.04 | 0.11 | 0.12 | 0.14 | 0.38 | 0.33 |

500 | 0.03 | 0.05 | 0.09 | 0.09 | 0.11 | 0.2 | 0.33 |

750 | 0.03 | 0.03 | 0.04 | 0.07 | 0.11 | 0.2 | 0.32 |

900 | 0.03 | 0.04 | 0.06 | 0.09 | 0.09 | 0.23 | 0.25 |

1000 | 0.03 | 0.03 | 0.08 | 0.08 | 0.09 | 0.17 | 0.28 |

The threshold is a parameter of the Gamma distribution tested and could therefore impact the results, as seen in Section 5.3.1. We reproduce a similar study for different threshold levels for two levels of correlation. Thus, we perform 1000 Anderson–Darling goodness-of-fit tests, each based on the realizations of 1000 quotients derived from 1000 simulations of Gaussian copulas with null correlation as well as different thresholds and numbers of observations.

First, we focus on the null correlation case, since it is the perfect case of independence. The results are given in Table 5, which shows the median $p$-values of the 1000 goodness-of-fit tests.

As we can see, the lower the threshold, the better the convergence to the Gamma distribution with small data sets. This must be due to the number of observations in the tail, which also contributes to the definition of ${q}_{u,n}$. As the principle of this method is to test tail independence, we maintain the high threshold that we first set arbitrarily to be the 80% quantile of a unit Fréchet distribution when testing on small data sets. With more observations ($n>500$) we can use higher thresholds, especially if we are inclined to reject the null hypothesis, since we observe extreme dependencies. Finally, there is a trade-off between the number of observations in the tail and the possible convergence to the Gamma distribution that requires us to distinguish between small and large data sets.

When the correlation grows (20% here), we obtain the results in Table 6. As we can see, when the correlation increases, even slightly, the convergence to the Gamma distribution is less marked. For small numbers of observations, we need to take small threshold values to tend to the correct distribution, which is logical since there would be very few values above this threshold otherwise. For larger numbers of observations, we can set higher thresholds.

#### 5.3.3 Acceptance level of the Gamma test

We propose for several levels of threshold $u$ ($u$ being the quantile of a unit Fréchet distribution) and several levels of correlation to study the Gamma test when the initial sample stems from a Gaussian copula, that is to say, when the null hypothesis of tail independence is true.

We then simulate 1000 Gaussian copulas, and test whether the null hypothesis is valid for two different levels of correlations, three different thresholds and small data sets. The results are given in Table 7.

Percentile level | Percentile level | Percentile level | |||||||
---|---|---|---|---|---|---|---|---|---|

($\rho \mathbf{=}\text{?}$) | ($\rho \mathbf{=}\mathbf{\text{0.2}}$) | ($\rho \mathbf{=}\mathbf{\text{0.8}}$) | |||||||

$?$ | 0.75 | 0.80 | 0.90 | 0.75 | 0.80 | 0.90 | 0.75 | 0.80 | 0.90 |

50 | 0.97 | 0.98 | 0.98 | 0.95 | 0.97 | 0.97 | 0.59 | 0.76 | 0.95 |

100 | 0.95 | 0.96 | 0.97 | 0.93 | 0.94 | 0.95 | 0.32 | 0.44 | 0.79 |

150 | 0.94 | 0.97 | 0.96 | 0.91 | 0.95 | 0.94 | 0.18 | 0.3 | 0.65 |

200 | 0.96 | 0.96 | 0.97 | 0.92 | 0.93 | 0.95 | 0.14 | 0.22 | 0.56 |

250 | 0.96 | 0.95 | 0.96 | 0.93 | 0.91 | 0.94 | 0.10 | 0.18 | 0.45 |

300 | 0.96 | 0.96 | 0.97 | 0.92 | 0.93 | 0.95 | 0.07 | 0.14 | 0.42 |

500 | 0.94 | 0.96 | 0.96 | 0.9 | 0.93 | 0.94 | 0.05 | 0.70 | 0.29 |

750 | 0.96 | 0.95 | 0.95 | 0.91 | 0.92 | 0.93 | 0.02 | 0.05 | 0.22 |

900 | 0.94 | 0.94 | 0.96 | 0.9 | 0.89 | 0.95 | 0.02 | 0.04 | 0.16 |

1000 | 0.96 | 0.94 | 0.97 | 0.91 | 0.91 | 0.94 | 0.02 | 0.03 | 0.15 |

These results show good performance, whatever the threshold or the number of observations. However, when the correlation increases, the acceptance level decreases and sometimes leads to the rejection of the null hypothesis for large numbers of observations.

### 5.4 Power of the statistical tests

#### 5.4.1 Testing tail dependence with tail trajectories

In Section 5.2, we stated that the minimal number of observations required to observe the tail dependence of $t$-copulas depends on the correlation level and degree of freedom. Nevertheless, the degree of freedom is always unknown. Consequently, testing tail dependence on copulas with very low tail dependence can be tough and may require a large data set. Thus, we propose to study the performance of our method by testing tail independence on very tail-dependent copulas. To do so, we first simulate $t$-copulas with correlation 50% and $\text{df}=3$ and compare the tail trajectories of these copulas with the confidence interval of Gaussian copulas’ tail trajectories. The results are shown in Figure 7.

For $n\le 500$, trajectories are included in the Gaussian confidence interval. Thus, we might wrongly conclude tail independence in these cases. The copula tested is very tail dependent; consequently, this test is inefficient at detecting tail independence. As a result, when tail dependence is expected to be lower than it is in this experiment, this method might perform poorly, even with 1000 observations.

We then carried out an alternative experiment, testing tail dependence on Gaussian copulas. We thus observed the trajectories of Gaussian copulas and checked whether these were included in tail trajectories of $t$-copulas with correlations of 0.5 and $\text{df}=3$. The results for different $n$ are given in Figure 8.

For $n\le 200$, the null hypothesis is accepted (tail dependence). This test performs better when $n$ increases, but as $t$-copulas with correlations of 0.5 and $\text{df}=3$ are assumed to be extremely tail dependent, we would expect stronger rejection of the null hypothesis. We showed that testing tail dependence due to tail trajectories is not reliable in most configurations and should not be used. We prefer to detect tail dependence by using tail trajectories as a graphical tool.

#### 5.4.2 Testing tail independence with the Gamma test

Percentile level | Percentile level | Percentile level | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

($\rho \mathbf{=}\text{?}$) | ($\rho \mathbf{=}\mathbf{\text{0.2}}$) | ($\rho \mathbf{=}\mathbf{\text{0.5}}$) | ||||||||||

$?$ | 0.75 | 0.80 | 0.90 | 0.95 | 0.75 | 0.80 | 0.90 | 0.95 | 0.75 | 0.80 | 0.90 | 0.95 |

50 | 4.5 | 4.4 | 2.3 | 0.0 | 7.7 | 6.2 | 3.0 | 0.0 | 18.6 | 12.2 | 4.0 | 0.0 |

100 | 7.9 | 9.4 | 3.8 | 2.0 | 13.6 | 13.9 | 5.3 | 2.2 | 35.1 | 30.7 | 11.7 | 4.0 |

150 | 8.1 | 8.7 | 4.2 | 3.2 | 14.8 | 14.9 | 7.6 | 4.1 | 38.7 | 36.0 | 19.5 | 8.1 |

200 | 9.3 | 10.5 | 7.1 | 3.9 | 16.6 | 17.1 | 10.9 | 5.2 | 43.2 | 40.9 | 24.5 | 12.5 |

250 | 11.0 | 11.3 | 6.9 | 3.2 | 20.0 | 18.3 | 11.9 | 5.7 | 48.1 | 43.6 | 30.4 | 13.8 |

300 | 10.5 | 11.3 | 8.3 | 7.5 | 19.2 | 20.0 | 14.0 | 10.7 | 50.9 | 42.8 | 34.8 | 22.3 |

500 | 12.8 | 13.6 | 9.0 | 8.3 | 24.2 | 21.9 | 14.5 | 13.5 | 57.5 | 51.1 | 40.1 | 31.1 |

1000 | 16.1 | 15.3 | 13.8 | 11.7 | 27.9 | 25.6 | 23.1 | 17.8 | 62.9 | 57.6 | 53.0 | 42.9 |

5000 | 16.8 | 18.3 | 17.7 | 18.1 | 31.1 | 32.2 | 31.5 | 29.5 | 70.8 | 69.1 | 65.7 | 62.9 |

We propose studying the Gamma test for different levels of threshold $u$ and different correlation levels when the initial sample is drawn from a $t$-copula. We then simulate 1000 $t$-copulas with $\text{df}=3$ (it gives high tail dependence and is far from the null hypothesis) for three levels of correlation, different thresholds and different numbers of observations, and perform a Gamma test on these simulated data sets. Note that we use different thresholds because the Gamma test can be sensitive to the number of observations required to model the tail of the distribution. For instance, the results for higher thresholds are mostly interesting when dealing with numerous observations. The results are given in Table 8.

The Gamma test rejects the null hypothesis mostly when the correlation and the number of observations are high. Indeed, for small data sets, the test tends mostly to accept the null hypothesis, whatever the correlation. The proportion of rejections is lower when the threshold is set to 90% or 95%. Nevertheless, we never get around 95% rejections, as we might expect when testing for tail independence with a 5% level of confidence. Moreover, when the number of observations is greater than 1000, the test still performs badly for small correlations. Consequently, the test is not very reliable, whatever the configuration, and should not be used in practice.

### 5.5 Does performance depend on the class of copulas?

Name | ${?}_{\text{?}}$ | ${?}_{\text{?}}$ | |
---|---|---|---|

Clayton | ($\theta >\text{0}$) | ${\text{2}}^{-\text{1}/\theta}$ | 0 |

Gumbel | ($\theta \ge \text{1}$) | 0 | $\text{2}-{\text{2}}^{\text{1}/\theta}$ |

Frank | ($\theta \ne \text{0}$) | 0 | 0 |

Until now, we have only focused on elliptical copulas. However, if dependence is asymmetric, would the previous conclusions still apply? We thus propose to apply some of our methods to the class of Archimedean copulas to complete the analysis in the previous sections. Under the Archimedean copula framework, we propose to study the copulas with the tail dependence detailed in Table 9.

Parameters of these copulas are directly linked to Kendall’s $\tau $. We propose to study the ability of our previous techniques to detect tail dependence on this class of copulas.

#### 5.5.1 Estimation methods

First, we simulate 1000 realizations of random variables described by a Clayton copula with parameters 1 and 0.3 (respectively, ${\lambda}_{\mathrm{L}}=0.5$ and ${\lambda}_{\mathrm{L}}=0.1$), and compute the tail index trajectories to study the potential impact of the copula class on the biases observed previously. We do the same with Gumbel copulas with the same tail coefficient parameter. The tail index trajectories are shown in Figure 9.

We only tested these copulas for small data sets when the tail index is assumed to be high, and for 500 observations when it is low. The aim of this study is not to search for optimal conditions according to the class of copulas but only to compare these results to those obtained with the elliptical copulas. It seems that when the tail index is high, the estimation performs quite well, with almost no bias. Nevertheless, when this coefficient is assumed to be small, even with 500 observations we observe some bias, although less than that of $t$-copulas for similar tail indexes and numbers of observations. Moreover, these graphics all exhibit tail dependence and can be used as an indicator of the type of copula to select.

We study the performances of the other estimates only on Clayton copulas, since the results for Gumbel copulas are expected to be similar. We then simulated 1000 copulas with a lower tail index of 0.5. The statistics for the other estimators are provided in Table 10.

${\widehat{?}}_{{?}_{\text{?}}}$ | ${\widehat{?}}_{{?}_{\text{?}}}$ | |||||||

Relative | Relative | Relative | Relative | |||||

$?$ | Mean | SD | bias (%) | RMSE (%) | Mean | SD | bias (%) | RMSE (%) |

50 | 0.28 | 0.15 | $-$44 | 53 | 0.34 | 0.19 | $-$32 | 50 |

100 | 0.28 | 0.12 | $-$45 | 51 | 0.33 | 0.15 | $-$33 | 45 |

150 | 0.31 | 0.07 | $-$39 | 43 | 0.38 | 0.12 | $-$25 | 34 |

200 | 0.31 | 0.07 | $-$38 | 41 | 0.38 | 0.09 | $-$24 | 30 |

300 | 0.30 | 0.07 | $-$40 | 42 | 0.37 | 0.09 | $-$24 | 30 |

500 | 0.33 | 0.05 | $-$35 | 36 | 0.41 | 0.06 | $-$19 | 22 |

750 | 0.31 | 0.04 | $-$37 | 36 | 0.39 | 0.04 | $-$22 | 23 |

900 | 0.31 | 0.04 | $-$38 | 38 | 0.39 | 0.05 | $-$23 | 25 |

1000 | 0.32 | 0.03 | $-$37 | 37 | 0.39 | 0.04 | $-$21 | 23 |

In this configuration, ${\widehat{\lambda}}_{{L}_{2}}$ seems to perform better than ${\widehat{\lambda}}_{{L}_{1}}$, which was not the case with elliptical copulas. Comparing this with the elliptical case, where tail dependence is assumed to be high, it seems that the relative bias for ${\widehat{\lambda}}_{{L}_{1}}$ is larger, and the accuracy is better for ${\widehat{\lambda}}_{{L}_{2}}$. Finally, the comparison between the two estimators is slightly different, but the results are both very biased. These estimators are both unreliable and should not be used in practice.

#### 5.5.2 Testing tail (in)dependence with the Gamma test

Percentile level | Percentile level | |||||||

(Gumbel copula) | (Frank copula) | |||||||

$?$ | 0.75 | 0.80 | 0.90 | 0.95 | 0.75 | 0.80 | 0.90 | 0.95 |

50 | 37 | 28 | 6 | 0 | 10 | 6 | 3 | 0 |

100 | 66 | 57 | 29 | 5 | 15 | 9 | 5 | 3 |

150 | 78 | 75 | 49 | 16 | 13 | 13 | 5 | 3 |

200 | 86 | 83 | 60 | 28 | 13 | 10 | 7 | 3 |

300 | 91 | 90 | 75 | 47 | 14 | 12 | 6 | 5 |

500 | 94 | 93 | 86 | 68 | 15 | 12 | 7 | 5 |

750 | 97 | 96 | 91 | 80 | 18 | 12 | 7 | 6 |

900 | 99 | 97 | 93 | 83 | 14 | 13 | 8 | 4 |

1000 | 97 | 98 | 94 | 86 | 13 | 13 | 9 | 6 |

For the Gamma test, we compute the quotient correlation for 1000 simulated Gumbel copulas, since it seems that this test is more appropriate for detecting upper tail dependence. We set the parameter to 1.7, which leads to an upper tail index of 0.5 and Kendall’s $\tau $ of 41%. Tail dependence is then comparable with a $t$-copula of $\text{df}=3$ and correlation 75%, ensuring a significant tail dependence and the best conditions to test tail independence. We also test the tail independence for 1000 Frank copulas with parameter 4, which is equivalent to Kendall’s $\tau $ of 38%, and is comparable with the Gumbel one.

We note that this test is more reliable when applied to Archimedean copulas than to elliptical ones. Indeed, the 5% confidence level of rejections is not always observed for the Frank copula but is almost reached. For the upper-tail-dependent Gumbel copula, the test performs well with a Fréchet percentile threshold of 75% or for $n\ge 500$ and shows better properties for fewer observations than with a $t$-copula. To be exhaustive, we should test different levels of correlation, since this seemed to impact the results. Nevertheless, the correlation parameter is high enough for us to form an opinion on the sensitivity of this test to the class of the underlying copula.

We will not study the test for tail dependence (or independence) relying on the tail trajectories, since we have shown that the estimation with this tool was only slightly better.

#### 5.5.3 Review of the simulation studies

Tail dependence analysis is linked to extreme value theory, and thus raises similar concerns when applying it in practice. Indeed, the statistics rely on the observation of phenomenons that are quite rare in their extreme realizations, and consequently even sparser in their simultaneous extreme realizations. Thus, even with large data sets, we encounter difficulties when setting the level of extreme values, and when calibrating them, since few extreme values are observed in practice. This is worse when data sets, eg, those of operational risk, are small, leading practitioners to observe between one and ten extreme events most of the time. Thus, requirements from regulators to use very tail-dependent structures and to calibrate them can be controversial and unrealistic. The analysis provided in this section pointed out the limited performance of our methods, especially the statistical test for tail (in)dependence. However, we also showed how estimation methods can be very useful in detecting tail dependence if it is really observed, whatever the size of the data set. We will thus apply the best performing methods to an empirical data set to understand how we can benefit from such techniques in practice.

## 6 Empirical results

### 6.1 The data set

Our study relies on operational risk losses that occurred in numerous financial institutions in Western Europe between 2002 and 2013. These were collected by the Operational Riskdata eXchange (ORX), a leading banking consortium. The risk profiles for different countries proved to be very different (see Cope and Labbi 2008), which explains why we focus on a specific region and reduce the entire perimeter provided by ORX. Moreover, we propose to model operational risks of a bank that performs only retail activities, as in Crénin et al (2015). Risks are consequently modeled on the following five business lines during forty-eight quarters:

- (1)
retail banking;

- (2)
commercial banking;

- (3)
payment and settlement;

- (4)
agency services;

- (5)
retail brokerage.

Thus, the conditions required to ensure the previous methods’ good properties are not met. However, some of the methods can still be used in practice, as seen earlier. If biases exist, ideas about the underlying dependence structure and demonstration of a tail index are useful for practitioners.

### 6.2 Studying tail dependence

Note that we study the pairwise tail dependencies between the different units of measure (UOM), ie, business lines that cut across risk types, also called risk cells. In practice, we need to calibrate the correlation matrix first. Thus, all the following studies to detect or even estimate tail-dependent behaviors will take as an input our estimated correlations and their potential biases. We thus calibrate the correlation matrix as detailed in Crénin et al (2015) and obtain correlations between 15% and 50%. The number of observations is too small for us to expect to obtain the best results, as explained in Section 5, but some high levels of correlation still allow promising analyses. Moreover, we intend to work under a real operational risk framework in order to propose the best tools to study tail dependence. Indeed, the simulation studies showed that the best way to detect tail dependence is to measure or observe it. Thus, we will not use the statistical tests described in the paper for a sample of such size and prefer the estimation methods that allow us to detect the kind of dependence structure that exists between UOM.

#### 6.2.1 Tail trajectory

We propose here to estimate and test tail dependence simultaneously thanks to the tail trajectory graphical tool. We do not plot all our results in Figure 10, just those that seem the most interesting for illustrating the potential use of this tool. We computed the confidence interval of a Gaussian structure with the same correlation as that observed between the UOM to test for tail independence.

We do not observe tail dependence between all the different pairs of UOM tested on our data set very often. This result is consistent with expert opinions on the unlikely tail contagion between all kinds of operational risks, and casts doubt on the regulators’ rationale of modeling operational risks with a $t$-copula. However, as we have very few observations and very small pairwise correlations between most risk cells, our conclusions must be drawn with care. Indeed, we saw earlier that the only result we can get from such an analysis is the potential existence of tail independence rather than direct evidence. We note very different behaviors in the examples provided in Figure 10. For instance, we do not expect any tail dependence between the risk cells “damage to physical assets of retail banking” and “employment practices and workplace safety of payment and settlement”, for which the correlation is 17%. However, when dealing with “business disruption and system failures of retail banking” and “internal fraud of commercial banking”, for which the correlation is 25%, it seems we can conclude from both the estimation and the test for tail dependence that there is a small tail dependence. Consequently, different kinds of tail dependence can exist between operational risks. This is not compatible with the EBA opinion of a $t$-like copula, where pairwise tail dependencies would be symmetrical and identical between all types of risk. Indeed, we observe symmetrical tail dependence in some cases, and upper tail dependence for the two other pairs of UOM for which correlations are very low.

Finally, this study enables us to show some quite different tail dependencies according to the pairs studied. We may therefore consider hierarchical dependence structures, such as vine copulas, to represent the dependencies that exist between the different operational risk cells rather than a $t$-copula one. If regulators were to insist on the modeling of upper tail dependency, structures such as the grouped $t$-copula should be considered. Indeed, it allows us to take into account upper tail dependencies with different degrees of severity according to what we observe via our graphical tools. We still need a validation in the case of experiment 2 in Figure 10 since the change of behavior in the left tail is not very clear.

#### 6.2.2 Estimations

We computed both ${\widehat{\lambda}}_{{L}_{1}}$ and ${\widehat{\lambda}}_{{L}_{2}}$ in the previous experiments to see if we could find similar values when lower tail dependence is expected. We did the same for upper tail dependence, as Figure 10 suggests an upper tail dependence in experiment 3. Here, estimation can be seen as a validation of the graphical analysis in order to select the right copula. We recall our a priori assumptions in the previous analysis. In experiment 1, we do not consider tail dependence, as the tail trajectory was included in the independent tail trajectories’ confidence interval. Moreover, no change of behavior can be detected. In experiment 2, we typically need a validation of the possible interpretation of the plot in Figure 10. Indeed, the lower tail trajectory escapes the confidence interval very slightly, and upper tail dependence can make us imagine a change in behavior, but it is not fully observable. Is there really a symmetric tail dependence, or are these strange results only reflecting noise? In experiment 3, left tail dependence is assumed because there seems to be a change in slope on the left-hand side of the plot that puts the tail trajectory beyond the tail-independent confidence interval. Moreover, there is a strong asymmetry between the lower and upper tail trajectories. In experiment 4, we assess right tail dependence in analogy with experiment 3. Results are provided in Table 12.

Experiment | ||||
---|---|---|---|---|

1 | 2 | 3 | 4 | |

${\widehat{\lambda}}_{{L}_{\text{1}}}$ | 0.04 | 0.01 | 0.26 | 0.04 |

${\widehat{\lambda}}_{{L}_{\text{2}}}$ | 0.07 | 0.04 | 0.36 | 0.08 |

${\widehat{\lambda}}_{{U}_{\text{1}}}$ | 0.04 | 0.00 | 0.16 | 0.21 |

${\widehat{\lambda}}_{{U}_{\text{2}}}$ | 0.09 | 0.00 | 0.19 | 0.21 |

We note that when we assumed symmetric tail dependence or independence based on graphical analysis the results were similar. However, when we assumed lower tail dependence, both lower and upper tail dependencies are deduced from these estimates. When upper tail dependence is suggested by the graphical analysis, all estimators obtain the same conclusion. We recall that these estimation methods perform better with Archimedean copulas. Thus, the inclination to point out upper tail dependency only when a low value is assumed raises issues about the analysis we can carry out in practice. In this specific case, practitioners should either seek expert opinion on the possible extreme dependencies, or select the most conservative dependence structure linking the two underlying UOM (here, for instance, it would be a symmetric tail-dependent structure).

Moreover, the estimates are biased compared with the values for which we observe breakpoints in the tail trajectories examples. Consequently, this estimation method cannot be fully trusted to estimate tail dependence but can enable practitioners to test the goodness-of-fit of an Archimedean copula, for instance, instead of testing every copulas.

#### 6.2.3 Tail dependence in practice

Finally, we showed that the best way to detect tail dependence is analysis of the tail trajectories. Even with few data, the test for tail independence rejects the null hypothesis when we observe breakpoints in the trajectory. On the ORX data set, we also pointed out the diversity of tail dependencies that coexist between pairs of operational risks. The most common behavior seems to be tail independence, which is the frequently discussed opinion of operational risk practitioners. In practice, such tools can help us to decide on the possible use of a vine structure that would allow us to model the different behaviors, instead of a $t$-copula that would lead to us applying the same tail dependency to all the operational risks together.

## 7 Conclusion

In this paper, we studied tail dependence by defining the conditions required for all the methods used to perform and to quantify their efficiency and accuracy. We focused on estimation methods and statistical tests aimed at assessing tail dependence or independence. Our estimation methods performed better than the statistical tests, but they produce biases, especially in small dimensions. These methods behave differently according to the class of copula, with slightly better results being obtained with Archimedean copulas. Indeed, tail trajectories demonstrate the asymmetry of joint distributions more easily on this class of copulas, and estimations tend to better validate the true distributions. In practice, we see that studying tail dependence is not always conclusive when the number of observations is low ($n\le 100$), as can be the case in operational risk. Nevertheless, for larger data sets, most methods provide reliable results, except for the Gamma test, which needs further investigation to understand its possible flaws.

Nevertheless, despite the size of the data set, we can obtain good insight into the different joint behaviors of some operational risks as shown by the tail trajectories. Therefore, we can get clues about which class of copula to select due to the tail trajectory graphical tool, allowing us to point out, for example, symmetric behaviors or a possible breakpoint leading to the assumption of tail dependence when it is greater than 0.2, regardless of the correlation parameter. When applying the tail dependence estimation methods to the ORX data sets, we noted different kinds of tail dependence (symmetric, high, etc), and deduced that vine or hierarchical dependence structures could model operational risk dependence better than the $t$-copula, for instance. Thus, we question the regulators’ rationale to select a $t$-copula with an identical tail dependence between every operational risk event type, as we showed that many different kinds of tail dependencies can be found when studying pairs of operational risks. Consequently, if tail dependence analysis seems more appropriate with large data sets, it is also an interesting tool but needs to be complemented with goodness-of-fit tests and other selection criteria when modeling extreme dependencies on smaller data sets.

## Declaration of interest

The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper. The views expressed in this paper are solely those of the author and should not be interpreted as reflecting the official positions of Société Générale.

## Acknowledgements

I would like to thank the two anonymous referees who gave me precious advice.

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