In this chapter we establish the mapping of the conditional mean in an AR(1) and ARMA(1, 1) model into the general framework. Further, the conditional variance and the conditional volatility in a GARCH(1, 1) and T-GARCH(1, 1) model, re- spectively, are shown to be encompassed in this setup. Further, the theoretical results of Chapter 2 are validated by verifying the corresponding assumptions for each model. Clearly, the list of nested models is non-exhaustive and can be ex- tended. For instance one could study higher order models such as the ARMA(p, q) or the GARCH(p, q) model with p, q ∈ N, which come at the cost of a more evolved analysis. Table 3.1 enlists four other GARCH-type extensions that are frequently encountered in the literature. The family of quadratic GARCH (Q-GARCH) mod- els has been proposed by Sentana (1995). Its Q-GARCH(1, 1) member is very similar to the GARCH(1, 1) model and can be verified in a similar fashion replac- ing αX2
t−1by αXt−12 + φXt. The GJR-GARCH(1, 1) model named after Glosten,
Jagannathan and Runkle (1993) is a variant of the T-GARCH(1, 1), which corre- sponds to squaring the variables involved. It can be easily verified along the lines of Section 3.6. The exponential GARCH (E-GARCH) model suggested by Nel- son (1991) and the non-linear GARCH (N-GARCH) introduced by Engle and Ng (1993) can also be embedded into the general framework. For example, the condi- tional variance in an N-GARCH(1, 1) given by σ2n+1= ω0+α0(Xn−φ0σn)2+β0σn2,
3.7 Concluding Remarks explicit expression for the conditional variance in terms of θ0and {Xt}t≤n is com-
plicated due to non-linearities in the recursive formula: e.g. σ2n+1depends on σn2
and σn in the N-GARCH(1, 1).
There are few GARCH extensions such as the fractionally integrated (FI- GARCH) of Baillie, Bollerslev, and Mikkelsen (1996) or the fractionally integrated EGARCH (FIE-GARCH) of Bollerslev and Mikkelsen (1996) that cannot be en- compassed in the framework at hand. The corresponding processes typically ex- hibit intermediate or long memory such that standard mixing results do not apply. Establishing the merging results on the basis of verifying Assumption 2.3.(iii) di- rectly, instead via some mixing result, is an interesting question, which demands further investigation.
Finally, we would like to emphasize that conditional risk measures such as conditional Value-at-Risk (VaR) can be mapped into the general framework. For instance in the T-GARCH(1,1) model of Section 3.6, the conditional VaR of Xn+1
given {Xt}t≤n at level a ∈ (0, 1) reduces to
V aRa(Xn+1|Xn, Xn−1, . . . ) = − ξa ∞ X k=0 β0k ω0+ α+0X + n−k+ α − 0X − n−k (3.26) with ξa = infτ ∈ R : P[εt≤ τ ] ≥ a ; see Francq and Zako¨ıan (2015) for details.
Fixing a and treating ξa as additional parameter, (3.26) is a function of {Xt}t≤n
and ϑ0 = (ω0, α+0, α −
0, β0, ξa)0 and hence is nested in the setup. Similarly, the
conditional Expected Shortfall (ES) of Xn+1given {Xt}t≤n at level a ∈ (0, 1)
ESa(Xn+1|Xn, Xn−1, . . . ) = − µa ∞ X k=0 β0k ω0+ α+0X + n−k+ α − 0X − n−k (3.27) with µa= −Eεt|εt< ξa can also be mapped into the general framework.
Chapter 4
A Residual Bootstrap for
Conditional Value-at-Risk
This chapter proposes a fixed-design residual bootstrap method for the two-step estimator of Francq and Zako¨ıan (2015) associated with the conditional Value-at- Risk. The bootstrap’s consistency is proven under mild assumptions for a general class of volatility models and intervals are constructed for the conditional Value-at- Risk. A simulation study reveals that the equal-tailed percentile bootstrap interval tends to fall short of its nominal value. In contrast, the reversed-tails bootstrap interval yields accurate coverage. We also compare the theoretically analyzed fixed-design bootstrap with the recursive-design bootstrap. It turns out that the fixed-design bootstrap performs equally well in terms of average coverage, yet leads on average to shorter intervals in smaller samples. An empirical application illustrates the interval estimation.1
4.1
Introduction
Risk management has tremendously developed in past decades becoming an in- creasing practice. With minimum capital requirements being enforced by current legislation (Basel III and Solvency II), financial institutions and insurance com- panies monitor risk by using conventional measures such as Value-at-Risk (VaR). Typically, the volatility dynamics are specified by a (semi-)parametric model lead- ing to conditional risk measure versions. For GARCH-type models the conditional VaR reduces to the conditional volatility scaled by a quantile of the innovations’ distribution. The latter is conventionally treated as additional parameter and forms together with the others the risk parameter (Francq and Zako¨ıan, 2015). The true parameters are generally unknown and need to be estimated to obtain an estimate for the conditional VaR. Clearly, this VaR evaluation is subject to estimation risk that needs to be quantified for appropriate risk management.
Whereas an estimator based on a single step is available after reparameteriza- tion (Francq and Zako¨ıan, 2015), a widely used approach is the following two-step estimation procedure. First, the parameters of the conditional volatility model are estimated. Arguably the most popular estimation method in a GARCH-type setting is the Gaussian quasi-maximum-likelihood (QML) method. Based on the model’s residuals the quantile is estimated by its empirical counterpart in a second step. For realistic sample sizes (e.g. 500 or 1,000 daily observations) the estimators are subject to considerable estimation risk. In particular, the estimation uncer- tainty associated with the quantile estimator is substantial for extreme quantiles (e.g. 5% or smaller).
To quantify the uncertainty around the point estimators, one traditionally relies on asymptotic theory while replacing the unknown quantities in the limiting distri- bution by consistent estimates. An alternative approach – frequently employed in practice – is based on a bootstrap approximation. Regarding the estimators of the GARCH parameters, various bootstrap methods have been studied to approximate the estimators’ finite sample distribution including the subsample bootstrap (Hall and Yao, 2003), the block bootstrap (Corradi and Iglesias, 2008), the wild boot- strap (Shimizu, 2010) and the residual bootstrap. The residual bootstrap method is particularly popular and can be further divided into recursive (Pascual et al., 2006; Hidalgo and Zaffaroni, 2007; Jeong, 2017) and fixed (Shimizu, 2010; Cava- liere, Pedersen, and Rahbek, 2018) design. Whereas in the former the bootstrap observations are generated recursively using the estimated volatility dynamics, the
4.1 Introduction latter design keeps the dynamics of the bootstrap samples fixed at the value of the original series.
The estimation of the quantile and the conditional VaR have received only se- lected attention in the bootstrap literature and proposed bootstrap methods have been, to the best of our knowledge, exclusively investigated by means of simu- lation. Christoffersen and Gon¸calves (2005) examine various quantile estimators and construct intervals for the conditional VaR using a recursive-design residual bootstrap method. In addition, Hartz, Mittnik, and Paolella (2006) presume the innovation distribution to be standard normal such that the quantile parameter is known; they propose a resampling method based on a residual bootstrap and a bias-correction step to account for deviations from the normality assumption. In contrast, Spierdijk (2016) develops an m-out-of-n without-replacement bootstrap to construct confidence intervals for ARMA-GARCH VaR.
We propose a fixed-design residual bootstrap method to mimic the finite sam- ple distribution of the two-step estimator and provides an algorithm for the con- struction of bootstrap intervals for the conditional VaR. The proposed bootstrap method is proven to be consistent for a general class of volatility models. In par- ticular, our framework does not only encompass GARCH but also several GARCH extensions such as the threshold GARCH (T-GARCH) of Zako¨ıan (1994) and the GJR-GARCH named after Glosten, Jagannathan and Runkle (1993). The boot- strap consistency is established under a set of mild assumptions, which relaxes moment conditions on the innovations imposed in the GARCH bootstrap litera- ture. To the best of our knowledge we are the first to theoretically validate the residual bootstrap for the quantile and the conditional VaR.
The remainder of the chapter is organized as follows. Section 4.2 specifies the model and the conditional VaR is derived. The two-step estimation procedure is described in Section 4.3 and asymptotic theory is provided under mild assump- tions. In Section 4.4, a fixed-design residual bootstrap method is proposed and proven to be consistent. Further, bootstrap intervals are constructed for the con- ditional VaR. A simulation study is conducted in Section 4.5 and an empirical application illustrates the interval estimation based on the fixed-design residual bootstrap. Section 4.6 concludes and auxiliary results are gathered in the Ap- pendix. Appendix 4.A contains lemmas and their proofs while Appendix 4.B is devoted to the related recursive-design residual bootstrap.