The benefit of using random matrix theory to fit high-dimensional t-copulas

Jiali Xu; Loïc Brin

Risk modelers often want to model dependencies in order to link several risks together by taking their dependence relationships into account. Copulas are today widely used to do this. Of these, the Gaussian copula is often preferred for its simplicity. Nonetheless, it does not take into account the so-called tail dependencies, that is, the dependence on extreme events. $t$ -copulas, on the other hand, have strictly positive tail dependencies, and thus can be considered as an interesting substitute. These require the fitting of two parameters: the correlation matrix and the degree of freedom.

McNeil et al (2005) suggested a procedure to estimate these two parameters, which provides better results than a simple maximization of the likelihood function when the dimension of the copula is high. It consists of estimating the correlation matrix separately (using the rank-based measure Kendall’s $\tau$ and the properties of elliptical copulas) and using this estimate as an input in the likelihood function to estimate the degree of freedom by maximizing it.

In this paper, we introduce a procedure derived from McNeil et al’s, which gives better empirical results even when the number of observations is not high in comparison with the dimension of the copula. We recall theoretical results about $t$ -copulas in Section 2. In Section 3, we introduce the three-stage procedure suggested by McNeil et al, and show the problems that may arise when using it and how to circumvent these using the random matrix theory (RMT)’s denoising technique. Section 4 applies the improved procedure to simulated data sets to observe how the bias of the degree of freedom’s estimator is reduced and how it is less variant. In Section 5, we give an example of fitting $t$ -copulas in operational risk modeling. Section 6 concludes.

2 Copulas

Elliptical copulas are simply the copulas of elliptically contoured distributions, eg, the Gaussian and $t$ -distributions. A key advantage of elliptical copulas is that they can model heterogeneous levels of correlation between the marginal distributions. These two copulas are widely used in finance or risk modeling.

In this section, we will introduce some basic concepts related to copulas and properties of $t$ -copulas.

2.1 Basic concepts related to copulas

2.1.1 The copula

A copula is a multivariate distribution function with margins that are uniformly distributed on $[0,1]$ . It was defined by Sklar (1959) as

C(u_{1},\dots,u_{d})=\mathbb{P}(U_{1}\leq u_{1},\dots,U_{d}\leq u_{d}),

where $C(\cdot)$ is the copula; $(U_{1},\dots,U_{d})$ , with $U_{i}\sim\mathcal{U}(0,1)$ for all $i=1,\dots,d$ , is a vector of random variables; and $\smash{(u_{1},\dots,u_{d})\in[0,1]^{d}}$ are realizations of $(U_{1},\dots,U_{d})$ .

2.1.2 Sklar’s theorem

Theorem 2.1.

Let $F$ be a joint distribution function with margins $F_{1},\dots,F_{d}$ . Then, there exists a copula $\smash{C\colon[0,1]^{d}\rightarrow[0,1]}$ such that, for all $x_{1},\dots,x_{d}$ ,

F(x_{1},\dots,x_{d})=C(F_{1}(x_{1}),\dots,F_{d}(x_{d})).

If the margins are continuous, then $C$ is unique. We define the density of a copula, $c$ , as

f(x_{1},\dots,x_{d})=c(F_{1}(x_{1}),\dots,F_{d}(x_{d}))\prod_{i=1}^{d}f_{i}(x_% {i}),

and we have

c(x_{1},\dots,x_{d})=\frac{\partial^{d}C}{\partial x_{1}\cdots\partial x_{d}}(% F_{1}(x_{1}),\dots,F_{d}(x_{d})).

According to the above theorem, the risk modeling process with copulas consists of two steps. First, we must determine the marginal distribution of each single risk component $F_{i}$ . Second, we need to model the dependence structure between these risk components via the copula function, $C(\cdot)$ .

2.2 $t$ -copulas

A classic copula often used in risk management is the $t$ -copula. This is defined by

	$\displaystyle C_{\bm{\varSigma},\nu}^{t}(u_{1},\dots,u_{d})$	$\displaystyle=\mathbb{P}(U_{1}\leq u_{1},\dots,U_{d}\leq u_{d})$
		$\displaystyle=\bm{t}_{\bm{\varSigma},\nu}(t_{\nu}^{-1}(u_{1}),\dots,t_{\nu}^{-% 1}(u_{d})),$

where $t_{\nu}$ is the distribution function of a standard univariate $t$ -distribution with degree of freedom $\nu$ , and $\bm{t}_{\bm{\varSigma},\nu}$ denotes the joint distribution function of a multivariate Student distribution $X\sim\bm{t}_{d}(0,\bm{\varSigma},\nu)$ .

The density of a $t$ -copula can be written as

	$\displaystyle c_{\bm{\varSigma},\nu}^{t}(u_{1},\dots,u_{d})$	$\displaystyle=\frac{1}{\sqrt{\det\bm{\varSigma}}}\varGamma(\tfrac{1}{2}(\nu+d)% )\varGamma(\tfrac{1}{2}\nu)^{d-1}\left[1+\frac{1}{\nu}\begin{pmatrix}t_{\nu}^{% -1}(u_{1})\\ \vdots\\ t_{\nu}^{-1}(u_{d})\end{pmatrix}^{\!\mathrm{T}}\cdot\bm{\varSigma}^{-1}\cdot% \begin{pmatrix}t_{\nu}^{-1}(u_{1})\\ \vdots\\ t_{\nu}^{-1}(u_{d})\end{pmatrix}\right]$
		$\displaystyle\qquad\times\bigg(\varGamma(\tfrac{1}{2}(\nu+1))^{d}\prod_{j=1}^{% d}\bigg(1+\frac{1}{\nu}t_{\nu}^{-1}(u_{j})^{2}\bigg)^{\!-(\nu+1)/2}\bigg)^{\!-% 1}.$

3 Estimation of parameters $\bm{\varSigma}$ and $\nu$ of a $t$ -copula

Given a $t$ -copula $\smash{C_{\bm{\varSigma},\nu}^{t}}$ , we have to estimate $d(d-1)/2$ parameters for the correlation matrix $\bm{\varSigma}$ and one parameter for the degree of freedom $\nu$ . When the data is low dimensional, we may use the maximum likelihood method to estimate these parameters. However, the optimization complexity with high-dimensional data is very difficult to handle, as it is hard to simultaneously optimize $\tfrac{1}{2}d(d-1)+1$ parameters. McNeil et al (2005) proposed a semi-parametric estimation method based on Kendall’s $\tau$ for high-dimensional $t$ -copulas. In order to simplify the optimization complexity, they first estimate the correlation matrix $\bm{\varSigma}$ of the $t$ -copula via Kendall’s $\tau$ and then maximize the likelihood function, taking $\smash{\hat{\bm{\varSigma}}}$ as an input to estimate the degree of freedom, $\nu$ .

3.1 Kendall’s canonical maximum likelihood estimator

Fantazzini (2010) talks about the “three-stage procedure”, referring to the semi-parametric estimation method proposed by McNeil et al (2005), as follows.

Let $\bm{L}=(L_{1},L_{2},\dots,L_{d})$ be a continuous random vector with margins $F_{1},F_{2},\dots,F_{d}$ .

Stage 1: transform the data set

In order to compute the likelihood function of a $t$ -copula, we transform the data set ( $n$ independent copies of $\bm{L}$ ) $(L_{11},L_{12},\dots,L_{1d}),\dots,(L_{n1},L_{n2},\dots,L_{nd})$ into uniform variables

\hat{\bm{U}}_{1}=(\hat{F}_{1}(L_{11}),\hat{F}_{2}(L_{12}),\dots,\hat{F}_{d}(L_% {1d})),\quad\dots,\quad\hat{\bm{U}}_{n}=(\hat{F}_{1}(L_{n1}),\hat{F}_{2}(L_{n2% }),\dots,\hat{F}_{d}(L_{nd}))

using the empirical distribution function $\smash{\hat{F}_{i}(\cdot)}$ , which is defined as

\hat{F}_{i}(x)=\frac{1}{n+1}\sum_{t=1}^{n}\bm{1}_{\{L_{ti}\leq x\}}.

Stage 2: PP-Kendall correlation matrix estimator for $\bm{\varSigma}$

$\smash{C_{\bm{\varSigma},\nu}^{t}}$ requires not only the specification of the estimate of the correlation matrix $\bm{\varSigma}$ , but also the degree of freedom, $\nu$ . In this case, we cannot build an estimator $\smash{\hat{\bm{\varSigma}}}$ with $\smash{\hat{\bm{\varSigma}}}_{ij}=\text{cor}(t_{\nu}^{-1}(F_{i}(L_{i})),t_{\nu% }^{-1}(F_{j}(L_{j})))$ , as we do not know the value of $\nu$ . McNeil et al (2005) introduced a robust estimator of $\bm{\varSigma}$ named the “pseudo-Pearson Kendall (PP-Kendall) correlation matrix estimator”, using the relationship between Kendall’s $\tau$ and Pearson’s $\rho$ in the case of elliptical copulas. When the variables $(L_{1},L_{2},\dots,L_{d})$ with marginal distribution functions $F_{1},F_{2},\dots,F_{d}$ are connected by an elliptical copula with correlation matrix $\bm{\varSigma}$ , Lindskog et al (2002) demonstrated the following relationship between the Kendall and Pearson coefficients:

\sin(\tfrac{1}{2}\pi\tau(L_{i},L_{j}))=\varSigma_{ij}.

As Kendall’s $\tau$ is a rank correlation coefficient, the PP-Kendall correlation matrix estimator is invariant to monotonic transformations:

\tau_{K}(L_{i},L_{j})=\tau_{K}(F_{i}(L_{i}),F_{j}(L_{j})).

Let $\smash{\hat{\bm{\varSigma}}}_{\text{PP-K}}$ denote the PP-Kendall correlation matrix estimator of $\bm{\varSigma}$ .

Stage 3: Kendall canonical maximum likelihood estimator for $\nu$

We look for the canonical maximum likelihood estimator of the degree of freedom $\nu$ by maximizing the loglikelihood function of the $t$ -copula density:

\hat{\nu}=\operatorname*{arg~{}max}_{\nu}\sum_{i=1}^{n}\log c_{\hat{\bm{% \varSigma}},\nu}^{t}(\hat{U}_{i}).

We will call this the “Kendall CML” method.

3.2 The correlation matrix estimate may not be positive semi-definite

The formulas in the second stage of Section 3.1 connect all the Kendall’s $\tau$ of the sample pairwise to their corresponding Pearson’s $\rho$ . Strictly speaking, the relation between Kendall’s $\tau$ and Pearson’s correlation $\rho$ found by Lindskog et al (2002) is valid for the bivariate case, but in practice we can build a correlation matrix from Kendall’s $\tau$ bivariate estimates when the dimension of the matrix is not too high. However, in the case of high-dimensional data, $\smash{\hat{\bm{\varSigma}}_{\text{PP-K}}}$ is a matrix close to $\bm{\varSigma}$ , whose entries contain some small residual “error”; this becomes relevant when such small errors are accumulated, making the deduced correlation matrix unreliable.

In addition, there is no guarantee that this componentwise transformation of the empirical Kendall’s $\tau$ matrix is positive semi-definite. Rousseeuw and Molenberghs (1993) proposed a simple technique to adjust $\smash{\hat{\bm{\varSigma}}}$ to obtain a positive definite matrix. They first compute the eigenvalues and the eigenvectors of the correlation matrix calculated from Kendall’s $\tau$ . Then, they replace the negative eigenvalues by arbitrarily chosen slightly positive ones ( $\varDelta$ ) and keep the eigenvectors unchanged. They rebuild the matrix with the new eigenvalues and the old eigenvectors. As the diagonal elements of the computed matrix will not necessarily be equal to $1$ , Rousseeuw and Molenberghs consider the computed matrix as a covariance matrix and transform it into its corresponding correlation matrix. They do not propose a slightly positive value for $\varDelta$ to use in the procedure. McNeil et al (2005) chose $\varDelta=0.001$ to estimate the $t$ -copula parameters. Considering the matrix itself, the choice of $\varDelta$ will not greatly affect the results of the PP-Kendall correlation matrix estimator. But, the spectrum of its matrix inverse, which appears in the density formula of a $t$ -copula, is much more affected by the choice of $\varDelta$ (indeed, $1/\varDelta$ is the highest eigenvalue of the inverse), and this choice induces bias when estimating the degree of freedom using the Kendall CML method.

An algorithm proposed by Higham (2002) can adjust $\smash{\hat{\bm{\varSigma}}}$ to obtain its “nearest” positive definite matrix. Here, the distance between matrixes is measured by the Frobenius norm. Higham (2002) uses the modified alternating projections method to compute the solution: a point in the intersection of two convex sets,

	$\displaystyle S$	$\displaystyle=\{\bm{Y}=\bm{Y}^{\mathrm{T}}\in\mathbb{R}^{d\times d}\colon\bm{Y% }\text{~{}is positive definite}\}$
and
	$\displaystyle U$	$\displaystyle=\{\bm{Y}=\bm{Y}^{\mathrm{T}}\in\mathbb{R}^{d\times d}\colon Y_{% ii}=1,~{}i=1,\dots,d\}.$

This technique finds a matrix that minimizes the Frobenius norm, but the smallest eigenvalue of this matrix may differ from the true value of the correlation matrix. Thus, this method will also bias the parameter’s estimation of $t$ -copulas.

We now present two simple examples to illustrate the existence of noise and the disadvantages of the Rousseeuw–Molenberghs and Higham methods.

Estimation of ν for 500 simulated data sets. d=200, n=1000 — Figure 1: Estimation of $\nu$ for 500 simulated data sets. $d=\text{200}$ , $n=\text{1000}$ .

Table 1: Bias and variance of Kendall CML estimator of

\nu

.

	No denoising
Bias of $\hat{\nu}$	1.10
Variance of $\hat{\nu}$	0.006

Example 3.1 (Existence of noise).

We simulate 500 data sets of $t$ -copulas with a uniform correlation matrix:

\bm{\varSigma}_{\text{uniform}}=\begin{pmatrix}1&\cdots&\rho\\ \vdots&\ddots&\vdots\\ \rho&\cdots&1\end{pmatrix},

and we set the dimension of the data set to $d=200$ , the sample size of the data set to $n=1000$ , the degree of freedom of the $t$ -copula to $\nu=5$ and the coefficient of the uniform correlation matrix to $\rho=0.2$ . Then, we estimate the value of the degree of freedom of the $t$ -copula by using the CML estimator. In Figure 1, we provide the results of the estimation for 500 simulated data sets. In Table 1, we give the values of the empirical bias and empirical variance of the estimator $\hat{\nu}$ .

According to these results, the Kendall CML estimator is clearly not an unbiased estimator of the degree of freedom for a high-dimensional $t$ -copula. In this configuration ( $Q=n/d=5$ is quite large), the PP-Kendall correlation matrix estimator of $\bm{\varSigma}$ is almost always positive definite, so the bias of the estimation is due to the “noise” in the data set, and not due to an arbitrary technique to make a nonpositive definite correlation matrix positive definite.

Example 3.2 (Bias of estimation).

We again simulate 500 data sets of $t$ -copulas with a uniform correlation matrix

\bm{\varSigma}_{\text{uniform}}=\begin{pmatrix}1&\cdots&\rho\\ \vdots&\ddots&\vdots\\ \rho&\cdots&1\end{pmatrix},

and this time we choose $d=200$ , $n=300$ , $\nu=5$ and $\rho=0.2$ .

In this configuration ( $Q=n/d=1.5$ is not very large), the PP-Kendall correlation matrix estimator $\smash{\hat{\bm{\varSigma}}}$ is almost never positive definite. Given a data set, the method we use to find a positive definite correlation matrix will greatly impact the results of the estimation.

Figure 2 shows that all of these estimations are biased. The results of estimation in Table 2 are very sensitive to the slightly positive value used in the Rousseeuw–Molenberghs method.

3.3 Denoising the correlation matrix to ensure a better estimate of high-dimensional $t$ -copulas

Because of the two sources of noise we exhibited in the previous section, we decide to denoise the PP-Kendall correlation matrix estimator.

3.3.1 Principle of the denoising technique

Estimation of ν for 500 simulated data sets. d=200, n=300 — Figure 2: Estimation of $\nu$ for 500 simulated data sets. $d=\text{200}$ , $n=\text{300}$ .

Table 2: Bias and variance of Kendall CML estimator of

\nu

.

	Rousseeuw and
	Molenberghs

	$\varDelta=\textbf{10}^{-\textbf{3}}$	$\varDelta=\textbf{10}^{-\textbf{5}}$	$\varDelta=\textbf{10}^{-\textbf{11}}$	Higham
Bias of $\hat{\nu}$	9.10	2.37	$-$ 0.67	1.01
Variance of $\hat{\nu}$	0.215	0.018	0.004	0.009

Laloux et al (2000) introduced the denoising technique to improve the estimation of the correlation matrix. This technique is based on results proved by the Marčenko–Pastur theorem in RMT, which provides a criterion to differentiate estimated eigenvalues of an empirical correlation matrix that carry a signal from others that might only be noise. In practice, it is necessary to compare the law of eigenvalues of an empirical matrix $\smash{\hat{\bm{\varSigma}}}$ to the law of eigenvalues of a correlation matrix obtained from purely random uncorrelated variables in the same configuration (dimension and sample size).

3.3.2 Marčenko–Pastur theorem and the denoising technique

We state the Marčenko–Pastur law as follows.

Let $\bm{X}$ denote an $n\times d$ random matrix

\bm{X}=\begin{pmatrix}X_{11}&\cdots&X_{1d}\\ X_{21}&\cdots&X_{2d}\\ \vdots&\ddots&\vdots\\ X_{n1}&\cdots&X_{nd}\end{pmatrix},

whose entries are independent and identically distributed random variables with mean $0$ and variance $1$ , ie,

\text{for all~{}}i,j,\quad\mathbb{E}(X_{ij})=0,\quad\operatorname{var}(X_{ij})% =1.

Let

\bm{Y}_{n}=\frac{1}{n}\bm{XX}^{\mathrm{T}},

and let $\lambda_{1},\lambda_{2},\dots,\lambda_{d}$ be the eigenvalues of $\bm{Y}_{n}$ . We define the random spectral measure of $\bm{Y}_{n}$ by

\mu_{d}(A)=\frac{1}{d}\sum\bm{1}_{\lambda_{i}\in A},\quad A\subset\mathbb{R}.

Finally, when $(n,d)\rightarrow(+\infty,+\infty)$ and the ratio $Q=n/d\geq 1$ , we obtain

\mu_{d}\to\mu\text{~{}almost surely},

where $\mu$ is a deterministic measure whose density is given by

\frac{\mathrm{d}\mu}{\mathrm{d}x}=Q\frac{\sqrt{(\lambda^{+}-x)(x-\lambda^{-})}% }{2\pi x},\quad\lambda^{-}\leq x\leq\lambda^{+},

with

\lambda^{-}=\bigg(1-\frac{1}{\sqrt{Q}}\bigg)^{\!2},\qquad\lambda^{+}=\bigg(1+% \frac{1}{\sqrt{Q}}\bigg)^{\!2}.

Note that the support of Marčenko–Pastur law is bounded. The interval $[\lambda^{-},\lambda^{+}]$ is called the “bulk”, and this gets narrower as $Q$ increases. In the case when $Q\rightarrow\infty$ , the bulk is limited to $\{1\}$ and the Marčenko–Pastur law is a Dirac mass in $1\colon\delta_{1}(x)$ .

The support of Marčenko–Pastur law, ie, the bulk, appears numerically to be the support of estimated eigenvalues, even in a non-asymptotic framework (Crenin et al 2015). From this, we see that eigenvalues that are out of the bulk can be supposed to carry a signal when others may not. According to Laloux et al (2000) and Crenin et al (2015), for finite sample sizes and finite dimensions there is a small probability of finding eigenvalues above $\lambda^{+}$ and below $\lambda^{-}$ when data sets are uncorrelated, and this probability naturally tends to $0$ as $n$ grows.

To clean an empirical correlation matrix, we would do the following:

(1)

estimate the correlation matrix and compute the eigenvalues and eigenvectors;
(2)

if necessary, apply Rousseeuw and Molenberghs’s technique with a slightly positive value $\varDelta$ to make it positive definite;
(3)

keep the estimated eigenvalues that are higher than the upper bound of the bulk unchanged, and substitute the others by their mean to keep the trace unchanged;
(4)

renormalize the matrix so that it is a correlation matrix (with $1$ on the diagonal).

3.3.3 The benefit of using the denoising technique to fit $t$ -copulas

There are two advantages when using this technique to clean the correlation matrix estimator.

(1)

It provides a criterion to differentiate signal and noise in the empirical correlation matrix, which depends only on the sample size and the dimension.
(2)

It is not greatly affected by the arbitrary choice of $\varDelta$ , as we replace all the eigenvalues lower than the upper bound of the bulk by their mean, which does not differ a lot for different choices of $\varDelta$ .

In addition, Crenin et al (2015) shows that this theory can still be applied when the dimension of the data set and the sample size are both not very large ( $d=20$ and $n=50$ ).

4 Simulation studies

In this section, we present some numerical examples to illustrate that the denoising technique can improve the estimate of the parameters of a $t$ -copula using the Kendall CML method. The denoising technique performs well in many cases.

Configuration 1: $d=\text{200}$ , $n=\text{1000}$

In this configuration, the dimension of the data set, $d$ , and the sample size, $n$ , are both large, and their ratio, $Q=n/d=5$ , is also large. As $n$ and $Q$ are very large, there are rarely negative eigenvalues in the PP-Kendall correlation matrix estimator $\smash{\hat{\bm{\varSigma}}}_{\text{PP-K}}$ , and each element of $\smash{\hat{\bm{\varSigma}}_{\text{PP-K}}}$ is close to the theoretical one.

Configuration 2: $d=\text{200}$ , $n=\text{300}$

In this configuration, the dimension of the data set, $d$ , and the sample size, $n$ , are both large, but the ratio $Q=n/d=1.5$ is not so large. As $Q$ is small, there are almost always negative eigenvalues in the PP-Kendall correlation matrix estimator $\smash{\hat{\bm{\varSigma}}}_{\text{PP-K}}$ .

Configurations 1 and 2 are often faced in portfolio management, market risk and credit risk.

Configuration 3: $d=\text{20}$ , $n=\text{50}$

In this configuration, the dimension of the data set, $d$ , the sample size, $n$ , and the ratio $Q=n/d=2.5$ are all not large.

We want to illustrate that the denoising technique can simultaneously clean the noise and the artefact due to the arbitrary choice of $\varDelta$ in Rousseeuw and Molenberghs’s technique.

Configuration 3 is met in operational risk or many cases of risk aggregation.

4.1 Simulation studies with correlation matrix depending on factors

Before simulating the data set whose dependence structure is modeled by a $t$ -copula, we should first build a theoretical correlation matrix. In the financial field, we often use common factors to explain market phenomena and the equilibrium of asset prices. We decided to build such a correlation matrix with the multifactor model.

Assume that random variables are modeled by common and idiosyncratic factors. Each random variable $L_{i}$ , $i\in\{1,\dots,d\}$ , is thus modeled as

L_{i}=\beta_{i,1}F_{1}+\beta_{i,2}F_{2}+\cdots+\beta_{i,k}F_{k}+\varepsilon_{i},

(4.1)

where $F_{j}$ , $j\in\{1,\dots,k\}$ , $k\in\mathbb{N}^{*}$ , are the factors assumed to be independent, with $\mathbb{E}(L_{i})=0$ and $\operatorname{var}(L_{i})=1$ . $\beta_{i,j}$ is the sensitivity of factor $L_{i}$ to factor $F_{j}$ , and $\varepsilon_{i}$ is the idiosyncratic factor, such that

\varepsilon_{i}\perp\varepsilon_{j},\quad F_{j}\perp F_{i}\quad\text{for all~{% }~{}}i\neq j\quad\text{and}\quad\varepsilon_{j}\perp F_{j}\quad\text{for all~{}}j.

In this modeling, the theoretical correlation matrix depends on the factor sensitivities and is given by the following:

\text{for all~{}}i\neq j,\quad\rho(L_{i},L_{j})=\varSigma(i,j)=\frac{\sum_{s=1% }^{k}\beta_{i,s}\beta_{j,s}}{\sqrt{(\sum_{s=1}^{k}\beta_{i,s}^{2}+1)(\sum_{s=1% }^{k}\beta_{j,s}^{2}+1)}}

for all $i\in\{1,\dots,d\}$ , $\varSigma(i,i)=1$ .

4.1.1 Numerical results

Table 3: Bias and variance of the Kendall CML estimator of

\nu

.

	No denoising	Denoising
Bias of $\hat{\nu}$	1.05	0.01
Variance of $\hat{\nu}$	0.005	0.002

Here, we only give the results of the case with a one-factor correlation matrix. To do so, we assume that

L_{i}=\beta_{i}F_{1}+\sqrt{\vphantom{\rule[0.0pt]{0.0pt}{8.0pt}}\smash{1-\beta% _{i}^{2}}}\varepsilon_{i},\quad i\in\{1,\dots,d\}.

The correlation between $L_{i}$ and $L_{j}$ is thus equal to

\rho(L_{i},L_{j})=\varSigma(i,j)=\beta_{i}\beta_{j}.

In order to have appropriate correlation coefficients, we simulate $\beta_{i}\sim\mathcal{U}(0,0.5)$ for all $i\in\{1,\dots,d\}$ , and we compute $\bm{\varSigma}$ with the above formulas. In this case, we will get a theoretical one-factor correlation matrix.

In each configuration, we simulate 500 data sets with the dimension of the data set equal to $d$ , the sample size equal to $n$ of a $t$ -copula with theoretical correlation matrix $\bm{\varSigma}$ and a degree of freedom of $\nu=5$ . Then, we estimate the value of the degree of freedom of a $t$ -copula using the CML estimator. We will compare the results of the estimation without the denoising technique with those of the estimation with the denoising technique.

Configuration 1: $d=\text{200}$ , $n=\text{1000}$

As we saw in Section 3.2, the CML estimator is a biased estimator of the degree of freedom of the $t$ -copula. In this configuration, there is no arbitrary choice of a slight positive value, and the bias is just the accumulated effect of noise in the correlation matrix estimator. From Figure 3, we can see that the CML estimator observed after denoising is a robust estimator of the degree of freedom of the $t$ -copula. Table 3 shows the empirical bias and the empirical variance of $\hat{\nu}$ estimated on the 500 simulated data sets. The denoising technique based on RMT reduces not only the bias but also the variance’s estimator.

Configuration 2: $d=\text{200}$ , $n=\text{300}$

In this configuration, the impact of the method chosen to adjust the empirical correlation matrix is huge. Higham’s method cannot give an unbiased estimation of the degree of freedom of the $t$ -copula, and we cannot find any slightly positive value that should be used in the Rousseeuw–Molenberghs method, as this would change in another configuration. From Figure 4 and Table 4, we can see that the denoising technique performs very well in this configuration.

Table 4: Bias and variance of Kendall CML estimator of

\nu

.

	Rousseeuw and
	Molenberghs

	$\varDelta=\textbf{10}^{-\textbf{3}}$	$\varDelta=\textbf{10}^{-\textbf{5}}$	$\varDelta=\textbf{10}^{-\textbf{11}}$	Higham	Denoising
Bias of $\hat{\nu}$	9.48	2.52	$-$ 0.59	0.92	0.06
Variance of $\hat{\nu}$	0.242	0.019	0.004	0.008	0.008

Configuration 3: $d=\text{20}$ , $n=\text{50}$

In this extreme configuration, which often appears in operational risk modeling, both the dimension and the number of observations are relatively small. We do the same test and always notice the benefits from using RMT to fit a $t$ -copula. In this case, the classical method cannot give a robust estimation of the degree of freedom for a $t$ -copula. Meanwhile, after applying the denoising technique, the noise in the estimation is much reduced (see Figure 5 and Table 5).

Estimation of ν for 500 simulated data sets — Figure 5: Estimation of $\nu$ for 500 simulated data sets. (a) No denoising. (b) Denoising. $d=\text{200}$ , $n=\text{50}$ .

Table 5: Bias and variance of the Kendall CML estimator of

\nu

.

	No denoising	Denoising
Bias of $\hat{\nu}$	05.60	0.22
Variance of $\hat{\nu}$	12.839	0.651

4.2 Simulation studies with correlation matrix having specified eigenvalues

Table 6: Bias and variance of the Kendall CML estimator of

\nu

.

	No denoising	Denoising
Bias of $\hat{\nu}$	1.23	0.19
Variance of $\hat{\nu}$	0.006	0.003

In this section, we want the theoretical correlation matrix to have specified eigenvalues. We use the method proposed by Marsaglia et al (1984) and Jones (2010) to generate a theoretical correlation matrix of dimension $d$ with the following configuration:

\bm{\lambda}=(\underset{d/4}{\underbrace{2,\dots,2}},\underset{d/2}{% \underbrace{0.75,\dots,0.75}},\underset{d/4}{\underbrace{0.5,\dots,0.5}}).

Unlike the correlation matrix generated by several factors in the previous section, the structure of the above correlation matrix is more sophisticated, and has more information lying in the bulk. We then simulate the data set and conduct the experiment introduced in Section 4.1.

4.2.1 Numerical results

Table 7: Bias and variance of Kendall CML estimator of

\nu

.

	Rousseeuw and
	Molenberghs

	$\varDelta=\textbf{10}^{-\textbf{3}}$	$\varDelta=\textbf{10}^{-\textbf{5}}$	$\varDelta=\textbf{10}^{-\textbf{11}}$	Denoising
Bias of $\hat{\nu}$	8.97	2.48	$-$ 0.56	0.30
Variance of $\hat{\nu}$	0.145	0.014	0.003	0.009

Table 8: Bias and variance of the Kendall CML estimator of

\nu

.

	No denoising	Denoising
Bias of $\hat{\nu}$	05.46	0.29
Variance of $\hat{\nu}$	14.765	0.690

•

Configuration 1: $d=\text{200}$ , $n=\text{1000}$ . See Figure 6 and Table 6.
•

Configuration 2: $d=\text{200}$ , $n=\text{300}$ . See Figure 7 and Table 7.
•

Configuration 3: $d=\text{20}$ , $n=\text{50}$ . See Figure 8 and Table 8.

As in the previous section, Figures 6–8 show that the denoising technique also performs very well in the case of a correlation matrix with specified eigenvalues. In each configuration of risk modeling that we have tested, the denoising technique reduced the bias and provided a more robust estimator of the degree of freedom of $t$ -copulas. Further, from Figures 6 and 7, we see that, after applying the denoising technique on simulation studies, the CML estimator seems to be slightly biased, and thus demonstrates the limit of the suggested technique, which might ignore some small true data lying in the bulk.

5 Empirical results in operational risk modeling

5.1 The data set

The consultative paper from European Banking Authority (2014) provides guidance on dependence modeling in operational risk and suggests that banks use $t$ -copulas. Thus, the estimation of the degree of freedom of $t$ -copulas has become the central issue in operational risk modeling. Our study is based on losses occurring in numerous financial institutions in Western Europe between 2002 and 2013 that were collected by the Operational Riskdata eXchange Association (ORX), a leading banking consortium. Actually, the localization proved to be discriminant in terms of risk profiles (see Cope and Labbi 2008), and thus we decided to reduce the perimeter provided by ORX. For that reason, we propose modeling operational risk on a bank that would perform only retail activities, as in Crenin et al (2015). Risks are consequently modeled on the following five business lines for forty-eight quarters:

(1)

retail banking;
(2)

commercial banking;
(3)

payment and settlement;
(4)

agency services;
(5)

retail brokerage.

The main idea here is to illustrate the application of RMT in operational risk when fitting $t$ -copulas. Consequently, the following results rely on thirty-five units of measure ( $5\times 7$ risk types as defined by Basel III’s advanced measurement approach model) and forty-eight quarters, leading to $Q=48/35=1.4$ .

5.2 RMT application

5.2.1 Estimation of the degree of freedom of $t$ -copulas without applying the denoising technique

Table 9: Estimation of

\nu

without applying the denoising technique.

	Rousseeuw and
	Molenberghs

	$\varDelta=\textbf{10}^{-\textbf{3}}$	$\varDelta=\textbf{10}^{-\textbf{5}}$	$\varDelta=\textbf{10}^{-\textbf{11}}$	Higham
$\hat{\nu}$ (no denoising)	5.51	3.08	1.93	2.34

As explained in Crenin et al (2015) and in Section 3.1, we first transform our observations into pseudo-observations in order to estimate the Kendall correlations. These are then transformed into Pearson correlations using the formula that links the two parameters, leading to what are called PP-Kendall correlations. When necessary, the estimated correlation matrix is made positive definite using the method suggested by Rousseeuw and Molenberghs (1993), with an arbitrarily chosen $\varDelta$ , or using the method of Higham (2002). Eventually, we estimate the degree of freedom of the $t$ -copula by maximization of the likelihood function. Table 9 shows the impact of the choice of method on the estimation of the degree of freedom of the $t$ -copula. The results of estimation of $\nu$ are very sensitive to the method or the value of $\varDelta$ used to make the estimated correlation matrix positive definite.

5.2.2 Estimation of the degree of freedom of $t$ -copulas after applying the denoising technique

In this section, we will show the result of the estimation of the degree of freedom of a $t$ -copula after applying the denoising technique. As explained in Section 3.3, we apply the denoising technique to clean the empirical correlation matrix, and then estimate the degree of freedom of the $t$ -copulas by maximizing the likelihood function.

Table 10: Estimation of

\nu

after applying the denoising technique.

	Rousseeuw and
	Molenberghs

	$\varDelta=\textbf{10}^{-\textbf{3}}$	$\varDelta=\textbf{10}^{-\textbf{5}}$	$\varDelta=\textbf{10}^{-\textbf{11}}$	Higham
$\hat{\nu}$ (denoising)	9.93	9.93	9.93	9.97

Table 10 shows that the results of the estimation of $\nu$ after applying the denoising technique are stable with the method used to make the estimated correlation matrix positive definite. The difference between Tables 9 and 10 suggests that the denoising technique is very necessary in order to get a robust estimation of the degree of freedom of $t$ -copulas.

6 Conclusion

We introduced bias into the Kendall canonical maximum likelihood estimator of the degree of freedom of a $t$ -copula and demonstrated a technique that may clean the noise from the estimated correlation matrix to give a more robust estimator of the degree of freedom.

We also showed, using simulation studies and an example of operational risk modeling, the necessity and the benefit of using RMT to fit high-dimensional $t$ -copulas in risk modeling, even when the number of observations is small. More studies are still to be conducted in order to further reduce the slight remaining bias, by identifying all the true information contained in sophisticated correlation matrixes.

Declaration of interest

The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the official positions of Société Générale.

Acknowledgements

We are grateful to Vivien Brunel, who gave us the idea of applying RMT to operational risk. We are also grateful to the anonymous reviewer, who provided very helpful comments. The usual disclaimer nonetheless applies, and all errors remain ours.

References

Cope, E., and Labbi, A. (2008). Operational loss scaling by exposure indicators: evidence from the ORX database. Working Paper, ORX Analytical Working Group.

Crenin, F., Cressey, D., Lavaud, S., Xu, J., and Clauss, P. (2015). Random matrix theory applied to correlations in operational risk. The Journal of Operational Risk 10(4), 45–71 (http://doi.org/brw3).

European Banking Authority (2014). Draft regulatory technical standards on assessment methodologies for the advanced measurement approaches for operational risk under Article 312 of Regulation (EU) No 575/2013. Consultation Paper, EBA.

Fantazzini, D. (2010). Three-stage semi-parametric estimation of $t$ -copulas: asymptotics, finite-samples, properties and computational aspects. Computational Statistics and Data Analysis 54(11), 2562–2579 (http://doi.org/bbqbq5).

Higham, N. J. (2005). Computing the nearest correlation matrix: a problem from finance. IMA Journal of Numerical Analysis 22, 329–343 (http://doi.org/d8gj5f).

Jones, J. A. (2010). GenCorr: an R routine to generate correlation matrices from a user-defined eigenvalue structure. Applied Psychological Measurement 34, 68–69 (http://doi.org/bhjb5s).

Laloux, L., Cizeau, P., Potters, M., and Bouchaud, J.-P. (2000). Random matrix theory and financial correlations. International Journal of Theoretical and Applied Finance 3(3), 391–397 (http://doi.org/dd7skx).

Lindskog, F., McNeil, A. J., and Schmock, U. (2002). Kendall’s tau for elliptical distributions. In Credit Risk: Measurement, Evaluation and Management, pp. 149–156. Contributions to Economics. Physica, Heidelberg.

Marsaglia, G., and Olkin, I. (1984). Generating correlation matrices. SIAM Journal on Scientific and Statistical Computing 5, 470–475 (http://doi.org/bgv7hw).

McNeil, A. J., Frey, R., and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press.

Rousseeuw, P., and Molenberghs, G. (1993). Transformation of non positive semidefinite correlation matrices. Communications in Statistics: Theory and Methods 22(4), 965–984 (http://doi.org/c5z7kh).

Sklar, A. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris 8, 229–231.

[1] Cope, E., and Labbi, A. (2008). Operational loss scaling by exposure indicators: evidence from the ORX database. Working Paper, ORX Analytical Working Group.

[2] Crenin, F., Cressey, D., Lavaud, S., Xu, J., and Clauss, P. (2015). Random matrix theory applied to correlations in operational risk. The Journal of Operational Risk 10(4), 45–71 (http://doi.org/brw3).

[3] European Banking Authority (2014). Draft regulatory technical standards on assessment methodologies for the advanced measurement approaches for operational risk under Article 312 of Regulation (EU) No 575/2013. Consultation Paper, EBA.

[4] Fantazzini, D. (2010). Three-stage semi-parametric estimation of $t$ -copulas: asymptotics, finite-samples, properties and computational aspects. Computational Statistics and Data Analysis 54(11), 2562–2579 (http://doi.org/bbqbq5).

[5] Higham, N. J. (2005). Computing the nearest correlation matrix: a problem from finance. IMA Journal of Numerical Analysis 22, 329–343 (http://doi.org/d8gj5f).

[6] Jones, J. A. (2010). GenCorr: an R routine to generate correlation matrices from a user-defined eigenvalue structure. Applied Psychological Measurement 34, 68–69 (http://doi.org/bhjb5s).

[7] Laloux, L., Cizeau, P., Potters, M., and Bouchaud, J.-P. (2000). Random matrix theory and financial correlations. International Journal of Theoretical and Applied Finance 3(3), 391–397 (http://doi.org/dd7skx).

[8] Lindskog, F., McNeil, A. J., and Schmock, U. (2002). Kendall’s tau for elliptical distributions. In Credit Risk: Measurement, Evaluation and Management, pp. 149–156. Contributions to Economics. Physica, Heidelberg.

[9] Marsaglia, G., and Olkin, I. (1984). Generating correlation matrices. SIAM Journal on Scientific and Statistical Computing 5, 470–475 (http://doi.org/bgv7hw).

[10] McNeil, A. J., Frey, R., and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press.

[11] Rousseeuw, P., and Molenberghs, G. (1993). Transformation of non positive semidefinite correlation matrices. Communications in Statistics: Theory and Methods 22(4), 965–984 (http://doi.org/c5z7kh).

[12] Sklar, A. (1959). Fonctions de répartition à $n$ dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris 8, 229–231.

Journal of Operational Risk

2 Copulas

2.1 Basic concepts related to copulas

2.1.1 The copula

2.1.2 Sklar’s theorem

Theorem 2.1.

2.2 t-copulas

3 Estimation of parameters ? and ν of a t-copula

3.1 Kendall’s canonical maximum likelihood estimator

Stage 1: transform the data set

Stage 2: PP-Kendall correlation matrix estimator for ?

Stage 3: Kendall canonical maximum likelihood estimator for ν

3.2 The correlation matrix estimate may not be positive semi-definite

Example 3.1 (Existence of noise).

Example 3.2 (Bias of estimation).

3.3 Denoising the correlation matrix to ensure a better estimate of high-dimensional t-copulas

3.3.1 Principle of the denoising technique

3.3.2 Marčenko–Pastur theorem and the denoising technique

3.3.3 The benefit of using the denoising technique to fit t-copulas

4 Simulation studies

Configuration 1: d=200, n=1000

Configuration 2: d=200, n=300

Configuration 3: d=20, n=50

4.1 Simulation studies with correlation matrix depending on factors

4.1.1 Numerical results

Configuration 1: d=200, n=1000

Configuration 2: d=200, n=300

Configuration 3: d=20, n=50

4.2 Simulation studies with correlation matrix having specified eigenvalues

4.2.1 Numerical results

5 Empirical results in operational risk modeling

5.1 The data set

5.2 RMT application

5.2.1 Estimation of the degree of freedom of t-copulas without applying the denoising technique

5.2.2 Estimation of the degree of freedom of t-copulas after applying the denoising technique

6 Conclusion

Declaration of interest

Acknowledgements

References

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2.2 $t$ -copulas

3 Estimation of parameters $\bm{\varSigma}$ and $\nu$ of a $t$ -copula

Stage 2: PP-Kendall correlation matrix estimator for $\bm{\varSigma}$

Stage 3: Kendall canonical maximum likelihood estimator for $\nu$

3.3 Denoising the correlation matrix to ensure a better estimate of high-dimensional $t$ -copulas

3.3.3 The benefit of using the denoising technique to fit $t$ -copulas

Configuration 1: $d=\text{200}$ , $n=\text{1000}$

Configuration 2: $d=\text{200}$ , $n=\text{300}$

Configuration 3: $d=\text{20}$ , $n=\text{50}$

Configuration 1: $d=\text{200}$ , $n=\text{1000}$

Configuration 2: $d=\text{200}$ , $n=\text{300}$

Configuration 3: $d=\text{20}$ , $n=\text{50}$

5.2.1 Estimation of the degree of freedom of $t$ -copulas without applying the denoising technique

5.2.2 Estimation of the degree of freedom of $t$ -copulas after applying the denoising technique