Sébastien Laurent
email: sebastien.laurent@univ-amu.fr.

Shuping Shi
email: shuping.shi@mq.edu.au.

Yacine Ait-Sahalia

Peter B C Phillips

Orimar Sauri

Volatility Estimation and Jump Detection for drift-diffusion Processes *

Keywords: Diffusion process, nonzero drift, finite sample theory, volatility estimation, jumps JEL classification: C12, C14

Logarithms of prices of financial assets are conventionally assumed to follow driftdiffusion processes. While the drift term is typically ignored in the infill asymptotic theory and applications, the presence of nonzero drifts is an undeniable fact. The finite sample theory and extensive simulations provided in this paper reveal that the drift component has a nonnegligible impact on the estimation accuracy of volatility and leads to a dramatic power loss of a class of jump identification procedures. We propose an alternative construction of volatility estimators and jump tests and observe significant improvement of both in the presence of nonnegligible drift. As an illustration, we apply the new volatility estimators and jump tests, along with their original versions, to 21 years of 5-minute log-returns of the NASDAQ stock price index.

Introduction

Consider the conventional setup of an Ito semimartingale process of log-prices y t such that

dy t = µ t dt + σ t dW t , (1)
where W t is an F t -adapted standard Brownian motion with {F t : t ∈ [0, T ]} being a rightcontinuous information filtration. With locally bounded coefficients µ t and σ t , the drift term is dominated by the diffusion process. For this reason, most infill asymptotics are unaffected by the presence of a drift. The drift component is therefore typically ignored in the high-frequency literature.

There is however substantial empirical evidence documenting that asset prices might have a nonzero drift component. In the low-frequency framework (such as monthly, weekly and daily), prolonged periods of mildly explosive trends were identified in many financial assets, for example, the stock market during the dot-com bubble period in the late 1990s (Phillips et al., 2011[START_REF] Phillips | Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500[END_REF][START_REF] Shi | Identifying speculative bubbles using an infinite hidden Markov model[END_REF] and the commodity markets over the last decade [START_REF] Etienne | Bubbles in food commodity markets: Four decades of evidence[END_REF][START_REF] Gutierrez | Speculative bubbles in agricultural commodity markets[END_REF]Phillips and Yu, 2011). Evidence of a non-negligible drift was also observed by [START_REF] Phillips | Detecting financial collapse and ballooning sovereign risk[END_REF] in log-prices of the S&P 500 index during the 2008 subprime mortgage crisis period and in bond yields and CDS spreads of most European countries during the 2010 debt crisis. Additionally, there is extensive literature documenting temporary deviations of log-prices from the random walk. 1 In the high-frequency framework, motivated by the large number of flash crashes, 2 [START_REF] Christensen | The drift burst hypothesis[END_REF] propose a drift bust hypothesis, which postulates the existence of short-lived locally explosive trends in log-prices. This hypothesis was tested using a nonparametric approach and with tick data. The authors find that drift bursts (especially negative drifts) form an integral part of price dynamics in equities, fixed income, currencies and commodities.

Our paper investigates the finite sample impact of nonzero drifts on the estimation ac-1 See, for example, [START_REF] Bekaert | Characterizing predictable components in excess returns on equity and foreign exchange markets[END_REF]; [START_REF] Bessembinder | Time-varying risk premia and forecastable returns in futures markets[END_REF]; [START_REF] Campbell | What moves the stock and bond markets? A variance decomposition for long-term asset returns[END_REF]; [START_REF] Campbell | Predictable stock returns in the United States and Japan: A study of long-term capital market integration[END_REF]; [START_REF] Lo | Stock market prices do not follow random walks: Evidence from a simple specification test[END_REF]; [START_REF] Fama | Permanent and temporary components of stock prices[END_REF]; [START_REF] Balvers | Mean reversion across national stock markets and parametric contrarian investment strategies[END_REF]; [START_REF] Chaudhuri | Random walk versus breaking trend in stock prices: Evidence from emerging markets[END_REF]. 2 See, for example, Nanex Research http://www.nanex.net/NxResearch/.

curacy of volatility in the high-frequency setting, namely the realized volatility [START_REF] Barndorff-Nielsen | Econometric analysis of realized volatility and its use in estimating stochastic volatility models[END_REF], the bipower variation [START_REF] Barndorff-Nielsen | Power and bipower variation with stochastic volatility and jumps (with discussion)[END_REF], and the noise-robust volatility estimator [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF]. We consider a linear drift-diffusion process, which captures an important form of nonzero drifts and has been studied extensively in the literature. See, for example, [START_REF] Lo | Implementing option pricing models when asset returns are predictable[END_REF]; [START_REF] Barndorff-Nielsen | Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics[END_REF]; [START_REF] Nicolato | Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type[END_REF]; [START_REF] Aalen | Survival models based on the Ornstein-Uhlenbeck process[END_REF]; [START_REF] Zhou | Asymptotic theory for linear diffusions under alternative sampling schemes[END_REF]; [START_REF] Wang | Double asymptotics for explosive continuous time models[END_REF]. The discrete-time model of this process allows for both the random walk and local-to-unity dynamics3 [START_REF] Phillips | Time series regression with a unit root[END_REF].

The theoretical literature on realized volatility merely focuses on the asymptotic properties and apart from some notable exceptions [START_REF] Meddahi | A theoretical comparison between integrated and realized volatility[END_REF][START_REF] Bandi | On the finite sample properties of kernel-based integrated variance estimators[END_REF] do not study their finite sample properties. One important contribution made by this paper is deriving the finite sample bias of the realized volatility under the linear drift diffusion process. The finite sample theory, together with extensive simulations, reveal that the nonzero drift causes a substantial bias in the volatility estimation. We propose a correction that leads to a significant improvement in the estimation accuracy.

Furthermore, we study the impact of nonzero drifts on the performance of jump tests.

Testing for jumps and precisely identifying their occurrences is of overwhelming importance in finance since jumps have implications in risk management, portfolio allocation and derivatives pricing [START_REF] Aït-Sahalia | Disentangling diffusion from jumps[END_REF]). Several tests have been proposed in the literature (see [START_REF] Mancini | Jumps[END_REF] for a survey). The most popular test is probably the test for finite activity jumps4 proposed independently by [START_REF] Andersen | No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects, jumps and iid noise: Theory and testable distributional implications[END_REF] and Lee and Mykland (2008, LM08 hereafter) and extended by Lee and Mykland (2012, LM12 hereafter) to account for the presence of microstructure noise. These tests (especially the LM08 test) have been shown to have the overall best performance by [START_REF] Dumitru | Identifying jumps in financial assests: a comparison between nonparametric jump tests[END_REF] in a comprehensive Monte-Carlo simulation comparing nine jump detection procedures available in the literature.

We demonstrate that the finite sample performance of the LM08 and LM12 tests is unsatisfactory if the drift is nonzero. Indeed, under this circumstance, the tests are severely undersized, which translates into a dramatic loss of power. To address this finite sample problem, we propose an alternative construction of the test statistic. Despite its ease of implementation, our test improves the finite sample performance significantly.

As an illustration, we apply the new bipower volatility estimator and the new LM08 jump test, along with their original versions, to 5-minute log-returns of the NASDAQ stock index from 1996 to 2016. The main conclusion is that the bipower variation tends to overestimate the volatility in the presence of nonzero drift by on average 2.5% but sometimes up to more than 40%. The proposed new jump test allows the identification of more jumps. These additional jumps occur during periods with upward or downward trends in log-prices.

The Drift-Diffusion Process

Let {0 < t 1 < • • • < t T < N } be a set of T equally spaced observation times spanning N days.
The distance between two consecutive observation times is denoted by ∆ = t i -t i-1 = N/T .

Assumption 2.1 Assume that the drift and diffusion coefficients in (1) do not change dramatically over a short time interval such that

(1) sup

i sup t i ≤s≤t i+1 |µ s -µ t i | = O p (∆ 1/2-) (2) sup i sup t i ≤s≤t i+1 |σ s -σ t i | = O p (∆ 1/2-)
for any ≥ 0.

These conditions are satisfied by most Ito processes and allow both the drift and stochastic volatility to depend on the log-price process itself. For the convenience of exposition, we focus on a linear drift-diffusion process with µ t and σ t specified as follows.

Assumption 2.2 The drift coefficient in (1) takes the form of µ t = θ(y t -m) with constants θ and µ.

Note that the drift coefficient µ t is a linear function of the log-price y t and satisfies Assumption 2.1-(1) because for any

t i ≤ s ≤ t i+1 , µ s -µ t i = θ 2 s t i y t dt -θ 2 m(s -t i ) + θ s t i σ t dW t = θ s t i σ t dW t {1 + o p (1)} = O p (∆ 1/2 ).

Assumption 2.3

The diffusion coefficient is given by

σ 2 t = p i=0 d i Π i,St ,
where p is an integer number, the d i 's are real numbers and the Π i,St 's are the eigenfunctions of the infinitesimal generator5 associated to the state variable S t whose dynamic is

dS t = µ (S t ) dt + σ(S t )dW t .
Assumption 2.3 includes the most popular volatility models such as the log-normal model, the squared root model and the GARCH(1,1) model of [START_REF] Nelson | ARCH models as diffusion approximations[END_REF] as special cases. See [START_REF] Meddahi | A theoretical comparison between integrated and realized volatility[END_REF] for more details. Note that Assumption 2.3 implies that E σ 2 t = d 0 .

The volatility process satisfies Assumption 2.1-(2) that σ s -

σ t i = O p (∆ 1/2 ) for any t i ≤ s ≤ t i+1 .
For example, the GARCH(1,1) volatility model is defined as

σ 2 t = S t and dS t = κ (ω -S t ) dt + √ 2λκS t dW t , (2)
where κ > 0, ω > 0, and 0 < λ < 1. The changes of the diffusion coefficient σ s -

σ t i = O p (∆ 1/2 ) because σ s + σ t i = O p (1); hence, σ 2 s -σ 2 t i = κω(s -t i ) -κ s t i σ 2 t dt + √ 2λκ s t i σ 2 t dW t = √ 2λκ s t i σ 2 t dW t {1 + o p (1)} = O p (∆ 1/2 ),
and σ s -

σ t i = (σ 2 s -σ 2 t i )/(σ s + σ t i ).
The linear drift-diffusion process is

dy t = θ (y t -m) dt + σ t dW t , (3)
with an exact discrete solution (Arnold, 1974, Corollary 8.2.4)

y t i+1 = g (θ) + α (θ) y t i + V t i+1 , (4)
where

g (θ) = µ [1 -exp (θ∆)], α (θ) = exp (θ∆), and V t i+1 = t i+1 t i e θ(t i +∆-s) σ s dW s . The inter- cept g (θ)
converges to zero at a rate of ∆. The autoregressive coefficient α (θ) = 1 if θ = 0;

hence, the log-price has a random walk dynamic. If θ = 0, we obtain that

α (θ) = exp (θ∆) = 1 + θ∆ + O p ∆ 2
converges to unity as ∆ → 0 at a rate of ∆. The order of magnitude of the autoregressive coefficient O p (∆) can be written as O p (1/T ), given that ∆ = N/T and N is a constant.

Therefore, the dynamic in ( 4) is local-to-unity [START_REF] Phillips | Time series regression with a unit root[END_REF] in the explosive direction if θ > 0 and in the stationary direction if θ < 0. One can show that V t i+1 follows a linear drift process such that

dV t i+1 = θV t i+1 ds + σ t i+1 dW t i+1 . (5)
Under (3), we have

t i +∆ t i σ s dW s = O p (∆ 1/2 ); hence, V t i+1 = O p (∆ 1/2
). See Phillips and Yu (2011) for the derivation.

The Return Process

The solution of the stochastic differential equation ( 3) is

y t = e θt y 0 + m(1 -e θt ) + t 0 e θ(t-s) σ s dW s , (6)
where y 0 is the initial value. The return process is denoted by r t i and defined as y t i -y t i-1 .

Lemma 2.1 Under the linear drift-diffusion process (3), r t i has the following properties:

(i) r t i can be written as

r t i = ∆ i A + t i t i-1 σ s dW s + t i t i-1 B(t i , s)σ s dW s + t i-1 0 [B(t i , s) -B(t i-1 , s)]σ s dW s (7) = t i t i-1 σ s dW s {1 + o p (1)},
where (ii) The expectation of r t i , denoted by m t i , is

∆ i A = A(t i ) -A(t i-1 ), A(t) = t 0 a(
m t i = ∆ i A = (y 0 -m)e θt i-1 (e θ∆ -1).
See Appendix A for the proof.

From Lemma 2.1(i), the return process is asymptotically dominated by the volatility component

t i t i-1 σ s dW s .
In other words, the drift term is asymptotically negligible and as

shown by [START_REF] Barndorff-Nielsen | Econometric analysis of realized volatility and its use in estimating stochastic volatility models[END_REF],

r 2 t i → t i t i-1 σ 2 s ds.
Although the drift component is negligible when the sampling interval ∆ approaches zero, very often, one has to rely on low-frequency data for estimations and hypothesis testing. This is because ultrahigh-frequency data for asset prices are not always available, or because lower-frequency data (such as 5-or 10-minute data) are preferred to mitigate the impact of microstructure noise [START_REF] Park | Realized volatility: Theory and application[END_REF].

Lemma 2.1(ii) suggests that the expected value of the log-return process depends on the values of m, θ, ∆, as well as the initial value y 0 . The mean of the average log-returns over N = 1 day is

∆ 1/∆ i=1 m t i = ∆(y 0 -m)(e θ -1). ( 8
)
Figure 1 shows the value of ( 8) for several combinations of values for y 0 and θ at the 1minute (∆ = 1/400) and 5-minute (∆ = 1/80) frequencies. We consider a wide range of values of θ and allow the initial value to vary from 0 to 7. We set m to zero for simplicity.

As we observe, when the process deviates from the random walk (i.e., with nonzero θ),

the drift component has an order of magnitude of 10 -3 . Additionally, the magnitude of the drift increases under three circumstances: 1) if the sampling frequency is lower, 2) if the initial value becomes larger, and 3) if θ moves away from zero.6

Figure 1: The average drift of log-returns over one day for various combinations of θ, y 0 and ∆. The parameter settings are empirically realistic. As a comparison, the daily median 7 of 5-minute log-returns of the NASDAQ stock market index is plotted in Figure 2 for the period 1996-2016. This figure shows that the daily median of 5-minute log-returns ranges from -2.2 × 10 -3 to 1.1 × 10 -3 , which is of the same order of magnitude as the drift component presented in Figure 1. In particular, the daily median of the 5-minute log-returns deviates from zero for a substantial period of time in early 2000 when the dot-com bubble burst and around the subprime mortgage crisis period.

The Realized Volatility Estimator

The realized volatility estimator (see Andersen andBollerslev, 1998 andBarndorff-Nielsen and[START_REF] Barndorff-Nielsen | Econometric analysis of realized volatility and its use in estimating stochastic volatility models[END_REF] among others) is defined as

RV t i (K) = i j=i-K+1 r 2 t j ,
where K is the number of observations included in the estimation. The bias of the realized volatility is therefore

E RV t i (K) - t i t i -K σ 2 u du .
Proposition 2.1 Under the linear diffusion process (3), the finite sample bias of the realized Section 3.1 for a detailed discussion).

volatility is zero if θ = 0. If θ = 0,

E RV t i (K) - t i t i -K σ 2 u du = E 1 + E 2 + E 3 , (9)
where

E 1 = (y 0 -m) 2 e θ∆ -1 2 e θt i-K 1 -e 2θ∆K 1 -e 2θ∆ , (10)
E 2 = d 0 K 1 2θ e 2θ∆ -1 + 2 θ 1 -e θ∆ + 1 , (11)
E 3 = d 0 2θ e θ∆ -1 2 e 2θt i-K 1 -e 2θ∆K 1 -e 2θ∆ -K . ( 12
)
The proof of the proposition is in Appendix A. One can see that although the drift term µ t is asymptotically negligible, it has a finite sample impact on the realized volatility estimator.

To minimize the impact of the drift term, we propose calculating the realized volatility from centered log-returns. We study two modified realized volatility estimators. The first one is infeasible because it is based on the unknown quantity m t j , while the second one is feasible because it relies on the sample mean of the K log-returns entering in the computation of the realized volatility, i.e.,

mt i (K) = 1 K i j=i-K+1 r t j .
The two modified realized volatility estimators are Infeasible:

RV † t i (K) = i j=i-K+1 (r t j -m t j ) 2 ; Feasible: RV * t i (K) = i j=i-K+1 [r t j -mt i (K)] 2 .
Proposition 2.2 Under the linear diffusion process (3), the finite sample bias of the infea-sible modified realized volatility estimator RV † is zero when θ = 0; while when θ = 0,

E RV † t i (K) - t i t i -K σ 2 u du = E 2 + E 3 . ( 13
)
Proposition 2.3 Under the linear diffusion process (3), the finite sample bias of the feasible modified realized volatility estimator RV * is zero when θ = 0; while when θ = 0,

E RV * t i (K) - t i t i -K σ 2 u du = (E 1 -E 4 ) + K + 1 K (E 2 + E 3 ) + d 0 ∆, ( 14
)
where

E 4 = 1 K (y 0 -m) 2 e θt i-K e θ∆K -1 2 .
The proofs of the above two propositions are in Appendix A. The first term in (9) disappears when the realized volatility is calculated from centered log-returns (i.e., r t j -m t j ).

When using the RV * estimator, while we observe additional noise terms arising from the estimation of the sample mean, the first bias term in ( 9) is reduced by E 4 . To visualize the relative magnitude of the bias, we simulate the quantities in Proposition 2.1-2.3 with the same parameter setting as for Figure 1 but also set the unconditional variance d 0 to 10 -4

(as in the subsequent simulations), K = 1/∆ and t i-K = 0.

The bias of the realized volatilities RV , RV † and RV * are plotted in Figure 3. It is obvious that the biases of the modified volatility estimators are much smaller than that of the original estimator in finite sample. The discrepancy becomes increasingly obvious as the process deviates further away from the random walk and when the initial value increases.

Additionally, the bias of the modified volatility estimator is extremely close to zero with an order of magnitude of 10 -9 for RV † and 10 -6 for RV * when ∆ = 1/80. This suggests that although Propositions 2.2 and 2.3 show that the biases of the modified estimators are nonzero when the process deviates from the random walk, they are negligible compared to that of the realized volatility estimator.

Furthermore, we plot each of the three terms characterizing the bias of RV * in Proposition 2.3 in Figure 4. First, the term (K + 1)/(E 2 + E 3 )K is the smallest among the three terms. The order of (K + 1)/(E 2 + E 3 )K, therefore E 2 + E 3 , is 10 -9 for the 5-minute data and 10 -10 for the 1-minute data, which is negligible compared with the total bias of RV (i.e., 10 -3 as shown in Figure 3). This allows us to infer that the bias of RV is mostly due to E 1 .

Second, the E 1 -E 4 term has the same order as (K +1)/(E 2 +E 3 )K and hence also negligible (although it increases with both |θ| and y 0 ). Lastly, the term that dominates the bias of RV * is d 0 ∆. It takes value 2.5 × 10 -7 and 1.25 × 10 -6 , respectively, when ∆ = 1/400 and 1/80, which are much smaller than the unconditional variance (recall that E(σ 2 t ) = 10 -4 ). The finite sample bias of the realized volatility estimator was first documented in [START_REF] Meddahi | A theoretical comparison between integrated and realized volatility[END_REF], where the drift coefficient is assumed to be a square integrable function of the state variable S t such that µ t = p i=0 g i Π i,St , where p i=0 |g i | < ∞. This specification includes, for example, the case where the drift coefficient is an affine function of the variance σ 2 t and also the case of constant drift.

Remark 2.1 Assume that the drift coefficient µ t = m. The log-price y t can be rewritten as

y t = y 0 + mt + t 0 σ s dW s and therefore r t i = m∆ + t i t i-1 σ s dW s .
The biases of RV , RV † and RV * are respectively

E RV t i (K) - t i t i-1 σ 2 u du = Km 2 ∆ 2 , E RV † t i (K) - t i t i-1 σ 2 u du = 0, E RV * t i (K) - t i t i-1 σ 2 u du = d 0 ∆.
The proof is given in Appendix A. See Meddahi (2002, Proposition 4.1) for the exact expression for the bias of the realized volatility estimator under the general assumption that

µ t = p i=0 g i Π i,St .
It is important to note that while the realized volatility estimator is biased, the modified estimator RV † is unbiased. The feasible RV * estimator is biased but the magnitude of its bias is marginal. 8

Jump Robust Drift and Volatility Estimators

There is ample evidence on the presence of jumps in asset prices. This questions the relevance of the empirical mean for the estimation of E(r t j ) and also calls for the use of robust to jumps estimators of the integrated variance. In this section, we study the finite sample properties of the median and the jump robust volatility estimators computed on centered log-returns via a comprehensive Monte-Carlo simulation.

8 Deriving a bias correction might be possible but is beyond the scope of this paper.

Jump Robust Drift Estimator

It is well known in the robust statistics literature that the median is less sensitive to outliers or jumps than is the sample mean. Indeed, the asymptotic breakdown point is 0 for the sample mean and 1/2 for the median (see Maronna et al., 2006, among others).

In presence of jumps, it is therefore natural to center the log-returns using the median of the past K log-returns, denoted by

m * t i (K) = median r t i-K+1 , • • • , r t i ,
rather than their empirical mean mt i (K).

To compare the accuracy of m * t i (K) and mt i (K) as estimators for the drift component in the presence of jumps, we consider a data generating process that generalizes (4) by allowing for p additive jumps:

y t i+1 = g(θ) + α(θ)y t i + p j=1 φ j t i+1 I j t i+1 + V t i+1 , (15)
where I j t i+1 is a dummy variable indicating the location of the j th jump (the occurrence of which is random), and φ j t i+1 is the corresponding jump size. The error term V t i+1 is specified by

V t i+1 = σ t i+1 √ ∆ε t i+1 (16)
σ 2 t i+1 = α 0 + σ 2 t i (β 1 + α 1 √ ∆v t i+1 ), (17)
where ε t i+1 and v t i+1 are two independent i.i.d.

∼ N (0, 1) random variables. Equation ( 16) is consistent with the specification of V t i+1 in (5) if the value of the error term within the time interval [t i , t i+1 ] is assumed to be fixed. The volatility dynamic ( 17) is a standard Euler discretization of the GARCH(1,1) diffusion process (2). The parameters are linked as follows: α 0 = κω∆, β 1 = 1 -κ∆, and α 1 = √ 2λκ.

To illustrate the impact of jumps on the empirical mean and the robustness of the median, we generate 24, 000 observations corresponding to one day of 1-second data of an asset traded during 6.5 hours per day. 9 We pick one observation every 60 and 300 data points to obtain the 1-minute and 5-minute data, respectively. That is, the time span of interest is one day (N = 1), the sample size T equals the number of observations per day, and the time interval is ∆ = 1/T . The settings of y 0 , µ and θ are as in Figure 1.

Two different settings for jumps are considered. There is either one single large negative jump within a day with φ 1 t i+1 = -1.5σ t i+1 or two small jumps with φ j t i+1 = -0.6σ t i+1 for j = 1, 2.10 For the volatility dynamic, we follow Andersen and Bollerslev (1998) and choose the parameters κ = 0.035 and λ = 0.296 to simulate a realistic log-price process with very persistent GARCH effects and set ω = 10 -4 such that E(σ 2 t i+1 ) = 10 -4 .

Figure 5: The bias of the sample mean and median in the presence of one single large jump (i.e. φ 1 The empirical mean and median of the last observation of each day, denoted respectively by mt T (T ) and m * t T (T ), are compared to the true value m t T . The superiority of the median is clear from Figures 5 and6, which shows that the bias is systematically smaller for the median than for the sample mean for all combinations of parameters considered in the simulation. 11 Therefore, we recommend the use of the median instead of the sample Figure 6: The bias of the sample mean and median in the presence of two small jumps (φ j t i+1 = -0.6σ t i+1 for j = 1, 2). mean in empirical applications if the presence of jumps is suspected.

t i+1 = -1.5σ t i+1 ).

Jump Robust Volatility Estimators

The bipower variation proposed by Barndorff-Nielsen and Shephard ( 2004) is probably the most popular jump-robust estimator of the integrated variance. It has been extended by [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF] to accommodate the presence of market microstructure noise in ultrahigh-frequency data. Despite the well-behaved asymptotic properties of these estimators, we demonstrate below that like the realized volatility estimator their finite sample performance is unsatisfactory in the presence of a nonzero drift. We propose an alternative construction of these estimators that leads to a significant improvement in the estimation accuracy.

The bipower variation of Barndorff-Nielsen and Shephard ( 2004) is defined as

BV t i (K) = 1 c 2 1 i j=i-K+2 |r t j ||r t j-1 |, (18)
where c 1 = E(|U |) = 0.7979 is a normalizing constant (U denoting the standard normal distribution). As ∆ → 0,

BV t i (K) → p t i t i-K σ 2 u du.
The new estimator of the integrated variance we propose, denoted by BV * t i (K), corresponds to the bipower variation computed from centered log-returns, i.e.,

BV * t i (K) = 1 c 2 1 i j=i-K+2 |r * t j ||r * t j-1 |, where r * t i = r t i -m * t i (K).
Since the drift component of asset returns is asymptotically dominated by the variance, using the centered log-returns r * t i for the calculation of the bipower variation will not alter its limiting property but should improve its finite sample properties, as for the modified realized volatility RV * .

To illustrate the robustness of our estimator, we consider the data generating process ( 15)-( 17) with the same parameter settings as in Section 3.1. For simplicity, we allow for one negative jump per day with the magnitude equal to 60% of the spot volatility (i.e.,

φ 1 t i+1 = -0.6σ t i+1
). We report in Figure 7 the bias of both the bipower variation and modified bipower variation for the last observation of the day (i.e., t T ) from observations over the past day (i.e., respectively BV t T (K) and BV * t T (K), with K = T ). First, if θ = 0, there is almost no difference in the estimation accuracy between the two estimators. Second, the bias of the bipower variation (BV ) increases dramatically as the sampling frequency decreases and as |θ| and y 0 become larger. On the other hand, as expected, BV * provides a much more accurate estimation of the integrated variance. Indeed, the bias of BV * is very small for all combinations of parameters considered in this simulation.

A similar correction can be applied to other estimators of the integrated variance. We have also performed a Monte-Carlo simulations with the MedRV of [START_REF] Andersen | Jump-robust volatility estimation using nearest neighbor truncation[END_REF] and the threshold realized variance of [START_REF] Mancini | Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps[END_REF]. Results are qualitatively the same as for BV and BV * in Figure 7 and are therefore not reported to save space.

The Jump and Noise Robust Volatility Estimators

One might expect that with ultrahigh-frequency data, the drift component will be extremely close to zero; hence, the discrepancy between the bipower variation and the modified bipower variation will diminish. We show in this section that this is unfortunately not true for the noise-robust volatility estimator of [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF].

Assume that the noise-contaminated log-price y † t i is

y † t i = y t i + η t i , (19)
where (η t i ) 1≤i≤T is a noise process with mean zero and variance q 2 , independent of y t i .

Assume also that the noise process is serially correlated with an order of s -1.

The estimator of [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF] is constructed as follows. Define the Γreturn r (Γ)

t i as r (Γ) t i = y † t i -y † t i-Γ ,
where Γ = γ 1 K 1/2 with γ 1 > 0. We divide the past K Γ-returns into B nonoverlapping blocks. The number of blocks is B = γ 2 Γ with γ 2 > 1, and the size of the blocks is

S = K/B. The average log-return of block b is rt i ,b = 1 S i-(B-b)S j=i-(B-b+1)S+1 r (Γ) t j .
(20) [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF] propose to compute the bipower variation for noise-contaminated data using preaveraged returns, i.e.,

BV N t i (K) = 1 c 2 1 B b=2 |r t i ,b ||r t i ,b-1 |. ( 21
)
One can obtain a consistent estimator for the integrated variance IV t i (K) by removing the variation induced by the market microstructure noise from the bipower variation such that

IV t i (K) := γ 1 γ 2 BV N t i (K) -v 2 q2 t i (K) v 1 → t i t i-K σ 2 u du, (22)
where q2 t i (K) is a consistent estimator of the noise variance q 2 and defined as

q2 t i (K) = 1 2(K -s) i j=i-K+s+1 (y † t j -y † t j-s ) 2 , v 1 = γ 1 [3γ 2 -4 + max{(2 -γ 3 2 ), 0}] 3(γ 2 -1) 2 and v 2 = 2 min{(γ 2 -1), 1} γ 1 (γ 2 -1) 2 .
While the drift component of log-returns is extremely close to zero in the ultrahighfrequency setting, the drift of the preaveraged returns rt i ,b (using which the bipower variation is computed) might be of a nonnegligible magnitude. For the sake of illustration, we consider a data generating process following ( 15), ( 16)-( 17) and ( 19). Parameter settings for the non-noise components remain unchanged. We assume i.i.d. noise (i.e., s = 1)

for simplicity. The variance of the noise is set to be proportional to the variance of the underlying process as in [START_REF] Bandi | Separating microstructure noise from volatility[END_REF]; [START_REF] Boudt | Positive semidefinite integrated covariance estimation, factorizations and asynchronicity[END_REF]; [START_REF] Lee | Jumps in equilibrium prices and market microstructure noise[END_REF]. Specifically, q 2 = 0.01 1 0 σ 4 (s)ds and hence q ≈ 0.1%. We simulate one day of 1-second data with one small jump per day (for a total of 10 4 replications).

Figure 8 displays the average of the median of rt i ,b (K), denoted by m *

t i (K) = median(r t i ,1 , • • • , rt i ,B ),
for various combinations of θ and y 0 . This figure shows that the median increases as the process deviates from the random walk, and the magnitude is comparable to the median of 5-minute returns plotted in Figure 1. As before, this deviation is expected to affect the performance of the integrated variance estimator.

As an alternative, we propose modifying the estimator of [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF] by computing the bipower variation using centered preaveraged returns, i.e.,

BV N * t i (K) = 1 c 2 1 B b=2 |r t i ,b -m * t i (K)||r t i ,b-1 -m * t i (K)|, (23)
and

IV * t i (K) := γ 1 γ 2 BV N * t i (K) -v 2 q2 t i (K) v 1 → t i t i-K σ 2 u du.
The consistency of IV * t i (K) follows directly from [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF].

With the same simulation setting, we compute the bias of the two noise and jumps robust estimators IV t i (K) and IV * t i (K) (see Figure 9). We observe patterns similar to those in Figure 7. While the estimation accuracy of the original volatility estimator deteriorates substantially as |θ| and y 0 deviate from zero, the new estimator is much more accurate.

Jump Tests

The popularity of the LM08 and LM12 tests is due to their ease of implementation, their good size and power under fairly general conditions (e.g., allowing both the drift and the volatility to be stochastic) and the fact that they allow for testing the presence of jumps within a short period of time (e.g., 5 minutes or 1 hour).

We show in this section that the nonzero drift component leads to a significant downward size distortion and power loss for the LM08 test when applied to relatively lowfrequency (e.g., 5-minute) data. Despite the fact that the LM12 test is designed for ultrahigh-frequency (e.g. 1-second) data and can be applied to very short time spans (e.g., 1-hour), it is also undersized if the log-price process has a nonzero drift. We propose a modification of both tests and show the importance of this correction for the finite sample performance of the tests through Monte-Carlo simulations.

The Lee and Mykland (2008) Tests

Andersen et al. ( 2007) and [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF] independently proposed a test statistic for jumps, denoted J t i below, for which they derived the asymptotic distribution in the zero drift case, while [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF] also proposed another test statistic, denoted by Jt i below, for the nonzero drift case. The two statistics are defined as follows:

J t i = r t i σt i (K)
and

Jt i = r t i -mt i (K) σt i (K) . ( 24
)
Lee and Mykland (2008) proposed estimating the instantaneous volatility σt i (K) using a rolling window of K log-returns as follows:

σt i (K) = 1 K -1 BV t i (K). ( 25
)
The construction of both test statistics J t i and Jt i involves the bipower variation. As discussed in the previous section, the finite sample performance of this volatility estimator is unsatisfactory if the process has a nonzero drift. Additionally, the demeaned test statistic J is based on the sample mean, the performance of which is inferior to that of the median in the presence of jumps as shown in Section 3.1. To improve the finite sample performance of the tests, we propose a correction to these jump test statistics. The new test statistic is denoted by J * t i and defined as

J * t i = r t i -m * t i (K) σ * t i (K) . ( 26
)
We replace mt i (K) by the median m * t i (K) and σt i (K) by σ * t i (K), an estimator of the instantaneous volatility based on the bipower variation computed from centered log-returns, i.e.,

σ * t i (K) = 1 K -1 BV * t i (K). (27)
The properties of both test statistics have been studied by [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF].

More specifically, they show that in the absence of jumps and if log-prices follow a diffusion process as in (1), and satisfying the two conditions stated in Assumption 2.1, J t i and Jt i converge to a standard normal distribution as the sampling interval ∆ tends to zero, provided that K is sufficiently large. Given that the drift component is asymptotically negligible, the proposed correction will not alter the limiting distribution of the test statistic. Therefore, if K = O p (∆ α ) with -1 < α < -0.5, we have

sup i∈{1,••• ,T } |S t i -U | = O p (∆ η ),
where S t i = J t i , Jt i , J * t i and -< η < 3 2 + α -for any ≥ 0.

The jump test is implemented for each individual observation within the day.

|U i | -C L S L → ξ (28)
where

C L = (2 log L) 1/2 -1 2 (2 log L) -1/2 [log π + log(log L)], S L = (2 log L) -1/2 ,
P max i |S t i | > cv L,β = 1 -exp -exp - cv L,β -C L S L = β,
and hence,

cv L,β = C L -S L log [-log (1 -β)] . (29)
Therefore, we declare that there is a jump at time t i according to the J t i (resp. Jt i and

J * t i ) statistic if |J t i | > cv L,β (resp. Jt i > cv L,β and J * t i > cv L,β
). If we set L to be the number of observations per day, the probability of finding at least one spurious jump (either positive or negative) within each day is 100β%.12

The Finite Sample Performance of the Jump Tests

In this section, we investigate the finite sample performance of the LM08 tests J and J, and our modified test J * . The data generating process used to study the size of the tests is (4) while we rely on (15) to study their power. The volatility dynamic is specified by ( 16)-( 17). The parameter settings are the same as in Section 3.1. Under the alternative, there is one jump per day with a magnitude of φ 1 t i+1 = -0.6σ t i+1 .

We generate 48, 000 observations corresponding to two days of 1-second data of an asset and aggregate them at the 1-minute and 5-minute frequencies as above. The first day is used as a burn-in period; we focus on the detection results of the second day.

Therefore, the time span of interest is one (N = 1), the sample size T equals the number of observations per day, and the time interval is ∆ = 1/T . The sample mean and median and the instantaneous volatility are estimated from a rolling window of K log-returns, which is required to be between √ T and T . We choose K to be closer to the upper bound (i.e., K = T -1) to ensure that there are sufficient observations to estimate the integrated variance before rescaling it to obtain an estimate of the instantaneous volatility of r t i .

For the critical values, we set L = T and β = 1% so that the probability of finding at least one spurious jump within each day is 1% or, equivalently, we expect that one out of 100 days contains at least one spurious jump. Therefore, the critical values used in this simulation are cv 24000,1% = 5.13, cv 400,1% = 4.37 and cv 80,1% = 4.07 when the tests are applied to 1-second, 1-minute and 5-minute data, respectively. Note that the asymptotic critical values (29) depend on the values chosen for both L and β. Here, we control for the overall size of the test over a day (L = T ) and set β = 1%. Alternatively, we could control the size of the test over a longer period (e.g., a month or a year). Interestingly, when setting L = 20T

and β = 10%, the critical value is 4.02 for 5-minute data and is slightly smaller than the one of the above setting (i.e., 4.07). The empirical sizes and powers (with 10 4 replications) of the LM08 statistics J and J, and the modified LM08 statistic J * are plotted in Figure 10 for two sampling frequencies (1-and 5-minute data). 13 We observe that if log-prices follow a random walk (i.e., θ = 0), the empirical sizes of the LM08 tests J and J are close to the nominal size of 1%. 14 On the contrary, we observe a significant downward size distortion for both tests as the dynamic of log-prices deviates from the random walk (i.e., θ = 0) and the initial value y 0 increases. 15 The undersize problem becomes even more severe if the sampling frequency is lower. Indeed, if the tests are applied to 5-minute data, the null hypothesis of no jump is almost 13 A table display of results in Figure 10, 12, and 13 are provided in an online supplement. 14 We obtained qualitatively the same results for the case of constant volatility and for other quantiles (5% and 0.1%). Results are not reported to save space. 15 When the LM08 tests are applied to 1-second data (without microstructure noise), both tests have an empirical size close to the nominal size of 1% for all combinations of parameters and a power of 100%. The results are not reported here to save space. never rejected for both tests16 if |θ| ≥ 0.02 and y 0 ≥ 3 (Figure 10b).

Figure 10c shows that the power of both tests is close to 100% for all specifications (except the bottom left corner) if the tests are applied to data at the 1-minute frequency.

For 5-minute data, power values are approximately 88% if asset prices follow a random walk with additive jumps. As expected, in the absence of microstructure noise, jumps are easier to detect if the sampling frequency increases. However, the assumption of no microstructure noise for data sampled at a frequency higher than 1-minute is unrealistic.

Therefore, the LM08 tests are usually applied to 5-minute data to reduce the impact of microstructure noise at the cost of a slight power loss. If 5-minute data are used (see Figure 10d), the downward size distortion problem of both tests translates into a dramatic loss of power if the process has a nonzero drift. In particular, both tests have a power close to 0 if |θ| ≥ 0.02 and y 0 ≥ 4. This result is consistent with our expectation that the nonnegligible mean of log-returns affects the performance of the LM08 tests in finite samples. Importantly, the demeaned version of the test (i.e., J), which relies on m and BV (rather than m * and BV * ), does not improve the performance of the test.

The new jump test has an outstanding performance. The empirical size of the test is reasonably close to the nominal size of 1% for both sampling frequencies. The empirical power of the test is 100% if the sampling frequency is high (i.e., 1-minute) and approximately 88% if applied to 5-minute data. This result is in sharp contrast to Figure 10, where both J and J tests suffer from serious size distortion towards 0 and a lack of power for 1/∆ = 80 and |θ| = 0.

Intraday Periodicity

For ease of exposition, we have so far ignored intraday periodicity effects in the spot volatility. However, it has been well known since the studies of [START_REF] Taylor | The incremental volatility information in one million foreign exchange quotations[END_REF] and Andersen and Bollerslev (1998b) that the opening, lunch period and closing of financial markets induce a strong periodic pattern in the volatility of high-frequency returns. More recently, [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF] proposed several nonparametric robust-to-jumps estimators of the intraday (or intraweek) periodicity and a correction to the LM08 jump statistics by allowing the spot volatility to depend on the estimated periodicity. They show that this modification helps increase the power to detect the relatively small jumps occurring at times of volatility being periodically low and to reduce the number of spurious jump detections at times when volatility is periodically high.

The most efficient nonparametric periodicity estimator of [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF], denoted ft i , is the weighted standard deviations (WSD). Assuming for simplicity that the length of the periodicity cycle is one day and that we dispose of 5-minute data, the WSD estimator corresponds to the standard deviation of weighted standardized log-returns J t i computed on all the observations belonging to the same 5-minute interval (across T days) and multiplied by a correction factor to ensure its consistency in the absence of jumps.

The observation J t i receives either a weight of either zero when J 2 t i is higher than a high quantile (e.g., 99%) of the χ 2 1 distribution (i.e., the distribution of J 2 t i in the absence of jumps) or one otherwise. We refer the reader to [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF] for details on the WSD estimator.17 It is important to note that this estimator of the intraday periodicity relies on the assumption that, in the absence of jumps, J t i follows a standard normal distribution, while we have observed in Section 5.2 that this assumption is likely to be violated in finite samples if the process has a nonzero drift.

Consequently, we propose a modified WSD estimator, denoted by f * t i , where the periodicity is estimated on J * t i rather than J t i . Finally, as [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF], we also modify our newly proposed J * statistic by multiplying σ * t i in ( 26) by the estimated intraday periodicity f * t i . The periodicity-adjusted jump test statistic is denoted by J * P t i and defined as

J * P t i = r t i -m * t i (K) f * t i σ * t i (K) . ( 30
)
Periodicity-adjusted jump test statistics J t i and Jt i can be obtained similarly, i.e., J P t i = J t i / ft i and JP t i = Jt i / ft i .

To study the finite sample properties of the modified WSD estimator and the periodicityadjusted jump test statistic, we extend the previous simulation by introducing intraperiodicity in addition to GARCH(1,1) dynamics in the conditional variance. To this end, we simulate data according to ( 15) and ( 17), where

V t i+1 = σ t i f t i √ ∆ε t i+1 , (31)
and f t i , i.e., the true periodicity, depicts the usual U-shaped pattern during the day and is restricted to be same on all days. We simulate 252 days of one-second data. The first 251 days serve as a burn-in period to estimate the periodicity using 5-minute data, while the last day is used to study the size and power of the J * P t i test statistic (also computed on 5-minute data). The burn-in period contains on average one jump per day (with size

φ 1 t i+1 = -0.6σ t i+1 f t i ).
The last day contains no jump under the null hypothesis and one jump under the alternative.

The true periodicity of the simulated 5-minute data (i.e., f t i ) is plotted in Figure 11 together with the averages (over 10 4 replications) of ft i and f * t i . To study the impact of a nonzero drift on the estimation of f t i , θ is set to a nonzero value for either 20 (left panel) or 50 (right panel) randomly chosen days (out of the first 251 days) and 0 on the remaining days. Figure 11 corresponds to the most extreme case, i.e. y 0 = 7 and θ = 0.05 for the 20 or 50 days. The results suggest that unlike that for ft i , the bias of the modified WSD estimator f * t i is negligible in both cases, even in this very extreme scenario.

Finally, we explore the performance of the periodicity-adjusted jump test in Figure 12.

The same DGP is used as above but now we consider the same range of values for θ and y 0 as in the previous simulations for the last day of the simulated sample (i.e., day 252).

We assume that 20 random days (out of the first 251 days) deviate from the random walk (i.e., θ = 0) in the burn-in period. Results are qualitatively the same for 50 days and are not reported here to save space. The general conclusion from Figure 12 is that our True Periodicity Averaged centered WSD Averaged non-centered WSD periodicity-adjusted jump test J * P behaves similarly to the unadjusted one using J * in the absence of periodicity. The test is slightly oversized but reasonably close to the nominal size, and the power is satisfactory.

The (Modified) Ultrahigh-Frequency Jump Test

Lee and Mykland (2012) extend the LM08 tests to the ultrahigh-frequency setting, allowing for the presence of market microstructure noise. Interestingly, examining noisecontaminated one-second data, we observe similar patterns of size distortion and power loss of the LM12 test if θ and y 0 deviate from zero and show that a similar correction to the test statistic dramatically improves its finite sample performance.

modification similar to the one in (26). The new test statistic is To study the finite sample properties of the three tests, we consider the same data generating process and parameter settings as in Section 4. However, for consistency with the simulations and application in [START_REF] Lee | Jumps in equilibrium prices and market microstructure noise[END_REF], we apply the tests to 1-second data over a very short period of 1 hour (rather than 1 day as in Sections 5.2 and 5.3).

L * b = √ M rb -m * b (K) V * b (K) , (34)
To this end, we simulate observations over a 2-hour period and use the first hour as a burn-in period. The rolling window size is set to T -1. Parameter C is set according to Table 5 of [START_REF] Lee | Jumps in equilibrium prices and market microstructure noise[END_REF]. The optimal value of C is 1/18 for a value of q ≈ 0.1%.

We set γ 1 = 1 and γ 2 = 1.6 as in [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF]. The nominal size of the test is again 1%.

The empirical performance of L b , Lb and L * b statistics is illustrated in Figure 13. First, there is no visible difference in the empirical performance (in both size and power) of the L b and Lb tests. Second, we observe a downward size distortion and a power deterioration in both tests as the process deviates from the random walk and the initial value y 0 increases, although the magnitudes of the size distortion and the power loss are smaller than those observed in the right panel of Figure 10, that correspond to the original J and J tests of [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF] applied to 5-minute data and the span of one day. Importantly, the size of the new test L * b fluctuates slightly below the nominal size of 1%, while its power is approximately 80% for all configurations of θ and y 0 . These results are not surprising given our discussion and findings in Section 4 regarding the ultrahigh-frequency volatility estimators.

Empirical Application

In this section, we compute the three J test statistics using 5-minute data of the NASDAQ stock price index during the period from 1996 to 2016. All trades before 9:30 am or after 4:00 pm and the first trade after 9:30 am are discarded, which is the usual method of avoiding the overnight effect. The choice of this series is dictated by the fact that several studies (see Phillips et al., 2011[START_REF] Homm | Testing for speculative bubbles in stock markets: a comparison of alternative methods[END_REF], and Shi and Song, 2016, among others) To compute the periodicity-adjusted J test statistics, we first estimate the intraday periodicity ft i and f * t i year-by-year with a cycle length of one week (to allow for different day-of-week effects as in [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF] with the procedures described in Section 5.3.

Results of the three tests (J P , JP and J * P ) are reported in Table 1. For the critical value cv L,β , we set L to the total number of 5-minute log-returns per day (i.e., L = 78 and β = 0.1%). The values reported in the three columns below 'Significant jumps' correspond to the number of jump statistics greater than the critical value, while those below 'Significant days' correspond to the number of days in which at least one significant jump is detected. The first conclusion we can draw from Table 1 is that for each year, the numbers of jumps detected by the three tests are not dramatically different. However, our proposed J * P statistic allows us to almost systematically detect more jumps. Over 21 years, 864, 883 and 984 jumps are detected using the statistics J P , JP and J * P , respectively, which correspond to 675, 689 and 759 days with at least one significant jump. This result is consistent with our simulation findings that if the log-price process has a nonzero drift, the newly proposed test J * P has a higher power than that of the other two tests.

We can identify 87 days on which a jump is only detected by the J * P statistic (i.e., J P * > cv 78,0.1% while J P and JP ≤ cv 78,0.1% ). To better understand the difference between the tests, the log-prices of 4 out of these 87 days (selected randomly) are plotted in Figure 16. Interestingly, these 4 days are characterized by a strong upward or downward trends that we attribute to a nonzero drift. Recall from Figure 10 that if the process has a nonzero drift, the original LM ( 2008) tests (both J and J) applied to 5-minute data are observed to be undersized and to have less power than J * , while J * has good size and very good power (the same comment applies to periodicity adjusted tests).

Figure 16: The log-prices on 4 days on which jumps are identified by J * P but not by J P and JP . The vertical lines indicate the arrival times of the detected jumps.

Conclusion

This paper derives the finite sample theory for the realized volatility under the driftdiffusion process. The finite sample theory and extensive simulations reveal that with a fixed sampling interval, the realized volatility but also the jump robust volatility estimators (including the bipower variation, the MedRV, and the threshold realized variance) tend to overestimate the integrated variance in the presence of a nonzero drift, and the bias rises with the magnitude of the drift term. Moreover, despite the drift term becoming extremely close to zero as the sampling frequency increases, our simulations show that the volatility estimator of [START_REF] Podolskij | Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[END_REF], which is robust to microstructure noise and designed for ultrahigh-frequency data, suffers from the same problem due to the use of preaveraged returns.

Consequently, procedures derived from these integrated variance estimators (such as the intraday periodicity estimator of [START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF][START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF][START_REF] Gutierrez | Speculative bubbles in agricultural commodity markets[END_REF] jump tests have unsatisfactory performance in finite samples when log-prices have a nonzero drift. In particular, we demonstrate that in the presence of a nonzero drift, the jump tests have strong size distortions and power losses. We propose an alternative construction of the integrated variance estimator, intraday periodicity estimator and jump test statistics, which leads to significant improvements of their performance.

The newly proposed estimators and the jump test, along with their original versions, are applied to 5-minute log-returns of the NASDAQ for the period from 1996 to 2016. For most observations, the new estimator provides a lower estimate of the daily integrated variance than the bipower variation, which is consistent with our theory and simulations. Furthermore, more jumps are detected using the new jump test. Interestingly, on days when jumps are detected only by the new test, log-prices exhibit clear upward or downward trend movements. We attribute these trends to the presence of a relatively large drift, which explains why the original tests of [START_REF] Lee | Jumps in financial markets: A new nonparametric test and jump dynamics[END_REF] fail to detect jumps on these days.

where a(t) = θ (y 0 -m) e θt and b(t, s) = θe θ(t-s) . The process y t can be rewritten as

y t = y 0 + t 0 µ s ds + t 0 σ s dW s = y 0 + A(t) + t 0 B(t, s)σ s dW s + t 0 σ s dW s ,
where A(t) = t 0 a(r)dr and B(t, s) = t s b(r, s)dr with t ≥ s.

(i) The return of the asset follows a dynamic of the type

r t i = ∆ i A + t i t i-1 σ s dW s + t i t i-1 B(t i , s)σ s dW s + t i-1 0 [B(t i , s) -B(t i-1 , s)]σ s dW s (35) = t i t i-1 σ s dW s {1 + o p (1)},
where

∆ i A = t i t i-1 a(r)dr.
The second equality arises from the fact that ∆ i A = O p (∆),

t i t i-1 σ s dW s = O p (∆ 1/2
), and the third and fourth terms on the right-hand side of the equation is O p (∆ 3/2 ).

(ii) Take expectation of both sides of the return process, we have

E(r t i ) = ∆ i A = (y 0 -m)e θt i-1 (e θ∆ -1).

Theoretical Properties of the Eigenfunctions

The eigenfunctions has the following properties: (1) The eigenfunctions are normalized such that Π 0,St = 1 and V ar (Π i,St ) = 1 for all i = 0.

(2) Any non-constant eigenfunction is centered such that

E [Π i,St ] = 0.
(3) Two eigenfunctions that are associated with two different eigenvalues are orthogonal

E [Π i,St Π j,St ] = 0 for i = j.
(4) Any eigenfunction is an autoregressive process of order one,

E Π i,S t+h |S τ , τ ≤ t = exp(-δ i h)Π i,St , ∀h > 0.
(5) Any square-integrable function g, i.e. E g (S t ) 2 < ∞, may be written as a linear combination of the eigenfunctions, i.e.

g (S t ) = ∞ i=0 d i Π i,St ,
where

d i = E [g (S t ) Π i.St ] and ∞ i=0 d 2 i = E g (S t ) 2 < ∞.

Proof of Proposition 2.1

Proof. The bias of the realized volatility estimator is

E   i j=i-K+1 r 2 t j - t i t i -K σ 2 u du   .
The first step is to calculate the squared return from (35) such that

r 2 t j = (∆ j A) 2 (1) + t j t j-1 σ s dW s 2 (2) + t j t j-1 B(t j , s)σ s dW s 2 (3) + t j-1 0 [B(t j , s) -B(t j-1 , s)σ s dW s 2 (4) + 2∆ j A t j t j-1 σ s dW s (5) + 2 ∆ j A + t j t j-1 σ s dW s t j t j-1 B(t j , s)σ s dW s (6) + 2 ∆ j A + t j t j-1 σ s dW s t j-1 0 [B(t j , s) -B(t j-1 , s]σ s dW s (7)
+ 2

t j t j-1 B(t j , s)σ s dW s t j-1 0 [B(t j , s) -B(t j-1 , s]σ s dW s (8)
, which is consists of eight terms. The first term

(∆ j A) 2 = t j t j-1 a(r)dr 2 .
We apply the Ito's lemma to the squared terms (2), ( 3) and ( 4). 18 The second term

t j t j-1 σ s dW s 2 = s t j t j-1 u t j-1 σ s dW s σ u dW u + t j t j-1 σ 2 u du;
the third term

t j t j-1 B(t j , s)σ s dW s 2 = 2 t j t j-1 u t j-1 B(t j , s)σ s dW s B (t j , u) σ u dW u + t j t j-1 B (t j , u) 2 σ 2 u du;
and the fourth term

t j-1 0 [B(t j , s) -B(t j-1 , s)σ s dW s 2 18 If η (x) = x 2 , we have dη = ∂η ∂x dx + 1 2 ∂ 2 η ∂x 2 (dx) 2 . +2 t j t j-1 E u 0 [B(t j , s) -B(t j-1 , s)]σ s dW s a(u)du.
From Assumption 2.3, we have

E σ 2 t = E p i=0 d i Π i,St = p i=0 d i E (Π i,St ) = d 0 .
Therefore, E r 2 t j -

t j t j-1 σ 2 u du = t j t j-1 a(r)dr 2 + d 0 t j t j-1 B (t j , u) 2 du +d 0 t j-1 0 [B(t j , u) -B(t j-1 , u)] 2 du.
By construction, 1 -e 2θ∆ -K when θ = 0 and equals zero when θ = 0.

Proof of Proposition 2.3

Proof. The bias of the feasible realized volatility estimator is

E    i j=i-K+1 r t j -mt i 2 - t i t i -K σ 2 u du    51 = E    i j=i-K+1 r 2 t j -2 mt i i j=i-K+1 r t j + K m2 t i - t i t i -K σ 2 u du    = E   i j=i-K+1
r 2 t j -

t i t i -K σ 2 u du   -2 i j=i-K+1
E mt i r t j + KE( m2 t i ).

The first term is the bias of the realized volatility estimator derived in Proposition 2.1. We only need to derive the second and third terms. From Lemma 2.1, r 2 t j -

mt i = 1 K t i t i -K a(
t i t i -K σ 2 u du   -2 i j=i-K+1
E mt i r t j + KE(r 2 t i ).

The sample average

mt i = m∆ + 1 K t i t i-K σ s dW s .
We can show that

-2 i j=i-K+1 E mt i r t j = -2Km 2 ∆ 2 , KE( m2 t i ) = Km 2 ∆ 2 + 1 K t i t i-K E σ 2 u du = Km 2 ∆ 2 + d 0 ∆.
Therefore,

E    i j=i-K+1 r t j -mt i 2 - t i t i -K σ 2 u du    = d 0 ∆.

  r)dr and B(t, s) = t s b(r, s)dr with t ≥ s, a(t) = θ (y 0 -m) e θt and b(t, s) = θe θ(t-s) ;

Figure 2 :

 2 Figure 2: Daily median of 5-minute log-returns of the NASDAQ stock index from 1996 to 2016.

Figure 3 :

 3 Figure 3: The bias of the (modified) realized volatility estimators. (a) 1-minute ∆ = 1/400

Figure 4 :

 4 Figure 4: Components of the bias of RV * . (a) ∆ = 1/400

Figure 7 :

 7 Figure 7: The estimation bias of the bipower and modified bipower variation.(a) 1-minute ∆ = 1/400

Figure 8 :

 8 Figure 8: Median of rt i ,b for various combinations of θ and y 0 .

Figure 9 :

 9 Figure 9: The estimation bias of IV t i (K) and IV * t i (K).

Figure 10 :

 10 Figure 10: Empirical performance of the (modified) LM08 tests. (a) Size: 1-minute ∆ = 1/400

Figure 11 :

 11 Figure 11: The simulated intraday periodicity and the periodicity estimators. (a) 20 days

Figure 12 :

 12 Figure 12: Empirical performance of the periodicity-adjusted jump test J * P . (a) Size: 5-minute ∆ = 1/80

  where m * b (K) = median(r b-B 0 +1 , • • • , rb ) and V * b (K) is defined similarly to Vb (K) with IV b (K) replaced by IV * b (K). All three test statistics (L b , Lb , and L * t b ) converge to the standard normal distribution under the null hypothesis of no jumps.

Figure 13 :

 13 Figure 13: Empirical performance of L, L and L * tests. (a) Size: L and L

  Figure 14: The five-minute log-prices of the NASDAQ stock index over the period of 1996-2016.

Figure 15 :

 15 Figure 15: Estimated bipower variation (BV ) of the NASDAQ stock market and the discrepancy between BV and the modified bipower variation (BV * ).

2 ,e

 2 0 -m) 2 e θt j -e θt j-1 which equal zero when θ = 0. By construction,B (t j , u) = t j u b (r, u) dr = t j u θe θ(r-u) dr = e θ(t j -u) -1, 2θ(t j -u) -2e θ(t j -u) + 1 du = d 0 t j t j-1 e 2θ(t j -u) du -2 t j t j-1 e θ(t j -u) du + (t j -u) -e θ(t j-1 -u) ] 2 du = d 0 [e θ∆ -1] 2 e 2θt j-1 e θ∆ -1 2 e 2θt j-1 -1 if θ = 0 0 if θ = 0.Therefore, the bias of the realized volatility is zero when θ = 0 and when θ = 0 0 -m) 2 e θ∆ -1 2 e θt i-K 1 -e 2θ∆K 1 -e 2θ∆

ee

  2θ(t j -u) -2e θ(t j -u) t j , u) -B(t j-1 , u)] 2 du = d (t j -u) -e θ(t j-1 -u) ] 2 du = d 0 [e θ∆ -1] 0 -m) 2 e θt i-K e θ∆ -1 2 1 -e 2θ∆K 1 -e 2θ∆ -2θt i-K 1 -e 2θ∆K1 -e 2θ∆ -K when θ = 0 and equals zero when θ = 0.

  To control for the size of multiple tests, while Andersen et al. (2007) use a Bonferroni correction, Lee and Mykland (2008) suggest using critical values based on the extreme value theory.Recall that the maximum of a set of L i.i.d. realizations of the absolute value of the standard normal random distribution U

i (for i = 1, . . . , L) follows asymptotically a Gumbel distribution (see, for example,

[START_REF] Aldous | Probability approximations via the Poisson clumping heuristic[END_REF][START_REF] Mutangi | On how to find the norming constants for the maxima of a folded normally distributed variable[END_REF]

, i.e., max i

Table 1 :

 1 Descriptive statistics of significant jumps.

	Significant jumps Significant days
	J P	JP	J * P	J P	JP	J * P
	1996 42	39	50	32	30	35
	1997 41	44	51	34	34	40
	1998 41	47	50	30	35	37
	1999 22	21	28	19	18	24
	2000 25	30	35	21	24	28
	2001 27	28	28	24	23	23
	2002 20	25	29	18	23	27
	2003 36	39	44	31	33	38
	2004 43	48	54	35	39	44
	2005 73	75	81	53	54	57
	2006 60	58	61	44	43	45
	2007 60	59	71	42	41	48
	2008 34	32	33	29	27	28
	2009 45	39	49	35	31	38
	2010 56	52	57	40	37	41
	2011 34	44	43	30	37	37
	2012 42	41	45	34	34	37
	2013 47	48	49	35	36	36
	2014 47	46	51	38	38	40
	2015 31	32	37	25	26	30
	2016 38	36	38	26	26	26
	Total 864 883 984 675 689 759

  [B(t j , s) -B(t j-1 , s)]σ s dW s [B(t j , s) -B(t j-1 , s)]σ s dW s B(t j , u)σ u dW u B(t j , s)σ s dW s [B(t j , u) -B(t j-1 , u)]σ u dW u .The expected value of the bias is therefore E r t j -m t j 2 -

	and
			t j				t j-1
			2	B(t j , s)σ s dW s
			t j-1				0
			t j	(u
		= 2
			t j-1	0
			t j-1	u
			+2
			0	t j-1
				t j	σ 2 u du =	t j	B (t j , u) 2 E σ 2 u du
				t j-1			t j-1
							t j-1
						+
	Therefore,
	E	  j=i-K+1 i	r t j -m t j	2 -	t i t i -K	 u du σ 2  = d 0 K	1 2θ	e 2θ∆ -1 +	2 θ	1 -e θ∆ + 1
								+	d 0 2θ	e θ∆ -1	2 1 -e 2θ∆K
						+d 0 K		1 2θ	e 2θ∆ -1 +	2 θ	1 -e θ∆ + 1
						+	d 0 2θ	e θ∆ -1	2	e 2θt i-K 1 -e 2θ∆K 1 -e 2θ∆ -K .

0 [B(t j , u) -B(t j-1 , u)] 2 E σ 2 u du.

  [B(t j , s) -B(t j-1 , s)]σ s dW s . [B(t j , u) -B(t j-1 ∆, u)] 2 du [B(t j , u) -B(t j-1 , u)] 2 E σ 2 u du.

						+	d 0 K	i j=i-K+1	0	t j-1
	since
						1 K	E		t i t i-K	σ s dW s	2	=	1 K	t i-K t i	E σ 2 u du;
	1 K	1 K Therefore, i j=i-K+1	i j=i-K+1 t j-1 + K E 1	r)dr + t j-1 t j B(t j , s)σ s dW s 1 K t i t i-K i t j-1 0 j=i-K+1	σ s dW s + 2 = 1 K K j=i-K+1 i i 1 j=i-K+1 2 i 1 = K j=i-K+1	t j t j-1 t j t i-j B (t j , u) 2 E σ 2 u du; B(t j , s)σ s dW s t j-1 0
	Following the proof of Proposition 2.1, we can show that E   i r t j -mt i 2 - t i σ 2 u du 
		= E	-2 j=i-K+1    i j=i-K+1 r 2 j=i-K+1 i t j -t i t i -K E mt i r t j = -2 t i -K  1 K σ 2 u du   -t i 1 t i -K K	t i t i -K a(r)dr a(r)dr 2 + d 0 ∆ 2 .
	Moreover,
		KE( m2 t i ) = + d 0 K i j=i-K+1	1 K t j t j-1	t i t i -K B (t j , u) 2 du + a(r)dr	2 K + j=i-K+1 K E 1 d 0 i	t i-K t i t j-1 σ s dW s	2
						+	K 1	j=i-K+1 i	E	t i-j t j	B(t j , s)σ s dW s	2
						+	K 1	j=i-K+1 i	0	t j-1	[B(t j , s) -B(t j-1 , s)]σ s dW s	2
				=	1 K		t i t i -K	a(r)dr	2	+ d 0 +	d 0 K	j=i-K+1 i	t j-1 t j	B (t j , u) 2 du
											52

0 [B(t j , s) -B(t j-1 , s)]σ s dW s 0 [B(t j , u) -B(t j-1 , u)] 2 du.

The local-to-unity process deviates mildly from the random walk in both stationary and explosive directions. It bridges the gap between a random walk and stationary/explosive processes.

 4 [START_REF] Lee | Detecting jumps from levy jump diffusion processes[END_REF] propose a test for the presence of infinite-activity jumps (i.e., a Levy jump diffusion process).

See Hansen and Scheinkman (1995) and[START_REF] Aıt-Sahalia | Operator methods for continuoustime markov models[END_REF] for a review and Appendix A for a summary of their theoretical properties.

If the sampling frequency is one second (∆ = 1/24000), the drift component is negligible for all combinations of parameters and is therefore not reported to save space.

 7 We calculate the median instead of the sample mean to reduce the impact of jumps in asset prices (see

This is the case for the NASDAQ stock market, which trades from 9:30 to 16:00.

The simulation results are qualitatively the same for positive jumps and jumps with a random sign.

Similar figures are available for the root mean squared errors (RMSE) but not reported here to save space and because they follow more or less closely the shape of the figures on the bias.

Given the rolling window calculation of the test statistics, the i.i.d assumption required by the extreme value theorem is likely to be violated for the jump tests. As a consequence, the multiplicity issue might not have been perfectly controlled for with the proposed critical values. One can see from Figure10(b) and 12(a) that there is a small upward size distortion remaining for the modified LM tests. This is, however, not the focus of this paper. A solid investigation of this problem is left for future work.

Importantly, a close-to-zero rate of rejection is also observed if using a higher critical value of β = 5%.

Note that, as[START_REF] Boudt | Robust estimation of intraweek periodicity in volatility and jump detection[END_REF], we normalize ft i such that f 2 t i averages to one over the length of the periodicity cycle (e.g., one day or one week).

If η (x, y) = 2xy, we have dη = ∂η ∂x dx + ∂η ∂y dy.

The noise-contaminated log-price is y † t i , as defined by (19). Recall that the noise is assumed to be serially correlated of order s. Let Γ = sM with M = CT 1/2 1 (C is a constant and T 1 = T /s). For every s th observation, we calculate the Γ-differenced log-return as r ( Γ)

Let us divide the sequence {r ( Γ)

t js } T 1 j=M +1 into B blocks of size M . For each block b, we calculate the average log-return over the block such that

The [START_REF] Lee | Jumps in equilibrium prices and market microstructure noise[END_REF] test statistics for the presence of jumps in the log-prices between t bsM and t (b+1)sM are

and Lb = M rb -mb (K) Vb (K)

where Vb (K) is an estimate of the variance of √ M rb and mb (K) is the empirical mean of rb .

Both Vb (K) and mb (K) are calculated using a rolling window of K returns (over D days).

Specifically, let

where IV b (K) and q2 b (K) are as in Section 4.

To improve the finite sample accuracy of the two test statistics in (32), we propose a

Appendix A

The proof of Lemma 2.1

Proof. Replacing y t in the drift coefficient µ t with (6), we have

Similarly, we apply the Ito's lemma to all the cross product terms ( 5), ( 6), ( 7) and ( 8). 19 The fifth term

the seventh term

and the eighth term

The expected value of r 2 t j -

Proof of Proposition 2.2

Proof. The bias of the infeasible realized volatility estimator is

where m t j = E(r t j ). From ( 35), the squared returns

, which is consists of six terms. From the proof of Proposition 2.1, the (1), ( 2) and (3) terms are

Similarly, the cross product terms (4), ( 5) and ( 6) are

Proof of Remark 2.1

Proof.

(1) The squared return of the constant drift process is

.

The first term

Therefore, E r 2 t j -

and the bias of RV is

(2) Similarly, the square term of the demeaned returns

Therefore, the bias of RV † t i (K) is zero.

(3) The bias of the feasible modified realized volatility estimator RV * is