Changing default risk dependencies during the subprime crisis: DJ iTraxx subindices and goodness-of-fit-testing for copulas

Abstract

This paper tests whether (and to what extent) default risk dependencies changed during the subprime crisis in 2007 and 2008. This is done by applying a Goodness-of-fit test, based on the Rosenblatt transformation, to test various null hypotheses with respect to the copula function that describes the stochastic dependence between daily returns of six sector-specific subindices of the Dow Jones iTraxx Credit Default Swap index for Europe. Overall, the results suggest that in the bivariate case, the t-copula is a better approximation to the true copula of returns of DJ iTraxx subindices than the normal copula or the generalized Clayton copula. On average, the number of degrees of freedom of the bivariate t-copula tends to decrease during the crisis. As expected, the correlation between the returns of the subindices increases significantly during the crisis. However, the multivariate analysis reveals that it is only before the crisis that the null hypothesis of a six-dimensional t-copula is not rejected. During the crisis, the multivariate stochastic dependence between the sector-specific DJ iTraxx subindices seems to change in such a complex way that it is no longer sufficiently described by a multivariate t-copula.

Appendices

Appendix A

1.1 GoF test for copulas based on the Rosenblatt transformation

The following bootstrap algorithm for computing the distribution function of the AD test statistic under the null hypothesis has been proposed by Dobrić and Schmid (2007):

• (1) Based on the originally observed random sample $(x_{1} ,y_{1} ),\, \ldots ,\,(x_{N} ,y_{N} )$, estimate the parameter vector $\hat{\lambda }$ of the parametric family of copulas C _λ that is tested in the null hypothesis H ₀: “(X, Y) has copula C _λ (u, v)”.

• (2) Repeat for $s = 1, \ldots ,S_{B}$ where S _B is the number of bootstrap simulations:

  • (2a) Generate i.i.d. observations $\left( {x_{1}^{(s)} ,y_{1}^{(s)} } \right),\, \ldots ,\,\left( {x_{N}^{(s)} ,y_{N}^{(s)} } \right)$ from $C_{{\hat{\lambda }}}$.

  • (2b) Estimate the parameter vector $\hat{\lambda }^{(s)}$ from $\left( {x_{1}^{(s)} ,y_{1}^{(s)} } \right),\, \ldots ,\,\left( {x_{N}^{(s)} ,y_{N}^{(s)} } \right)$, compute $\hat{S}^{(s)} \left( {x_{1}^{(s)} ,y_{1}^{(s)} } \right),\, \ldots ,\,\hat{S}^{(s)} \left( {x_{N}^{(s)} ,y_{N}^{(s)} } \right)$ and calculate the value AD ^(s) of the AD test statistic.

• (3) Based on $AD^{(1)} , \ldots ,AD^{{(S_{B} )}}$, the empirical distribution function of the AD test statistic under the null hypothesis can be computed. The critical value corresponding to an error probability of the first kind α equals the (1 − α)-percentile of the empirical distribution function of the AD test statistic. The null hypothesis H ₀: “(X, Y) has copula $C_{\lambda } (u,v)$” is rejected if the AD test statistic computed from the originally observed random sample $(x_{1} ,y_{1} ),\, \ldots ,\,(x_{N} ,y_{N} )$ is larger than that critical value.

When C _λ is a bivariate normal copula (see Tables 4, 5), only the correlation parameter ρ has to be estimated. This is done based on the estimated value $\hat{\rho }_{Sp}$ of Spearman’s coefficient of correlation and the relationship $\rho = 2\sin \left( {{{\pi \rho_{Sp} } \mathord{\left/ {\vphantom {{\pi \rho_{Sp} } 6}} \right. \kern-\nulldelimiterspace} 6}} \right)$ [see McNeil et al. (2005, p. 230)]. When C _λ is a t-copula (see Tables 6, 7, 8), the computation of ρ is based on the estimated value $\hat{\tau }$ of Kendall’s tau and the relationship $\rho = \sin \left( {{{\pi \tau } \mathord{\left/ {\vphantom {{\pi \tau } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)$ [see McNeil et al. (2005, p. 231)]. When C _λ is the generalized Clayton copula (see Tables 9, 10), the parameter δ is estimated for a given parameter θ by employing the relationship $\tau = {{((2 + \theta )\delta - 2)} \mathord{\left/ {\vphantom {{((2 + \theta )\delta - 2)} {(2 + \theta )\delta }}} \right. \kern-\nulldelimiterspace} {(2 + \theta )\delta }}$ [see McNeil et al. (2005, p. 222)].

Appendix B

2.1 ARMA models with GARCH errors

To remove autocorrelation and conditional heteroscedasticity in the univariate time series of returns, an ARMA model with GARCH errors is fitted to the raw returns $(r_{n,t} )_{{t \in \mathbb{Z}}}$ of each sector-specific DJ iTraxx subindex $n \in \{ 1, \ldots ,6\}$. First, an ARMA(p _n , q _n )-model is fitted to the data:

$$ r_{n,t} = \mu_{n,t} + \varepsilon_{n,t}, $$

$$\mu_{n,t} = \mu_{n} + \sum\limits_{i = 1}^{{p_{n} }} {\theta_{n,i} (r_{n,t - i} - \mu_{n} ) + \sum\limits_{j = 1}^{{q_{n} }} {\theta_{n,j} \varepsilon_{n,t - j} } }. $$

Afterwards, a GARCH(r _n , s _n )-model is fitted to the residuals $\hat{\varepsilon }_{n,t} = r_{n,t} - \hat{\mu }_{n,t}$:

$$ \varepsilon_{n,t} = \sigma_{n,t} \eta_{n,t}, $$

$$ \sigma_{n,t}^{2} = \alpha_{n,0} + \sum\limits_{i = 1}^{{r_{n} }} {\alpha_{n,i} \varepsilon_{n,t - i}^{2} } + \sum\limits_{j = 1}^{{s_{n} }} {\beta_{n,j} \sigma_{n,t - j}^{2} } $$

where $(\eta_{n,t} )_{{t \in \mathbb{Z}}}$ is strict white noise with mean zero and variance one, and $\alpha_{n,0} > 0,\,\alpha_{n,i} \ge 0,\,$ i = 1, …, r _n , $\beta_{n,j} \ge 0$, j = 1, …, s _n , and $\sum\nolimits_{i = 1}^{{r_{n} }} {\alpha_{n,i} } + \sum\nolimits_{j = 1}^{{s_{n} }} {\beta_{n,j} } < 1$ [see McNeil et al. (2005, pp. 148)]. Finally, the GoF test for copulas is applied to the filtered returns $\hat{\eta }_{n,t} = {{\hat{\varepsilon }_{n,t} } \mathord{\left/ {\vphantom {{\hat{\varepsilon }_{n,t} } {\hat{\sigma }_{n,t} }}} \right. \kern-\nulldelimiterspace} {\hat{\sigma }_{n,t} }} = {{\left( {r_{n,t} - \hat{\mu }_{n,t} } \right)} \mathord{\left/ {\vphantom {{\left( {r_{n,t} - \hat{\mu }_{n,t} } \right)} {\hat{\sigma }_{n,t} }}} \right. \kern-\nulldelimiterspace} {\hat{\sigma }_{n,t} }}$. Table 11 shows the specifications of the ARMA(p _n ,q _n )- and GARCH(r _n ,s _n )- models that are necessary to remove autocorrelation and conditional heteroscedasticity in the raw returns $(r_{n,t} )_{{t \in \mathbb{Z}}}$ of each DJ iTraxx subindex $n \in \left\{{1,\ldots,6} \right\}$ before and during the crisis.

Table 11 Specification of the ARMA(p _n , q _n )- and GARCH(r _n , s _n )-models

For estimating the parameters of the GARCH-models by maximum likelihood, the innovations $(\eta_{n,t} )_{{t \in \mathbb{Z}}}$ are assumed to have a standardized t-distribution. A Kolmogorow–Smirnov and Anderson-Darling test applied to $\left( {\hat{\eta }_{n,t} } \right)_{{t \in \mathbb{Z}}}$ do not reject the null hypothesis of a standardized t-distribution. In contrast, the Jarque-Bera test rejects the null hypothesis that the innovations $\left( {\hat{\eta }_{n,t} } \right)_{{t \in \mathbb{Z}}}$ come from a normal distribution. Applying the Ljung-Box test and Engle’s Lagrange multiplier (LM) test to the filtered returns $\left( {\hat{\eta }_{n,t} } \right)_{{t \in \mathbb{Z}}}$ (both up to lag 10), the null hypothesis of no autocorrelation and no ARCH effects, respectively, cannot be rejected any more. The model fitting has been done by using the statistics toolbox of Matlab and additionally some functions of Kevin Sheppard’s freely available GARCH toolbox.¹⁴