3.1 Robust LAR

Analogous to the practice of sequential jump tests in the literature (e.g., Lee and Mykland, Reference Lee and Mykland2008, Reference Lee and Mykland2012; Laurent and Shi, Reference Laurent and Shi2020), we test for jumps, respectively, over time windows whose lengths shrink as n grows, and the union of these windows basically covers $(0, T)$ for large $n.$ Specifically, the ith window is $\left [t_{(i-1)K_1}, t_{(i+1)K_1}\right ]$ with $i\in \mathcal {S}\!:=\left \{1, \dots , \lfloor (n-1)/K_1-1\rfloor \right \},$ where $\lfloor x\rfloor $ is the largest integer not larger than $x, K_1=\lfloor cn^{c_1}\rfloor $ for some constant $c_1\!\in \!(0, 1),$ and c is a generic positive constant which may take different values at different appearances throughout this article. For each $i\in \mathcal S,$ we compute the robust LAR $\bar {\phi }_i$ based on observations within the ith window, where

$$ \begin{align*} \bar{\phi}_i=\arg\hspace{1mm}\underset{\phi\in\mathbb{R}}{\min}\sum_{j=0}^{K_1}\ell_{\beta_0}\left(y_{iK_1+j}-y_{(i-1)K_1+j}-\phi\right), \end{align*} $$

$\beta _0=cn^{\bar {c}_0}$ for some constant $\bar {c}_0\!\in \!(0, c_1),$ and

(3.1) $$ \begin{align} \ell_{\beta_0}(y)=\begin{cases} 2\beta_0|y|-\beta_0^2, & \text{ if } |y|>\beta_0, \\ y^2, & \text{ if } |y|\leq \beta_0. \end{cases} \end{align} $$

Here, $\ell _{\beta _0}(\cdot )$ is the Huber loss function (Huber, Reference Huber1964) and recently has been widely used in the literature for the construction of statistics robust to fat-tailed noises (e.g., Fan et al., Reference Fan, Li and Wang2017; Fan and Kim, Reference Fan and Kim2018; Fan, Guo, and Jiang, Reference Fan, Guo and Jiang2022). When $\ell _{\beta _0}(y)$ is simply chosen as $y^2, \bar {\phi }_i$ would equal the LAR $\phi _i$ given by

$$ \begin{align*}\phi_i=\sum_{j=0}^{K_1}(y_{iK_1+j}-y_{(i-1)K_1+j})/(K_1+1).\end{align*} $$

For a generic $i,$ the impact of the noise on $\phi _i$ vanishes in probability as n increases. However, in this article, we need to show that the maximum impact of the noise on the (robust) LAR for all the windows is negligible. Lee and Mykland (Reference Lee and Mykland2012) deal with such a problem under the assumption that the noise process is stationary in distribution, but in reality, $u_i$ ’s can be heteroskedastic (see, e.g., the discussions in Kalnina, Reference Kalnina2011 and Taniai et al., Reference Taniai, Usami, Suto and Taniguchi2012). Then, following Sun (Reference Sun2024), we adopt the robust LAR $\bar {\phi }_i$ to build the local statistics for jump tests.

Compared to the choice of $\beta _0 :=\infty $ (which makes $\ell _{\beta _0}(y)=y^2$ as defined in (3.1)), employing a finite $\beta _0$ can relieve the impact of outliers in the noise on $\bar {\phi }_i$ ’s, as $2\beta _0|y|-\beta _0^2\leq y^2,$ and noise outliers can induce abnormally large values in the absolute intraday returns $\left |y_{iK_1+j}-y_{(i-1)K_1+j}\right |.$ On the other hand, $\beta _0$ needs to be sufficiently large for us to control the bias in approximating the efficient LAR (i.e., the LAR of the efficient price) with $\bar {\phi }_i.$ Then choosing $\beta _0$ as a quantity growing jointly with n helps us to show the theoretical validity of $\bar {\phi }_i$ in approximating the efficient LAR uniformly across i in the presence of possibly fat-tailed noises.

Conditional on values of $\beta _0$ and observed prices, we define the functions $\bar {\ell }_i(\phi )=\sum _{j=0}^{K_1}\ell _{\beta _0}(y_{iK_1+j}-y_{(i-1)K_1+j}-\phi )$ and

$$ \begin{align*}\tilde{\ell}_i(\phi)=\partial\bar{\ell}_i(\phi)/\partial\phi=-2\beta_0\sum_{j=0}^{K_1}f\left(\beta_0^{-1}\left(y_{iK_1+j}-y_{(i-1)K_1+j}-\phi\right)\right)\end{align*} $$

for $f(x)=\text { sign }(x)\min \left (\left |x\right |, 1\right )$ and $\phi \in \mathbb {R}.$ Notice that $\tilde {\ell }_i\left (\phi \right )\tilde {\ell }_i\left (-\phi \right )<0$ when $\left |\phi \right |$ is sufficiently large. Then there exists a solution to the equation $\tilde {\ell }_i(\phi )=0,$ since $\tilde {\ell }_i(\phi )$ is a continuous function. As $\bar {\ell }_i(\phi )$ is convex, $\tilde {\ell }_i(\bar {\phi }_i)=0$ according to the first-order condition. By Lemma S1 in the Supplementary Material, we show that the solution to the equation $\tilde {\ell }_i(\phi )=0$ is unique when $K_1$ is an even integer. Then we assume that $K_1$ is even to guarantee that $\bar {\phi }_i$ is well defined for theoretical analysis. Since minimizing $\bar {\ell }_i(\phi )$ belongs to the type of optimization problems considered by Qin and Fang (Reference Qin and Fang2023, Expression (2)), the value of $\bar {\phi }_i$ can be obtained with the iteratively reweighted least-squares (IRLS) approach as specified in that work.

Sun (Reference Sun2024) requires the noise terms to be mutually independent and variance-stationary in demonstrating the similarity between the robust LAR and the LAR of the efficient price, while we allow $u_i$ ’s to be M-dependent and have time-varying variances. Then with more general regularity conditions on the noise, we show the property of the robust LAR in Lemma 1, which is analogous to Theorem 1 in Sun (Reference Sun2024) and is proved with techniques used in that work.

Lemma 1. Under Assumptions 1–5, for any fixed $\epsilon>0,$ we have

(3.2) $$ \begin{align} \underset{i\in\mathcal S}{\max} \ \left|\bar{\phi}_i-\sum_{j=0}^{K_1}(x_{iK_1+j}-x_{(i-1)K_1+j})/(K_1+1)\right|=O_{\mathrm{P}}\left(n^{\max(-\bar{c}_0, \bar{c}_0-c_1)+\epsilon}\right). \end{align} $$

Since the continuous increments of the efficient price are small within short intervals, Lemma 1 implies that for large $n, \left |\bar {\phi }_i\right |$ is small in probability when there is no jump on $\left (t_{(i-1)K_1}, t_{(i+1)K_1}\right ],$ which is the intuition of Equation (3.3) in Corollary 1 below. Notice that under the alternative hypothesis $H_1$ and Assumptions 2, 3, and 5, for $\mathrm {P}$ -almost all $\omega ,$ there exists some $n^*(\omega ),$ such that when $n>n^*(\omega )$ , $t_{K_1}(\omega )$ is smaller than the arrival time of the first jump on $(0, T)$ (i.e., $\tau _1(\omega )>t_{K_1}(\omega )),$ since $t_{K_1}(\omega )$ tends to zero as n grows. The similar reasoning yields that as $n\rightarrow \infty ,$ there is almost surely no jump on $(t_{n-K_1}, T),$ and thus $\left [t_{K_1}, t_{n-K_1}\right ]$ almost surely contains all jumps on $(0, T).$ On the other hand, any jump on $\left [t_{K_1}, t_{n-K_1}\right ]$ is contained in two consecutive intervals in the form of $\left [t_{(i-1)K_1}, t_{(i+1)K_1}\right ]$ for $i\in \mathcal S$ (see Figure 1). Thus, a jump on $(0, T)$ can affect the values of two $\bar {\phi }_i$ ’s, and Corollary 1 implies that with high probability, it significantly increases the absolute value of at least one $\bar {\phi }_i.$

Figure 1

This figure shows that for a jump which occurs at time $t_{iK_1}+\triangle t$ with $0\leq \triangle t\leq t_{(i+1)K_1}-t_{iK_1},$ it is contained in both blocks $\left [t_{(i-1)K_1}, t_{(i+1)K_1}\right ]$ and $\left [t_{iK_1}, t_{(i+2)K_1}\right ].$

Corollary 1. (a) Define $\mathcal S_0=\Big \{\ell : \ell \in \mathcal S, \ \text { and } \ \Delta J_t=0 \ \text { for all } \ t\in (t_{(\ell -1)K_1}, t_{(\ell +1)K_1}]\Big \}$ and $\tilde {c}_0=\min (\bar {c}_0, c_1-\bar {c}_0, (1-c_1)/2).$ Then, for any fixed $\epsilon>0,$ under Assumptions 1–5, we have

(3.3) $$ \begin{align} \underset{i\in\mathcal S_0}{\max} \ \left|\bar{\phi}_i\right|=O_{\mathrm{P}}\left(n^{-\tilde{c}_0+\epsilon}\right). \end{align} $$

(b) Suppose that Assumptions 2, 3, and 5 hold. Then, for $\mathrm {P}$ -almost all $\omega ,$ there exists some $\bar {n}(\omega ),$ such that when $n>\bar {n}(\omega ),$

$$ \begin{align*} \underset{i\in\mathcal S}{\max} \ (N_{t_{(i+1)K_1}}(\omega)-N_{t_{(i-1)K_1}}(\omega))\leq 1. \end{align*} $$

(c) Under Assumptions 1–5 and the alternative hypothesis $H_1,$ it holds that

(3.4) $$ \begin{align} \mathrm{P}\left(\underset{i\in\{1,\ldots , N_{T-}\}}{\cap} \ \left\{h_i\in\mathcal S, \ \text{ and } \ |\bar{\phi}_{h_i}|>\tilde{c}_1|\Delta J_{\tau_i}|\right\}\right)\overset{n\rightarrow\infty}{\longrightarrow} 1 \end{align} $$

for any constant $\tilde {c}_1<1/2,$ where $N_{T-}=\underset {s<t, s\rightarrow T}{\lim } N_s$ and $h_i\in \left \{0,\dots , \text { card }(S)+1\right \}$ is defined such that $\tau _i\in \tilde {L}_i :=(t_{\lceil (h_i-1/2)K_1\rceil \vee 0}, t_{\lceil (h_i+1/2)K_1\rceil \wedge (n+1)}]$ for each $i\in \{1, \ldots , N_{T-}\},$ with $x\wedge y$ and $x\vee y,$ respectively, representing $\min (x,y)$ and $\max (x,y),$ and $\lceil x\rceil $ being the smallest integer not smaller than $x.$

Corollary 1(a) indicates that the absolute values of the $\bar {\phi }_i$ ’s unaffected by jumps are small in probability for large $n.$ Corollary 1(b) says that there is almost surely at most one jump on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ for each $i\in \mathcal S$ as $n\rightarrow \infty .$ Corollary 1(c) suggests that under $H_1,$ with high probability, there exists a $\bar {\phi }_{h_i}$ for each $i\in \{1,\ldots , N_{T-}\},$ such that the absolute value of $\bar {\phi }_i$ is substantially enlarged by the ith jump on $(0, T).$ Notice that $h_i\in \mathcal S$ for each $i\in \{1, \ldots , N_{T-}\},$ when there is no jump on $(0, t_{K_1})\cup (t_{n-K_1}, T).$ Then the discussion above Corollary 1 indicates that the event $\underset {i\in \{1,\ldots , N_{T-}\}}{\cap } \ \{h_i\in \mathcal S\}$ holds almost surely as $n\rightarrow \infty .$ In proving Corollary 1(c), we show that the jump size $\Delta J_{\tau _i}$ is likely to be dominating in determining the value of $\bar {\phi }_{h_i},$ since the impact of the continuous component in the price on $\bar {\phi }_i$ is vanishing in probability as $n\rightarrow \infty .$ Given Lemma 1 which reveals the similarity between the robust LAR and the LAR of the efficient price, the contribution by the ith jump to $\bar {\phi }_{h_i}$ can be roughly expressed as $\psi _i\Delta J_{\tau _i},$ and the value of $\psi _i$ is dependent on $\tau _i$ and the (left and right) boundaries of the time window $\tilde {L}_i.$ We let $\tilde {c}_1$ in (3.4) be smaller than $1/2,$ since $1/2$ is the infimum of $\psi _i$ as $\tau _i$ varies within $\tilde {L}_i$ and $n\rightarrow \infty .$ The essential reason is that as $\tau _i\in \tilde {L}_i,$ numbers of price observations on intervals $[t_{(h_i-1)K_1}, \tau _i] $ and $[\tau _i, t_{(h_i+1)K_1}]$ are both not smaller than $\lfloor K_1/2\rfloor -1$ conditional on $h_i\in \mathcal S.$ In contrast, if we consider non-overlapping intervals over which we test for jumps, respectively, with the current assumptions, we cannot roughly exclude the possibility that there are not many price observations between the arrival time of a jump and the left/right boundary of the time window that contains this jump. Then with non-overlapping windows, it would be more difficult to demonstrate that a jump is likely to exert significant impact on some robust LAR. Moreover, in the literature, it is typical to consider overlapping time windows for sequential jump tests constructed with noisy data (e.g., Lee and Mykland, Reference Lee and Mykland2012; Sun, Reference Sun2019, Reference Sun2024).

3.2 Randomized Tests

Given Corollary 1, we can utilize differences in the magnitudes of $\bar {\phi }_i$ ’s to detect jumps with randomized tests. As mentioned before, the significance level of this test is defined as the limiting probability of the GTE, which means that there exists some $i\in \mathcal S_0,$ such that we mistakenly believe that there is a jump on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ due to the test result. In addition to controlling the significance level, we aim to let the likelihood of missing any jump on $(0, T)$ vanish as $n\rightarrow \infty .$ Then we propose procedures as follows to obtain the randomized statistics $\eta _i$ across $i\in \mathcal S$ :

1. 1. Define $\bar {g}_n=cn^{-\tilde {c}_2}$ for some constant $\tilde {c}_2\in (0,\tilde {c}_0).$ Compute $\bar {\eta }_i=\exp \left (-\left |\bar {g}_n/\bar {\phi }_i\right |\right )$ for $i\in \mathcal S.$

2. 2. For each $i\in \mathcal S,$ independently generate $L_n$ i.i.d. random variables $\left \{z_{i,j}\right \}_{j=1}^{L_n},$ with $z_{i, j}\sim \mathcal D$ for some distribution $\mathcal D$ and $L_n=cn^{c_2}$ for some positive constant $c_2>2(1-c_1).$

3. 3. For each $i\in \mathcal S,$ compute

   (3.5) $$ \begin{align} \eta_i :=2L_n^{-1/2}\sum_{j=1}^{L_n}\left(\boldsymbol 1_{\left\{\bar{\eta}_i\geq z_{i,j}\right\}}-2^{-1}\right). \end{align} $$

We denote the CDF of $\mathcal D$ as $\tilde {F}(\cdot ),$ and state regularity conditions concerning $\tilde {F}(\cdot )$ in Assumption 6, which is satisfied when $\mathcal D$ is, for example, the centered uniform or Gaussian distribution.

For $i\in \mathcal S,$ we believe that there is a jump (or no jump) on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ if $\eta _i$ is larger (or smaller) than the test threshold $\mathcal T_n.$ Then the GTE refers to the event that $\underset {i\in \mathcal S_0}{\max } \ \eta _i$ is larger than $\mathcal T_n,$ and thus knowledge on the asymptotic behavior of $\underset {i\in \mathcal S_0}{\max } \ \eta _i$ is needed in order to set the value of $\mathcal T_n$ given the prescribed significance level. On the other hand, Corollary 1(a) indicates that in the absence of jumps, with high probability $\bar {\eta }_i$ ’s are uniformly small, which implies that whether $\bar {\eta }_i$ is larger than $z_{i, j}$ is basically determined by the sign of $z_{i,j}.$ Consequently, the difference between $\eta _i$ and $\tilde {\eta }_i$ is vanishing in probability uniformly for all $i\in \mathcal S_0,$ where

$$ \begin{align*}\tilde{\eta}_i=2L_n^{-1/2}\sum_{j=1}^{L_n}\left(\boldsymbol 1_{\left\{0\geq z_{i,j}\right\}}-2^{-1}\right),\end{align*} $$

which is asymptotically standard Gaussian by the central limit theorem. The methodology developed by Galambos (Reference Galambos1978) indicates that the asymptotic distribution of the maximum among many i.i.d. random variables can be derived based on the tail behavior of the CDF of a generic individual variable. Then we investigate the asymptotic distribution of $\underset {i\in \mathcal S_0}{\max } \ \eta _i$ based on the derived similarity between the tail probabilities of $\tilde {\eta }_i$ and the standard Gaussian variable, and the similarity between $\eta _i$ and $\tilde {\eta }_i.$ In particular, we utilize the characteristic function to study the tail probability of $\tilde {\eta }_i$ in the technical proofs. Now, we describe the asymptotic behavior of $\underset {i\in \mathcal S_0}{\max } \ \eta _i$ in Theorem 1, which indicates that $\underset {i\in \mathcal S_0}{\max } \ \eta _i$ has the same asymptotic distribution as the maximum of $\lfloor n/K_1 \rfloor $ independent standard Gaussian variables (see, e.g., Chapter 2 in Galambos, Reference Galambos1978).

Theorem 1. Under Assumptions 1–5, for any fixed $\bar {x}\in \mathbb R,$ we have

$$ \begin{align*}\mathrm{P}\left(\underset{i\in\mathcal S_0}{\max} \ \left(\eta_i-\bar{a}_n\right)/\bar{b}_n\leq\bar{x}\right)\rightarrow \exp\left(-e^{-\bar{x}}\right),\end{align*} $$

as $n\rightarrow \infty ,$ where

$$ \begin{align*}\bar{a}_n=\left(2\ln\left(n/K_1\right)\right)^{1/2}-\frac{2^{-1}\left(\ln\left(\ln\left(n/K_1\right)\right)+\ln(4\pi)\right)}{\left(2\ln\left(n/K_1\right)\right)^{1/2}},\end{align*} $$

and $\bar {b}_n=\left (2\ln (n/K_1)\right )^{-1/2}.$

Theorem 1 suggests that given the significance level $\alpha>0,$ we can set the test threshold $\mathcal T_n$ as

(3.6) $$ \begin{align} \mathcal T_n :=\bar{a}_n-\bar{b}_n\ln\left(-\ln(1-\alpha)\right), \end{align} $$

so that

$$ \begin{align*}\mathrm{P}\left(\underset{i\in\mathcal S_0}{\max} \ \eta_i>\mathcal T_n\right)\overset{n\rightarrow\infty}{\longrightarrow}\alpha.\end{align*} $$

On the other hand, detecting a given jump with our test means that there exists some $i\in \mathcal S,$ such that the arrival time of this jump falls on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ and $\eta _i>\mathcal T_n.$ Corollary 2 below suggests that each jump on $(0, T)$ is likely to cause one $\eta _i$ associated with a window containing this jump to be at the order of $n^{c_2/2}.$ Then since $\mathcal T_n$ is at the order of $\sqrt {\ln (n)}$ under (3.6), based on the comparison between $\eta _i$ ’s and $\mathcal T_n$ , we can let the limiting probability of the GTE tend to $\alpha ,$ and detect all jumps on $(0, T)$ with probability tending to one as $n\rightarrow \infty .$

Corollary 2. Recall the definition of $h_i$ in Corollary 1. Under Assumptions 1–5 and the alternative hypothesis $H_1,$ there exists some positive constant $c,$ such that

$$ \begin{align*}\mathrm{P}\left(\underset{i\in\{1,\ldots , N_{T-}\}}{\min} \ \eta_{h_i}>cn^{c_2/2}\right)\overset{n\rightarrow\infty}{\longrightarrow}1.\end{align*} $$

Moreover, we can conveniently construct a randomized test to examine whether there is a jump at one or multiple given time points. The corresponding test and the associated theories are presented in Section 1 in the Supplementary Material.

With a mild abuse of notation, an alternative choice of $\bar {\eta }_i$ is to let $\bar {\eta }_i=\left |\bar {\phi }_i/\bar {g}_n\right |$ in computing the randomized statistics $\eta _i,$ in which case it still holds that $\underset {i\in \mathcal S_0}{\max } \ \bar {\eta }_i\overset {\mathrm {P}}{\rightarrow } 0^+$ as $n\rightarrow \infty $ by Corollary 1(a). However, as $\exp \left (-x^{-1}\right )/x\overset {x\rightarrow 0^+}{\longrightarrow } 0^+,$ the exponential transformation that we adopt in Step 1 of the randomization procedures can substantially decrease values of $\bar {\eta }_i$ for $i\in \mathcal S_0,$ compared to the choice of $\bar {\eta }_i=\left |\bar {\phi }_i/\bar {g}_n\right |.$ Then, with high probability, the exponential transformation would make $\eta _i$ more similar to $\tilde {\eta }_i$ for $i\in \mathcal S_0,$ which facilitates the derivation of the asymptotic distribution of $\underset {i\in \mathcal S_0}{\max } \ \eta _i.$ Moreover, in constructing randomized tests, it is a common practice to apply the exponential transformation to the original statistics (see, e.g., Trapani, Reference Trapani2018; Barigozzi and Trapani, Reference Barigozzi and Trapani2022; Sun, Xu, and Zhang, Reference Sun, Xu and Zhang2023).

Alternatively, we may also choose $\bar {\eta }_i$ as

$$ \begin{align*} \bar{\eta}_i=\exp\left(\left|\bar{g}_n/\bar{\phi}_i\right|\right), \end{align*} $$

in which case $\underset {i\in \mathcal S_0}{\min } \ \bar {\eta }_i\overset {\mathrm {P}}{\rightarrow } \infty $ as $n\rightarrow \infty $ by Corollary 1(a), and accordingly, the final statistic $\eta _i$ can be redefined as

(3.7) $$ \begin{align} \eta_i :=\int_U\left(2L_n^{-1/2}\sum_{j=1}^{L_n}\left(\boldsymbol 1_{\left\{\bar{\eta}_iz_{i,j}\leq |u|\right\}}-1/2\right)\right)^2 dF(u) \end{align} $$

for $i\in \mathcal S,$ where U and $F(u)$ are, respectively, the support and CDF of some chosen distribution. Actually, the statistics (after the exponential transformation) to which the randomization technique is applied in Trapani (Reference Trapani2018), Barigozzi and Trapani (Reference Barigozzi and Trapani2022), and Sun et al. (Reference Sun, Xu and Zhang2023) tend to infinity in probability under the null hypothesis, and their randomized statistics are constructed in similar fashion as shown in (3.7). Given Corollary 1(a), it is straightforward to verify that under certain regularity conditions, $\eta _i$ given by (3.7) is asymptotically subject to the distribution of $\chi _1^2$ for each $i\in \mathcal S_0$ as $n\rightarrow \infty .$ However, we employ the randomized statistic given by (3.5) rather than the one defined in (3.7) to build the test, since in this way, we do not need to choose the distribution characterized by U and $F(u).$

From now on, we stick to the choices of $\bar {\eta }_i=\exp \left (-\left |\bar {g}_n/\bar {\phi }_i\right |\right )$ and $\eta _i$ given by (3.5). When Assumption 6(a) is changed to the weaker assumption that $\tilde {F}(0)\in (0, 1),$ Corollary 1(a) suggests that under certain regularity conditions (including Assumption 6), the statistic

$$ \begin{align*}\sum_{j=1}^{L_n}\left(\boldsymbol 1_{\left\{\bar{\eta}_i\geq z_{i,j}\right\}}-\tilde{F}(0)\right)/\sqrt{L_n\tilde{F}(0)\left(1-\tilde{F}(0)\right)}\end{align*} $$

is asymptotically standard Gaussian for $i\in \mathcal S_0.$ However, we need the condition that $\tilde {F}(0)=1/2$ (as assumed in Assumption 6(a)) for analyzing the tail probabilities of $\tilde {\eta }_i$ ’s (or $\eta _i$ ’s for $i\in \mathcal S_0$ ) in the technical proofs.

3.3 Inferences on the Number and Sizes of Jumps

As a jump on $(0, T)$ can be contained in two consecutive intervals in the form of $\left [t_{(i-1)K_1}, t_{(i+1)K_1}\right ]$ for $i\in \mathcal S,$ we can overestimate the number of jumps if we record the number of all the $\eta _i$ ’s that are larger than $\mathcal T_n$ as the estimator. If both $\eta _i$ and $\eta _{i+1}$ are larger than the threshold, under Assumptions 2, 3, and 5, it is likely to be the same jump on $(t_{iK_1}, t_{(i+1)K_1})$ that increases the values of these two statistics, since, with high probability, there is at most one jump on $[t_{(i-1)K_1}, t_{(i+2)K_1}]$ for large $n.$ Accordingly, we propose the estimator for the number of jumps on $(0, T)$ as $\hat {N}=\text { card }(\mathcal S_1),$ where

$$ \begin{align*}\mathcal S_1=\left\{i: i\in\mathcal S, i>1, \eta_{i-1}\leq\mathcal T_n, \text{ and } \ \eta_{i}>\mathcal T_n\right\}.\end{align*} $$

When the threshold $\mathcal T_n$ is given by (3.6), Theorem 1 indicates that the limiting probability of the GTE tends to $\alpha \in (0, 1).$ Then it can only be shown that $\hat {N}$ equals $N_{T-}$ with probability approaching $1-\alpha $ instead of 1. However, if $\mathcal T_n$ is redefined such that

(3.8) $$ \begin{align} \mathcal T_n/\sqrt{\ln(n)}\rightarrow\infty, \ \text{ and } \ \mathcal T_n/n^{c_2/2}\rightarrow 0 \end{align} $$

as $n\rightarrow \infty ,$ which holds when, for example, $\mathcal T_n=cn^{\tilde {b}_3}$ for some constant $\tilde {b}_3\in (0, c_2/2),$ given Theorem 1 and Corollary 2, it is straightforward to verify that $\hat {N}$ converges to $N_{T-}$ in probability, since the probability of committing the GTE or missing any jump on $(0, T)$ tends to zero. Now, we state these results in Corollary 3.

Theorem 1 and Corollary 2 suggest that when (3.8) holds, with probability approaching one, the proposed method can identify all jumps on $(0, T)$ without committing the GTE. However, in line with the literature on sequential jump tests (e.g., Lee and Mykland, Reference Lee and Mykland2008, Reference Lee and Mykland2012; Laurent and Shi, Reference Laurent and Shi2020), we aim to let the probability of the GTE converge to some prescribed significance level $\alpha \in (0, 1),$ and thus assume that $\mathcal T_n$ is given by (3.6) in the remainder of this article.

Figure 2 indicates that when $K_2\geq 2K_1, (i+1)K_1-K_2\geq 0,$ and $(i-1)K_1+K_2\leq n$ with $K_2\in \mathbb {N}^+,$ an interval $[t_{(i-1)K_1}, t_{(i+1)K_1}]$ is contained in $K_2-2K_1+1$ intervals in the form of $[t_\ell , t_{\ell +K_2}]$ for $\ell \in \{(i+1)K_1-K_2,\ldots , (i-1)K_1\}.$ Then when the test detects a jump on $[t_{(i-1)K_1}, t_{(i+1)K_1}],$ we can use the LAR $\bar {y}_i$ given by

$$ \begin{align*}\bar{y}_i=\sum_{k=0}^{K_2-2K_1}\left(y_{((i+1)K_1+k)\wedge n}-y_{((i+1)K_1+k-K_2)\vee 0}\right)/(K_2-2K_1+1)\end{align*} $$

to estimate the jump size, where $K_2-2K_1=cn^{c_3}$ for some constant $c_3\in [c_1, 1).$ Since the length of an interval $[t_\ell , t_{\ell +K_2}]$ is $O\left (n^{c_3-1}\right )$ by Assumptions 2 and 3, the contribution to $\bar {y}_i$ from the continuous component in the price can be bounded at a small level in probability uniformly for all $i\in \mathcal S.$ In addition, for a generic i with $ (i+1)K_1-K_2\geq 0$ and $(i-1)K_1+K_2\leq n,$ the effect of the noise on $\bar {y}_i$ can be roughly smoothed away by the averaging scheme. Then since the occurrence of jumps is independent of the noise, $\bar {y}_i$ is a consistent jump size estimator for each $i\in \mathcal S_2$ under the alternative hypothesis $H_1,$ where $\mathcal S_2=\left \{i:i\in \mathcal S, \ \text { and } \ N_{t_{(i+1)K_1}}-N_{t_{(i-1)K_1}}>0\right \}.$ On the other hand, we can obtain reliable information about the jump arrival times from the test result, which facilitates the estimation of jump sizes with $\bar {y}_i.$ Now, we summarize these results in Corollary 4.

Figure 2

The figure shows that there are $K_2-2K_1+1$ intervals in the form of $[t_\ell , t_{\ell +K_2}]$ that contain the interval $[t_{(i-1)K_1}, t_{(i+1)K_1}].$

Corollary 4. Under the alternative hypothesis $H_1$ and Assumptions 1–5, it holds that:

(3.9) $$ \begin{align} \mathrm{P}\left(\hat{N}=N_{T-}, \ \text{ and } \ \ \underset{i\in\mathcal S_0}{\max} \ \eta_i\leq\mathcal T_n\right)\overset{n\rightarrow\infty}{\longrightarrow}1-\alpha, \ \ \text{ and } \end{align} $$

(3.10) $$ \begin{align} \underset{i\in\mathcal S_2}{\max} \ \left|\bar{y}_i-\sum_{\ell=1}^{N_{T-}}\Delta J_{\tau_\ell}\boldsymbol 1_{\left\{\tau_\ell\in (t_{(i-1)K_1}, t_{(i+1)K_1}]\right\}}\right|=O_{\mathrm{P}}\left(n^{-c_3/2}\vee \left(n^{(c_3-1)/2}\sqrt{\ln(n)}\right)\right). \end{align} $$

The convergence in (3.9) implies that when $\alpha $ is small, with high probability, the proposed test can locate the arrival time of each jump on $(0, T)$ within some interval in the form of $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ without spurious jump detection, while (3.9) can be proved with steps that are essentially similar to those in the proof of Corollary 3(a). Equation (3.10) indicates that the difference between $\bar {y}_i$ and the jump size on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ can be bounded at a small level in probability for $i\in \mathcal S_2.$

4.1 Asymptotic Results in the Presence of Infinite Activity Jumps

The previous analyses rely on the assumption that jumps only have FA. However, the empirical studies by, for example, Aït-Sahalia and Jacod (Reference Aït-Sahalia and Jacod2011, Reference Aït-Sahalia and Jacod2014), Jing et al. (Reference Jing, Kong, Liu and Mykland2012), Todorov (Reference Todorov2015), and Kong (Reference Kong2019) suggest the presence of IA jumps in financial markets. Accordingly, we extend our analyses to the condition where jumps may have IA in this section. Specifically, we suppose that $J_t$ is given by the sum of an FA jump process and a Lévy process that possibly has IA of jumps, as stated in Assumption 7.

Assumption 7. There exist an FA pure jump process $J_{t}^1$ and a Lévy process $J_{t}^2,$ which are adapted to $\mathcal F_t,$ such that $J_{t}=J_{t}^1+J_{t}^2$ for $t\in [0, T].$ The two processes $\left \{J_t^1\right \}$ and $\left \{J_t^2\right \}$ are independent, and $J_{t}^2$ is given by

(4.1) $$ \begin{align} J_t^2=\int_0^t\int_{|x|>1}x\mu(dx, ds)+\int_0^t\int_{|x|\leq 1}x(\mu(dx, ds)-\nu(dx)ds), \end{align} $$

where $\mu $ is a Poisson random measure on $[0, T]\times \mathbb {R}$ with intensity measure $\nu (dx)dt.$

Assumption 7 allows the process $J_t^2$ to have infinite number of jumps within a finite interval, and regularity conditions similar to this assumption have been widely employed in the literature to characterize jumps in asset prices (see, e.g., Mancini, Reference Mancini2009, Reference Mancini2017; Cont and Mancini, Reference Cont and Mancini2011; Sun et al., Reference Sun, Xu and Zhang2023; Shin, Kim, and Fan, Reference Shin, Kim and Fan2023). On the right-hand side of Equation (4.1), the component $\int _0^t\int _{|x|>1}x\mu (dx, ds),$ which is a compound Poisson process, is driven by jumps whose absolute sizes are larger than one, and $\int _0^t\int _{|x|\leq 1}x(\mu (dx, ds)-\nu (dx)ds)$ is the compensated sum of jumps with sizes not larger than one in absolute value, while $\tilde {\mu }(dx,ds):=\mu (dx, ds)-\nu (dx)ds$ is the compensated Poisson random measure associated with $\mu .$ Under Assumption 7 and some other regularity conditions, we can show that $\bar {\phi }_i$ is still valid in approximating the LAR of the efficient price, as indicated in Lemma 2.

Lemma 2. Under Assumptions 1–4 and 7, we have

$$ \begin{align*}\underset{i\in\mathcal S}{\max} \ \left|\bar{\phi}_i-\sum_{j=0}^{K_1}(x_{iK_1+j}-x_{(i-1)K_1+j})/(K_1+1)\right|=O_{\mathrm{P}}\left(n^{\max(-\bar{c}_0, \bar{c}_0-c_1)+\epsilon}\right)\end{align*} $$

for any fixed $\epsilon>0.$

The convergence rate shown in Lemma 2 is the same as in Lemma 1, as $J_t^2$ does not substantially affect the robustness of $\bar {\phi }_i$ to the noise. Under Assumption 7, the magnitude of jump sizes that may cause the corresponding $\eta _i$ ’s to be larger than $\mathcal T_n$ is related to the vibrancy of small jumps in $J_t^2,$ which can be measured by the $Blumenthal$ – $Getoor$ (BG) index defined as

$$ \begin{align*}\tilde{\gamma} := \inf \ \left\{\gamma\geq 0, \int_{|x|\leq 1}|x|^\gamma\nu(dx)<\infty\right\}\leq 2.\end{align*} $$

The BG index has been widely considered in the literature (e.g., Todorov, Reference Todorov2017; Kong, Reference Kong2019; Park and Wang, Reference Park and Wang2021), and estimation methods for this index can be found in Aït-Sahalia and Jacod (Reference Aït-Sahalia and Jacod2009) and Jing et al. (Reference Jing, Kong, Liu and Mykland2012). A larger value of $\tilde {\gamma }$ suggests that small jumps in $J_t^2$ are more vibrant. In particular, $\tilde {\gamma }=0$ when $J_t^2$ is an FA jump process, and $\tilde {\gamma }\leq 1$ when $J_t^2$ is of finite variation. In expressing our theories, we treat $c/\tilde {\gamma }$ as infinity when $\tilde {\gamma }=0.$ Now, we use Corollary 5 to shed some light on asymptotic behaviors of some relevant statistics with Assumption 7.

Corollary 5. (a) Suppose that Assumptions 2, 3, and 7 hold, and define $\bar {N}_{t,x} :=\sum _{s\in (0, t]}\boldsymbol 1_{\left \{\left |\Delta J_s\right |\geq x\right \}}.$ Then, for any fixed $\epsilon \in (0, (1-c_1)/(2\tilde {\gamma })),$ we have

(4.2) $$ \begin{align} \mathrm{P}\left(\underset{i\in\mathcal S}{\max} \ \left(\bar{N}_{t_{(i+1)K_1}, n^{-\epsilon}}-\bar{N}_{t_{(i-1)K_1}, n^{-\epsilon}}\right)\leq 1\right)\overset{n\rightarrow\infty}{\longrightarrow}1. \end{align} $$

(b) For any constants $\epsilon \in (0, \min ((1-c_1)/\max (2\tilde {\gamma }, 1+\tilde {\gamma }), \bar {c}_0, c_1-\bar {c}_0))$ and $\tilde {c}_1<1/2,$ under the alternative hypothesis $H_1$ and Assumptions 1–4 and 7, the event

$$ \begin{align*} \left\{\bar{N}_{T-,n^{-\epsilon}}>0, \ \text{ and } \ \underset{i\in\{1,\ldots , \bar{N}_{T-,n^{-\epsilon}}\}}{\cap}\left\{\bar{h}_{i,n^{-\epsilon}}\in\mathcal S, \ \text{ and } \ |\bar{\phi}_{\bar{h}_i, n^{-\epsilon}}|>\tilde{c}_1|\Delta J_{\bar{\tau}_{i,n^{-\epsilon}}}|\right\}\right\} \end{align*} $$

holds with probability approaching 1 as $n\rightarrow \infty ,$ where $\bar {N}_{T-,x}=\underset {s<T, s\rightarrow T}{\lim } \bar {N}_{s,x},$ and $\bar {h}_{i,x}$ is defined such that $\bar {h}_{i, x}\in \{0, \dots , \text { card }(\mathcal S)+1\}$ and $\bar {\tau }_{i,x}\in (t_{\lceil (\bar {h}_{i,x}-1/2)K_1\rceil \vee 0}, t_{\lceil (\bar {h}_{i,x}+1/2)K_1\rceil \wedge (n+1)}]$ for each $i\in \{1, \ldots , \bar {N}_{T-,x}\},$ with $\bar {\tau }_{i,x}\in \left \{\tau : \tau \in (0, T), \ \text { and } \ |\Delta J_\tau |\geq x\right \}$ and $\bar {\tau }_{i,x}<\bar {\tau }_{j,x}$ if $i<j.$

(c) Suppose that Assumptions 1–4 and 7 hold. For any fixed $\epsilon \in (0, (1-c_1)/\tilde {\gamma })$ and $\tilde {\epsilon }>0,$ it follows that

(4.3) $$ \begin{align} \sum_{i\in\mathcal S}\boldsymbol 1_{\left\{\bar{N}_{t_{(i+1)K_1}, n^{-\epsilon}}-\bar{N}_{t_{(i-1)K_1}, n^{-\epsilon}}>0\right\}}=o_{\mathrm{P}}(n^{1-c_1}), \end{align} $$

and the event

$$ \begin{align*} \underset{i\in\mathcal S}{\cap}\left\{\boldsymbol 1_{\left\{\bar{N}_{t_{(i+1)K_1}, n^{-\epsilon}}-\bar{N}_{t_{(i-1)K_1}, n^{-\epsilon}}>0\right\}}\geq\boldsymbol 1_{\left\{\left|(K_1+1)^{-1}\sum_{j=0}^{K_1}(J_{t_{iK_1+j}}-J_{t_{(i-1)K_1+j}})\right|\geq n^{(\tilde{\gamma}/2-1)\epsilon+\tilde{\epsilon}}\right\}}\right\} \end{align*} $$

holds with probability approaching 1 as $n\rightarrow \infty .$

Corollary 5(a) roughly suggests that with probability approaching one, each local window contains at most one jump whose size is at an order larger than $n^{(c_1-1)/(2\tilde {\gamma })}.$ Given Corollary 1(a) which shows the order of the robust LAR’s unaffected by jumps, Corollary 5(b) intuitively indicates that relatively large jumps, whose absolute sizes are bigger than some frequency-specific threshold, can substantially enlarge the corresponding robust LAR’s in absolute value, while (4.2) guarantees that each of these jumps affects a distinct robust LAR in probability. We prove Corollary 5(b) by verifying that a relatively large jump is likely to be dominating in determining the value of the corresponding robust LAR. Corollary 5(c) intuitively suggests that with high probability, most windows contain only small jumps, and for such windows, the impact of jumps on the associated robust LAR’s can be bounded at a small level when $\tilde {\gamma }<2,$ as Lemma 2 shows that the effects of jumps on $\bar {\phi }_i$ can be roughly expressed as $(K_1+1)^{-1}\sum _{j=0}^{K_1}(J_{t_{iK_1+j}}-J_{t_{(i-1)K_1+j}}).$

Lemma 2 and Corollary 5 imply that under certain regularity conditions including Assumption 7, with the proposed test, it is hopeful to detect relatively large jumps whose absolute sizes are allowed to decrease as n increases, since these jumps have substantially larger impact on the robust LAR than the continuous component and small jumps in the price. Moreover, in the presence of IA jumps, we can derive the extreme value theory in similar fashion as shown in Theorem 1 to describe the asymptotic behavior of the maximum of the $\eta _i$ ’s affected by jumps whose absolute sizes are not larger than some frequency-specific threshold. Now, we summarize these findings in Corollary 6.

Corollary 6. Suppose that Assumptions 1–4 and 7 hold.

(a) When $\tilde {c}_2\leq (1-c_1)/\max ((2\tilde {\gamma }), 1+\tilde {\gamma }),$ for any fixed $\epsilon \in (0, \tilde {c}_2),$ the event

$$ \begin{align*} \left\{\bar{N}_{T-,n^{-\tilde{c}_2+\epsilon}}>0, \ \text{ and } \ \underset{i\in\{1,\ldots , \bar{N}_{T-,n^{-\tilde{c}_2+\epsilon}}\}}{\cap}\left\{\bar{h}_{i,n^{-\tilde{c}_2+\epsilon}}\in\mathcal S, \ \text{ and } \ \eta_{\bar{h}_i, n^{-\tilde{c}_2+\epsilon}}>\mathcal T_n\right\}\right\} \end{align*} $$

holds with probability approaching 1 under the alternative hypothesis $H_1.$

(b) Under $\tilde {\gamma }<2$ and $\tilde {c}_2<(2-\tilde {\gamma })(1-c_1)/(2\tilde {\gamma }),$ for any constant $\bar {c}_1\in \left (\tilde {c}_2/(1-\tilde {\gamma }/2), (1-c_1)/\tilde {\gamma }\right )$ , we have

(4.4) $$ \begin{align} \mathrm{P}\left(\underset{i\in\tilde{\mathcal S}(n^{-\bar{c}_1})}{\max} \ \eta_i\leq\mathcal T_n\right)\overset{n\rightarrow\infty}{\longrightarrow} \exp\left(-e^{-\bar{x}}\right), \end{align} $$

where $\tilde {\mathcal S}(a) :=\left \{i: i\in \mathcal S, \ \text { and } \ \bar {N}_{t_{(i+1)K_1}, a}-\bar {N}_{t_{(i-1)K_1}, a}=0\right \}$ for $a>0.$

Corollary 6(a) states that the relatively large jumps are likely to cause the null hypothesis $H_0$ to be rejected, and the order of the lower bound of the absolute jump sizes under consideration is roughly $n^{-\tilde {c}_2}.$ The condition assumed for $\tilde {c}_2$ in Corollary 6(a) intuitively implies that jumps which are detectable by the proposed test are required to be larger if the small jumps are more vibrant. This implication is not surprising, since the impact of a relatively large jump on $\eta _i$ may be compromised by many small jumps. Corollary 6(b) presents the extreme value theory to describe the asymptotic behavior of the maximum of the $\eta _i$ ’s associated with windows only containing jumps whose absolute sizes are smaller than $n^{-\bar {c}_1}.$ This statement implies that once we observe $\eta _i>\mathcal T_n$ for some $i,$ under certain regularity conditions, we can reasonably expect that there exists a jump on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ whose absolute size is, roughly speaking, at least at the order of $n^{\tilde {c}_2/(\tilde {\gamma }/2-1)}.$ In this case, if $\tilde {\gamma }$ is sufficiently small (so that $\tilde {c}_2/(1-\tilde {\gamma }/2)<(1-c_1)/(2\tilde {\gamma })$ ), by Corollary 5(a), we can further expect that the size of this jump is at a larger order than sizes of the other jumps on $(t_{(i-1)K_1}, t_{(i+1)K_1}];$ moreover, if both $\eta _i$ and $\eta _{i+1}$ are larger than $\mathcal T_n,$ it is likely to be the same jump within $[t_{iK_1}, t_{(i+1)K_1}]$ that causes the largeness of these two test statistics, since there is likely to be only one detectable jump on $(t_{(i-1)K_1}, t_{(i+2)K_1}]$ (as Corollary 5(a) still holds when $K_1$ is replaced with $2K_1$ ). Then we can see that under certain regularity conditions, an $\eta _i$ which is larger than $\mathcal T_n$ signals the presence of a distinctively large jump on $(t_{(i-1)K_1}, t_{(i+1)K_1}],$ and the jump size can be estimated with the robust LAR, as indicated in Corollary 7 below.

The asymptotic distribution shown in (4.4) is derived on the basis that with high probability, most elements in $\mathcal S$ belong to $\tilde {\mathcal S}(\bar {c}_1).$ Then we require $\bar {c}_1$ to be smaller than $(1-c_1)/\tilde {\gamma }$ in using the threshold $n^{-\bar {c}_1}$ for jump sizes, since roughly speaking, Equation (4.3) implies that most windows only contain jumps whose absolute sizes are at most at the order of $n^{(c_1-1)/\tilde {\gamma }}.$ On the other hand, $\bar {c}_1$ needs to be larger than $\tilde {c}_2/(1-\tilde {\gamma }/2),$ so that given Lemma 2 and Corollary 5(c), the term which absorbs the impact of jumps on $\bar {\phi }_i$ for $i\in \tilde {\mathcal S}(\bar {c}_1)$ is probably at an order smaller than $n^{-\tilde {c}_2}$ (i.e., the order of $\bar {g}(n)$ ). Also, letting $\tilde {c}_2$ be smaller than $(2-\tilde {\gamma })(1-c_1)/(2\tilde {\gamma }),$ as required by Corollary 6(b), ensures that $\tilde {c}_2/(1-\tilde {\gamma }/2)<(1-c_1)/\tilde {\gamma }.$

As mentioned before, when observing $\eta _i>\mathcal T_n$ , we can reasonably expect that there is a relatively large jump on $(t_{(i-1)K_1}, t_{(i+1)K_1}],$ and the size of this jump can be estimated with the robust LAR $\tilde {\phi }_i$ constructed with data on $[t_{((i+1)K_1-K_2)\vee 0}, t_{((i-1)K_1+K_2)\wedge n}].$ Different from our practice in Section 3.3, here we employ the robust instead of the ordinary LAR to estimate the jump sizes. This is because the number of detected jumps may tend to infinity as $n\rightarrow \infty $ in the presence of IA jumps, and the robust estimation technique provides the convenience to bound the extreme estimation error in probability. Specifically, $\tilde {\phi }_i$ is given by

$$ \begin{align*}\tilde{\phi}_i=\text{ arg }\ \underset{\phi}{\min}\sum_{j=0}^{K_2-2K_1}\ell_{\beta_0}\left(y_{((i+1)K_1+j)\wedge n}-y_{((i+1)K_1+j-K_2)\vee 0}-\phi\right),\end{align*} $$

and, from now on, $K_2$ is assumed to be at the same order as $K_1$ in order to simplify the technical analysis, as the impact of jumps on $\tilde {\phi }_i$ hinges on the length of the interval $[t_{(i+1)K_1-K_2}, t_{(i-1)K_1+K_2}].$ That is, we let $K_2= \bar {c}_2K_1$ with some constant integer $\bar {c}_2>2.$ Now, we use Corollary 7 to indicate that for each jump whose absolute size is, roughly speaking, at an order larger than $n^{(c_1-1)/(2\tilde {\gamma })},$ the difference between its size and the robust LAR is small in probability.

Corollary 7. Under the alternative hypothesis $H_1$ and Assumptions 1–4 and 7, for any constants $\bar {c}_3\in \left (0, (1-c_1)/(2\tilde {\gamma })\right )$ and $\epsilon>0,$ we have

$$ \begin{align*}\mathrm{P}\left(\bar{N}_{T-,n^{-\bar{c}_3}}>0\right)\overset{n\rightarrow\infty}{\longrightarrow}1,\end{align*} $$

and conditional on $\bar {N}_{T-,n^{-\bar {c}_3}}>0,$

$$ \begin{align*}\underset{i\in\bar{\mathcal S}\left(n^{-\bar{c}_3}\right)}{\max} \ \left|\tilde{\phi}_i-\sum_{i=1}^{\bar{N}_{T-,n^{-\bar{c}_3}}}\Delta J_{\bar{\tau}_{i, n^{-\bar{c}_3}}}\boldsymbol 1_{\left\{t_{(i-1)K_1}< \bar{\tau}_{i, n^{-\bar{c}_3}}\leq t_{(i+1)K_1}\right\}}\right|=O_{\mathrm{P}}\left(n^{-\breve{c}_0+\epsilon}\right),\end{align*} $$

where $\bar {\mathcal S}(a) :=\left \{i: i\in \mathcal S, \ \text { and } \ \bar {N}_{t_{(i+1)K_1}, a}-\bar {N}_{t_{(i-1)K_1}, a}>0\right \}$ for $a>0,$ and $\breve {c}_0=\min (\tilde {c}_0, 1-c_1-\bar {c}_3\tilde {\gamma }, (1-c_1)/2+(1-\tilde {\gamma })\bar {c}_3).$

As Corollary 5(a) suggests that there is likely to be at most one jump on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ whose absolute size is larger than $n^{-\bar {c}_3}$ for each $i\in \mathcal S,$ Corollary 7 indicates that the difference between $\tilde {\phi }_i$ and the largest absolute jump size on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ is approximately at the order of $n^{-\breve {c}_0}.$ The appearance of terms $1-c_1-\bar {c}_3\tilde {\gamma }$ and $(1-c_1)/2+(1-\tilde {\gamma })\bar {c}_3$ in the definition of $\breve {c}_0$ is due to the effects of small jumps. It is obvious that $\breve {c}_0>0$ as $\bar {c}_3<(1-c_1)/(2\tilde {\gamma }),$ and a smaller $\tilde {\gamma }$ may yield a larger $\breve {c}_0,$ which intuitively suggests that errors of estimating the relatively large sizes of jumps tend to become smaller when the small jumps are less vibrant.

4.2 Asymptotic Results for Large T

In this section, we reveal the theoretical properties of the proposed test under the condition where T goes to infinity jointly with $n.$ In the case of a fixed T and FA jumps, we can treat both the number of jumps and jump sizes roughly as constants, which provides the convenience for us to prove the theories in Section 3. However, when T goes to infinity as n increases, the number of jumps on $(0, T)$ may grow as well, and consequently, the smallest and largest sizes of jumps on $(0, T)$ may, respectively, tend to zero and infinity, even when the process $J_t$ is assumed to have only FA of jumps. Then an interesting extension is to derive asymptotic theories for the proposed test under the condition where T diverges jointly with $n.$ In this case, we consider the null and alternative hypotheses as

$$ \begin{align*}\tilde{H}_0: \Delta J_t=0 \ \text{ for all } \ t\in(0, \infty), \ \ \text{ and } \ \ \tilde{H}_1: \Delta J_{\tau}\neq 0 \ \text{ for some } \ \tau\in(0, \infty),\end{align*} $$

since different from hypotheses $H_0$ and $H_1$ , $\tilde {H}_0$ and $\tilde {H}_1$ do not depend on some specific value of $T.$

Given the growth rate of the sample size $n, T$ cannot tend to infinity at an arbitrarily fast rate, and we employ Assumption 8 below to bound the magnitude of T relative to $n.$ Furthermore, we impose some regularity condition on the jump counting process $N_t$ with Assumption 9 as follows.

Assumption 9. $J_t$ is an FA pure jump process, and the process $N_t$ as given in (2.2) has independent increments. Moreover, there exists some constant c such that

$$ \begin{align*} \mathrm{P}(N_{T_2}-N_{T_1}>0)<c(T_2-T_1) \end{align*} $$

for all $0\leq T_1<T_2<\infty .$

In the high-frequency econometrics literature where both n and T go to infinity, it is typical to assume that the observational times are equally spaced and derive asymptotics under assumptions involving $\Delta _n :=t_i-t_{i-1}$ and T jointly (e.g., Kanaya and Kristensen, Reference Kanaya and Kristensen2016; Kanaya, Reference Kanaya2017; Park and Wang, Reference Park and Wang2021; Andersen et al., Reference Andersen, Fusari, Todorov and Varneskov2019). In this article, a relation between $\bar {\Delta }_n :=\underset {i}{\max } \ (t_i-t_{i-1})$ and T is introduced by Assumptions 3 and 8, which suggest that $\bar {\Delta }_n=O(T^{1-c_4^{-1}})$ . Assumption 9 is satisfied when $N_t$ follows, for example, the Poisson process, and it implies that uniformly for each $i\in \mathcal S,$ there is at most one jump on interval $[t_{(i-1)K_1}, t_{(i+1)K_1}]$ with probability approaching one (see Lemma S3 in the Supplementary Material). Then it is unlikely that some jump is not detected due to the disturbance from another jump near it.

When $T\rightarrow \infty ,$ Assumption 1 is not enough for us to bound $a_t$ and $\sigma _t$ (in probability) uniformly over the whole time span. Then we consider Assumption 10 to restrict the magnitudes of $ \underset {t\in [0, T]}{\sup }\left |a_t\right |$ and $\underset {t\in [0, T]}{\sup }\left |\sigma _t\right |.$

The orders of $\underset {t}{\sup } \ |a_t|$ and $\underset {t}{\sup } \ |\sigma _t|$ for $t\in [0, T]$ can be derived with some regularity conditions concerning the degree of continuity of the processes $a_t$ and $\sigma _t.$ In particular, the validity of the condition about $\sigma _t$ in Assumption 10 can be justified under Assumption S1 which has been employed by Kanaya and Kristensen (Reference Kanaya and Kristensen2016) for the estimation of spot volatilities in the case where $T\rightarrow \infty .$

Assumption S1. There exist positive constants $c, q_1$ , and $q_2$ such that for all $s,t\in [0, \infty ),$

$$ \begin{align*}\mathbb{E}\left(\left|\sigma_t^2-\sigma_s^2\right|{}^{q_1}\right)\leq c\left|t-s\right|{}^{1+q_2}.\end{align*} $$

Assumption S1 imposes a smoothness condition on the volatility process. As pointed out by Kanaya and Kristensen (Reference Kanaya and Kristensen2016), this assumption is satisfied when $\left \{\sigma _t\right \}$ follows a drift–diffusion process with the drift and diffusion coefficients subject to certain moment conditions; in particular, Assumption S1 holds when $\left \{\sigma _t\right \}$ is an Ornstein–Uhlenbeck or CIR process. Moreover, Lemma A.1 derived by Kanaya and Kristensen (Reference Kanaya and Kristensen2016) suggests that under Assumption S1, the process $\sigma _t^2$ is almost surely locally Hölder continuous with exponent d that can take any value on $(0, q_2/q_1).$ Then it is natural to see that when $q_2/q_1>1, \sigma _t^2$ is almost surely time-invariant over the infinite interval $[0, \infty ),$ as illustrated in Lemma 3. Moreover, in Lemma 3, we bound the growth rate of $\underset {t\in [0, T]}{\sup } |\sigma _t^2|$ almost surely as $T\rightarrow \infty $ when $q_2/q_1\leq 1.$

Lemma 3. Under Assumption S1, it holds that: (a) if $q_2/q_1>1,$

$$ \begin{align*} \mathrm{P}\left(\underset{t\in[0, \infty)}{\sup} \ \left|\sigma_t^2-\sigma_0^2\right|=0\right)=1; \end{align*} $$

(b) if $q_2/q_1\leq 1,$ almost surely

$$ \begin{align*} \underset{t\in[0, T]}{\sup} |\sigma_t|<T\ln(T)^{1-q_2/q_1+\epsilon} \end{align*} $$

for any fixed $\epsilon>0,$ as $T\rightarrow \infty .$

Lemma 3 suggests that $\underset {t\in [0, T]}{\sup } |\sigma _t|=O_{\mathrm {P}}\left (T^{1+\epsilon }\right )$ for any fixed $\epsilon>0$ under Assumption S1. Then Assumption 10 is satisfied under Assumption S1 and the regularity condition for $a_t$ assumed in similar fashion as Assumption S1, that is,

$$ \begin{align*}\mathbb{E}\left(\left|a_t^2-a_s^2\right|{}^{q_1}\right)\leq c\left|t-s\right|{}^{1+q_2}\end{align*} $$

for all $s, t\in [0, \infty ).$

Now, we show in Lemma 4 that in the context of growing $T, \bar {\phi }_i$ can be still regarded as a valid approximation of the LAR of the efficient price, when the interval $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ does not contain a jump whose absolute size is larger than some threshold increasing in $n.$

Lemma 4. Under Assumptions 3, 4, and 8–10, for any fixed $\epsilon \in (0, \min (\bar {c}_0/2, (c_1-\bar {c}_0)/2)),$ we have

$$ \begin{align*}\underset{i\in\tilde{\mathcal S}\left(n^{\bar{c}_0/2-\epsilon}\right)}{\max} \left|\bar{\phi}_i -\sum_{j=0}^{K_1}(x_{iK_1+j}-x_{(i-1)K_1+j})/(K_1+1)\right|=O_{\mathrm{P}}\left(n^{-2\epsilon}\right)\end{align*} $$

as $n, T\rightarrow \infty .$

Comparing Lemmas 1 and 4, we can see that in the absence of jumps, the difference between $\bar {\phi }_i$ and $\sum _{j=0}^{K_1}(x_{iK_1+j}-x_{(i-1)K_1+j})/(K_1+1)$ is approximately at the order of $n^{\max (-\bar {c}_0, \bar {c}_0-c_1)},$ no matter whether T goes to infinity or not. However, when $T\rightarrow \infty ,$ we can reveal the order of the difference between $\bar {\phi }_i$ and $\sum _{j=0}^{K_1}(x_{iK_1+j}-x_{(i-1)K_1+j})/(K_1+1)$ only for the condition where the order of the possible jump size that affects $\bar {\phi }_i$ is smaller than $n^{\bar {c}_0/2},$ as shown in Lemma 4. Nevertheless, a jump whose absolute size is not smaller than $n^{\bar {c}_0/2}$ is still likely to substantially enlarge some robust LAR, according to Corollary 8, which describes the behaviors of $\bar {\phi }_i$ in the context of growing $T.$

Corollary 8. (a) For any fixed $\breve {\epsilon }_0>0$ and $\epsilon>0,$ under Assumptions 3, 4, and 8–10, we have

(4.5) $$ \begin{align} \underset{i\in\tilde{\mathcal S}\left(n^{-\tilde{c}_2-\breve{\epsilon}_0}\right)}{\max} \left|\bar{\phi}_i\right|=O_{\mathrm{P}}\left(n^{-\tilde{c}_3+\epsilon}\right) \end{align} $$

for $\tilde {c}_3=\min \left (\bar {c}_0, c_1-\bar {c}_0, \tilde {c}_2+\breve {\epsilon }_0, (1-c_1-3c_4)/2\right ),$ as $n, T\rightarrow \infty .$

(b) Recall the definitions of $\bar {N}_{t,x}, \bar {h}_{i,x}$ , and $\bar {\tau }_{i,x}$ given in Corollary 5. Under the alternative hypothesis $\tilde {H}_1$ and Assumptions 3, 4, and 8–10, for any constants $\bar {\epsilon }_0>0, \bar {\epsilon }_1\in (0, \tilde {c}_3) $ and $\tilde {c}_1<1/2,$ we have

(4.6) $$ \begin{align} \mathrm{P}\left(\bar{N}_{T-,n^{-\tilde{c}_3+\bar{\epsilon}_1}}>0, \ \text{ and } \ \underset{i\in\{1,\ldots , \bar{N}_{T-,n^{-\tilde{c}_3+\bar{\epsilon}_1}}\}}{\min}\bar{f}_i=1 \right)\overset{n, T\rightarrow\infty}{\longrightarrow}1 \end{align} $$

with $\bar {f}_i :=\boldsymbol 1_{\left \{\bar {h}_{i,n^{-\tilde {c}_3+\bar {\epsilon }_1}}\in \mathcal S, \ \text { and } \ |\bar {\phi }_{\bar {h}_{i,n^{-\tilde {c}_3+\bar {\epsilon }_1}}}|>\tilde {c}_1\left (|\Delta J_{\bar {\tau }_{i, n^{-\tilde {c}_3+\bar {\epsilon }_1}}}|\wedge n^{\bar {c}_0/2-\bar {\epsilon }_0}\right )\right \}}$ .

Corollary 8(a) uniformly bounds the magnitudes of the $\bar {\phi }_i$ ’s which are unaffected by jumps or affected by jumps whose sizes are at a smaller order than $\bar {g}_n$ in probability. The convergence in (4.6) indicates that for a jump on $(0, T)$ with a non-negligible size, it is likely to significantly increase the magnitude of at least one $\bar {\phi }_i.$ We have to add the threshold $n^{\bar {c}_0/2-\bar {\epsilon }_0}$ in the definition of $\bar {f}_i,$ given the difficulty in bounding the difference between $\bar {\phi }_i$ and the corresponding LAR due to overly large jumps. Then a limitation of Corollary 8 is that it fails to provide an explicit relation between the size of a jump which is at an order not smaller than $n^{\bar {c}_0/2}$ and the corresponding robust LAR. However, from (4.6), it can be seen that such jumps are likely to cause the corresponding $|\bar {\phi }_i|$ ’s to be large.

Given the results in Corollary 8, we illustrate the properties of $\eta _i$ ’s to justify the sizes and powers of our test in Corollary 9. Analogous to Theorem 1 and Corollary 6(b), Corollary 9 reveals the asymptotic behavior of the maximum of the $\eta _i$ ’s associated with the windows that do not contain jumps or contain only small jumps. Moreover, Corollary 9 roughly suggests that the jumps whose sizes are at an order larger than $n^{-\tilde {c}_2}$ are likely to be detected by the proposed test.

Corollary 9. Suppose that $\tilde {c}_2<(1-c_1-3c_4)/2$ and Assumptions 3, 4, and 8–10 hold. For any fixed $\bar {x}\in \mathbb {R}$ and $\epsilon>0,$ we have

(4.7) $$ \begin{align} \mathrm{P}\left(\underset{i\in\tilde{\mathcal S}\left(n^{-\tilde{c}_2-\epsilon}\right)}{\max} \left(\eta_i-\bar{a}_n\right)/\bar{b}_n\leq\bar{x}\right)\overset{n\rightarrow\infty}{\longrightarrow} \exp\left(-e^{-\bar{x}}\right), \end{align} $$

and under the alternative hypothesis $\tilde {H}_1,$

(4.8) $$ \begin{align} \mathrm{P}\left(\bar{N}_{T-,n^{-\tilde{c}_2+\bar{\epsilon}_2}}>0, \ \text{ and } \ \underset{i\in\{1,\ldots , \bar{N}_{T-,n^{-\tilde{c}_2+\bar{\epsilon}_2}}\}}{\cap}\bar{\mathcal{A}}_i \right)\overset{n, T\rightarrow\infty}{\longrightarrow}1 \end{align} $$

for any fixed $\bar {\epsilon }_2\in (0, \tilde {c}_2),$ where $\bar {\mathcal A}_i:=\{\bar {h}_{i,n^{-\tilde {c}_2+\bar {\epsilon }_2}}\in \mathcal S, \ \text { and } \ \eta _{\bar {h}_{i,n^{-\tilde {c}_2+\bar {\epsilon }_2}}}>\mathcal T_n\}.$

We may not detect all jumps on $(0, T)$ as $n, T\rightarrow \infty ,$ since there may exist jumps whose absolute sizes are too small to substantially enlarge the corresponding $\eta _i$ ’s. Accordingly, (4.7) concerns jumps whose sizes are at an order smaller than $n^{-\tilde {c}_2}.$ Given the proof of (4.7), it is trivial to verify that this convergence still holds when the set $\tilde {\mathcal S}\left (n^{-\tilde {c}_2-\epsilon }\right )$ is replaced with $\mathcal S_0.$ Then a comparison between (4.7) and the result in Theorem 1 suggests that under certain regularity conditions, letting $T\rightarrow \infty $ does not change the asymptotic behavior of the maximum $\eta _i$ for $i\in \mathcal S_0.$ On the other hand, (4.8) implies that as $n, T\rightarrow \infty ,$ the lower bound of detectable jump sizes in absolute value is allowed to approach zero roughly at the rate of $n^{-\tilde {c}_2}.$

Based on steps similar to those in the proof of Lemma S3 in the Supplementary Material, it can be checked that with probability approaching one, for each $i\in \mathcal S,$ there is no more than one jump within the interval $\left [t_{(i-1)K_1}, t_{((i+2)K_1)\wedge n}\right ]$ as $n, T\rightarrow \infty .$ Thus, when both $\eta _i$ and $\eta _{i+1}$ are larger than $\mathcal T_n,$ it is likely to be the same jump within $[t_{iK_1}, t_{(i+1)K_1}]$ that causes the largeness of these two statistics. Accordingly, we can still use $\hat {N}$ as the estimator for the number of detected jumps. Since $N_T$ may go to infinity when $T\rightarrow \infty ,$ we can use the robust version of the LAR $\bar {y}_i$ ’s, which are $\tilde {\phi }_i$ ’s given by

$$ \begin{align*}\tilde{\phi}_i=\text{ arg }\ \underset{\phi}{\min}\sum_{j=0}^{K_2-2K_1}\ell_{\beta_0}\left(y_{((i+1)K_1+j)\wedge n}-y_{((i+1)K_1+j-K_2)\vee 0}-\phi\right),\end{align*} $$

to estimate the sizes of detected jumps. It can be readily shown that the difference between $\tilde {\phi }_i$ and the jump size on $(t_{(i-1)K_1}, t_{(i+1)K_1}]$ is small in probability uniformly across $i\in \mathcal S_2,$ as illustrated in Corollary 10 below.

Corollary 10. Under the alternative hypothesis $\tilde {H}_1$ and Assumptions 3, 4, and 8–10, we have

$$ \begin{align*} \underset{i\in\mathcal S_2}{\max} \ \left|\tilde{\phi}_i-\sum_{\ell=1}^{N_{T-}}\Delta J_{\tau_\ell}\boldsymbol 1_{\left\{\tau_\ell\in (t_{(i-1)K_1}, t_{(i+1)K_1}]\right\}}\right|=O_{\mathrm{P}}\left(n^{-\tilde{c}_3+\epsilon}\right) \end{align*} $$

for any fixed $\epsilon>0.$

Corollary 10 is proved based on the union of two principles: First, in the absence of jumps, $\tilde {\phi }_i$ ’s would be approximately at the order of $n^{-\tilde {c}_3}$ uniformly across $i\in \mathcal S.$ Second, with high probability, the effect of a jump on $[t_{(i-1)K_1}, t_{(i+1)K_1}]$ is to increase or decrease (depending on the sign of this jump) $\tilde {\phi }_i$ by the absolute jump size.