A Test Statistic for Identifying Jump Variations in High-Frequency Time Series

1. Introduction

Historically, the volatility of financial time series has been treated in two ways. The first approach is parametric and postulates a latent variable via conditional heteroscedastic models to describe the volatility, such as the ARCH (Autoregressive Conditional Heterocedasticity) family models proposed by [10] and the stochastic volatility models, proposed by [18]. There is a huge literature on this approach and for details, see, for example, [17,21].

The main motivation for using the nonparametric approach is because parametric models often fail to adequately capture the movements of intraday volatility, see [1]. A nonparametric approach consists of constructing the daily realized volatility (RV) of a financial series using intraday returns, sampled at intervals $\Delta t$, of the order of 5 or 15 minutes, for example. RV is obtained from sums of squares of intraday returns, see [1] and [6]. Another method is to construct the realized bi-power variance (RVBP), from the sums of cross products of absolute adjacent, properly scaled returns, see [7]. For details, see [5]. [7,12] and [2] derived test statistics for detecting the existence of jumps. Other authors established tests for the necessity of adding a Brownian force, see [3,14,16,20] and [19]. [4] studied whether the jump component is of finite activity when the Brownian force is present.

In recent years, efforts have been devoted to study the distribution function of test statistics in the presence of jumps. [15] studied whether it is necessary to add an infinite variation jump term in addition to a continuous local martingale, using a Kolmogorov-Smirnov (KS) type test statistic. [8] derived the exact and asymptotic distribution of Cramér-von mises statistic when the empirical distribution function is a uniform distribution function. [20] suggested using other measures of discrepancy between distributions, as the Cramér-von mises test for the presence of diffusion component in a process $X_t$ which usually represents the price (or log-prices) of some financial asset. These studies motivated us to propose a test statistic using a Cramér-Von Mises type statistic.

It would be important to obtain the distribution of our proposed test statistic in the presence of jumps, however this is not an easy task, so we will employ numerical methods to approximate this distribution.

The paper is organized as follows. In Section 2 we establish the set up for our problem, define the hypotheses of interest and discuss the choice of the proposed test statistic. Section 3 discuss an approximation to the true distribution of the statistics, whereas Section 4 presents a simulation study to assess its performance. Section 5 applies our proposed test statistic to real data and Section 6 concludes with comments on the usefulness of the proposed methodology and on recommendations for future works.

2. Setup

The standard jump-diffusion model used for modeling many stochastic processes is given by the following differential equation:

$$d{X}_t={\alpha }_{t}dt+{\sigma }_{t}d{W}_t+d{Z}_t,$$

(1)

where ${\alpha }_t$ and ${\sigma }_t$ are processes with càdlàg paths, $W_t$ is a standard Brownian motion, and $Z_t$ is an Itô semimartingale process of pure-jump type. [20] generalized Eq. (1) to accommodate the alternative hypothesis that $X_t$ can be of pure-jump type. Itô semimartingale plays an important role in stochastic calculus and the following model plays a major role.

Suppose that $Y_t$ follows a non-parametric volatility model

$$\begin{array}{c}\hfill Y_t=X_t+{ϵ}_t,t\in [0,1].\end{array}$$

(2)

where $X_t$ is a continuous Itô semimartingale, that is,

$$X_t=X_0+{\int }_0^t{b}_{s}ds+{\int }_0^t{\sigma }_{s}d{W}_s+Z_t,$$

(3)

where ${\int }_0^t{b}_{s}ds$ is the drift term with $b_s$ being an optional and càdlàg process, ${\int }_0^t{\sigma }_{s}d{W}_s$ is a continuous local martingale with ${\sigma }_s$ being an adapted process, $W_s$ is a standard Brownian motion, and $Z_t$ is a skewed $\beta$-stable Lévy process.

[15] provided a theoretical test for the presence of infinite variation jumps in the simultaneous presence of a diffusion term and a jump component of finite variation and established the asymptotic theory of the empirical distribution of the “devolatilized" increments of Itô semimartingale with infinitely active or even infinite variation jumps. There are other methods to estimate the spot volatility, see, for instance, [20] and [11].

Recently, [9] discussed a volatility functional model and showed that jumps asymptotically impact the volatility estimate and presented a jump detection model based on wavelets.

We now consider the following hypotheses

$$\begin{array}{ccc}\hfill H_0:& {\Delta }_i^{n}X& \text{follows a standard normal model,}\hfill \\ \hfill H_1:& {\Delta }_i^{n}X& \text{follows a jump-diffusion model,}\hfill \end{array}$$

where ${\Delta }_i^{n}X=X_{t_i}-X_{t_{i-1}}$ denotes the ith one-step increment for $1\le i\le n$ and we assume that the available data set $\{X_{t_j};0\le j\le n\}$ are discretely equally spaced variables sampled from $X_t$, in the fixed interval $[0,T]$, i.e., $t_j=j{\Delta }_n$ with ${\Delta }_n=T/n$ for $0\le j\le n$. The empirical process is given by:

$$\begin{array}{c}\hfill {\widehat{Y}}_n\left(\tau \right)=\sqrt{⌊n/2k_n⌋m_n}(F_n\left(\tau \right)-\mathsf{\Phi }\left(\tau \right)),\tau \in \mathbb{R},\end{array}$$

(4)

for a finite sample size n, $k_n$ and $m_n$ are some integer depending on n, $k_n$ should be smaller than $\sqrt{n}$, $m_n$ should be smaller than $k_n$ and $\lfloor ·\rfloor$ denotes the integer part. Here, $\mathsf{\Phi }\left(\tau \right)$ denotes the distribution function (df) of a standard normal random variable and $F_n\left(\tau \right)$ is the empirical distribution function (edf) of the devolatilized increments. We used the local estimator proposed by [20] to estimate ${\sigma }_t^2$. On each of the blocks the local estimator of ${\sigma }_t^2$ is given by

$${\widehat{V}}_j^n={\displaystyle \frac{\pi }{2}}{\displaystyle \frac{n}{k_n-1}}\sum _{i=(j-1)k_n+2}^{j{k}_n}|{\Delta }_{i-1}^{n}X\left|\right|{\Delta }_i^{n}X|,j=1,\dots ,⌊n/2k_n⌋,$$

which is the bipower variation for measuring the quadratic variation of the diffusion component of $X_t$. [20] removed the high-frequency increments that contain big jumps. The total number of increments used in their statistic is thus given by

$$\begin{array}{c}\hfill N^n(\alpha ,\overline{w})=\sum _{j=1}^{⌊n/2k_n⌋}\sum _{i=(j-1)k_n+1}^{(j-1)k_n+m_n}I\left(|{\Delta }_i^{n}X|\le \alpha \sqrt{{\widehat{V}}_j^n}{n}^{\overline{w}}\right),\end{array}$$

where $\alpha >0$ and $\overline{w}\in (0,1/2)$. They use a time-varying threshold in the truncation to account for the time varying ${\sigma }_t$. The scaling of every high-frequency increment is done after adjusting ${\widehat{V}}_j^n$ to exclude the contribution of that increment in its formation:

$${\widehat{V}}_j^n\left(i\right)=\left\{\begin{array}{c}{\displaystyle \frac{k_n-1}{k_n-3}}{\widehat{V}}_j^n-{\displaystyle \frac{\pi }{2}}{\displaystyle \frac{n}{k_n-3}}|{\Delta }_i^{n}X\left|\right|{\Delta }_{i+1}^{n}X|,fori=(j-1)k_n+1,\hfill \\ {\displaystyle \frac{k_n-1}{k_n-3}}{\widehat{V}}_j^n-{\displaystyle \frac{\pi }{2}}{\displaystyle \frac{n}{k_n-3}}\left(\right|{\Delta }_{i-1}^{n}X\left|\right|{\Delta }_i^{n}X|+|{\Delta }_i^{n}X\left|\right|{\Delta }_{i+1}^{n}X\left|\right),\hfill \\ fori=(j-1)k_n+2,\dots ,j{k}_n-1\hfill \\ {\displaystyle \frac{k_n-1}{k_n-3}}{\widehat{V}}_j^n-{\displaystyle \frac{\pi }{2}}{\displaystyle \frac{n}{k_n-3}}|{\Delta }_{i-1}^{n}X\left|\right|{\Delta }_i^{n}X|,fori=j{k}_n.\hfill \end{array}\right.$$

(5)

Then, they define

$${\widehat{F}}_n\left(\tau \right)={\displaystyle \frac{1}{N^n(\alpha ,\overline{w})}}\sum _{j=1}^{⌊n/2k_n⌋}\sum _{i=(j-1)k_n+1}^{(j-1)k_n+m_n}I\left({\displaystyle \frac{\sqrt{n}{\Delta }_i^{n}X}{\sqrt{{\widehat{V}}_j^n\left(i\right)}}}\le \tau \right)I_{\left(|{\Delta }_i^{n}X|\le \alpha \sqrt{{\widehat{V}}_j^n}{n}^{\overline{w}}\right)},$$

(6)

which is simply the edf of the devolatilized increments that do not contain any big jumps. In the jump-diffusion case of Eq. (1), ${\widehat{F}}_n\left(\tau \right)$ should be approximately the df of a standard normal random variable. [20] use an alternative estimator of the volatility that is the truncated variation defined as

$$\begin{array}{c}\hfill \widehat{C_j^n}={\displaystyle \frac{n}{k_n}}\sum _{i=(j-1)k_n+1}^{j{k}_n}{\left|{\Delta }_i^{n}X\right|}^{2}I\left(|{\Delta }_i^{n}X|\le \alpha n^{\overline{w}}\right),j=1,\dots ,\lfloor n/2k_n\rfloor ,\end{array}$$

(7)

where $\alpha >0,\overline{w}\in (0,1/2)$ and, the corresponding one excluding the contribution of the ith increment for $i=(j-1)k_n+1,\dots ,j{k}_n$, is

$$\begin{array}{c}\hfill \widehat{C_j^n}\left(i\right)={\displaystyle \frac{k_n}{k_n-1}}{\widehat{C}}_j^n-{\displaystyle \frac{n}{k_n-1}}{\left|{\Delta }_i^{n}X\right|}^{2}I\left(|{\Delta }_i^{n}X|\le \alpha n^{\overline{w}}\right),j=1,\dots ,\lfloor n/2k_n\rfloor .\end{array}$$

(8)

The edf of the devolatilized (and truncated) increments is given by,

$$\begin{array}{c}\hfill \widehat{F_n^{\prime }}\left(\tau \right)={\displaystyle \frac{1}{N^{\prime n}(\alpha ,\overline{w})}}\sum _{j=1}^{[n/k_n]}\sum _{i=(j-1)k_n+1}^{(j-1)k_n+m_n}I\left({\displaystyle \frac{\sqrt{n}{\Delta }_i^{n}X}{\sqrt{{\widehat{C}}_j^n\left(i\right)}}}\le \tau \right)I_{\left(|{\Delta }_i^{n}X|\le \alpha n^{\overline{w}}\right)},\end{array}$$

(9)

where $\alpha >0,\overline{w}\in (0,1/2)$. Here, the total number of increments is defined as

$$\begin{array}{c}\hfill N^{\prime n}(\alpha ,\overline{w})=\sum _{j=1}^{[n/k_n]}\sum _{i=(j-1)k_n+1}^{(j-1)k_n+m_n}I\left(|{\Delta }_i^{n}X|\le \alpha n^{\overline{w}}\right).\end{array}$$

(10)

We define the test statistic as:

$$T_A^n=d^2\sqrt{⌊n/2k_n⌋m_n},$$

(11)

where A is a compact set in $\mathbb{R}$ and $d^2={\int }_{-\infty }^{\infty }{[F_n\left(x\right)-F\left(x\right)]}^{2}dF\left(x\right)$. Here, $F_n\left(x\right)$ is an edf and we assume that $F\left(x\right)$ is the df of a standard normal random variable and use the notation $\mathsf{\Phi }\left(x\right)$ for it. So, we have

$$\begin{array}{c}\hfill d^2={\int }_{-\infty }^{\infty }{[F_n\left(x\right)-\mathsf{\Phi }\left(x\right)]}^{2}d\mathsf{\Phi }\left(x\right).\end{array}$$

The critical region for the test is given by

$$C{R}_n=\left\{d^2\sqrt{⌊n/2k_n⌋m_n}>q_n(\alpha ,A)\right\},$$

(12)

where A is a compact set in $\mathbb{R}$ and $q_n(\alpha ,A)$ is the $(1-\alpha )$-quantile of the distribution of the statistic. We evaluate $q_n(\alpha ,A)$ via simulation. The test rejects $H_0$ if $T_A^n>q_n(\alpha ,A)$. The main purpose is to develop a test statistic that has better statistical properties than the existing KS test statistic for identifying jump variations in high-frequency time series. We will use the notation $TA1$ and $TA2$ for $T_A^n$ when using Eqs. (6) and (9), respectively. To determine the critical region of the test we will perform extensive simulations, using the R language, version $3.6.3$. All data and codes are available upon request to the authors.

2.1. Performance of Test Statistics

The idea is to observe which test statistic performs better using the two df mentioned in the previous section. A simulation was performed with $10,000$ replications and different n, $k_n$ and $m_n$ values, as shown in the following panel.

n	$(k_n,m_n)$
$1,000$	$(4,2)$, $(6,3)$, $(8,4)$, $(10,5)$, $(12,5)$, $(14,8)$, $(16,7)$, $(18,11)$,
	$(20,9)$, $(22,11)$, $(24,12)$, $(26,13)$, $(28,13)$, $(30,16)$
$2,000$	$(4,2)$, $(6,3)$, $(8,4)$, $(10,5)$, $(12,5)$, $(14,8)$, $(16,7)$, $(18,11)$
	$(20,9)$, $(22,11)$, $(24,12)$, $(26,13)$, $(28,13)$, $(30,16)$, $(32,15)$
	$(34,18)$, $(36,17)$, $(38,20)$, $(40,20)$, $(42,26)$, $(44,21)$
$5,000$	$(4,2)$, $(6,3)$, $(8,4)$, $(10,5)$, $(12,5)$, $(14,8)$, $(16,7)$, $(18,11)$
	$(20,9)$, $(22,11)$, $(24,12)$, $(26,13)$, $(28,13)$, $(30,16)$, $(32,15)$,
	$(34,18)$, $(36,17)$, $(38,20)$, $(40,20)$, $(42,26)$, $(44,21)$, $(46,25)$
	$(48,23)$, $(50,28)$, $(52,27)$, $(54,34)$, $(56,31)$, $(58,30)$, $(60,34)$
	$(62,33)$, $(64,30)$, $(66,40)$, $(68,35)$, $(70,33)$.

Figure 1 shows the behavior of $TA1$ and $TA2$ test statistics against $k_n$, for the jump-diffusion model. From the plots, we observe the existence of outliers. Also, the values of $TA1$ and $TA2$ test statistics increase as $k_n$ increases. In cases where $k_n=4$ and $k_n=6$, the values of the test statistics present some inconsistencies.

Figure 2 shows the behavior of $TA1$ and $TA2$ test statistics against $k_n$, for the standard normal model.

The first simulations were based on $10,000$ replicas with $n=1,000$ and the same pairs of $k_n$ and $m_n$ as mentioned above. The mean squared distance (MSD) is the difference between the values observed of test statistics from the two models used. It can be observed in Figure 3 that, for the jump-diffusion model, the MSD of test statistic increases as $k_n$ increases. On the other hand, for the normal model, on the average, the MSD of test statistics stabilizes, that is, the values of test statistics are close as $k_n$ increases.

One purpose is to estimate the best critical value of test statistics to choose which test performs better.

The ROC (Receiver Operating Characteristic) curve is widely used to determine the cutoff point. The best critical value is obtained using simulation. A ROC curve can be described as the shape of the trade-off between the sensitivity and the specificity of a test, for every possible decision rule cutoff (or threshold) between 0 and 1. The optimal cutoff point is the one closer to the $(1,1)$ point, and this gives the values of the specificity and sensitivity of the test. The area under the ROC curve (AUC) gives the accuracy of the test. The larger the area, the larger the accuracy of the test. In our case, we will call the sensitivity by detection power and 1-specificity the error rate.

2.2. The Best Critical Value

We test the critical value q between 1 and 3, and used the ROC curve to find the cutoff point. We compare the detection power and error rate for these values and a fixed critical value. Table 1 shows that, for $TA2$ test statistic with a fixed critical value $1.5$, the error rate for $k_n=30$ is around $2.8\%$ and the detection power is $84\%$. In contrast, for $TA1$ test statistic, the error rate is around $3\%$ and the detection power is $60\%$. It can also be seen that, for low $k_n$ values of $TA1$, there are low error rates but with low detection power. For example, for $q=1.5$, with $k_n=20$, there is an error rate around $1.4\%$ and a detection power of $34\%$ and, for $TA2$, we observe $1.01\%$ of error rate and $44\%$ of detection power. Moreover, for the optimal critical value $q=1.2322$, with $k_n=30$, we have an error rate of $5\%$ and a detection power of $90\%$ for $TA2$ and, for $TA1$, an error rate of $6\%$ and $70\%$ of detection power.

Figure 4 shows the detection power and error rate for a fixed critical value $1.5$.

Figure 5 shows the detection power and error rate of a simulated critical value $q=1.2322$, that were obtained through the respective ROC curves in Figure 6. We repeat our simulation for $n=2,000$ and the same pairs of $k_n$ and $m_n$ already mentioned above. The conclusion is similar to $n=1,000$.

The MSD of test statistics, are shown in Figure 7.

Figure 8 shows the detection power and error rate for a fixed critical value $1.5$.

Table 2 shows the detection powers and error rates for two critical values, a fixed and simulated, which lead us to conclude that $TA2$ test statistic is better than $TA1$ test statistic. For example, for $q=1.3663$ with $k_n=44$, we have an error rate of $5\%$ and a detection power of $91\%$ for $TA2$ test statistic whereas $TA1$ test statistic presents an error rate of $5.8\%$ and a low detection power of $66\%$.

Figure 9 shows the detection power and error rate for $q=1.3663$, obtained by simulation, with the ROC curves in Figure 10. It can be seen that as $k_n$ increases the error rate for $TA1$ is higher than $TA2$ and, for the detection power, as $k_n$ increases, $TA2$ has a higher detection power than $TA1$. Again, $TA2$ test statistic seems to be better than $TA1$ test statistic.

We can observe in Figure 11 that the MSD of the test statistics has a similar behavior to the previous analysis.

We are interested in the largest squared distance, so the pairs of $(k_n,m_n)$ for $n=5,000$, we will be used for our study for the aforementioned reasons.

Figure 12 shows the detection power and error rate for a fixed critical value $1.5$.

Figure 13 shows the detection power and error rate considering a simulated critical value, obtained by ROC curve, see Figure 14.

Table 3 shows that, for a fixed value $1.5$, the error rate for $TA2$ test statistic, with $k_n=70$ is around $3.9\%$ and the detection power is $90\%$. In contrast, for the test statistic $TA1$, the error rate is around $4.3\%$ and detection power of $56\%$. It can also be seen that, for low $k_n$ values for $TA1$ test statistic, there are low error rates but with low detection power. For example, taking $k_n=48$, for $TA1$ test statistic, there is an error rate around $1.2\%$ and $32\%$ of detection power. On the other hand, for the simulated critical value $q=1.3943$ obtained by ROC curve and $k_n=70$, we have an error rate of $4.9\%$ and a detection power of $91\%$ for $TA2$ test statistic and for $TA1$, we observe an error rate of $5.5\%$ and a detection power of $59\%$.

Remark: Note that; Table 1, Table 2 and Table 3 for $k_n=4$ and $k_n=6$ values, the error rate and detection power show inconsistent values, due to a high false alarm rate and not so much detection power. Therefore, we do not recommend using low values of $k_n$.

We will use $TA2$ test statistic, since it presented better results in relation to low error rate and high detection power. The ideal is to choose a $k_n$ that shows a low error rate and high detection power.

In contrast, for $TA1$ test statistic, there is high error rate and low detection power for most $k_n$ values. See also Table 4. Also, the area under the ROC curve of $TA2$ shows a larger area, and therefore the larger the area, the greater the accuracy of the test, which leads us to conclude that our $TA2$ test is better than $TA1$.

3. Aproximating the edf of TA2

As we remarked before, the purpose would be to derive the df of $TA2$. However, this is very hard theoretically and therefore we will address the problem using numerical methods. Our simulation shows that the edf of $TA2$ test statistic is approximately a gamma distribution, with values in the good agreement, see Figure 15.

Figure 16 shows that we have a very good approximation with the true density. For the gamma density with parameters a and b, given that $a=E\left(X\right)/b$ and $b=Var\left(X\right)/E\left(X\right)$, its df is $F\left(x\right)={\left[\mathsf{\Gamma }\left(a\right)\right]}^{-1}{\int }_0^x{(t/b)}^{a-1}{e}^{-t/b}dt$, which can be written as $F\left(x\right)=\gamma (a,x)/\mathsf{\Gamma }\left(a\right)$, where $\gamma (a,x)$ is the incomplete gamma function. For large sample sizes (see Table 5), we have $a\approx 2$ and $b\approx 0.25$, so we can write, approximately, $TA2{\sim }^{a}Gamma(2,0.25)$, with $E\left[TA2\right]=0.5$ and Var$\left[TA2\right]=0.125$.

In Table 6 we can find the quantiles of gamma and empirical distribution for different probabilities. Note that the theoretical quantiles compared with the empirical quantile are closer for probabilities higher than $0.90$.

4. A simulation Study

We conduct a simulation study to check the performance of the $TA2$ test statistic. Figure 17 shows data generated from the jump-diffusion and standard normal models that we used in our study.

The models are defined as follows: the jump-diffusion model is given by,

$$X_t=X_0+{\int }_0^t\sqrt{c_s}d{W}_s+0.5Z_t,0\le t\le T,$$

$$c_t=c_0+{\int }_0^{t}0.03(1.0-c_s)ds+0.15{\int }_0^t\sqrt{c_s}d{W}_s^{\prime }$$

where $Z_t$ is a skewed $\beta -stable$ Lévy process. The volatility $c_t$ is a square root diffusion process which is widely used in financial applications. The parameter in $c_t$ is specified as in [13]. The standard normal model is $\widetilde{X_t}=X_{t-1}+a_t,a_t\sim N(0,1)$, where $X_0$ is the initial value that should be defined.

We considered $10,000$ replicas for sample sizes $n=500;1,000;2,000;5,000$ and $10,000$. For the truncation of the increments, as is typical in the literature, we set $\alpha =3.0$ and $\overline{w}=0.49$. Hence for the sampling frequencies mentioned in Section 2, we set the pairs of $(k_n,m_n)$ as: $(22,12)$, $(30,19)$, $(44,22)$, $(70,34)$ and $(100,54)$ with $k_n/m_n$ ranging from $1.5$ to $2.15$ and having an increasing trend.

In Table 7 we can observe that the $TA2$ test power increases as the sample size increases and also, for n fixed, $TA2$ test power is bigger for largest $k_n$ values. The $TA2$ test power was calculated with the gamma quantile values, and we note that these values are very close to the empirical quantiles. To evaluate the performance of our test, we compare with the KS test, which also measures discrepancy between distributions.

In Table 8 we note that the $TA2$ test statistic shows better results. Moreover, the KS test power decreases and the error rate increases as n increases. For $n=500$, KS test power is bigger than $TA2$ test power, however $TA2$ test statistic has an error rate of $5\%$ and KS test has an error rate bigger than $76\%$. For larger sample sizes, the $TA2$ test statistic shows better test power with low error rate. Besides, KS test power decreases and the error rate increases as n increases. For $n=500$, KS test power is bigger than $TA2$ test power, however $TA2$ test statistic has an error rate of $5\%$ and KS test has an error rate bigger than $76\%$.

For larger sample sizes, the $TA2$ test statistic shows better power with low error rate.

5. Real Data Analysis

We collect intraday transaction prices of the Apple, Google and Goldman Sachs (GS) stocks, respectively, from November 11th to November 12th in 2014, with a sampling frequency of 15 seconds. The transaction records that are outside the ranges of quotes, 9:30 a.m to 4:00 p.m, are excluded. There are, in total $3,122$; $2,989$ and $2,819$ stock of prices, respectively. We aim to plot the observed test statistics against different values of $k_n$ and $m_n$. Figure 18 shows the price series for market data of Apple, Google and GS stocks.

Figure 19 shows the values of the $TA2$ test statistic, for different $k_n$ values.

Additionally, we illustrate the quantile values at the significance level of $5\%$ and $10\%$. The plots (a), (c) and (e) of Figure 19 consider the values of $k_n$ between $[4,55]$, $[4,54]$ and $[4,53]$ for the three stocks considered, respectively. The plots (b), (d) and (f) of Figure 19 show $k_n$ values starting from 44, that is, we show an enlarged version of the plots (a), (c) and (e) of Figure 19. It can be seen in Table 9 that, at a level of significance of $5\%$, Apple stock rejects null hypothesis, whereas Google and GS stocks do not.

On the other hand, when we increase the level of significance to $10\%$, the percentage increases for all stocks, with Apple still rejecting $H_0$, while Google and GS do not. Here we have used all $k_n$ values. For sufficiently large values of $k_n$, our test statistic tends to detect more precisely the dynamics of the Apple and GS series as can be seen in Table 10.

6. Conclusions

In this paper we discussed the approximate distribution of the proposed $TA2$ test statistic when the empirical distribution function is approximated by a gamma distribution function. This result is important, because it enables us to determine the critical region for the null hypothesis. Also, the real data analysis shows that our proposed test statistic is useful to identify that increments in time series like Apple and GS stocks, follow a jump-diffusion model for high-frequency data of prices for the largest $k_n$ values. Market activity on a trading day records containing a lot of price information between transactions is very important because it allows us to know the behavior of the micro-structure of market prices.

As a future work, we can explore the behavior of our proposed test statistic considering other financial time series, for different time records between transactions, i.e., with a sampling rates of 1, 2 or 5 seconds. In finance and econometrics it is still an open problem, being able to anticipate and/or measure a threshold, i.e., the size of the big losses that will occur on some day of trading on the stock exchange, or of big gains, given the need to mitigate the risk of any portfolio in question. Finally, another challenge for future research is to find the exact distribution of the $TA2$ test statistic.