Adapting extreme value statistics to financial time series: dealing with bias and serial dependence
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Abstract
We handle two major issues in applying extreme value analysis to financial time series, bias and serial dependence, jointly. This is achieved by studying bias correction methods when observations exhibit weak serial dependence, in the sense that they come from \(\beta\)mixing series. For estimating the extreme value index, we propose an asymptotically unbiased estimator and prove its asymptotic normality under the \(\beta\)mixing condition. The bias correction procedure and the dependence structure have a joint impact on the asymptotic variance of the estimator. Then we construct an asymptotically unbiased estimator of high quantiles. We apply the new method to estimate the valueatrisk of the daily return on the Dow Jones Industrial Average index.
Keywords
Hill estimator Bias correction \(\beta\)mixing condition Tail quantile processMathematics Subject Classification
62G32 60G70JEL Classification
C141 Introduction
In financial risk management, a key concern is on modelling and evaluating potential losses occurring with extremely low probabilities, i.e., tail risks. For example, the Basel committee on banking supervision suggests regulators to require banks holding adequate capital against the tail risk of bank assets measured by the valueatrisk (VaR). The VaR refers to a high quantile of the loss distribution with an extremely low tail probability.^{1} Estimating such risk measures thus relies on modeling the tail region of distribution functions of asset values. To serve such a purpose, statistical tools stemming from extreme value theory (EVT) are obvious candidates. By investigating data in an intermediate region close to the tail, extreme value statistics employs models to extrapolate intermediate properties to the tail region. Although this attractive feature of extreme value statistics makes it a popular tool for evaluating tail events in many scientific fields such as meteorology and engineering, it has not yet emerged as a dominating tool in financial risk management. This is potentially due to some crucial critiques on applying EVT to financial data; see e.g. [4]. The critiques are mainly on two issues: the difficulty in selecting the intermediate region in estimation, and the validity for financial data of the maintained assumptions in EVT. This paper tries to deal with the two critiques simultaneously and provide adapted EVT methods that overcome the two issues jointly.
We start with explaining the problem on selecting the intermediate region in estimation. Extreme value statistics usually uses only observations in an intermediate region. This has been achieved by selecting the highest (or lowest, when dealing with the lower tail) \(k=k(n)\) observations in a sample with size \(n\). The problem of selecting \(k\) is sometimes referred to as “selecting the cutoff point”. Theoretically, the statistical properties of EVTbased estimators are established for \(k\) such that \(k\to\infty\) and \(k/n\to 0\) as \(n\to\infty\). In applications with a finite sample size, it is necessary to investigate how to choose the number of high observations used in estimation. For financial practitioners, two difficulties arise: firstly, there is no straightforward procedure for the selection; secondly, the performance of the EVT estimators is rather sensitive to this choice. More specifically, there is a bias–variance tradeoff: with a low level of \(k\), the estimation variance is at a high level which may not be acceptable for the application; by increasing \(k\), i.e., using progressively more data, the variance is reduced, but at the cost of an increasing bias.
Recent developments in extreme value statistics provide two types of solutions for selecting the cutoff point. The first aims to find the optimal cutoff point that balances the bias and variance assuming that the bias term in the asymptotic distribution is finite; see e.g. [3, 8] and [12]. The second type corrects the bias under allowing that the bias term in the asymptotic distribution is at an infinite level; see e.g. [10]. In comparison with the optimal cutoff point method, the bias correction procedure usually requires additional assumptions, such as a third order condition. Nevertheless, it is preferred to the optimal cutoff point approach because of the following relative advantages. First, since bias correction methods allow an infinite bias term in the asymptotic distribution, they correspondingly allow choosing a higher level of \(k\) than that chosen in the optimal cutoff point approach. Second, by choosing a larger \(k\), bias correction methods result in a lower level of estimation variance with no asymptotic bias. Third, in practice, bias correction procedures lead to estimates that are less sensitive to the choice of \(k\). This mitigates the difficulty in the selection of the cutoff point.
The other criticism on applying extreme value statistics to financial data is on the fact that most existing EVT methods require independent and identically distributed (i.i.d.) observations, whereas financial time series exhibit obvious serial dependence features such as volatility clustering. This issue has been addressed in works dealing with weak serial dependence; see e.g. [15] and [5]. The main message from these studies is that usual EVT methods are still valid; only the asymptotic variance of estimators may differ from that in the i.i.d. case.
Since the selection of the cutoff point and the serial dependence in data have been handled separately, the literatures addressing these two issues are mutually exclusive. In the bias correction literature, it is always assumed that the observations form an i.i.d. sample; in the literature on dealing with serial dependence, the choice of \(k\) is assumed to be sufficiently low such that there is no asymptotic bias. Therefore, it is still an open question whether we can apply the bias correction technique to datasets that exhibit weak serial dependence. This is what we intend to address in this paper.
We consider a bias correction procedure on estimating the extreme value index and high quantiles for \(\beta\)mixing stationary time series with common heavytailed distribution. The bias term stems from the approximation of the tail region of distribution functions. In EVT, a second order condition is often imposed to characterize such an approximation. Such a condition is almost indispensable for establishing asymptotic properties of estimators. To correct the bias, one needs to estimate the second order scale function, the function \(A\) in (2.3) below. The existing literature is restricted to the case \(A(t)=Ct^{\rho}\) with constants \(C\neq0\) and \(\rho<0\). The estimation of \(C\) requires extra conditions. Instead, we estimate the function \(A\) in a nonparametric way which makes the analysis and application smoother.
The asymptotically unbiased estimator we obtain has the following advantages. Firstly, it allows serial dependence in the observations. Secondly, one may apply the unbiased estimator with a higher value of \(k\), which reduces the asymptotic variance and ultimately the estimation error thanks to the bias correction feature. Thirdly, the theoretical range of potential choices of \(k\) is larger for our asymptotically unbiased estimators than for the original estimators. This makes the choice of \(k\) less crucial. All these features become apparent in simulation and application.
The paper is organized as follows. Under a simplified model without serial dependence, Sect. 2 presents the bias correction idea for the Hill estimator. Section 3 presents the general model with serial dependence, and in particular the regularity conditions we are dealing with. Section 4 defines the asymptotically unbiased estimators of the extreme value index and quantiles. In addition, we state the main theorems on the asymptotic normality of these two estimators. The bias correction procedure and the serial dependence structure have a joint impact on the asymptotic variances of the estimators. Section 5 discusses such a joint impact for several examples. Section 6 demonstrates finite sample performance of the asymptotically unbiased estimators based on simulation. An application to estimate the VaR of daily returns on the Dow Jones Industrial Average index is given in Sect. 7. All proofs are postponed to the Appendix.
2 The idea of bias correction under independence
For the sake of simplicity, we first introduce our bias correction idea under the assumption of independent and identically distributed (i.i.d.) observations in this section. We show later that our bias correction procedure also works for \(\beta\)mixing series.
2.1 The origin of bias
2.2 Estimating the bias term
Comparing our asymptotically unbiased estimator with the original Hill estimator, the \(k\) sequences used for estimation are at different level. The conditions on \((k_{n})\) and \((k_{\lambda})\) imply that \(k_{n}/k_{\lambda}\to+\infty\) as \(n\to\infty\). Since the asymptotic variance of both the asymptotically unbiased estimator and the original Hill estimator is of an order \(1/k\), using a sequence \((k_{n})\) increasing faster than \((k_{\lambda})\) leads to a lower asymptotic variance of our asymptotic unbiased estimator compared to that of the original Hill estimator.
In addition, the \(k\) sequence used for the asymptotically unbiased estimator is more flexible in the following sense. The third order condition (2.6) implies that \(A\) and \(B\) are regularly varying functions with index \(\rho\) and \(\rho'\), respectively. Consider the special case that \(A(t)\sim Ct^{\rho}\) and \(B(t)\sim Dt^{\rho'}\) as \(t\to\infty\) for some constant \(C\) and \(D\). Then the condition that \(\sqrt{k_{\lambda}}A(n/k_{\lambda})\to\lambda\) restricts the level of \(k_{\lambda}\) as \(k_{\lambda}=O(n^{\frac{2\rho}{2\rho1}})\), whereas condition (2.8) implies that \(k_{n}=O\left(n^{\tau}\right)\) for any \(\tau\in(\frac{2\rho}{2\rho1},\frac{2(\rho+\max(\rho,\rho '))}{2(\rho+\max(\rho,\rho'))1})\).
3 The serial dependence conditions
 (a)
\(\frac{\beta(\ell)}{\ell}n+\ell k^{1/2}\log^{2}k \to0\);
 (b)
\(\frac{n}{\ell k}{\mathrm{Cov}}(\sum_{i=1}^{\ell}{\mathbf{1}}_{\{ X_{i}>F^{1}(1kx/n)\}}, \sum_{i=1}^{\ell}{\mathbf{1}}_{\{X_{i}>F^{1}(1k y/n)\}})\to r(x,y)\), for any \(0\leq x,y\leq1+\varepsilon\);
 (c)For some constant \(C\),for any \(0\leq x< y\leq1+\varepsilon\) and \(n\in\mathbb{N}\).$$\begin{aligned} \frac{n}{\ell k}\mathbb{E}\Bigg[\bigg(\sum_{i=1}^{\ell}{\mathbf{1}}_{\{ F^{1}(1k y/n)< X_{i}\leq F^{1}(1k x/n)\}}\bigg)^{4}\Bigg]\leq C(yx), \end{aligned}$$
We intend to correct the bias while allowing the observations to follow the \(\beta\)mixing condition and the regularity conditions. Since the asymptotic bias of the original Hill estimator under serial dependence has the same form as in (2.5), we can construct an asymptotically unbiased estimator for \(\beta\)mixing sequences with exactly the same form as in the case of independence. Nevertheless, due to the serial dependence, the asymptotic property of the estimator has to be reestablished. This is what we do in the next section.
4 Main results
We start by introducing the estimator of the second order parameter. Then we state our main results on the asymptotic properties of the asymptotic unbiased estimators of the extreme value index and high quantiles.
4.1 Estimating the second order parameter
4.2 Asymptotically unbiased estimator of the extreme value index
The following theorem shows the asymptotic normality of our asymptotically unbiased estimator for \(\beta\)mixing series. The consistency of the estimator could be obtained under the second order condition without requiring the third order condition.
Theorem 4.1
Compared to the original Hill estimator, we use different \(k\) sequences, namely \((k_{n})\) and \((k_{\rho})\), in the asymptotically unbiased estimator \(\hat{\gamma}_{k_{n},k_{\rho},\alpha}\). These \(k\) sequences are compatible with the regularity conditions. Recall that the third order condition (2.6) implies that \(A^{2}\) and \(AB\) are regularly varying functions with index \(2\rho\) and \(\rho+\rho'\), respectively. Conditions (2.7) and (2.8) ensure that \(k_{n},k_{\rho}\) are \(o(n^{\zeta})\) for some \(\zeta<1\) and consequently yield the compatibility of these two sequences with the regularity conditions. In general, as long as the original sequence \((k_{\lambda})\) is compatible with the regularity conditions, so are \((k_{n})\) and \((k_{\rho})\).
4.3 Asymptotically unbiased estimator of high quantiles
We consider the estimation of high quantiles. High quantile here refers to the quantile at a probability level \(1p\), where the tail probability \(p=p_{n}\) depends on the sample size \(n\): as \(n\to \infty\), \(p_{n}=O(1/n)\). The goal is to estimate the quantile \(x(p)=U(1/p)\). In extreme cases such that \(np_{n}<1\), it is not possible to have a nonparametric estimate of such a quantile.
The following theorem shows the asymptotic normality of the quantile estimator \(\hat{x}_{k_{n},k_{\rho},\alpha}(p)\).
Theorem 4.2
5 Examples

The \(k\)dependent process and the autoregressive (AR) process AR(1); see [19, 7, 20].

The AR(\(p\)) processes and the infinite moving averages (MA) processes; see [18, 6].

The autoregressive conditional heteroskedasticity process ARCH(1); see [6, 7].

The generalized autoregressive conditional heteroskedasticity (GARCH) processes; see [21, 5].
We review some simple cases of these processes and provide a comparison of the asymptotic variance under dependence to that under independence, and to that of the original Hill estimator under serial dependence.
5.1 Autoregressive model
Differently, we observe from Fig. 1(b) that the variation of the second ratio is mainly due to that of \(\rho\). Although this ratio is greater than one, it does not imply that the asymptotically unbiased estimator has a higher asymptotic variance because the current comparison is conducted using the same \(k\) level for both estimators, whereas the \(k\) value used in the asymptotically unbiased estimator can be at a much higher level than that used for the Hill estimator. Theoretically, the conditions on \((k_{n})\) and \((k_{\lambda})\) guarantee that \(k_{n}/k_{\lambda}\to+\infty\). Thus the variance of our estimator is at a lower level asymptotically. Practically, if we consider the example \(\rho=1\), then the ratio is between 5 and 7. Under such an example, if we use in the asymptotically unbiased estimator a \(k_{n}\) seven times higher than \(k_{\lambda}\) used for the original Hill estimator, we get an estimator with lower variance. If the level of \(\rho\) is closer to zero, then the ratio will be at a higher level. Correspondingly, one needs a higher level of \(k_{n}\) to offset the higher ratio. Nevertheless, together with the fact that the asymptotically unbiased estimator does not suffer from the bias issue, it may still perform better in terms of having a lower root mean squared error. Such a feature will show up in the simulation studies in Sect. 6 below.
5.2 Moving average model
In the second row of Fig. 1, we plot the variations of these ratios with respect to the extreme value index \(\gamma\) for different values of the parameters \(\theta\) and \(\rho\). The general feature is comparable to that observed from the first row. A notable difference between Figs. 1(a) and 1(c) is that although the ratios are both increasing in \(\theta\) and the absolute value of \(\rho\), their convexities with respect to \(\gamma\) are different in the two models: we observe a concave (resp., convex) relation in \(\gamma\) under the MA(1) (resp., AR(1)) model.
5.3 Generalized autoregressive conditional heteroskedasticity model
In addition, the stationary \(\mbox{GARCH}(1,1)\) series satisfies the \(\beta \)mixing condition and the regularity conditions; see [21] and [5]. Thus, it can be considered as an example for which we can apply the asymptotically unbiased estimators. Since it is difficult to explicitly calculate the \(r(\cdot,\cdot)\) function and consequently the asymptotic variance, we opt to use simulations to show the performance of the asymptotically unbiased estimator under the GARCH model.
6 Simulation
6.1 Data generating processes

Model 4. \(\mbox{GARCH}(1,1)\): \(X_{t}\) given as in Sect. 5.3 with \(\lambda_{0}=8.26\times10^{7}\), \(\lambda_{1}=0.052\), \(\lambda _{2}=0.941\). The innovation term follows the standardized Student\(t\) distribution with degree of freedom \(\nu=5.64\).
Simulated theoretical values of \(x(0.001)\) under Models 1–4
Model 1  Model 2  Model 3  Model 4 

749.80  1072.26  972.85  0.0592 
6.2 Estimation procedure
 Estimate the second order index \(\rho\) by (4.1) with \(\alpha=2\).
 Denote by \(m\) the number of positive observations in the sample. For each \(k\) satisfying \(k\leq\min(m1,\frac{2m}{\log\log m})\), calculate the statistic$$\begin{aligned} S_{k}^{(2)}=\frac{3}{4}\frac{(M_{k}^{(4)} 24 (M_{k}^{(1)})^{4}) (M_{k}^{(2)}2(M_{k}^{(1)})^{2})}{M_{k}^{(3)}6 (M_{k}^{(1)})^{3}}. \end{aligned}$$
 If \(S_{k}^{(2)}\in[2/3,3/4]\), then let$$\begin{aligned} \hat{\rho}_{k}=\frac{4+6 S_{k}^{(2)}+\sqrt{3 S_{k}^{(2)}2}}{4 S_{k}^{(2)}3}. \end{aligned}$$

If \(S_{k}^{(2)}<2/3\) or \(S_{k}^{(2)}>3/4\), then \(\hat{\rho }_{k}\) does not exist.
 The parameter \(\rho\) is estimated as \(\hat{\rho}_{k_{\rho}}\) with$$\begin{aligned} k_{\rho}=\sup\bigg\{ k:k\leq\min\left(m1,\frac{2m}{\log\log m}\right) \ \mbox{and } \hat{\rho}_{k} {\mbox{ exists}}\bigg\} . \end{aligned}$$


Estimate the extreme value index by (4.2) for various values of \(k_{n}\),^{3} using \(\hat{\rho}_{k_{\rho}}\).
Next, we estimate the high quantile \(x(0.001)\) by both the original Weissman estimator and the asymptotically unbiased estimator as in Sect. 4.3. When applying the asymptotically unbiased estimator for high quantiles, we use the same \(\hat{\rho}_{k_{\rho}}\) as above.
6.3 Results
Regarding the estimation of the extreme value index, we observe that even with a rather high level of \(k\), our asymptotically unbiased estimator does not suffer from a significant bias, at least for the first three models; see Figs. 2–4. In Model 4, the bias term increases with respect to \(k\), but still stays at a lower level than that of the original Hill estimator; see Fig. 5. In addition, we compare the reduction of RMSE when switching from the original Hill estimator to the asymptotically unbiased estimator. Across the first three models, the best levels of RMSE are reached for the largest values of \(k\). In Model 4, the RMSE has a different pattern as \(k\) increases. However, the reduction is the most significant in Model 4. Although the lowest achieved RMSE for the asymptotically unbiased estimator is at a comparable level as the lowest RMSE for the original Hill estimator for Models 2 and 3, the decrease of the RMSE with respect to \(k\) demonstrated by the asymptotically unbiased estimator allows a more flexible choice of \(k\) compared to the Ushaped RMSE demonstrated by the original Hill estimator.
Regarding the estimation of high quantiles, we observe from Figs. 6–9 that our goal in reducing the bias is well illustrated on a finite sample when using large \(k\) values. In addition, the RMSE of our asymptotically unbiased quantile estimator stays at a lower level than that of the original Weissman estimator for high levels of \(k\). It is remarkable that the reduction in RMSE is higher for dependent series than for independent series.
To conclude, the simulation studies show that under bias correction, the estimators for extreme value index and high quantiles remain stable for a wider range of \(k\) values even if the dataset exhibits serial dependence. The bias correction method under serial dependence thus helps to tackle the two major critiques for applying extreme value statistics to financial time series.
7 Application
Our goal is to estimate the valueatrisk of the return series at the \(99.9~\%\) level, which corresponds to a high quantile with tail probability \(0.1~\%\), i.e., \(x(0.001)\). From 8088 daily observations, a nonparametric estimate can be obtained by taking the eighth highest order statistic. We thus get \(7.16~\%\) as the empirical estimate.
Next, we apply both the original Hill estimator and the asymptotically unbiased estimator to estimate the extreme value index of the loss return series. We start with estimating the second order parameter \(\rho\). Following the estimation procedure in Sect. 6.2, we choose \(k_{\rho}=3515\) and obtain that \(\hat{\rho}=0.611\). Next we apply both estimators for \(k_{n}=50,51,\dots,2000\). Since we do not employ a parametric model for the time series, there is no explicit formula for calculating the asymptotic variance of the two estimators. Therefore, we opt to use a block bootstrapping method to construct the confidence interval for the extreme value index.
From the two figures, we observe that the estimates using the bias correction technique stay stable for a larger range of \(k\) values. In contrast, the estimates based on the original Hill estimator suffer from a large bias starting from \(k\geq400\). When applying the original EVT estimators, it is possible to choose \(k\) only around 250, which corresponds to \(3~\%\) of the total sample. Correspondingly, we obtain an estimated extreme value index at 0.349 from the Hill estimator and an estimated VaR at 0.06549 from the Weissman estimator. With our asymptotically unbiased estimators, we can take \(k=1000\) and obtain an estimated extreme value index at 0.280 with an estimated VaR at 0.05898. Note that the point estimates of the VaR are below, but close to, the empirical estimate. In addition to the point estimation, we investigate the confidence intervals of the estimated VaR. The Weissman estimator results in a 95 % confidence interval as \([0.04268, 0.08831]\), while the confidence interval obtained from our asymptotically unbiased estimator is \([0.04219,0.07577]\). Hence we conclude that the bias correction procedure helps to obtain a more accurate estimate with a narrower confidence interval.
Footnotes
 1.
In the revised Basel II accord and the subsequent Basel III accord, the VaR measures for risks on both trading and banking books must be calculated at a 99.9 % level.
 2.
In order to get a unit variance, we simply normalize the standard Student\(t\) distribution with degree of freedom \(\nu\) by its standard deviation \(\sqrt{\nu/(\nu2)}\).
 3.
For Models 1–3, we use \(k_{n}=10,11,\ldots, 700\), while for Model 4, we use \(k_{n}=10,11,\ldots, 450\) due to a lower number of positive observations.
Notes
Acknowledgements
The authors would like to thank two anonymous referees and the Associate Editor for their helpful comments. Laurens de Haan’s research was partially supported by ENES project EXPL/MATSTA/0622/2013. Cécile Mercadier and Chen Zhou’s research was partially supported by the EURfellowship grant.
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