The BNS nonparametric jump test

Barndorff-Nielsen and Shephard [3], [4] proposed a nonparametric test statistic (henceforth the BNS test statistic) which utilizes realized variance and bi-power variation to test jump components of price processes which have continuous sample paths. To begin with, we briefly introduce some important theoretical results of the jump test procedure in financial econometrics. We say a random variable belongs to the Brownian semimartingale plus jump class if(2)where and are assumed to be càdlàg, is a standard Brownian Motion, is the quantity of the th jump within , and is total number of the jumps occurring within . Here, we assume the number of jumps occurring within the interval , is finite for all . If (2) is without the jump term , we say belongs to the Brownian semimartingale without jump class. Some statistical assumptions can be made on and for purpose of simplifying the analysis. For example, in empirical finance literatures, magnitude of jump is often assumed to follow a normal distribution, and the number of jumps within the interval is often assumed to follow a counting process with (finite) intensity parameter which may be time varying.

The realized variance and the realized bi-power variation in period are defined asrespectively, where

is the intra-period log return in the th sub-interval of period and is the asset price at time point . Assume that for belongs to the Brownian semimartingale plus jump class. Then it can be shown that under some regularity conditions,

(3)(4)

as . The termin (3) is called the quadratic variation for the cumulative (log) return process and it is a sum of contributions due to the continuous log price process and the jump process The result of (3) follows from the theory of quadratic variation (e.g., [11]) and the result of (4) follows from the theory of power variation process which is a generalized version of the theory of the quadratic variation process [3]. Here can hold without any further assumptions on the jump process, the joint distribution of the jump process and Finally, if belongs to the Brownian semimartingale without jump class, it is easy to see that both and will converge in probability to as

Barndorff-Nielsen and Shephard [3], [4] showed thatcan consistently estimate the quantity In practice, to guarantee nonnegativity of the estimation, some truncation rules can be applied on for example using or a shrinkage type estimator like (15) in our empirical analysis. To construct a test statistic to test whether the jump term presents, now suppose for belongs to the Brownian semimartingale without jump class, and the following conditions hold,

1. The process of is pathwise bounded away from 0.
2. The joint process of and is independent of the Brownian motion term of the log price process,

then conditioning on the quadratic variation and realized bi-power variation process, Barndorff-Nielsen and Shephard [3], [4] showed that joint distribution of and will converge asymptotically to a bivariate normal distribution. Then under the null hypothesis when no jumps are present on period , it can be shown that(5)where The term in the denominator of (5) is called the integrated quarticity, and to consistently estimate it, we can use the realized tri-power quarticity,

where and

In the following simulation study and empirical applications, instead of using the test statistic shown in (5), we will use three improved test statistics to obtain better performances. The first one is proposed by [3], [4], which uses the transformation and is defined aswhere

The second one is the Box-Cox transformed test statistic with parameter , which is defined asHere the Box-Cox transformation for a positive number is defined as

The third one is the ratio type test statistic [12]:

Under the null hypothesis that there is no jump occurring in period , the test statistics and will have a standard normal distribution as their limiting joint distribution. When jumps occur in period , the test statistics will approach to infinity as For more discussions on theoretical properties of the test statistics under the alternative (when jump presents), please see [13].

The FDR and BH procedure

Let and denote the th null hypothesis and the corresponding -value, . Among the hypotheses, suppose there are true hypotheses and false hypotheses. Note that and are generally unknown to researchers, so they are assumed to be random variables. On contrary, the total number of hypotheses is generally known in advance and so is assumed to be nonrandom. Table 1 shows different situations when a multiple testing is performed. The numbers of hypotheses we reject and do not reject are denoted by and . The notations and denote the numbers of hypotheses we correctly accept, falsely accept, falsely reject and correctly reject, respectively. The false discovery rate (FDR) is then defined as the expectation of the false discovery proportion (FDP), i.e.

where

In testing jumps, controlling the FDR has several advantages over controlling other compound error rates. First, if the price process really does not have a jump component, i.e., all the null hypotheses are true, then controlling the FDR will be equivalent to controlling , the familywise error rate (FWER). Second, if the intensity of the jump process , as time goes on ( increases), the proportion of false hypotheses among all hypotheses will be a nonzero constant with a high probability. Although such proportion may not be large, one may still expect the more (fewer) rejections one has, the more (fewer) erroneous rejections are allowed to occur; or the number of rejections should be proportional to . In this situation, controlling compound error rates associated with proportion of erroneous rejections, like the FDR, makes sense. In addition, rejection criterion of some compound error rates such as the FWER, are sometimes too stringent to get rejections when the number of hypotheses becomes large. The criterion of the FDR is less conservative in this aspect. Finally, controlling the FDR currently seems to be more acceptable than controlling other compound error rates in many different research fields [14].

Let be the ordered 's and be the corresponding null hypotheses. Benjamini and Hochberg [9] proposed a stepwise procedure to control the FDR at the required level . The BH procedure can be simplified as the following two-step decision rule:

1. Obtain .
2. Reject for all

Some controlling procedures for the FDR need a resampling scheme to construct the rejection region, which relies on intensive computations. The BH procedure, however, requires far less computational sources than those computational intensive methods. As shown above, the only computational burden of the BH procedure is to rank the -values. Such advantage becomes even more obvious when the number of hypotheses becomes very large.

It can be shown that there is a relationship between the type I error and the FDR. That is, if we reject as it is possible to know what level of the FDR is controlled for the hypotheses multiple testing. For example, if the hypotheses are identical and the test statistics are all independent, given the type I error , the following estimator [25]can be used to estimate the corresponding FDR. Here is a turning parameter.

How the BH procedure performs relies on dependence structure of the test statistics. Benjamini and Yekutieli [16] showed that the BH procedure can still control the FDR when the test statistics are not independent, but the positive regression dependency (PRDS) for each test statistic under the true null hypotheses is satisfied. In addition, simulation studies in [14] showed that even if the PRDS condition is violated (e.g., there exist negative common correlations between the test statistics or the covariance matrix has an arbitrary structure), the BH procedure can still provide a satisfactory control of the FDR. Finally, if the test statistics have an arbitrary dependence structure, Benjamini and Yekutieli [16] showed that the BH procedure still guarantees thatA more detailed discussion on the theoretical properties of the FDR and the BH procedure is provided in next section.

Asymptotically results

Let be a vector of samples defined on a probability space Let be the smallest sub- field of such that for is measurable. Let A test statistic for testing marginal hypothesis with sample size is a function , and is measurable. Let denote the vector of the test statistics for testing hypotheses. Given each , suppose that there exists a vector of random variables such that for is measurable. Also assume for each as Let be the limiting distribution function of the test statistic under the null hypothesis . For one-sided test, -value of the th one-sided hypothesis is defined by for the th two-sided hypothesis). Let A feasible estimated -value for hypothesis is then given by

In our case of testing jumps, since our null hypotheses are homogeneous, is the c.d.f. of for all

Let and . Let be the exact distribution of under the null hypothesis . The -value under such distribution for hypothesis is For as if If are continuous random variables, thenfor If are also continuous random variables, then for

Before we proceed to the main results, we need to introduce some definitions. Let denote the Borel set. Let define the fold products of the real line with the Borel sets . Letdenote the symmetric set of permutations of integers . Let be a probability measure on where

Discussions on the asymptotic results

The two theorems say that under some regularity conditions, we can asymptotically control FDR. A key condition making the two theorems different is the requirement on the dependence structure of elements in vector and . If the dependence structure of satisfies PRDS on , it ensures that Here we only require the PRDS should hold on , and the dependence structure of on can be arbitrary. Marginal distributions of and converging with the rate simultaneously for all is also needed for the consistent control. In addition, we also require the convergence of the joint distribution of the ordered -values. But as stated in Theorem 2, such condition can be ignored if other conditions hold.

The approximation error essentially vanishes to zero when . Magnitude of , as shown in our proof, is bounded by a non-decreasing function of This property indicates that the more false null, the better the convergence.

We then have a look of condition 1 in Theorem 1. This is a sufficient condition to ensure that exists as It is due to Kolmogorov's extension theorem [19, p.50]: An extension of any consistent family of probability measures on to a probability measures on necessarily exists and is unique. Conversely, if we have a probability measure on we can induce a family of finite-dimensional distributions on and these induced finite-dimensional distributions all satisfy consistency for multivariate distribution.

Condition 2 in Theorem 1 requires that the joint distribution of the - values should satisfy PRDS on the subset It is a sufficient condition for when we implement the BH procedure with Since our purpose is to control FDR with if we can guarantee that only the distribution of satisfying the condition is needed. For practically using the BH procedure, Benjamini and Yekutieli [16] listed many situations when the condition holds. For example, if where and is a covariance matrix with element . Suppose for each and each then the distribution of is PRDS on , regardless what the covariance structure of is. Mutual independence of can be easily seen as a special case of PRDS on As for the nonparametric jump test in this paper, since the limiting distribution of the test statistics is a multivariate normal with for each and each it implies PRDS on

The condition that for is called the distribution of is stochastically dominated by the Uniform If it is called that the distribution of is stochastically dominated by the Uniform distribution asymptotically. In order to control FDR with the BH method asymptotically, we at least need that for and The condition is more liberal than that has the exact Uniform distribution for and applies to the case when the test statistics are discrete random variables.

As shown in the proof of Theorem 1,(10)(11)(12)

In the first equality, is the probability that in addition to rejecting the hypothesis we also reject other hypotheses. Sarkar [23] showed that if then

Therefore if (10) becomes(13)

Equation (13) is the difference between two familywise error rates (FWER, the probability that we at least have one false rejection) which are obtained respectively from using and under the BH procedure. The result is not surprising since when all null hypotheses are true, FDRFWER.

To make (11) vanish as (9) in condition 5 of Theorem 1 is one of the sufficient conditions. However, as shown in Theorem 2, such condition is redundant when test statistics are independent and continuous.

We finally have a look of the assumption:(14)

The assumption says that the convergence in law should hold simultaneously at the points for and for all Such convergence is reasonable for test statistics with limiting normal distribution if we set Note that if and are continuous,

If and are asymptotically normal, and satisfy for an integer then by theory of Edgeworth expansion of the distributions of and [20, pg.76],

So can converge to with the rate of

In our nonparametric jump test, standard normal is used to approximate under the null. There are several methods to improve the approximation, for example, the bootstrap approximation and the Box-Cox transformation. Some theoretical results about how the methods perform have been established. Goncalves and Meddahi [21] showed that when no jump presents, distribution of the test statistic for standardised realized volatility can be approximated by with the rate of convergence They also documented that under some situations, the bootstrap approximation is better than the standard normal approximation, and the error rate can be reduced to For the Box-Cox transformation, if there is no jump component, the skewness of the test statistic for realized volatility can be efficiently reduced by optimally choosing the parameter for the Box-Cox transformation [22].

When and both go to infinity

In practice, the number of samples within a hypothesis, may be less than the number of hypotheses How such a large small (or in statisticians' view: Large (number of dimensions), small (number of samples)) situation affects statistical inferences has been intensively studied recently, especially in simultaneously convergence of the test statistics. For example, when the samples are i.i.d., sufficient conditions for uniformly for all already was provided by [23]. Clarke and Hall [24] documented that the difficulties caused by dependence of test statistics can be alleviated when grows, but the result subjects to that distributions of test statistics should have light tails such as normal or Student's t. Fan et al. [25] proved that if normal or Student's t distribution is used to approximate the exact null distribution, the rejection area is accurate when but if the bootstrap methods are applied, then is sufficient to guarantee the asymptotic-level accuracy.

In practice, high frequency returns might not be i.i.d. distributed. Instead of assuming that samples have certain distributional properties, here we assume that (14) needs to hold. However, by jointly restricting growth rates of and and together with some mild conditions, (14) can also be achieved. It can be seen in the following proposition.

Simulation study

For the simulation study, we consider the following stochastic volatility plus jump model (SVJ):where and follow the standard Brownian motion and follows the CIR process. follows a Compound Poisson Process (CPP) with a constant intensity , and is the number of jumps occurring within the small interval We set correlation between and equal to zero (no leverage effect). We use the following parameter values for the simulation:

In the simulation, the unit of a period is one day. We vary the (daily) jump intensity at five different levels: and 0.2. Note that the intensity parameter here is the expected number of jumps occurring per day. Different values of tend to have different numbers of jump days over the whole sampling period, therefore result in different numbers of false null hypotheses. This allows us to see how such differences affect outcomes of the simulation.

We mimic the U.S. stock market and generate one minute intradaily log prices over hours each day. Thus in our simulation, and After obtaining a sample path, the jump test statistics and and their corresponding -values are calculated. We test hypothesis (1) with the test statistics and control the FDR at the level with the BH procedure.

Simulation results

We first focus on the case when the FDR control level and the number of null hypotheses Figures 1, 2, 3, 4 and 5 show the plots of average values of relevant quantities from 1000 simulation runs. Figure 1 is for performances of the three different test statistics when the FDR is controlled with the BH procedure. In the top left panel, we show the realized FDR. The solid horizontal line is at the level It can be seen that the realized FDR of is almost around or under the required level, while has the largest realized FDR for all different values of Overall, as increases, no matter which test statistic we use, the desired FDR level can be achieved.

Let denote the realized number of correct rejections. We use to measure the ability of the test statistics to correctly reject the false hypotheses. As shown in the top right panel of Figure 1, the three test statistics have small differences in It also can been seen that increases only slightly as increases.

In the bottom left panel of Figure 1, we can see that the significance level obtained from the BH procedure increases as increases. As goes up, the number of false hypotheses tends to increase, and we have less possibility that the test statistic will signal a true null as a false one. Consequently, we do not need a more stringent to prevent the false rejections, and more rejections can be obtained.

The average number of rejections made by the BH procedure is constantly less than the average value of as shown in the bottom right panel of Figure 1. It might be due to that is too restricted to obtain more rejections. A remedy is that we can use a more liberal level ( or ), but tolerate more false rejections. One thing worth to note here is that the average values of would in general be less than their corresponding since there may be more than one jump on a day, and this becomes even more obvious when becomes large.

We then compare performances of the BH procedure with the conventional procedure of controlling type I error in each hypothesis: is rejected if its realized -value is no greater than Here we specify are two frequently used levels: and Relevant results are shown in Figure 2. As can be seen in the first row, when different test statistics are used, the conventional procedure results in a high realized FDR, especially when the jump intensity is small (the number of the false null hypotheses tends to be relatively low in the situation). An extremely case is that when there is no jump rejecting when (or ) results in false rejections. It says that the probability we at least make one false rejection (the familywise error rate, FWER) is one as we follow the conventional procedure. The reason is that when all the null are true and the test statistics for each hypothesis are almost serially independent, if we reject when on average we would reject hypotheses, and all of these rejections are wrong. However, the BH procedure performs far better in this situation. Even in the worst case, on average it only takes about probability to make such an error.

Since the specified are on average greater than it is expected that more rejections can be obtained under the conventional procedure than the BH procedure. This can be seen in the second row of Figure 2. of the conventional procedure tends to be higher than that of the BH procedure, but as goes up, their gap becomes small.

Figure 3 shows performances of the method when lower frequency (5-min, 10-min and 15-min) data is used. still has the best ability to satisfy the required FDR levels, but it suffers the greatest loss of when the data frequency goes lower. does not perform better than the case when 1-min data is used, no matter in satisfying the required FDR level or For its performance still is in the middle, but overall its performance is more stable than the other two competitors.

We then have a look at how the method performs when the number of hypotheses changes. We vary at several different levels, ranging from 50 to 2000 and keep The results are shown in Figure 4. It can be seen that when and is large (no less than 100), the realized FDR and are stable over different

How does the method perform when FDR is controlled at different required levels? Figure 5 shows different required levels and the realized FDR. The thick line is a 45-degree line, and the vertical dotted line is for Ideally the realized FDR needs to be equal or below the 45-degree line. For and the method performs well, especially when goes large. However, when there is a significant difference between the three test statistics, and the required FDR level becomes difficult to achieve in this situation.

The above results suggest that performances of the hybrid method are positively related to sampling frequency and the intensity parameter Although the BH procedure results in quite stringent rejection criteria, it still can keep at a satisfying level. Fixing rejection region at and indeed can have better but it can suffer far higher false rejections when the number of true null is large. In sum, the simulation shows that combining the BNS test with the BH procedure, the FDR can be well controlled and the test statistics also can keep substantial ability to correctly identify jump components. Finally, we also conduct a simulation study with the stochastic volatility plus jump model (SV1FJ) used in [12]. The results can be found in the supplementary materials (Figures S1, S2, S3, S4 and S5 in the supplementary materials) and they are qualitatively similar to those of the SVJ case shown here.

Real data applications

In the following we present some empirical results with real data. The raw data used for the empirical applications are one minute recorded prices of S&P500 (SPC500) index in cash and Dow Jones Industrial Average (DJIA) index. The sample period spans from Jan-02-2003 to Dec-31-2007. In order to reduce estimation errors caused by microeconomic structure noises, we use five minute log returns to estimate , and and the jump test statistics. Figure S6 and S7 in the supplementary materials show volatility signature plots for detecting microstructure noise and time series plots of the price variations. A detail description of the data and discussion on the microstructure issue can be found in the supplementary materials.

Table 2 shows summary statistics of the price variations, different types of , their corresponding and mutual correlations of these quantities of the two indices. Results of the Ljung-Box test (denoted by LB.10) indicate that the price variations are highly serially correlated. However, for and , the Ljung-Box test instead indicates that they exhibit almost no serial correlation, which suggests that the BH procedure may efficiently control the FDR in this case.

The daily test statistics of the two indices have high mutual correlations. This property is quite different from the daily test statistics between individual stocks and the market index. As shown in [2], the jump test statistics of individual stocks and the market index almost have no mutual correlation, even though their returns are highly correlated. Such low correlation is due to a large amount of idiosyncratic noises in the individual stock returns, which causes a low signal-to-noise ratio in the nonparametric jump test statistics. The high mutual correlation between the jump test statistics of the two benchmark indices suggests that the idiosyncratic noises of returns is not significant and we may have more reliable results when we perform the jump test at the market level.

Common jump days

To measure daily price variation induced by jumps, we use sum of squared intradaily jumps, which can be estimated by the following estimator:(15)

where . Table 3 shows summary statistics of when FDR is controlled at level and . The mean and standard deviation of (15) shown here are conditional on . The conditional mean is around 0.14 to 0.22 for SPC500 and 0.13 to 0.16 for DJIA. For SPC500 and DJIA, the significant levels for the three statistics are all below 0.006 when the FDR control level . Depending on different test statistics, the proportion of identified jump days among all days, is around 1.5% to 11.6% for SP500 and around 2.4% to 8.6% for DJIA.

Common components in two highly correlated asset prices are often one of the most widely studied issues in empirical finance. Here we document some relevant empirical findings. Figure 6 shows the time series plots of the identified on the common jump days, and Table 4 shows their summary statistics. The term common jump days used here only means that the two indices both have jumps on these days. It does not necessarily mean that the two indices jump exactly at the same time within these days. Since the daily BNS test statistic is obtained by integrated quantities over one day, it cannot tell us how many and what exact time the jumps occur within that day. Nevertheless such test at least let us know what common days they have jumps, and this information is still valuable for further research.

It can be seen that the results from the two methods are very similar. When the FDR control level , proportion of the common jump days among all jump days is around 41% for SPC500. This proportion varies from 31% to 51% for DJIA when different test statistics are used. Comparing magnitudes of the variations in Table 4 with those in Table 3, the two indices tend to have larger jumps on the common days. The result seems to imply that a common shock such as announcements of macroeconomic news, may induce a larger jump than other idiosyncratic shocks such as announcements of news of individual stocks.

Jump intensity estimation

Jump intensity of an asset price process is a very crucial parameter for evaluating risks of the asset. As shown in [26] and [27], the jump intensity seems to change over time, which implies that clustering of jump variations is time varying. The time varying jump intensity also demonstrates very different dynamic behavior across different assets. In the previous literatures, the time varying jump intensity is estimated via moving average of the number of identified jump days, but the threshold for identifying these jump days is a fixed type I error. Here, rather than controlling the fixed type I error over the whole sampling period, we try to incorporate the FDR control into the rolling window estimation.

The simple moving average (rolling window) intensity estimator for the th day is defined as

where is a threshold, and is length of the rolling window. The estimator can serve as a local approximation for the true intensity of the jump process, if we assume that number of jumps occurring at most once per day. In the following analysis, we set , and is chosen based on two different ways: The first one is the FDR criterion using the whole hypotheses, and the second one is the FDR criterion using the hypotheses within that window with the required FDR level .

While the first method always has fixed, the later method leads to an adaptive FDR criterion which may change over time, since including a new may make a different FDR criterion. Time series plots for the estimations with the three different jump test statistics are illustrated in Figure 7. In the left panel are plots for the SPC500 and the right panel are plots for the DJIA. It can be seen that with , tends to be constantly lower than those with the other two test statistics. When is chosen adaptively over the whole sampling period, is more volatile; and it tends to be higher (lower) when more (less) jump days are identified. This phenomenon holds no matter which test statistic is used. On the other hand, with fixed, is less sensitive to inform such large price movements. Finally, one should note that adaptively choosing is only meaningful if the control procedure can lead to a different choice of as different information appended, which is possible for the BH procedure but can never be achieved via the conventional type I error control.