The Fixed Volatility Bootstrap for a Class of Arch(q) Models

Abstract

The 'fixed regressor' -or 'fixed design' -bootstrap is usually considered in the context of classic regression, or conditional mean (autoregressive) models, see for example, Goncalves and Kilian, 2004). We consider here inference for a general class of (non)linear ARCH models of order q, based on a 'Fixed Volatility' bootstrap. In the Fixed Volatility bootstrap, the lagged variables in the conditional variance equation are kept fixed at their values in the original series, while the bootstrap innovations are, as is standard, resampled with replacement from the estimated residuals based on quasi maximum likelihood estimation. We derive a full asymptotic theory to establish validity for the Fixed Volatility bootstrap applied to Wald statistics for general restrictions on the parameters. A key feature of the Fixed Volatility bootstrap is that the bootstrap sample, conditional on the original data, is an independent sequence. Inspection of the proof of bootstrap validity reveals that such conditional independence simplifies the asymptotic analysis considerably. In contrast to other bootstrap methods, one does not have to take into account the conditional dependence structure of the bootstrap process itself. We also investigate the finite sample performance of the Fixed Volatility bootstrap by means of a small scale Monte Carlo experiment. We find evidence that for small sample sizes, the Fixed Volatility bootstrap test is superior to the asymptotic test, and to the recursive bootstrap-based test. For large samples, both bootstrap schemes and the asymptotic test share properties, as expected from the asymptotic theory. Its appealing theoretical properties, together with its good finite sample performance, suggest that the proposed Fixed Volatility bootstrap may be an important tool for the analysis of the bootstrap in more general volatility models. FIXED VOLATILITY BOOTSTRAP 921With respect to the fixed design bootstrap in regression models, which keep the conditional mean of the bootstrap data fixed among bootstrap repetitions, when dealing with volatility models the fixed design bootstrap maintains the conditional volatility fixed among bootstrap repetitions. This requires generating heteroskedasticity in the bootstrap data by recolouring the (conditionally i.i.d.) bootstrap innovations with some estimates of the volatility process which depend on the original data only. Hence, and in contrast to other bootstrap methods such as the recursive bootstrap, this method -which we label the 'Fixed Volatility' bootstrap in what follows -generates bootstrap samples which, conditional on the original data, features the same conditional volatility process.The recursive bootstrap for ARCH type model has been explored in the existing literature. For instance, Hidalgo and Zaffaroni (2007) propose a recursive bootstrap scheme for model specification tests in ARCH(∞) models. Corradi and Iglesias (2008) compare recursive bootstrap with block bootstrap methods in terms of asymptotic refinements. Pascual et al. (2006) exploit the rec...

Bootstrap inference for conditional risk measures

“…We refer to Appendix 4.B for computational details. In comparison to the fixed-design approach (see Table 4.1) we find that the recursive-design method performs similarly in terms of average coverage for each interval type, which corresponds to the simulation results of Cavaliere et al (2018). It is striking, however, that the intervals' average lengths are larger in the recursive-design than in the fixed-design setup.…”

“…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 bootstrap (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;Cavaliere, Pedersen, and Rahbek, 2018) design. Whereas in the former the bootstrap observations are generated recursively using the estimated volatility dynamics, the We propose a fixed-design residual bootstrap method to mimic the finite sample distribution of the two-step estimator and provides an algorithm for the construction of bootstrap intervals for the conditional VaR.…”

“…We refer to Appendix 4.B for a complete description. See also Cavaliere et al (2018) for more theoretical insights on the difference in the design in an ARCH(q).…”

“…An alternative approach is based on bootstrap approximations. Regarding the estimators of the volatility models parameters, several bootstrap methods have been examined, among which the sub-sample bootstrap (Hall and Yao, 2003), the block bootstrap 5.2 Model (Corradi and Iglesias, 2008), the wild bootstrap (Shimizu, 2010) and the residual bootstrap, both with recursive (Pascual et al, 2006;Hidalgo and Zaffaroni, 2007) and fixed design (Shimizu, 2010;Cavaliere et al, 2018). However, the estimation of the conditional ES has received only limited attention in the bootstrap literature.…”

“…It would also be interesting to compare their recursive-design bootstrap method with its fixed-design counterpart and study different interval types. Concerning the bootstrap consistency, it however seems to be easier to show the asymptotic validity for the fixed-design bootstrap method (Cavaliere et al, 2018).…”