Bootstrapping volatility functionals: a local and nonparametric perspective

Bootstrapping volatility functionals: a local and nonparametric perspective SUMMARY Volatility functionals are widely used in financial econometrics. In the literature, they are estimated with realized volatility functionals using high-frequency data. In this paper we introduce a nonparametric local bootstrap method that resamples the high-frequency returns with replacement in local windows shrinking to zero. While the block bootstrap in time series (Hall et al., 1995) aims to reduce correlation, the local bootstrap is intended to eliminate the heterogeneity of volatility. We prove that the local bootstrap distribution of the studentized realized volatility functional is first-order accurate. We present Edgeworth expansions of the studentized realized variance with and without local bootstrapping in the absence of the leverage effect and jumps. The expansions show that the local bootstrap distribution of the studentized realized variance is second-order accurate. Extensive simulation studies verify the theory. 1. Introduction Volatility functionals are widely used in financial econometrics, and recent years have seen increasing interest in estimating them in the Itô process framework using high-frequency data. The first functional studied was the integrated volatility. Barndorff-Nielsen & Shephard (2002) established the central limit theorem for the realized variance, the most efficient estimator of the integrated volatility. Jacod & Rosenbaum (2013) considered the estimator of a general class of volatility functionals and proved its efficiency. However, microstructure noise poses a challenge to the empirical performance of the aforementioned estimators. Zhang et al. (2005) showed that the realized variance estimator inflates when the sampling frequency increases. Since then there have been many papers devoted to eliminating the effect of microstructure noise (Zhang et al., 2005; Jacod et al., 2009). Another commonly used approach is to sample sparsely from tick-by-tick data (Aït-Sahalia et al., 2005). In this paper, we adopt the latter approach. The drawback, however, is that the effective sample size is reduced, leading to poor normal approximation. Motivated by this, efforts have recently been made to demonstrate the validity of the distribution of the randomly weighted realized variance (Goncalves & Meddahi, 2009; Hounyo & Veliyev, 2016). Hounyo & Varneskov (2017) studied the distribution of the randomly weighted jump activity index estimator for the pure jump process. The random weighting method depends on tuning the weight distribution to match the moments of the returns. While it performs satisfactorily in the case of the realized variance, it is not adapted to a general class of volatility functionals, since the weight distribution is specific to the functional form, and hence one has to deal with different functionals case by case. In this paper, we introduce a local nonparametric bootstrap method that works for a general class of functional forms. We first split the data into non-overlapping blocks that shrink to zero, and then within each block we resample the returns with replacement to form a bootstrap estimate of the local spot volatility. Finally, we integrate functional values of local spot volatilities across all blocks, which yields a bootstrap estimate of the volatility functional. The rationale is that within each local window, the increments of the log price process can be regarded as approximately independent and identically distributed random variables conditional on the history up to the start of the window. But the increments vary a lot across windows. While the block bootstrap in time series (Hall et al., 1995) aims to reduce correlation, the local bootstrap here attempts to eliminate the heterogeneity of volatility. Our results show that the local bootstrap distribution of the realized volatility functionals after studentization is first-order accurate in approximating the distribution of the studentized realized volatility functionals for Itô processes with jumps of finite variation. Assuming independence of the volatility process and the driving forces, we obtain the uniform convergence rate of the distribution functions of the studentized realized volatility functionals with and without local bootstrapping. The rate demonstrates that the jump-truncation error and the volatility-discretization error dominate in the principal correction term, leading to an accuracy rate below $$\Delta_n^{1/2}$$. In particular, for the studentized realized variance with and without local bootstrapping in the absence of the leverage effect and jumps, we present Edgeworth expansions. We show in this particular case that the local bootstrap distribution has second-order accuracy while the limiting normal distribution is only first-order accurate. Nevertheless, although theoretically second-order accuracy requires a linear functional form and that the leverage effect and jumps be absent, simulation shows that the local bootstrap distribution still outperforms the limiting normal distribution in general scenarios. 2. Bootstrapping volatility functionals 2.1. Preliminary results on volatility functionals The continuous-time univariate Itô process defined on the filtered probability space $$(\Omega, \mathcal{F}, \mathcal{F}_t, P)$$ is \begin{eqnarray}\label{model} X_t&=&X_0+\int^t_0b_s\,{\rm d}s+\int^t_0\sigma_s\,{\rm d}W_s+\int^t_0\int_R\delta(s, x)I\{|\delta(s, x)|\leq 1\}(\mu-\nu)({\rm d}s, {\rm d}x)\nonumber\\ &&+\int^t_0\int_R\delta(s, x)I\{|\delta(s, x)|>1\}\mu({\rm d}s, {\rm d}x), \end{eqnarray} (1) where $$X_0$$ is the initial value, $$b$$ and $$\sigma$$ are locally bounded adapted processes representing the spot drift and volatility, $$W$$ is a standard Brownian motion, $$\delta$$ is a predictable process and $$\mu$$ is a Poisson random measure with compensator $$\nu({\rm d}x, {\rm d}s)={\rm d}x\otimes {\rm d}s$$. The volatility functional is defined as $$V(g)_t=\int^t_0g(c_s)\,{\rm d}s$$ where $$g$$ is a real function and $$c_s=\sigma_s^2$$. In particular, $$g(x)=x$$ and $$g(x)=x^2$$ correspond to the integrated volatility and quarticity, respectively. We assume that the dataset $$\{X_{t_0}, \ldots, X_{t_n}: 0=t_0<t_1<\cdots<t_n=T\}$$ is discretely sampled from $$X_t$$, where $$T$$ is a fixed time horizon and the $$t_i$$ are equally spaced. The time lag is denoted by $$\Delta_n=t_{i}-t_{i-1}$$ and $$\Delta^n_iX=X_{t_{i}}-X_{t_{i-1}}$$. We assume that $$\Delta_n\rightarrow 0$$ as $$n\rightarrow \infty$$ and $$T$$ is fixed. Let $$\bar{g}(x)=g(x)-g^{(2)}(x)x^2/k_n$$ and $$h(x)=2\{g^{\prime}(x)\}^2x^2$$. A natural estimate of $$V(g)_t$$ is $$V(g)^n_t=\sum^{\lfloor t/(k_n\Delta_n)\rfloor}_{i=0}\bar{g}(\hat{c}^n_{ik_n+1})k_n\Delta_n$$ where $$\lfloor x \rfloor$$ stands for the largest integer smaller than or equal to $$x$$, and $$\hat{c}^n_i=k_n^{-1}\sum^{k_n-1}_{j=0}(\Delta^n_{i+j}X)^2\Delta_n^{-1}I(|\Delta^n_{i+j}X|\leq u_n)$$ where $$u_n=\alpha\Delta_n^{\varpi}$$ for some positive constants $$\alpha$$ and $$\varpi$$. The threshold $$u_n$$ removes the jumps in the price dynamics. The jump truncation method comes from Mancini (2009). Clearly $$\hat{c}^n_i$$ is an estimate of $$\sigma^2_{t_{i-1}}$$. To derive the theoretical properties of the local nonparametric bootstrap, we need to make the following technical assumptions. Assumption 1. There is a sequence, $$J_n$$, of nonnegative bounded integrable functions and a sequence, $$\tau_n$$, of stopping times increasing to $$\infty$$ such that $$t<\tau_n \ \mbox{implies} \ |b_t(\omega)|\leq n \ \mbox{and} \ |\sigma_t(\omega)|\leq n,$$ and $$t\leq \tau_n \ \mbox{implies} \ |\delta(\omega, t, x)|^r\wedge 1 \leq J_n(x)\text{.}$$ Assumption 2. The process $$X$$ satisfies Assumption 1 with $$r<1$$. The volatility process $$c$$ is an Itô process of the form (1), with the notation $$b$$, $$\sigma$$, $$\delta$$ and $$\mu$$ replaced by $$\tilde{b}$$, $$\tilde{\sigma}$$, $$\tilde{\delta}$$ and $$\tilde{\mu}$$, respectively, satisfying Assumption 1 with $$r=2$$. Assume that $$b_t$$ and $$b^{\prime}_t=b_t-\int\delta(t, x)I(|\delta(t, x)|\leq 1)\,{\rm d}x$$ are càdlàg. Assumptions 1 and 2 are the same as Assumption (H-r) and Assumption (A-r) in Jacod & Rosenbaum (2013). These assumptions imply that $$X$$ is a process with jumps of finite variation while $$c$$ might contain infinite-variation jumps. Next we present assumptions on the function $$g$$ that are also required in Jacod & Rosenbaum (2013). Assumption 3. The function $$g$$ is a $$C^3$$ function such that $$|g^{(j)}(x)|\leq C(1+x^{p-j})$$$$(\,j=0, 1, 2, 3)$$ for some constants $$C>0$$ and $$p\geq 3$$. In addition $$(2p-1)\{2(2p-r)\}^{-1}\leq \varpi < 1/2$$. We need the notion of stable convergence in law to state our main results. The notion applies to a sequence of random variables $$Z_n$$, all defined on the same probability space $$(\Omega, \mathcal{F}, P)$$ and taking values in the same state space $$(E, \mathcal{E})$$ assumed to be a Polish space. We say that $$Z_n$$ converges stably if there is a probability measure $$\eta$$ on the product of $$(\Omega\times E, \mathcal{F}\times \mathcal{E})$$ such that $$E\{Yf(Z_n)\}\rightarrow \int Y(\omega)f(x)\eta({\rm d}\omega, {\rm d}x)$$ for all bounded continuous functions $$f$$ on $$E$$ and all bounded random variables $$Y$$ on $$(\Omega, \mathcal{F})$$. For more details, refer to Jacod & Protter (2012). Jacod & Rosenbaum (2013) established the following central limit theorem for $$V(g)^n_t$$. Lemma 1. Assuming $$k_n^3\Delta_n\rightarrow \infty$$, $$k_n^2\Delta_n\rightarrow 0$$, and Assumptions 1–3, $$T_n= \{V(g)^n_t-V(g)_t\}\{\Delta_n V(h)^n_t\}^{-1/2}$$ converges to $$N(0, 1)$$ stably. 2.2. Local nonparametric bootstrap To introduce the local bootstrap method, we split the sample $$\mathcal{X}= \{\Delta^n_{1}X,\ldots, \Delta^n_nX\}$$ into $$\lfloor n/k_n \rfloor$$ non-overlapping blocks, each containing $$k_n$$ data points. For the $$i$$th block, let $$\mathcal{X}_i = \{\Delta^n_{(i-1)k_n+1}X, \ldots, \Delta^n_{ik_n}X\}$$. We randomly draw elements from $$\mathcal{X}_i$$ with replacement to obtain the bootstrap sample $$\mathcal{X}_i^*=\{\Delta^n_{(i-1)k_n+1}X^*,\ldots, \Delta^n_{ik_n}X^*\}$$. Let $$\hat{c}^{n*}_{ik_n+1}=k_n^{-1}\sum^{k_n-1}_{j=0}(\Delta^n_{i+j}X^*)^2\Delta_n^{-1}I(|\Delta^n_{i+j}X^*|\leq u_n)$$. The rationale for local random drawing is that the $$\Delta^n_{(i-1)k_n+j}X$$ are approximately independent and identically distributed random variables conditional on $$\mathcal{F}_{(i-1)k_n\Delta_n}$$. Thus the bootstrap versions of $$V(g)_t$$ and $$V(g)^n_t$$ are respectively $$ V^*(g)_t=\sum^{\lfloor t/(k_n\Delta_n)\rfloor -1}_{i=0}g(\hat{c}^n_{ik_n+1})k_n\Delta_n, \quad V^*(g)^n_t=\sum^{\lfloor t/(k_n\Delta_n) \rfloor-1}_{i=0}\bar{g}(\hat{c}^{n*}_{ik_n+1})k_n\Delta_n\text{.} $$ Standardizing the difference between $$V^*(g)_t^n$$ and $$V^*(g)_t$$ gives the locally bootstrapped studentized realized volatility functional $$T_n^*=\{V^*(g)^n_t-V^*(g)_t\}\{\Delta_nV^*(h)^n_t\}^{-1/2},$$ where $$V^*(h)^n_t=\sum^{\lfloor t/(k_n\Delta_n)\rfloor}_{i=0}\bar{h}(\hat{c}^{n*}_{ik_n+1})k_n\Delta_n\text{.}$$ 3. Main results Theorem 1. Suppose Assumptions 1–3 hold. If $$k_n^3\Delta_n\rightarrow \infty$$ and $$k_n^{2}\Delta_n\rightarrow 0$$ as $$\Delta_n\rightarrow 0$$, then conditional on $$\mathcal{X}$$, $$T_n^*$$ converges to $$N(0, 1)$$ stably as $$n\rightarrow\infty$$. Remark 1. Theorem 1 demonstrates that $$\sup_{x\in R}|H_n^*(x)-\Phi(x)|=o_{\rm p}(1),$$ and hence by Lemma 1 $$\sup_{x\in R}|H_n^*(x)-H_n(x)|=o_{\rm p}(1),$$ where $$H_n^*(x)$$ is the local bootstrap distribution of $$T_n^*$$ conditional on $$\mathcal{X}$$ and $$H_n(x)$$ is the distribution of $$T_n$$. Thus $$H_n^*(x)$$ is a close approximation of $$H_n(x)$$ for all $$g$$ satisfying Assumption 3, and hence statistical inference, like constructing the confidence interval or testing hypotheses on $$V(g)$$ for a given path, based on $$H_n^*(x)$$ is reliable. Goncalves & Meddahi (2009) and Hounyo & Veliyev (2016) proved the first-order accuracy of the globally and locally randomly weighted realized variance, but not for a general class of volatility functionals with jumps contained in $$X$$. Next, we refine the central limit theorem by giving the uniform approximation rates of $$H_n(x)$$ and $$H_n^*(x)$$ for general functionals and the Edgeworth expansions of $$H_n(x)$$ and $$H_n^*(x)$$ for the studentized realized variance. To this end, we need one more assumption. Assumption 4. The volatility process $$c$$ is independent of $$(W, \mu, \delta)$$. Assumption 4 implies that there is no leverage effect for $$X$$, which is far from reality. However, our numerical studies show that the local bootstrap distribution outperforms the limiting normal approximation even in the presence of a leverage effect. Assumption 4 is also needed in Goncalves & Meddahi (2009), Hounyo & Veliyev (2016) and Zhang et al. (2011) in order to derive the Edgeworth expansions of the studentized realized variance. Theorem 2. Suppose Assumptions 1–4 hold. If $$k_n^3\Delta_n\rightarrow \infty$$ and $$k_n^{5/2}\Delta_n\rightarrow 0$$ as $$\Delta_n\rightarrow 0$$, then as $$n\rightarrow\infty$$, \begin{equation} \sup_{x\in R}|H_n(x)-\Phi(x)|=O_{\rm p}(k_n{\Delta_n}^{1/2}+b_n), \quad \sup_{x\in R}|H_n^*(x)-\Phi(x)|=O_{\rm p}\{(k_n^3\Delta_n)^{-1/2}+b_n\}, \end{equation} (2)where $$b_n=a_n\Delta_n^{(2q-r)\varpi+1-q}/{\Delta_n}^{1/2}$$ for some $$q>0$$ and some sequence of constants $$a_n$$ converging to $$0$$. If, further, $$g(x)=x$$, $$\delta=0$$ and $$b= 0$$, then \begin{eqnarray} \sup_{x\in R}|H_n^*(x)-\Phi(x)-{\Delta_n}^{1/2}q(x)\phi(x)|&=&o_{\rm p}(\surd{\Delta_n}), \\ \end{eqnarray} (3) \begin{eqnarray} \sup_{x\in R}|H_n(x)-\Phi(x)-{\Delta_n}^{1/2}q(x)\phi(x)|&=&o_{\rm p}(\surd{\Delta_n}), \end{eqnarray} (4)where $$q(x)=(\int^t_0c_s^2\,{\rm d}s)^{-1/2}-\{3(\int^t_0c_s^2\,{\rm d}s)^{3/2}\}^{-1}5\int^t_0c_s^3\,{\rm d}s(x^2-1)$$. Remark 2. Equation (2) implies that $$\sup_{x\in R}|H_n^*(x)-H_n(x)|=O_{\rm p}\{(k_n^3\Delta_n)^{-1/2}+b_n\},$$ which provides a uniform approximation rate for $$H_n^*(x)$$. The terms $$b_n$$ and $$(k_n^3\Delta_n)^{-1/2}$$ are due to jump truncation and volatility discretization, respectively. In particular, if jumps are absent, the rate is $$(k_n^3\Delta_n)^{-1/2}$$. For the realized variance, if $$b=0$$, (3) and (4) demonstrate that $$H_n^*(x)$$ is second-order accurate in approximating $$H_n(x)$$. The condition $$b= 0$$ is assumed in Goncalves & Meddahi (2009), for the reason that the drift term is negligible at high frequencies. We end this section with practical guidelines on choosing the tuning parameters $$\alpha$$, $$\varpi$$ and $$k_n$$. The threshold $$u_n$$ plays the role of eliminating the jumps. If $$u_n$$ is large, many jumps are missed and left in the realized volatility functionals. If $$u_n$$ is small, increments available mainly due to continuous diffusion are removed, which decreases the efficiency of estimating the spot volatilities. Since $$\alpha$$ and $$\varpi$$ are not independent, we fix $$\varpi$$ and adjust $$\alpha$$. Mancini (2009) suggests setting $$\varpi$$ smaller than and close to $$1/2$$, and hence we set $$\varpi=0.49$$ in our numerical analysis. By Lévy’s continuity theorem, a volatility adaptive upper bound for $$|\Delta^n_iX|$$ is $$\{2c_{t_{i-1}}\Delta_n\log{(1/\Delta_n)}\}^{1/2}$$. Thus most likely the increments larger than $$\alpha\Delta_n^{0.49}\surd{c_{t_{i-1}}}>\{2\Delta_n\log{(\Delta_n^{-1})}c_{t_{i-1}}\}^{1/2}$$ are due to jumps, so we choose $$\alpha$$ such that $$u_n=\alpha\Delta_n^{0.49}\surd{c_{t_{i-1}}}$$ is above and close to $$\{2\Delta_n\log{(\Delta_n^{-1})}c_{t_{i-1}}\}^{1/2}$$ to strike a balance between efficiency and jump thresholding. The spot volatility $$c_{t_{i-1}}$$ can be consistently estimated in moving windows or empirically replaced by the estimated average integrated volatility $$\rm{\small{\hat{{IV}}}}_T/T$$, which can be realized by the tuning-free bipower variation technique (Barndorff-Nielsen & Shephard, 2004). We choose the estimated average integrated volatility to replace all $$c_{t_{i-1}}$$ because it is independent of $$i$$. This selection is consistent with the form of $$u_n$$, which is a universal threshold for all increments. Inspired by Lemma 1 and Theorem 1, we can choose $$k_n$$ within the interval $$(n^{1/3}, n^{1/2})$$. Our experience from the simulation study is that the local bootstrap distributions of the studentized realized variance and quarticity are closer to the true sampling distributions than the limiting normal distribution uniformly in $$k_n$$ selected from the proposed integer set. 4. Numerical studies In this section, we conduct simulations to check the local bootstrap procedure. We generate data from the stochastic volatility model \begin{eqnarray*} {\rm d}X_t&=&c_s{^{1/2}}\,{\rm d}W_s+\theta Y_t, \quad 0\leq t\leq 1, \\ c_t&=&c_0+\int^t_0a_0(1-c_s)\,{\rm d}s+b_0\int^t_0 c_s{^{1/2}}\,{\rm d}W^{\prime}_s, \end{eqnarray*} where $$c_0=4.06$$, $$a_0=0.1$$, $$b_0=0.15$$, $$Y_t$$ is a symmetric $$0.5$$-stable Lévy process and $$W^{\prime}$$ is a second Brownian motion satisfying corr$$(W, W^{\prime})=\rho$$. Although we only proved the superiority of the local bootstrap distribution for the studentized realized variance without the leverage effect or jumps, we consider a general data process in this simulation study, and thus we set $$(\theta, \rho)=(1, -$$0$$\cdot$$5$$)$$. First we consider the studentized realized variance, that is, $$g(x)=x$$. We sample the data every five minutes in a day. Then $$\{2\Delta_n\log{(1/\Delta_n)}\}^{1/2}= 0.3342$$. By the rule of thumb presented in § 3, we choose $$\alpha$$ as 3 with $$\varpi=0.49$$ and $$k_n$$ from $$\{4,\ldots,9\}$$. Figure 1(a) displays the local bootstrap distribution, the true sampling distribution and the limiting standard normal distribution of the studentized realized variance for $$(k_n, \alpha)=(6, 3)$$. The local bootstrap distribution is closer to the true sampling distribution than the limiting normal distribution. In the Supplementary Material, we tested other values of $$\alpha$$ around 3 and $$k_n$$ around 6. The local bootstrap distributions are more accurate than the limiting normal distribution uniformly in $$\alpha$$ and $$k_n$$, especially in the tails. Fig. 1. View largeDownload slide The true sampling distribution (dashed), local bootstrap distribution (stars), and limiting standard normal distribution (solid) of the studentized realized volatility functional: (a) the studentized realized variance and (b) the studentized realized quarticity. Fig. 1. View largeDownload slide The true sampling distribution (dashed), local bootstrap distribution (stars), and limiting standard normal distribution (solid) of the studentized realized volatility functional: (a) the studentized realized variance and (b) the studentized realized quarticity. Next, we consider the studentized realized quarticity, that is, $$g(x)=x^2$$. We sample the data every 100 seconds in a day. Then $$\{2\Delta_n\log{(1/\Delta_n)}\}^{1/2}= 0.2159$$. Again, we choose $$\alpha=3$$. The parameter $$k_n$$ can be chosen from $$\{6,\ldots,16\}$$ by the empirical rule. However, when we tested $$k_n<13$$, $$V(h)^n_t$$ would sometimes be negative, making $$T_n$$ meaningless. So we choose $$k_n\geq 13$$. Figure 1(b) displays the local bootstrap distribution, the true sampling distribution and the limiting standard normal distribution of the studentized realized quarticity for $$(k_n, \alpha)=(13, 3)$$. The local bootstrap distribution performs better than the limiting distribution. The Supplementary Material also contains simulation studies for other values of $$\alpha$$ around 3 and $$k_n=14, \ 15$$, which lead to the same conclusion. Table 1 reports the errors of the tail probabilities of the local bootstrap and limiting normal distributions relative to those of the true sampling distribution. In most cells, especially in the left tails, the local bootstrap tail probabilities are more accurate than the normal ones. Table 1. Errors of the tail probabilities of the local bootstrap and limiting standard normal distributions relative to those of the true distribution. Upper part: $$n=78$$ and $$k_n=6$$; lower part: $$n=234$$ and $$k_n=14$$. The negative and positive signs of the first row of numbers denote the left $$(<t)$$ and right $$(>t)$$ tails, respectively; $$(\theta, \rho)=(1, -0{\cdot}5)$$ and $$\alpha=3$$  t $$-$$3$$\cdot$$5 $$-$$3 $$-$$2$$\cdot$$5 $$-$$2 $$-$$1$$\cdot$$5 1$$\cdot$$5 2 2$$\cdot$$5 Realized B $$-$$0$$\cdot$$102 $$-$$0$$\cdot$$147 $$-$$0$$\cdot$$006 $$-$$0$$\cdot$$003 $$-$$0$$\cdot$$150 $$-$$0$$\cdot$$572 $$-$$0$$\cdot$$715 $$-$$0$$\cdot$$703 variance N $$-$$0$$\cdot$$976 $$-$$0$$\cdot$$929 $$-$$0$$\cdot$$812 $$-$$0$$\cdot$$628 $$-$$0$$\cdot$$358 $$-$$0$$\cdot$$599 $$-$$0$$\cdot$$703 $$-$$0$$\cdot$$693 Realized B $$-$$0$$\cdot$$185 $$-$$0$$\cdot$$141 $$-$$0$$\cdot$$167 $$-$$0$$\cdot$$109 $$-$$0$$\cdot$$062 $$-$$0$$\cdot$$181 $$-$$0$$\cdot$$301 $$-$$0$$\cdot$$454 quarticity N $$-$$0$$\cdot$$998 $$-$$0$$\cdot$$991 $$-$$0$$\cdot$$966 $$-$$0$$\cdot$$905 $$-$$0$$\cdot$$782 $$-$$0$$\cdot$$664 $$-$$0$$\cdot$$838 $$-$$0$$\cdot$$934 t $$-$$3$$\cdot$$5 $$-$$3 $$-$$2$$\cdot$$5 $$-$$2 $$-$$1$$\cdot$$5 1$$\cdot$$5 2 2$$\cdot$$5 Realized B $$-$$0$$\cdot$$102 $$-$$0$$\cdot$$147 $$-$$0$$\cdot$$006 $$-$$0$$\cdot$$003 $$-$$0$$\cdot$$150 $$-$$0$$\cdot$$572 $$-$$0$$\cdot$$715 $$-$$0$$\cdot$$703 variance N $$-$$0$$\cdot$$976 $$-$$0$$\cdot$$929 $$-$$0$$\cdot$$812 $$-$$0$$\cdot$$628 $$-$$0$$\cdot$$358 $$-$$0$$\cdot$$599 $$-$$0$$\cdot$$703 $$-$$0$$\cdot$$693 Realized B $$-$$0$$\cdot$$185 $$-$$0$$\cdot$$141 $$-$$0$$\cdot$$167 $$-$$0$$\cdot$$109 $$-$$0$$\cdot$$062 $$-$$0$$\cdot$$181 $$-$$0$$\cdot$$301 $$-$$0$$\cdot$$454 quarticity N $$-$$0$$\cdot$$998 $$-$$0$$\cdot$$991 $$-$$0$$\cdot$$966 $$-$$0$$\cdot$$905 $$-$$0$$\cdot$$782 $$-$$0$$\cdot$$664 $$-$$0$$\cdot$$838 $$-$$0$$\cdot$$934 B, local bootstrap distribution; N, standard normal distribution. Table 1. Errors of the tail probabilities of the local bootstrap and limiting standard normal distributions relative to those of the true distribution. Upper part: $$n=78$$ and $$k_n=6$$; lower part: $$n=234$$ and $$k_n=14$$. The negative and positive signs of the first row of numbers denote the left $$(<t)$$ and right $$(>t)$$ tails, respectively; $$(\theta, \rho)=(1, -0{\cdot}5)$$ and $$\alpha=3$$  t $$-$$3$$\cdot$$5 $$-$$3 $$-$$2$$\cdot$$5 $$-$$2 $$-$$1$$\cdot$$5 1$$\cdot$$5 2 2$$\cdot$$5 Realized B $$-$$0$$\cdot$$102 $$-$$0$$\cdot$$147 $$-$$0$$\cdot$$006 $$-$$0$$\cdot$$003 $$-$$0$$\cdot$$150 $$-$$0$$\cdot$$572 $$-$$0$$\cdot$$715 $$-$$0$$\cdot$$703 variance N $$-$$0$$\cdot$$976 $$-$$0$$\cdot$$929 $$-$$0$$\cdot$$812 $$-$$0$$\cdot$$628 $$-$$0$$\cdot$$358 $$-$$0$$\cdot$$599 $$-$$0$$\cdot$$703 $$-$$0$$\cdot$$693 Realized B $$-$$0$$\cdot$$185 $$-$$0$$\cdot$$141 $$-$$0$$\cdot$$167 $$-$$0$$\cdot$$109 $$-$$0$$\cdot$$062 $$-$$0$$\cdot$$181 $$-$$0$$\cdot$$301 $$-$$0$$\cdot$$454 quarticity N $$-$$0$$\cdot$$998 $$-$$0$$\cdot$$991 $$-$$0$$\cdot$$966 $$-$$0$$\cdot$$905 $$-$$0$$\cdot$$782 $$-$$0$$\cdot$$664 $$-$$0$$\cdot$$838 $$-$$0$$\cdot$$934 t $$-$$3$$\cdot$$5 $$-$$3 $$-$$2$$\cdot$$5 $$-$$2 $$-$$1$$\cdot$$5 1$$\cdot$$5 2 2$$\cdot$$5 Realized B $$-$$0$$\cdot$$102 $$-$$0$$\cdot$$147 $$-$$0$$\cdot$$006 $$-$$0$$\cdot$$003 $$-$$0$$\cdot$$150 $$-$$0$$\cdot$$572 $$-$$0$$\cdot$$715 $$-$$0$$\cdot$$703 variance N $$-$$0$$\cdot$$976 $$-$$0$$\cdot$$929 $$-$$0$$\cdot$$812 $$-$$0$$\cdot$$628 $$-$$0$$\cdot$$358 $$-$$0$$\cdot$$599 $$-$$0$$\cdot$$703 $$-$$0$$\cdot$$693 Realized B $$-$$0$$\cdot$$185 $$-$$0$$\cdot$$141 $$-$$0$$\cdot$$167 $$-$$0$$\cdot$$109 $$-$$0$$\cdot$$062 $$-$$0$$\cdot$$181 $$-$$0$$\cdot$$301 $$-$$0$$\cdot$$454 quarticity N $$-$$0$$\cdot$$998 $$-$$0$$\cdot$$991 $$-$$0$$\cdot$$966 $$-$$0$$\cdot$$905 $$-$$0$$\cdot$$782 $$-$$0$$\cdot$$664 $$-$$0$$\cdot$$838 $$-$$0$$\cdot$$934 B, local bootstrap distribution; N, standard normal distribution. 5. Conclusion and discussion Our results imply that the local bootstrap quantiles, especially those in the tails, are likely to yield more accurate confidence intervals of the volatility functionals for a given path, or more accurate critical regions in hypothesis testing on volatility functionals. The local bootstrap method can be applied to testing for jumps. By locally resampling the high-frequency data, one can obtain the local bootstrap distribution of the test statistic based on the power variation (Aït-Sahalia & Jacod, 2009) under the null hypothesis that the jumps are absent. One can expect the local bootstrapped critical region under the null hypothesis to be more accurate than that constructed from the standard normal distribution. Acknowledgement The authors would like to thank the editor, the associate editor and two referees for their careful and detailed comments which have led to great improvements of the manuscript. 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Google Scholar CrossRef Search ADS © 2018 Biometrika Trust This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Biometrika Oxford University Press

Bootstrapping volatility functionals: a local and nonparametric perspective

Biometrika , Volume Advance Article (2) – Mar 1, 2018

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© 2018 Biometrika Trust
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Abstract

SUMMARY Volatility functionals are widely used in financial econometrics. In the literature, they are estimated with realized volatility functionals using high-frequency data. In this paper we introduce a nonparametric local bootstrap method that resamples the high-frequency returns with replacement in local windows shrinking to zero. While the block bootstrap in time series (Hall et al., 1995) aims to reduce correlation, the local bootstrap is intended to eliminate the heterogeneity of volatility. We prove that the local bootstrap distribution of the studentized realized volatility functional is first-order accurate. We present Edgeworth expansions of the studentized realized variance with and without local bootstrapping in the absence of the leverage effect and jumps. The expansions show that the local bootstrap distribution of the studentized realized variance is second-order accurate. Extensive simulation studies verify the theory. 1. Introduction Volatility functionals are widely used in financial econometrics, and recent years have seen increasing interest in estimating them in the Itô process framework using high-frequency data. The first functional studied was the integrated volatility. Barndorff-Nielsen & Shephard (2002) established the central limit theorem for the realized variance, the most efficient estimator of the integrated volatility. Jacod & Rosenbaum (2013) considered the estimator of a general class of volatility functionals and proved its efficiency. However, microstructure noise poses a challenge to the empirical performance of the aforementioned estimators. Zhang et al. (2005) showed that the realized variance estimator inflates when the sampling frequency increases. Since then there have been many papers devoted to eliminating the effect of microstructure noise (Zhang et al., 2005; Jacod et al., 2009). Another commonly used approach is to sample sparsely from tick-by-tick data (Aït-Sahalia et al., 2005). In this paper, we adopt the latter approach. The drawback, however, is that the effective sample size is reduced, leading to poor normal approximation. Motivated by this, efforts have recently been made to demonstrate the validity of the distribution of the randomly weighted realized variance (Goncalves & Meddahi, 2009; Hounyo & Veliyev, 2016). Hounyo & Varneskov (2017) studied the distribution of the randomly weighted jump activity index estimator for the pure jump process. The random weighting method depends on tuning the weight distribution to match the moments of the returns. While it performs satisfactorily in the case of the realized variance, it is not adapted to a general class of volatility functionals, since the weight distribution is specific to the functional form, and hence one has to deal with different functionals case by case. In this paper, we introduce a local nonparametric bootstrap method that works for a general class of functional forms. We first split the data into non-overlapping blocks that shrink to zero, and then within each block we resample the returns with replacement to form a bootstrap estimate of the local spot volatility. Finally, we integrate functional values of local spot volatilities across all blocks, which yields a bootstrap estimate of the volatility functional. The rationale is that within each local window, the increments of the log price process can be regarded as approximately independent and identically distributed random variables conditional on the history up to the start of the window. But the increments vary a lot across windows. While the block bootstrap in time series (Hall et al., 1995) aims to reduce correlation, the local bootstrap here attempts to eliminate the heterogeneity of volatility. Our results show that the local bootstrap distribution of the realized volatility functionals after studentization is first-order accurate in approximating the distribution of the studentized realized volatility functionals for Itô processes with jumps of finite variation. Assuming independence of the volatility process and the driving forces, we obtain the uniform convergence rate of the distribution functions of the studentized realized volatility functionals with and without local bootstrapping. The rate demonstrates that the jump-truncation error and the volatility-discretization error dominate in the principal correction term, leading to an accuracy rate below $$\Delta_n^{1/2}$$. In particular, for the studentized realized variance with and without local bootstrapping in the absence of the leverage effect and jumps, we present Edgeworth expansions. We show in this particular case that the local bootstrap distribution has second-order accuracy while the limiting normal distribution is only first-order accurate. Nevertheless, although theoretically second-order accuracy requires a linear functional form and that the leverage effect and jumps be absent, simulation shows that the local bootstrap distribution still outperforms the limiting normal distribution in general scenarios. 2. Bootstrapping volatility functionals 2.1. Preliminary results on volatility functionals The continuous-time univariate Itô process defined on the filtered probability space $$(\Omega, \mathcal{F}, \mathcal{F}_t, P)$$ is \begin{eqnarray}\label{model} X_t&=&X_0+\int^t_0b_s\,{\rm d}s+\int^t_0\sigma_s\,{\rm d}W_s+\int^t_0\int_R\delta(s, x)I\{|\delta(s, x)|\leq 1\}(\mu-\nu)({\rm d}s, {\rm d}x)\nonumber\\ &&+\int^t_0\int_R\delta(s, x)I\{|\delta(s, x)|>1\}\mu({\rm d}s, {\rm d}x), \end{eqnarray} (1) where $$X_0$$ is the initial value, $$b$$ and $$\sigma$$ are locally bounded adapted processes representing the spot drift and volatility, $$W$$ is a standard Brownian motion, $$\delta$$ is a predictable process and $$\mu$$ is a Poisson random measure with compensator $$\nu({\rm d}x, {\rm d}s)={\rm d}x\otimes {\rm d}s$$. The volatility functional is defined as $$V(g)_t=\int^t_0g(c_s)\,{\rm d}s$$ where $$g$$ is a real function and $$c_s=\sigma_s^2$$. In particular, $$g(x)=x$$ and $$g(x)=x^2$$ correspond to the integrated volatility and quarticity, respectively. We assume that the dataset $$\{X_{t_0}, \ldots, X_{t_n}: 0=t_0<t_1<\cdots<t_n=T\}$$ is discretely sampled from $$X_t$$, where $$T$$ is a fixed time horizon and the $$t_i$$ are equally spaced. The time lag is denoted by $$\Delta_n=t_{i}-t_{i-1}$$ and $$\Delta^n_iX=X_{t_{i}}-X_{t_{i-1}}$$. We assume that $$\Delta_n\rightarrow 0$$ as $$n\rightarrow \infty$$ and $$T$$ is fixed. Let $$\bar{g}(x)=g(x)-g^{(2)}(x)x^2/k_n$$ and $$h(x)=2\{g^{\prime}(x)\}^2x^2$$. A natural estimate of $$V(g)_t$$ is $$V(g)^n_t=\sum^{\lfloor t/(k_n\Delta_n)\rfloor}_{i=0}\bar{g}(\hat{c}^n_{ik_n+1})k_n\Delta_n$$ where $$\lfloor x \rfloor$$ stands for the largest integer smaller than or equal to $$x$$, and $$\hat{c}^n_i=k_n^{-1}\sum^{k_n-1}_{j=0}(\Delta^n_{i+j}X)^2\Delta_n^{-1}I(|\Delta^n_{i+j}X|\leq u_n)$$ where $$u_n=\alpha\Delta_n^{\varpi}$$ for some positive constants $$\alpha$$ and $$\varpi$$. The threshold $$u_n$$ removes the jumps in the price dynamics. The jump truncation method comes from Mancini (2009). Clearly $$\hat{c}^n_i$$ is an estimate of $$\sigma^2_{t_{i-1}}$$. To derive the theoretical properties of the local nonparametric bootstrap, we need to make the following technical assumptions. Assumption 1. There is a sequence, $$J_n$$, of nonnegative bounded integrable functions and a sequence, $$\tau_n$$, of stopping times increasing to $$\infty$$ such that $$t<\tau_n \ \mbox{implies} \ |b_t(\omega)|\leq n \ \mbox{and} \ |\sigma_t(\omega)|\leq n,$$ and $$t\leq \tau_n \ \mbox{implies} \ |\delta(\omega, t, x)|^r\wedge 1 \leq J_n(x)\text{.}$$ Assumption 2. The process $$X$$ satisfies Assumption 1 with $$r<1$$. The volatility process $$c$$ is an Itô process of the form (1), with the notation $$b$$, $$\sigma$$, $$\delta$$ and $$\mu$$ replaced by $$\tilde{b}$$, $$\tilde{\sigma}$$, $$\tilde{\delta}$$ and $$\tilde{\mu}$$, respectively, satisfying Assumption 1 with $$r=2$$. Assume that $$b_t$$ and $$b^{\prime}_t=b_t-\int\delta(t, x)I(|\delta(t, x)|\leq 1)\,{\rm d}x$$ are càdlàg. Assumptions 1 and 2 are the same as Assumption (H-r) and Assumption (A-r) in Jacod & Rosenbaum (2013). These assumptions imply that $$X$$ is a process with jumps of finite variation while $$c$$ might contain infinite-variation jumps. Next we present assumptions on the function $$g$$ that are also required in Jacod & Rosenbaum (2013). Assumption 3. The function $$g$$ is a $$C^3$$ function such that $$|g^{(j)}(x)|\leq C(1+x^{p-j})$$$$(\,j=0, 1, 2, 3)$$ for some constants $$C>0$$ and $$p\geq 3$$. In addition $$(2p-1)\{2(2p-r)\}^{-1}\leq \varpi < 1/2$$. We need the notion of stable convergence in law to state our main results. The notion applies to a sequence of random variables $$Z_n$$, all defined on the same probability space $$(\Omega, \mathcal{F}, P)$$ and taking values in the same state space $$(E, \mathcal{E})$$ assumed to be a Polish space. We say that $$Z_n$$ converges stably if there is a probability measure $$\eta$$ on the product of $$(\Omega\times E, \mathcal{F}\times \mathcal{E})$$ such that $$E\{Yf(Z_n)\}\rightarrow \int Y(\omega)f(x)\eta({\rm d}\omega, {\rm d}x)$$ for all bounded continuous functions $$f$$ on $$E$$ and all bounded random variables $$Y$$ on $$(\Omega, \mathcal{F})$$. For more details, refer to Jacod & Protter (2012). Jacod & Rosenbaum (2013) established the following central limit theorem for $$V(g)^n_t$$. Lemma 1. Assuming $$k_n^3\Delta_n\rightarrow \infty$$, $$k_n^2\Delta_n\rightarrow 0$$, and Assumptions 1–3, $$T_n= \{V(g)^n_t-V(g)_t\}\{\Delta_n V(h)^n_t\}^{-1/2}$$ converges to $$N(0, 1)$$ stably. 2.2. Local nonparametric bootstrap To introduce the local bootstrap method, we split the sample $$\mathcal{X}= \{\Delta^n_{1}X,\ldots, \Delta^n_nX\}$$ into $$\lfloor n/k_n \rfloor$$ non-overlapping blocks, each containing $$k_n$$ data points. For the $$i$$th block, let $$\mathcal{X}_i = \{\Delta^n_{(i-1)k_n+1}X, \ldots, \Delta^n_{ik_n}X\}$$. We randomly draw elements from $$\mathcal{X}_i$$ with replacement to obtain the bootstrap sample $$\mathcal{X}_i^*=\{\Delta^n_{(i-1)k_n+1}X^*,\ldots, \Delta^n_{ik_n}X^*\}$$. Let $$\hat{c}^{n*}_{ik_n+1}=k_n^{-1}\sum^{k_n-1}_{j=0}(\Delta^n_{i+j}X^*)^2\Delta_n^{-1}I(|\Delta^n_{i+j}X^*|\leq u_n)$$. The rationale for local random drawing is that the $$\Delta^n_{(i-1)k_n+j}X$$ are approximately independent and identically distributed random variables conditional on $$\mathcal{F}_{(i-1)k_n\Delta_n}$$. Thus the bootstrap versions of $$V(g)_t$$ and $$V(g)^n_t$$ are respectively $$ V^*(g)_t=\sum^{\lfloor t/(k_n\Delta_n)\rfloor -1}_{i=0}g(\hat{c}^n_{ik_n+1})k_n\Delta_n, \quad V^*(g)^n_t=\sum^{\lfloor t/(k_n\Delta_n) \rfloor-1}_{i=0}\bar{g}(\hat{c}^{n*}_{ik_n+1})k_n\Delta_n\text{.} $$ Standardizing the difference between $$V^*(g)_t^n$$ and $$V^*(g)_t$$ gives the locally bootstrapped studentized realized volatility functional $$T_n^*=\{V^*(g)^n_t-V^*(g)_t\}\{\Delta_nV^*(h)^n_t\}^{-1/2},$$ where $$V^*(h)^n_t=\sum^{\lfloor t/(k_n\Delta_n)\rfloor}_{i=0}\bar{h}(\hat{c}^{n*}_{ik_n+1})k_n\Delta_n\text{.}$$ 3. Main results Theorem 1. Suppose Assumptions 1–3 hold. If $$k_n^3\Delta_n\rightarrow \infty$$ and $$k_n^{2}\Delta_n\rightarrow 0$$ as $$\Delta_n\rightarrow 0$$, then conditional on $$\mathcal{X}$$, $$T_n^*$$ converges to $$N(0, 1)$$ stably as $$n\rightarrow\infty$$. Remark 1. Theorem 1 demonstrates that $$\sup_{x\in R}|H_n^*(x)-\Phi(x)|=o_{\rm p}(1),$$ and hence by Lemma 1 $$\sup_{x\in R}|H_n^*(x)-H_n(x)|=o_{\rm p}(1),$$ where $$H_n^*(x)$$ is the local bootstrap distribution of $$T_n^*$$ conditional on $$\mathcal{X}$$ and $$H_n(x)$$ is the distribution of $$T_n$$. Thus $$H_n^*(x)$$ is a close approximation of $$H_n(x)$$ for all $$g$$ satisfying Assumption 3, and hence statistical inference, like constructing the confidence interval or testing hypotheses on $$V(g)$$ for a given path, based on $$H_n^*(x)$$ is reliable. Goncalves & Meddahi (2009) and Hounyo & Veliyev (2016) proved the first-order accuracy of the globally and locally randomly weighted realized variance, but not for a general class of volatility functionals with jumps contained in $$X$$. Next, we refine the central limit theorem by giving the uniform approximation rates of $$H_n(x)$$ and $$H_n^*(x)$$ for general functionals and the Edgeworth expansions of $$H_n(x)$$ and $$H_n^*(x)$$ for the studentized realized variance. To this end, we need one more assumption. Assumption 4. The volatility process $$c$$ is independent of $$(W, \mu, \delta)$$. Assumption 4 implies that there is no leverage effect for $$X$$, which is far from reality. However, our numerical studies show that the local bootstrap distribution outperforms the limiting normal approximation even in the presence of a leverage effect. Assumption 4 is also needed in Goncalves & Meddahi (2009), Hounyo & Veliyev (2016) and Zhang et al. (2011) in order to derive the Edgeworth expansions of the studentized realized variance. Theorem 2. Suppose Assumptions 1–4 hold. If $$k_n^3\Delta_n\rightarrow \infty$$ and $$k_n^{5/2}\Delta_n\rightarrow 0$$ as $$\Delta_n\rightarrow 0$$, then as $$n\rightarrow\infty$$, \begin{equation} \sup_{x\in R}|H_n(x)-\Phi(x)|=O_{\rm p}(k_n{\Delta_n}^{1/2}+b_n), \quad \sup_{x\in R}|H_n^*(x)-\Phi(x)|=O_{\rm p}\{(k_n^3\Delta_n)^{-1/2}+b_n\}, \end{equation} (2)where $$b_n=a_n\Delta_n^{(2q-r)\varpi+1-q}/{\Delta_n}^{1/2}$$ for some $$q>0$$ and some sequence of constants $$a_n$$ converging to $$0$$. If, further, $$g(x)=x$$, $$\delta=0$$ and $$b= 0$$, then \begin{eqnarray} \sup_{x\in R}|H_n^*(x)-\Phi(x)-{\Delta_n}^{1/2}q(x)\phi(x)|&=&o_{\rm p}(\surd{\Delta_n}), \\ \end{eqnarray} (3) \begin{eqnarray} \sup_{x\in R}|H_n(x)-\Phi(x)-{\Delta_n}^{1/2}q(x)\phi(x)|&=&o_{\rm p}(\surd{\Delta_n}), \end{eqnarray} (4)where $$q(x)=(\int^t_0c_s^2\,{\rm d}s)^{-1/2}-\{3(\int^t_0c_s^2\,{\rm d}s)^{3/2}\}^{-1}5\int^t_0c_s^3\,{\rm d}s(x^2-1)$$. Remark 2. Equation (2) implies that $$\sup_{x\in R}|H_n^*(x)-H_n(x)|=O_{\rm p}\{(k_n^3\Delta_n)^{-1/2}+b_n\},$$ which provides a uniform approximation rate for $$H_n^*(x)$$. The terms $$b_n$$ and $$(k_n^3\Delta_n)^{-1/2}$$ are due to jump truncation and volatility discretization, respectively. In particular, if jumps are absent, the rate is $$(k_n^3\Delta_n)^{-1/2}$$. For the realized variance, if $$b=0$$, (3) and (4) demonstrate that $$H_n^*(x)$$ is second-order accurate in approximating $$H_n(x)$$. The condition $$b= 0$$ is assumed in Goncalves & Meddahi (2009), for the reason that the drift term is negligible at high frequencies. We end this section with practical guidelines on choosing the tuning parameters $$\alpha$$, $$\varpi$$ and $$k_n$$. The threshold $$u_n$$ plays the role of eliminating the jumps. If $$u_n$$ is large, many jumps are missed and left in the realized volatility functionals. If $$u_n$$ is small, increments available mainly due to continuous diffusion are removed, which decreases the efficiency of estimating the spot volatilities. Since $$\alpha$$ and $$\varpi$$ are not independent, we fix $$\varpi$$ and adjust $$\alpha$$. Mancini (2009) suggests setting $$\varpi$$ smaller than and close to $$1/2$$, and hence we set $$\varpi=0.49$$ in our numerical analysis. By Lévy’s continuity theorem, a volatility adaptive upper bound for $$|\Delta^n_iX|$$ is $$\{2c_{t_{i-1}}\Delta_n\log{(1/\Delta_n)}\}^{1/2}$$. Thus most likely the increments larger than $$\alpha\Delta_n^{0.49}\surd{c_{t_{i-1}}}>\{2\Delta_n\log{(\Delta_n^{-1})}c_{t_{i-1}}\}^{1/2}$$ are due to jumps, so we choose $$\alpha$$ such that $$u_n=\alpha\Delta_n^{0.49}\surd{c_{t_{i-1}}}$$ is above and close to $$\{2\Delta_n\log{(\Delta_n^{-1})}c_{t_{i-1}}\}^{1/2}$$ to strike a balance between efficiency and jump thresholding. The spot volatility $$c_{t_{i-1}}$$ can be consistently estimated in moving windows or empirically replaced by the estimated average integrated volatility $$\rm{\small{\hat{{IV}}}}_T/T$$, which can be realized by the tuning-free bipower variation technique (Barndorff-Nielsen & Shephard, 2004). We choose the estimated average integrated volatility to replace all $$c_{t_{i-1}}$$ because it is independent of $$i$$. This selection is consistent with the form of $$u_n$$, which is a universal threshold for all increments. Inspired by Lemma 1 and Theorem 1, we can choose $$k_n$$ within the interval $$(n^{1/3}, n^{1/2})$$. Our experience from the simulation study is that the local bootstrap distributions of the studentized realized variance and quarticity are closer to the true sampling distributions than the limiting normal distribution uniformly in $$k_n$$ selected from the proposed integer set. 4. Numerical studies In this section, we conduct simulations to check the local bootstrap procedure. We generate data from the stochastic volatility model \begin{eqnarray*} {\rm d}X_t&=&c_s{^{1/2}}\,{\rm d}W_s+\theta Y_t, \quad 0\leq t\leq 1, \\ c_t&=&c_0+\int^t_0a_0(1-c_s)\,{\rm d}s+b_0\int^t_0 c_s{^{1/2}}\,{\rm d}W^{\prime}_s, \end{eqnarray*} where $$c_0=4.06$$, $$a_0=0.1$$, $$b_0=0.15$$, $$Y_t$$ is a symmetric $$0.5$$-stable Lévy process and $$W^{\prime}$$ is a second Brownian motion satisfying corr$$(W, W^{\prime})=\rho$$. Although we only proved the superiority of the local bootstrap distribution for the studentized realized variance without the leverage effect or jumps, we consider a general data process in this simulation study, and thus we set $$(\theta, \rho)=(1, -$$0$$\cdot$$5$$)$$. First we consider the studentized realized variance, that is, $$g(x)=x$$. We sample the data every five minutes in a day. Then $$\{2\Delta_n\log{(1/\Delta_n)}\}^{1/2}= 0.3342$$. By the rule of thumb presented in § 3, we choose $$\alpha$$ as 3 with $$\varpi=0.49$$ and $$k_n$$ from $$\{4,\ldots,9\}$$. Figure 1(a) displays the local bootstrap distribution, the true sampling distribution and the limiting standard normal distribution of the studentized realized variance for $$(k_n, \alpha)=(6, 3)$$. The local bootstrap distribution is closer to the true sampling distribution than the limiting normal distribution. In the Supplementary Material, we tested other values of $$\alpha$$ around 3 and $$k_n$$ around 6. The local bootstrap distributions are more accurate than the limiting normal distribution uniformly in $$\alpha$$ and $$k_n$$, especially in the tails. Fig. 1. View largeDownload slide The true sampling distribution (dashed), local bootstrap distribution (stars), and limiting standard normal distribution (solid) of the studentized realized volatility functional: (a) the studentized realized variance and (b) the studentized realized quarticity. Fig. 1. View largeDownload slide The true sampling distribution (dashed), local bootstrap distribution (stars), and limiting standard normal distribution (solid) of the studentized realized volatility functional: (a) the studentized realized variance and (b) the studentized realized quarticity. Next, we consider the studentized realized quarticity, that is, $$g(x)=x^2$$. We sample the data every 100 seconds in a day. Then $$\{2\Delta_n\log{(1/\Delta_n)}\}^{1/2}= 0.2159$$. Again, we choose $$\alpha=3$$. The parameter $$k_n$$ can be chosen from $$\{6,\ldots,16\}$$ by the empirical rule. However, when we tested $$k_n<13$$, $$V(h)^n_t$$ would sometimes be negative, making $$T_n$$ meaningless. So we choose $$k_n\geq 13$$. Figure 1(b) displays the local bootstrap distribution, the true sampling distribution and the limiting standard normal distribution of the studentized realized quarticity for $$(k_n, \alpha)=(13, 3)$$. The local bootstrap distribution performs better than the limiting distribution. The Supplementary Material also contains simulation studies for other values of $$\alpha$$ around 3 and $$k_n=14, \ 15$$, which lead to the same conclusion. Table 1 reports the errors of the tail probabilities of the local bootstrap and limiting normal distributions relative to those of the true sampling distribution. In most cells, especially in the left tails, the local bootstrap tail probabilities are more accurate than the normal ones. Table 1. Errors of the tail probabilities of the local bootstrap and limiting standard normal distributions relative to those of the true distribution. Upper part: $$n=78$$ and $$k_n=6$$; lower part: $$n=234$$ and $$k_n=14$$. The negative and positive signs of the first row of numbers denote the left $$(<t)$$ and right $$(>t)$$ tails, respectively; $$(\theta, \rho)=(1, -0{\cdot}5)$$ and $$\alpha=3$$  t $$-$$3$$\cdot$$5 $$-$$3 $$-$$2$$\cdot$$5 $$-$$2 $$-$$1$$\cdot$$5 1$$\cdot$$5 2 2$$\cdot$$5 Realized B $$-$$0$$\cdot$$102 $$-$$0$$\cdot$$147 $$-$$0$$\cdot$$006 $$-$$0$$\cdot$$003 $$-$$0$$\cdot$$150 $$-$$0$$\cdot$$572 $$-$$0$$\cdot$$715 $$-$$0$$\cdot$$703 variance N $$-$$0$$\cdot$$976 $$-$$0$$\cdot$$929 $$-$$0$$\cdot$$812 $$-$$0$$\cdot$$628 $$-$$0$$\cdot$$358 $$-$$0$$\cdot$$599 $$-$$0$$\cdot$$703 $$-$$0$$\cdot$$693 Realized B $$-$$0$$\cdot$$185 $$-$$0$$\cdot$$141 $$-$$0$$\cdot$$167 $$-$$0$$\cdot$$109 $$-$$0$$\cdot$$062 $$-$$0$$\cdot$$181 $$-$$0$$\cdot$$301 $$-$$0$$\cdot$$454 quarticity N $$-$$0$$\cdot$$998 $$-$$0$$\cdot$$991 $$-$$0$$\cdot$$966 $$-$$0$$\cdot$$905 $$-$$0$$\cdot$$782 $$-$$0$$\cdot$$664 $$-$$0$$\cdot$$838 $$-$$0$$\cdot$$934 t $$-$$3$$\cdot$$5 $$-$$3 $$-$$2$$\cdot$$5 $$-$$2 $$-$$1$$\cdot$$5 1$$\cdot$$5 2 2$$\cdot$$5 Realized B $$-$$0$$\cdot$$102 $$-$$0$$\cdot$$147 $$-$$0$$\cdot$$006 $$-$$0$$\cdot$$003 $$-$$0$$\cdot$$150 $$-$$0$$\cdot$$572 $$-$$0$$\cdot$$715 $$-$$0$$\cdot$$703 variance N $$-$$0$$\cdot$$976 $$-$$0$$\cdot$$929 $$-$$0$$\cdot$$812 $$-$$0$$\cdot$$628 $$-$$0$$\cdot$$358 $$-$$0$$\cdot$$599 $$-$$0$$\cdot$$703 $$-$$0$$\cdot$$693 Realized B $$-$$0$$\cdot$$185 $$-$$0$$\cdot$$141 $$-$$0$$\cdot$$167 $$-$$0$$\cdot$$109 $$-$$0$$\cdot$$062 $$-$$0$$\cdot$$181 $$-$$0$$\cdot$$301 $$-$$0$$\cdot$$454 quarticity N $$-$$0$$\cdot$$998 $$-$$0$$\cdot$$991 $$-$$0$$\cdot$$966 $$-$$0$$\cdot$$905 $$-$$0$$\cdot$$782 $$-$$0$$\cdot$$664 $$-$$0$$\cdot$$838 $$-$$0$$\cdot$$934 B, local bootstrap distribution; N, standard normal distribution. Table 1. Errors of the tail probabilities of the local bootstrap and limiting standard normal distributions relative to those of the true distribution. Upper part: $$n=78$$ and $$k_n=6$$; lower part: $$n=234$$ and $$k_n=14$$. The negative and positive signs of the first row of numbers denote the left $$(<t)$$ and right $$(>t)$$ tails, respectively; $$(\theta, \rho)=(1, -0{\cdot}5)$$ and $$\alpha=3$$  t $$-$$3$$\cdot$$5 $$-$$3 $$-$$2$$\cdot$$5 $$-$$2 $$-$$1$$\cdot$$5 1$$\cdot$$5 2 2$$\cdot$$5 Realized B $$-$$0$$\cdot$$102 $$-$$0$$\cdot$$147 $$-$$0$$\cdot$$006 $$-$$0$$\cdot$$003 $$-$$0$$\cdot$$150 $$-$$0$$\cdot$$572 $$-$$0$$\cdot$$715 $$-$$0$$\cdot$$703 variance N $$-$$0$$\cdot$$976 $$-$$0$$\cdot$$929 $$-$$0$$\cdot$$812 $$-$$0$$\cdot$$628 $$-$$0$$\cdot$$358 $$-$$0$$\cdot$$599 $$-$$0$$\cdot$$703 $$-$$0$$\cdot$$693 Realized B $$-$$0$$\cdot$$185 $$-$$0$$\cdot$$141 $$-$$0$$\cdot$$167 $$-$$0$$\cdot$$109 $$-$$0$$\cdot$$062 $$-$$0$$\cdot$$181 $$-$$0$$\cdot$$301 $$-$$0$$\cdot$$454 quarticity N $$-$$0$$\cdot$$998 $$-$$0$$\cdot$$991 $$-$$0$$\cdot$$966 $$-$$0$$\cdot$$905 $$-$$0$$\cdot$$782 $$-$$0$$\cdot$$664 $$-$$0$$\cdot$$838 $$-$$0$$\cdot$$934 t $$-$$3$$\cdot$$5 $$-$$3 $$-$$2$$\cdot$$5 $$-$$2 $$-$$1$$\cdot$$5 1$$\cdot$$5 2 2$$\cdot$$5 Realized B $$-$$0$$\cdot$$102 $$-$$0$$\cdot$$147 $$-$$0$$\cdot$$006 $$-$$0$$\cdot$$003 $$-$$0$$\cdot$$150 $$-$$0$$\cdot$$572 $$-$$0$$\cdot$$715 $$-$$0$$\cdot$$703 variance N $$-$$0$$\cdot$$976 $$-$$0$$\cdot$$929 $$-$$0$$\cdot$$812 $$-$$0$$\cdot$$628 $$-$$0$$\cdot$$358 $$-$$0$$\cdot$$599 $$-$$0$$\cdot$$703 $$-$$0$$\cdot$$693 Realized B $$-$$0$$\cdot$$185 $$-$$0$$\cdot$$141 $$-$$0$$\cdot$$167 $$-$$0$$\cdot$$109 $$-$$0$$\cdot$$062 $$-$$0$$\cdot$$181 $$-$$0$$\cdot$$301 $$-$$0$$\cdot$$454 quarticity N $$-$$0$$\cdot$$998 $$-$$0$$\cdot$$991 $$-$$0$$\cdot$$966 $$-$$0$$\cdot$$905 $$-$$0$$\cdot$$782 $$-$$0$$\cdot$$664 $$-$$0$$\cdot$$838 $$-$$0$$\cdot$$934 B, local bootstrap distribution; N, standard normal distribution. 5. Conclusion and discussion Our results imply that the local bootstrap quantiles, especially those in the tails, are likely to yield more accurate confidence intervals of the volatility functionals for a given path, or more accurate critical regions in hypothesis testing on volatility functionals. The local bootstrap method can be applied to testing for jumps. By locally resampling the high-frequency data, one can obtain the local bootstrap distribution of the test statistic based on the power variation (Aït-Sahalia & Jacod, 2009) under the null hypothesis that the jumps are absent. One can expect the local bootstrapped critical region under the null hypothesis to be more accurate than that constructed from the standard normal distribution. Acknowledgement The authors would like to thank the editor, the associate editor and two referees for their careful and detailed comments which have led to great improvements of the manuscript. Kong’s work was supported in part by the Natural Science Foundation of China. Zhou’s research was partially supported by the Ministry of Education Tier 2 grant at the National University of Singapore. Supplementary material Supplementary material available at Biometrika online contains additional numerical studies, real data analysis and the proofs of the main results. References Aït-Sahalia Y. & Jacod J. ( 2009 ). Testing for jumps in a discretely observed process Ann. Statist. 37 , 184 – 222 . Google Scholar CrossRef Search ADS Aït-Sahalia Y. , Mykland P. A. & Zhang L. ( 2005 ). How often to sample a continuous-time process in the presence of market microstructure noise? Rev. Finan. Studies 18 , 351 – 416 . Google Scholar CrossRef Search ADS Barndorff-Nielsen O. & Shephard N. ( 2002 ). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Statist. Soc. B 64 , 253 – 80 . Google Scholar CrossRef Search ADS Barndorff-Nielsen O. & Shephard N. ( 2004 ). Power and bipower variation with stochastic volatility and jumps J. Finan. Economet. 2 , 1 – 37 . Google Scholar CrossRef Search ADS Goncalves S. & Meddahi N. ( 2009 ). Bootstrapping realized volatility Econometrica 77 , 283 – 306 . Google Scholar CrossRef Search ADS Hall P. , Horowitz J. & Jing B. Y. ( 1995 ). On blocking rules for the bootstrap and dependent data Biometrika 82 , 561 – 74 . Google Scholar CrossRef Search ADS Hounyo U. & Varneskov R. ( 2017 ). A local stable bootstrap for power variations of pure-jump semimartingales and activity index estimation J. Economet. 198 , 10 – 28 . Google Scholar CrossRef Search ADS Hounyo U. & Veliyev B. ( 2016 ). Validity of Edgeworth expansions for realized volatility estimators. Economet. J. 19 , 1 – 32 . Google Scholar CrossRef Search ADS Jacod J. , Li Y. , Mykland P.A. , Podolskij M. & Vetter M. ( 2009 ). Microstructure noise in the continuous case: The pre-averaging approach Stoch. Proces. Appl. 119 , 2249 – 76 . Google Scholar CrossRef Search ADS Jacod J. & Protter P. ( 2012 ). Discretization of Processes . Heidelberg : Springer . Google Scholar CrossRef Search ADS Jacod J. & Rosenbaum M. ( 2013 ). Quarticity and other functionals of volatility: Efficient estimation Ann. Statist. 41 , 1462 – 84 . Google Scholar CrossRef Search ADS Mancini C. ( 2009 ). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps Scand. J. Statist. 36 , 270 – 96 . Google Scholar CrossRef Search ADS Zhang L. , Mykland P. A. & Ait-Sahalia Y. ( 2005 ). A tale of two time scales: Determining integrated volatility with noisy high-frequency data J. Am. Statist. Assoc. 100 , 1394 – 411 . Google Scholar CrossRef Search ADS Zhang L. , Mykland P. A. & Ait-Sahalia Y. ( 2011 ). Edgeworth expansions for realized volatility and related estimators J. Economet. 160 , 190 – 203 . Google Scholar CrossRef Search ADS © 2018 Biometrika Trust This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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BiometrikaOxford University Press

Published: Mar 1, 2018

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