Business Time Sampling Scheme with Applications to Testing Semi-Martingale Hypothesis and Estimating Integrated Volatility

Abstract

We propose a new method to implement the Business Time Sampling (BTS) scheme for high-frequency financial data. We compute a time-transformation (TT) function using the intraday integrated volatility estimated by a jump-robust method. The BTS transactions are obtained using the inverse of the TT function. Using our sampled BTS transactions, we test the semi-martingale hypothesis of the stock log-price process and estimate the daily realized volatility. Our method improves the normality approximation of the standardized business-time return distribution. Our Monte Carlo results show that the integrated volatility estimates using our proposed sampling strategy provide smaller root mean-squared error.

1. Introduction

In high-frequency financial data analysis, researchers usually do not use all available data but would select a subgrid of transactions. To choose the subgrid, two issues have to be considered: selecting the sampling scheme and choosing the target average sampling frequency. Three sampling schemes are commonly used in the literature: Calendar Time Sampling (CTS), Tick Time Sampling (TTS) and Business Time Sampling (BTS). Under the CTS scheme, transactions are selected by regularly spaced calendar time, such as every 5 s/min. The TTS scheme selects transactions with regularly spaced number of ticks, e.g., every 5 or 10 ticks. The BTS transactions are often selected to ensure approximately equal volatility for the returns over each interval. Thus, the CTS and TTS schemes are implemented based on explicit criteria (i.e., regular calendar-time length or number of ticks, respectively). In contrast, the BTS scheme depends on the unobserved volatility. As a result, the CTS and TTS schemes have been used widely in the literature, while the BTS scheme is used less frequently.

The BTS scheme possesses some desirable properties for high-frequency financial data analysis. It dates back to Dacorogna et al. (1993), and see also Zhou (1998), Peters and De Vilder (2006), and Mykland (2012). In particular, the BTS scheme yields independently and identically distributed (iid) normal returns for a semimartingale price process even when there is leverage or feedback effect.1 In contrast, due to the leverage effect or varying volatility, the calendar-time returns may not be iid normal even if the price process is a continuous local martingale. The assumption of iid Gaussian returns is required for the Gaussian-likelihood approach Nowman (1997) and several widely used integrated volatility estimates, including the multipower variation (MPV) estimate of Barndorff-Nielsen et al. (2006), the quantile realized volatility (QRV) method of Christensen et al. (2010) and the nearest neighbor truncation method of Andersen et al. (2012, 2014). However, in the implementation, researchers often sample transactions under the CTS or TTS scheme. Andersen et al. (2007, 2010) find that the normalized daily and weekly returns sampled in business time accord well with the standard normal distribution. To sample the time points, they use a sequential method to include intraday returns until the cumulative squared returns exceeds the average daily or weekly realized volatility.

One drawback of the method in Andersen et al. (2007, 2010) is that its performance in obtaining returns with approximately equal volatility over each interval deteriorates as the sampling frequency increases. In addition, researchers need to choose a threshold to obtain sampled returns at the target frequency. In this paper, we propose a new method to implement the BTS scheme, which has better performance as the sampling frequency increases and needs no threshold. We estimate the intraday integrated volatility using the jump-robust Tripower Realized Volatility (TRV) method of Barndorff-Nielsen et al. (2006) and calculate a time-transformation (TT) function over the investigated time period. The value of the TT function corresponds to the cumulative increments of the estimated intraday integrated volatility over time and the sampled BTS transactions are obtained using the inverse of the TT function.

We test the semi-martingale hypothesis on the BTS returns of 40 stocks selected from the New York Stock Exchange (NYSE). Our proposed BTS method improves the normality approximation of the empirical standardized return distribution. We further explore the use of the BTS scheme in estimating integrated volatility. First, we consider the Realized Volatility (RV) estimates for daily integrated volatility when the returns are sampled using the BTS, CTS and TTS schemes. Our Monte Carlo simulation shows that the TRV estimates (with and without subsampling) using the BTS returns provide the smallest root mean-squared error (RMSE). Second, we modify the ACD-ICV method of Tse and Yang (2012), making use of the BTS scheme. Our modified ACD-ICV estimator performs better than the Realized Kernel (RK) estimates Barndorff-Nielsen et al. (2008) and the method of Tse and Yang (2012), which uses price-event sampling.

The rest of this paper is as follows. Section 2 outlines our proposed implementation of the BTS scheme. Section 3 reports some empirical results on testing the semi-martingale hypothesis. In Section 4, we outline the estimation methods of daily volatility examined in this paper. We report the results of our Monte Carlo study in Section 5 and draw conclusions in Section 6. The Appendix A provides further details of the jump detection procedure and computation of the BT time-transformation function. Some additional results can be found in the accompanying online supplementary material.

2. Intraday Periodicity and the BTS Scheme

The stylized fact of intraday periodicity is a well known phenomenon in high-frequency financial data analysis. Trading activities are usually high at the beginning and close of the trading day and low around lunch time. Modifications are needed to get rid of this pattern before data are fitted into high-frequency models. To adjust for the intraday periodicity of transaction activity, Tse and Dong (2014) use a time-transformation function computed by pooling all transactions over all trading days in the sample. The TT function transforms all observed transactions from the calendar time to a transformed time for which transactions are evenly observed. However, the time-transformation function proposed by Tse and Dong (2014) will not be appropriate for BT sampling if volatility and transaction activity exhibit different intraday periodicity patterns.

Figure 1 presents the intraday periodicity in volatility and trading activity of the stock JP Morgan (JPM) from January 2010 to April 2013. Figure 1A plots the means of the 1 min intraday realized volatilities over all trading days in the sample period, expressed in annualized standard deviation in percent, while Figure 1B plots the total number of transactions at each second from 9:30 a.m. to 4:00 p.m. over all trading days in the sample. We observe that the realized volatility at the beginning of the trading day is approximately two to three times larger than that near the end of the trading day. In contrast, the number of transactions at the end of the trading day is two to three times larger than that in the morning. This finding is quite regular across other stocks in our sample, which shows that intraday periodicity patterns in volatility and trading activity are quite different.2 Thus, TTS returns in the morning will have larger volatility than those in the afternoon. In contrast, by construction, BTS returns will have nearly constant volatility over the whole trading day.

To implement the BTS scheme, Peters and De Vilder (2006) select transactions based on increments of estimated quadratic volatility. To alleviate the microstructure noise effect, transactions are selected sparsely, such as at 2 min frequency, to calculate the volatility. One drawback of this method is that it assumes away the price jumps, although this assumption is often rejected in the literature (see, e.g., Barndorff-Nielsen and Shephard (2004, 2006), Huang and Tauchen (2005), Lee and Mykland (2008) and Boudt et al. (2011)).3 Another drawback of this method (also in Andersen et al. 2010) is that researchers need to select a threshold value to obtain transactions at the target sampling frequency. The procedure of choosing the threshold has to be iterated, especially when the target frequency is high.

In this paper, we adopt the time-transformation approach to implement the BTS scheme. Let T denote the calendar-time length in seconds aggregated over all trading days in the sample. The time-transformation function $Q\left(t\right)$ at calendar time t (in sec), for $0\le t\le T$, is computed as the empirical proportion of the intraday integrated volatility up to time t.4 The diurnally transformed time corresponding to calendar time t is denoted by $\tilde{t}$, with $\tilde{t}=T\times Q\left(t\right)$. Thus, returns over equal intervals of diurnally transformed time will have approximately equal integrated volatility. To obtain BTS transactions at a given frequency, we can select equally spaced transactions at the diurnally transformed time and choose the corresponding BTS transactions using the inverse function $Q^{-1}\left(t\right)$. In contrast to the methods in the literature, we do not need to choose a threshold value when implementing the BTS scheme.

We outline our method to obtain the BTS transactions as follows. Consider a sequence of estimated intraday integrated volatility $V_k$ for K consecutive time intervals over the period $(0,T]$, with the end point in each time interval represented by $t_k$, for $k=1,\cdots ,K$.5 Denote the collection of points $t_k$ by ${\mathcal{H}}_V$$(t_0=0)$ and define $N_{t_0}=0$ and $N_{t_k}={\sum }_{i=1}^k{V}_i$ for $k=1,\cdots ,K$. The time-transformation function is calculated as $Q\left(t_k\right)=N_{t_k}/N_{t_K}$, for $t_k\in {\mathcal{H}}_V$. $Q\left(t\right)$ at any calendar-time point t can then be computed using a cubic interpolation that preserves monotonicity in t.6 To sample a sequence of calendar-time points with BTS duration h, we take equally spaced diurnally transformed BT points ${\tilde{t}}_j,j=0,\cdots ,L$, with ${\tilde{t}}_j-{\tilde{t}}_{j-1}=h$ and $L=[T/h]$. Then, $t_j=Q^{-1}({\tilde{t}}_j/T)$ are the required calendar-time points for the BTS scheme.

Figure 2A,B present the time-transformation functions for stock JPM based on intraday volatility and trading activity, respectively. These functions are computed by merging the data over the complete sample period, resulting in representative one-day time-transformation functions.7 Note that the two transformation functions exhibit different intraday patterns, with the compression of diurnally transformed time during market open more prominent for volatility than for trading activity.

3. Testing the Semi-Martingale Hypothesis Using BTS Returns

As discussed above, BTS returns are iid normal for a semi-martingale price process even when there is leverage and/or feedback effect, whereas the CTS returns may not be iid normal even if the price process is a continuous local martingale. We now examine empirically the behavior of the BTS returns following the study of Andersen et al. (2010).

3.1. The Semi-Martingale Hypothesis

Let $r_k=Y_{t_k}-Y_{t_{k-1}}$ be the jump-adjusted returns over the time interval $(t_{k-1},t_k)$.8 If the log-price follows a jump-diffusion process9 with no leverage and volatility feedback, $r_k$ standardized by the integrated volatility will follow a standard normal distribution, i.e.,

$$\frac{r_k}{\left({\int }_{t_{k-1}}^{t_k}{\sigma }^2\left(\tau \right)d\tau \right)}\sim N(0,1),k=1,2,\cdots ,$$

(1)

where $\sigma (·)$ is the instantaneous volatility function. The above result, however, will not hold if $\sigma (·)$ exhibits correlation over time (feedback effect) or with the log-price innovation (leverage effect). On the other hand, if the jump-adjusted returns are sampled over business time so that, over each business-time interval $({\tilde{t}}_{k-1},{\tilde{t}}_k)$, we have

$${\tilde{t}}_k=\underset{s>{\tilde{t}}_{k-1}}{\inf }\left\{{\int }_{{\tilde{t}}_{k-1}}^s{\sigma }^2\left(\tau \right)d\tau >{\overline{\sigma }}^2\right\}$$

(2)

for a given volatility threshold ${\overline{\sigma }}^2$,10 Then, the jump-adjusted returns ${\tilde{r}}_k$ over the business-time intervals (${\tilde{r}}_{k-1},{\tilde{r}}_k$) satisfy

$$\frac{{\tilde{r}}_k}{\overline{\sigma }}\sim N(0,1),k=1,2,\cdots .$$

(3)

To sample a sequence of BTS returns, Andersen et al. (2010) include intraday returns until the cumulative squared 5-min returns exceed a threshold, defined as the average daily or weekly realized volatility in calendar time (ABFN method hereafter). They report improved accuracy of the normal approximation under this sampling scheme.

3.2. Empirical Results of the Tests

To examine empirically the performance of our proposed BTS method versus the ABFN method, we use data of the top 40 market-capitalization stocks from the NYSE in 2010. We extract the tick-by-tick transaction data of these stocks from the TAQ database from January 2010 to April 2013. To clean the raw data, we follow the steps described in Tse and Dong (2014). Using the sequential jump-detection procedure of Andersen et al. (2010), we investigate the proportion of detected jumps (number of the detected jumps over the total number of sampled returns) when different sampling intervals and sampling schemes are used.11

We test the normality assumption of the drift-corrected and jump-adjusted BTS returns (returns with jumps removed). For each trading day, all BTS returns with jumps are deleted and the jump-adjusted 30-min, daily or weekly BTS returns are then computed by summing the remaining consecutive jump-adjusted BTS returns. We apply the Lilliefors (1967) test for normality to the jump-adjusted CTS returns, ABFN returns, and BTS returns. The results are reported in

Table 1. Test of normality hypothesis for no-jump returns of 40 New York Stock Exchange (NYSE) stocks.

Frequency	CTS		ABFN		BTS
	5%	1%	5%	1%	5%	1%
Weekly	15	9	3	1	4	0
Daily	40	40	12	5	6	4
30 min	40	40	40	40	30	26

Notes: The figures are the numbers of stocks (out of 40) for which the normality hypothesis at the 5% and 1% levels of significance are rejected based on the Lilliefors test implemented for 30 min, daily and weekly jump-adjusted returns, 2010–2013. The returns are constructed over 5 min sampling frequency using the Calendar Time Sampling (CTS) method, the method of Andersen et al. (2010) (ABFN) and the method proposed in this paper (BTS). The total number of trading days of each stock ranges from 830 to 853.

. It can be seen that the BTS method and the ABFN method substantially improve the normality approximation of the standardized return distribution over the Calendar-Time method. While the results for the weekly data are similar for the BTS method and the ABFN method, the BTS sampling scheme at higher frequencies restores normality for several stocks.

We further compare the performance of various methods by computing the moments of 30 min returns with jump-adjustment. In addition to using the Lilliefors (1967) test for normality, we also examine the skewness and kurtosis of the sampled returns of the 40 stocks and compute the average of the absolute difference of the calculated skewness and kurtosis versus 0 and 3, respectively. The results are reported in

Table 2. Mean of the absolute difference for no-jump 30 min returns of 40 New York Stock Exchange (NYSE) stocks.

Measures of shape	CTS	TTS	ABFN	BTS
Skewness (diff.)	0.1272	0.2091	0.1105	0.0384
Kurtosis (diff.)	5.3745	10.7893	2.3749	0.2792

Notes: The figures are the average of the absolute difference of the calculated skewness and kurtosis versus 0 and 3, respectively. To calculate the skewness and kurtosis, we use 30 min jump-adjusted returns (2010–2013) of 40 stocks. The 30 min returns are constructed over 5 min sampling frequency using the Calendar Time Sampling (CTS) method, the Tick Time Sampling (TTS) method, the method of Andersen et al. (2010) (ABFN) and the method proposed in this paper (BTS). The total number of trading days of each stock ranges from 830 to 853.

. We observe that, for high-frequency returns, the BTS method performs the best in restoring normality, while the TTS method performs the worst. This observation confirms our finding in Section 2 that we cannot use the TTS scheme to approximate the BTS scheme to select intraday returns. For illustration, Figure 3 presents the QQ (Quantile-Quantile) plots of the 30 min jump-adjusted ABFN returns and BTS returns for the JPM stock data. Our proposed BTS method performs better than the ABFN method in restoring normality for returns sampled at high frequency.12

4. Estimation of Integrated Volatility

We now examine the use of the BTS scheme for estimating integrated volatility. We consider two methods of estimating daily volatility: Realized Volatility (RV) method and Autoregressive Conditional Duration-Integrated Conditional Volatility (ACD-ICV) method. The literature on RV estimation has grown tremendously since its inception. In this paper, we select the Tripower Realized Volatility (TRV) estimate of Barndorff-Nielsen et al. (2006) for its robustness to price jumps.13 We compare the performance of the TRV estimates when returns are sampled by BTS, CTS and TTS schemes, with and without subsampling. With the same estimator used, the performance of the estimates is only differentiated by the sampling method. We also consider the use of the ACD-ICV approach, with some modifications based on the BTS methodology.

4.1. Integrated Volatility Estimation Using BT Returns

For a given subgrid $\mathcal{H}$, the TRV estimate is computed as

$$V_T={\xi }_{\frac{2}{3}}^{-3}\left[\sum _{i=2}^{|\mathcal{H}|-1}|r_{i,-}{|}^{\frac{2}{3}}|r_i{|}^{\frac{2}{3}}{\left|r_{i,+}\right|}^{\frac{2}{3}}\right],$$

(4)

where $r_i=(Y_{i,+}-Y_i)$ for $Y_{i,+},Y_i\in \mathcal{H}$ and ${\xi }_k=2^{\frac{k}{2}}\Gamma ((k+1)/2)/\Gamma (1/2)$ for $k>0$, with $\Gamma (·)$ denoting the gamma function and $Y_{i,+}$ denoting the elements following $Y_i$ in $\mathcal{H}$. We define $\left|\mathcal{H}\right|$ as number of points in grid $\mathcal{H}$ minus 1. For the CTS scheme, the previous-tick method is adopted when there is no transaction at the selected time point. For the TTS scheme, the number of subsampling grids S is selected to ensure that each subgrid has transactions at the target sampling frequency. To implement the subsampling method under the BTS scheme, we select BTS transactions with the sampling frequency being twice the average transaction duration. Subsampling is then implemented to obtain subgrids at the target sampling frequency.

4.2. Integrated Volatility Estimation Using the Modified ACD-ICV Method

Tse and Yang (2012) propose the ACD-ICV method to estimate daily and intraday volatility by modeling the price durations parametrically. They point out that, for short intraday intervals, such as an hour or 15 min, the RV methods use only local data for the period of interest. Thus, the infill sample size may not be large enough to justify the applicability of the asymptotics of the RV estimates. In contrast, the ACD-ICV method estimates the conditional volatility using data beyond the period of interest and produces better estimates of volatility over short intraday intervals. Their simulation results show that the ACD-ICV method performs better than other methods (such as the Realized Kernel method of Barndorff-Nielsen et al. (2008) in estimating the daily, 1 h and 15 min integrated volatility.

The ACD-ICV method samples observations from the observed transaction data based on a pre-specified price threshold $\delta$. That is, transactions are selected whenever the absolute price change exceeds the threshold $\delta$, which are the price events. Suppose ${\mathcal{H}}_{PE}=\{t_0,t_1,t_2,\cdots ,t_N\}$ is the selected price events and the ith price duration is $x_i=t_i-t_{i-1},i=1,\cdots ,N$. Let ${\Phi }_i$ denote the information set upon the price event at time $t_i$. Denote ${\psi }_{i+1}=E\left(x_{i+1}\right|{\Phi }_i)$ as the conditional expectation of the price duration and assume that the standardized durations ${ϵ}_i=x_i/{\psi }_i,i=1,\cdots ,N$ are iid positive random variables with a mean of unity. Given the information ${\Phi }_i$ at time $t_i$, the conditional instantaneous return variance per unit time at time $t>t_i$, denoted by ${\sigma }^2\left(t\right|{\Phi }_i)$, is

$${\sigma }^2\left(t\right|{\Phi }_i)=\frac{{\delta }^2}{{\psi }_{i+1}}\lambda \left(\frac{t-t_i}{{\psi }_{i+1}}\right),$$

(5)

where $\lambda (·)$ is the hazard function of ${ϵ}_i$. Assuming ${ϵ}_i$ to be iid standard exponential distributed, the integrated conditional variance (ICV) over time period $[t_{n_1},t_{n_2+1}]$ is calculated as

$$\mathrm{ICV}={\delta }^2\sum _{i=n_1}^{n_2}\left[\frac{t_{i+1}-t_i}{{\psi }_{i+1}}\right].$$

(6)

The conditional expectation of the price durations ${\psi }_i$ can be estimated by various methods, such as the ACD method of Engle and Russell (1998) or the Augmented ACD method of Fernandes and Grammig (2006).

The original ACD-ICV method models the price durations obtained by the threshold $\delta$ by assuming that these durations have equal volatility, and this assumption may not be true. In this paper, we modify the ACD-ICV method in two ways. First, instead of modeling the durations of the price events, we model the BTS durations that are obtained based on volatility change (instead of the absolute price change). The BTS returns are sampled as in Section 2. Second, we replace ${\delta }^2$ in Equation (6) by an estimate of the mean volatility over each sampled BTS return. Suppose there are K BTS returns over m trading days, and the estimated integrated volatility over these m trading days is equal to $V_m$. Then, each BTS return has an approximately constant integrated volatility of $V_D=V_m/K$. Instead of using ${\delta }^2$ as an approximation of the integrated volatility of each price event as in Tse and Yang (2012), we use $V_D$ to replace ${\delta }^2$ in Equation (6) to obtain a new ACD-ICV estimate.

Thus, for the ACD-ICV approach, we consider three variations of estimates. We first select transactions using the price-event sampling method (where transactions are selected based on absolute price change) and estimate the daily integrated volatility using the ACD-ICV method as in Tse and Yang (2012). We denote this method by ME1. We then replace ${\delta }^2$ in equation (6) by $V_D$, which is the integrated volatility estimated using TRV with subsampling at 3-min sampling frequency. We call this method ME2.14 Finally, we sample data using the BTS scheme (not by price events) and repeat the computation as in ME2, which is called ME3. We compare the daily volatility estimates using the ACD-ICV methods against the RK method.15

5. Monte Carlo Study

We conduct a Monte Carlo (MC) study to examine the performances of different integrated volatility estimates. Our MC set-up draws upon other models in the literature.

5.1. Simulation Models

We consider five simulation models, which are summarized in

Table 3. Summary of simulation models.

Model	Code	Description of Model	Description of Model Parameters
Heston Model (high volatility)	MD1	$d\log X\left(t\right)=\left(\mu -{\displaystyle \frac{{\sigma }^2\left(t\right)}{2}}\right)dt+\sigma \left(t\right)d{W}_1\left(t\right)$, $d{\sigma }^2\left(t\right)=\kappa (\alpha -{\sigma }^2\left(t\right))dt+\gamma \sigma \left(t\right)d{W}_2\left(t\right).$	$\mu =0.05$, $\kappa =5$, $\alpha =0.25$, $\gamma =0.5$ and Corr$(d{W}_1\left(t\right),d{W}_2\left(t\right))=-0.5.$
Heston Model (low volatility)	MD2	Same as MD1.	$\alpha =0.04$, all remaining parameters same as MD1.
Two-factor affine stochastic volatility model with U-shape intraday volatility pattern	MD3	$d\log X\left(t\right)={\sigma }_u\left(t\right){\sigma }_{sv}\left(t\right)d{W}_1\left(t\right),$${\sigma }_{sv}^2\left(t\right)={\sigma }_1^2\left(t\right)+{\sigma }_2^2\left(t\right),$$d{\sigma }_1^2\left(t\right)={\kappa }_1[{\theta }_1-{\sigma }_1^2\left(t\right)]dt+{\eta }_1{\sigma }_1\left(t\right)d{W}_{21}\left(t\right),$$d{\sigma }_2^2\left(t\right)={\kappa }_2[{\theta }_2-{\sigma }_2^2\left(t\right)]dt+{\eta }_2{\sigma }_2\left(t\right)d{W}_{22}\left(t\right),$ ${\sigma }_u\left(t\right)=C+A{e}^{-at}+B{e}^{-b(1-t)}$, $t\in [0,1].$	${\kappa }_1=0.6$, ${\kappa }_2=0.1$, ${\theta }_1=0.09$, ${\theta }_2=0.04$, ${\eta }_1=0.2$, ${\eta }_2=0.1,$ ${\rho }_1=\mathrm{Corr}(d{W}_1\left(t\right),d{W}_{21}\left(t\right))=0.9,$ $\mathrm{and}{\rho }_2=\mathrm{Corr}(d{W}_1\left(t\right),d{W}_{22}\left(t\right))=-0.4.$$A=0.75$, $B=0.25$, $C=0.88929198$, $a=10$ and $b=10$.
Deterministic volatility model with U-shape intraday volatility pattern	MD4	$d\log X\left(u\right)=\sigma \left(u\right)dW\left(u\right),$$\sigma \left(u\right)=\sigma (t,\tau )={\sigma }_1\left(t\right){\sigma }_2\left(\tau \right)$, where t is the day of trade and $\tau$ is the intraday time. ${\sigma }_1\left(t\right)$ is the volatility of day t, ${\sigma }_2\left(\tau \right)$ is the intraday variations at time $\tau$ of each day.	${\sigma }_1\left(t\right)=20\%$ for $t=1$ with ${\sigma }_1\left(t\right)$ increasing linearly in t over 20 days to reach $30\%$. It then remains level for the next 20 days and decreases linearly in t to $20\%$ over 20 days. ${\sigma }_2\left(\tau \right)$ is computed as in Tse and Yang (2012) using the IBM tick-by-tick transaction data in 2012.
MD1 with price jumps	MD5	$d\log X\left(t\right)=\left(\mu -{\displaystyle \frac{{\sigma }^2\left(t\right)}{2}}\right)dt+\sigma \left(t\right)d{W}_1\left(t\right)+J\left(t\right)dP\left(t\right),$ $d{\sigma }^2\left(t\right)=\kappa (\alpha -{\sigma }^2\left(t\right))dt+\gamma \sigma \left(t\right)d{W}_2\left(t\right).$	$P\left(t\right)$ is a Poisson process with on average one price jump every two days. $J\left(t\right)$ is the size of the jumps with $J\left(t\right)\sim \mathrm{N}(0.02,0.004)$.

. Models MD1 and MD2 are the Heston models (Heston (1993) and Aït-Sahalia and Mancini (2008) with some modifications) with high and low volatility, respectively. MD3 is the two-factor afine stochastic volatility model with an intraday U-shape pattern Hasbrouck (1999) and Andersen et al. (2012). MD4 is a deterministic volatility set-up Tse and Yang (2012). Finally, MD5 is MD1 with price jumps.

For all set-ups above, we set the initial price to 60 and the initial value of $\sigma$ to 30%. We introduce sparsity of trade to the data by simulating exponentially distributed calendar-time transaction durations.16 We first simulate transactions second by second and then generate exponentially distributed transaction durations with mean equal to 5 s, 10 s and 20 s, respectively. For simplicity, we only investigate iid market microstructure noise with constant noise-signal ratio (NSR). Based on the findings in Dong and Tse (2017), we consider cases with NSR = 0.005%, 0.01% and 0.02%. We introduce a 0.01 price rounding error in all simulations. The intraday duration periodicity is adjusted by the time-transformation method in Tse and Dong (2014) before we fit all price durations and BTS durations to the ACD model. Each model is simulated over 60 trading days, with the simulation repeated 1000 times.

5.2. Simulation Results

We first report our results on the TRV estimates. For each model set-up, sampling frequencies of 1, 2, 3, 5 and 10 min are considered. We compute TRV based on the CTS, TTS and BTS returns, with and without subsampling. To save space, we present only the results for the BTS scheme with subsampling in

Table 4. Mean error (ME) and root mean-squared error (RMSE) of estimates of daily volatility using the Tripower Realized Volatility (TRV) method under the Business Time Sampling (BTS) scheme with subsampling.

Sparsity	NSR	Model	ME					RMSE
			1-min	2-min	3-min	5-min	10-min	1-min	2-min	3-min	5-min	10-min
5-s	0.005%	MD1	−0.4286	0.0470	0.0031	0.0004	−0.1777	1.3869	1.7228	2.1610	2.8165	4.0975
		MD2	−0.4321	−0.0500	−0.0513	−0.0338	−0.1309	0.9964	1.1706	1.4665	1.9242	2.7797
		MD3	−0.4435	0.0200	0.0123	−0.0019	−0.1916	1.2425	1.5657	1.9431	2.5511	3.6823
		MD4	−0.2403	−0.0913	0.0126	0.0055	−0.2069	1.0574	1.3723	1.6849	2.2278	3.2164
		MD5	−0.0687	0.6282	0.7732	1.0940	1.6463	1.4163	2.0196	2.5587	3.4485	5.2161
	0.01%	MD1	0.3382	0.4486	0.2625	0.1532	−0.1142	1.3520	1.7646	2.1562	2.8077	4.0850
		MD2	0.0786	0.2172	0.1192	0.0675	−0.0884	0.9049	1.1831	1.4608	1.9152	2.7695
		MD3	0.3291	0.4002	0.2555	0.1439	−0.1301	1.2126	1.6016	1.9521	2.5460	3.6734
		MD4	0.5904	0.3128	0.2734	0.1532	−0.1474	1.1809	1.4043	1.7059	2.2373	3.2230
		MD5	0.7280	1.0331	1.0365	1.2492	1.7128	1.5989	2.1671	2.6391	3.4936	5.2306
	0.02%	MD1	1.8463	1.2523	0.7789	0.4499	0.0126	2.2762	2.1053	2.2506	2.8148	4.0590
		MD2	1.0802	0.7479	0.4603	0.2651	−0.0059	1.4503	1.3864	1.5109	1.9145	2.7529
		MD3	1.7862	1.1644	0.7397	0.4265	−0.0167	2.1440	1.9166	2.0542	2.5558	3.6603
		MD4	2.2616	1.1036	0.7889	0.4437	−0.0291	2.4966	1.7643	1.8609	2.2798	3.2357
		MD5	2.2713	1.8503	1.5649	1.5559	1.8424	2.7080	2.6518	2.8753	3.5982	5.2587
10-s	0.005%	MD1	−0.9375	−0.0435	−0.0307	−0.0333	−0.1880	1.6335	1.8357	2.1600	2.8457	4.1021
		MD2	−0.8123	−0.1034	−0.0778	−0.0560	−0.1417	1.2304	1.2500	1.4735	1.9394	2.7833
		MD3	−0.8436	−0.0958	−0.0423	−0.0340	−0.2088	1.4854	1.6584	1.9742	2.5751	3.7013
		MD4	−0.5797	−0.2436	−0.0830	−0.0584	−0.2432	1.2715	1.4404	1.7121	2.2547	3.2308
		MD5	−0.5986	0.5429	0.7290	1.0637	1.6334	1.5659	2.0991	2.5573	3.4711	5.2171
	0.01%	MD1	0.0713	0.3397	0.2356	0.1224	−0.1251	1.3501	1.8561	2.1611	2.8355	4.0856
		MD2	−0.1345	0.1520	0.1010	0.0460	−0.0992	0.9467	1.2544	1.4661	1.9323	2.7757
		MD3	0.1028	0.2815	0.2097	0.1090	−0.1529	1.2390	1.6800	1.9774	2.5686	3.6909
		MD4	0.3144	0.1905	0.1929	0.0949	−0.1830	1.1801	1.4321	1.7293	2.2613	3.2356
		MD5	0.4190	0.9362	0.9980	1.2244	1.6968	1.5275	2.2311	2.6363	3.5126	5.2337
	0.02%	MD1	2.0479	1.1137	0.7770	0.4235	−0.0051	2.5005	2.1285	2.2661	2.8358	4.0650
		MD2	1.1815	0.6675	0.4586	0.2426	−0.0175	1.5885	1.4198	1.5277	1.9301	2.7608
		MD3	1.9622	1.0222	0.7158	0.3933	−0.0340	2.3465	1.9448	2.0766	2.5855	3.6804
		MD4	2.1082	1.0387	0.7207	0.3933	−0.0644	2.4205	1.7728	1.8823	2.2997	3.2520
		MD5	2.4111	1.7259	1.5437	1.5352	1.8277	2.8718	2.6669	2.8772	3.6225	5.2623
20-s	0.005%	MD1	−1.9043	−0.4650	−0.1715	−0.1586	−0.3052	2.5154	1.9081	2.3632	2.8884	4.1176
		MD2	−1.4513	−0.3972	−0.1718	−0.1551	−0.2189	1.8465	1.3232	1.6142	1.9728	2.8010
		MD3	−1.7349	−0.4945	−0.1976	−0.1617	−0.3245	2.2880	1.7421	2.0996	2.6517	3.7326
		MD4	−1.3673	−0.5413	−0.2730	−0.2261	−0.3586	1.9372	1.6463	1.8101	2.3044	3.2759
		MD5	−1.5684	0.0539	0.6149	0.9467	1.5263	2.3452	2.0185	2.7000	3.4925	5.2178
	0.01%	MD1	−0.7727	0.0595	0.0795	−0.0012	−0.2474	1.8388	1.8658	2.3619	2.8691	4.1079
		MD2	−0.6930	−0.0470	−0.0025	−0.0515	−0.1798	1.3436	1.2747	1.6111	1.9649	2.7898
		MD3	−0.6715	0.0110	0.0681	−0.0090	−0.2678	1.6684	1.6857	2.1131	2.6393	3.7245
		MD4	−0.3601	−0.0577	0.0292	−0.0623	−0.3056	1.4354	1.5473	1.7854	2.3140	3.2870
		MD5	−0.4358	0.5872	0.8854	1.1072	1.5868	1.8245	2.1169	2.7876	3.5323	5.2285
	0.02%	MD1	1.4062	1.0834	0.5790	0.3089	−0.1229	2.2543	2.1942	2.4346	2.8722	4.0765
		MD2	0.7467	0.6389	0.3302	0.1612	−0.0953	1.4646	1.4674	1.6470	1.9576	2.7740
		MD3	1.3598	0.9923	0.5629	0.2802	−0.1573	2.1106	1.9812	2.2102	2.6445	3.7064
		MD4	1.6100	0.9172	0.6508	0.2535	−0.1811	2.1813	1.8069	1.9153	2.3599	3.2984
		MD5	1.7450	1.6299	1.4063	1.4237	1.7176	2.5623	2.6474	3.0191	3.6372	5.2592

Notes: ME and RMSE are the mean error and root mean-squared error, respectively, of the volatility estimates in annualized standard deviation in percentage. The average true daily integrated volatility is around 40% for MD1, 27% for MD2, 36% for MD3, 28% for MD4 and 40% for MD5. MD1 and MD2 are the Heston model at different volatility level. MD3 is the two-factor stochastic volatility model with intraday volatility periodicity. MD4 is the deterministic volatility model with intraday volatility periodicity and MD5 is the Heston model (MD1) with price jumps. The first column indicates the average duration of the observed simulated transactions. NSR is the noise-signal ratio. Results for average transaction frequency of 1 min, 2 min, 3 min, 5 min and 10 min are reported. The sampling frequency of the BTS scheme equals twice the average transaction duration.

.

Table 5. Comparison of the Tripower Realized Volatility (TRV) estimates under different sampling schemes with subsampling.

Average RMSE Difference of the TRV Estimates	1-min	2-min	3-min	5-min	10-min
Avg. (RMSE(CTS)-RMSE(BTS))	0.3043	0.1666	0.1891	0.2215	0.1926
Avg. (RMSE(CTS)-RMSE(BTS))/RMSE(BTS) (%)	17.8262	9.8515	9.6493	8.8626	5.7456
Avg. (RMSE(TTS)-RMSE(BTS))	−0.0237	0.1261	0.1721	0.2240	0.2072
Avg. (RMSE(TTS)-RMSE(BTS))/RMSE(BTS) (%)	−0.1011	7.9977	9.0710	9.1005	6.2101

Notes: RMSE is the root mean-squared error of the daily TRV estimates in annualized standard deviation in percent. The first row is the average of the RMSE of the TRV estimates (over all simulation models) based on the CTS scheme over that based on the BTS scheme. The second row is the average relative difference in percentage. The third and fourth rows are similarly presented for the TTS scheme compared to the BTS scheme.

summarizes the average difference in RMSE of the CTS and TTS schemes versus the BTS scheme over all models and parameter set-ups, with subsampling, in both absolute and relative terms. All results can be found in the supplementary material (Tables S4–S9). The BTS returns perform the best in reporting generally smaller root mean-squared error (RMSE), especially for methods with no subsampling. When the subsampling method is used, the RMSE decreases substantially. Generally, the BTS scheme still performs the best and its advantage is especially obvious for MD3 and MD4 when there is intraday volatility periodicity in the simulated price process.17 Under all sampling schemes, the TRV estimates suffer from the market microstructure noise problems when the NSR is large, resulting in high mean error (ME) at high sampling frequency. When sparsity and NSR are both low, high sampling frequency at 1 min interval produces the lowest RMSE. Overall, the BTS returns outperform the CTS and TTS returns in estimating the integrated volatility.

We now turn to the results of the ACD-ICV method.

Table 6. Mean error (ME) and root mean-squared error (RMSE) of daily volatility estimates of the realized kernel (RK) and Autoregressive Conditional Duration-Integrated Conditional Variance (ACD-ICV) methods for Model MD1.

			ME								RMSE
Sparsity	NSR	RK	ACD-ICV	Avg. Sampling Frequency (ACD-ICV)					RK	ACD-ICV	Avg. Sampling Frequency (ACD-ICV)
				1-min	3-min	5-min	10-min	15-min			1-min	3-min	5-min	10-min	15-min
5-s	0.005%	−0.1087	ME1	−5.8450	−3.4421	−2.6778	−1.8270	−1.3546	2.2978	ME1	6.2142	3.8881	3.3149	2.9738	2.9941
			ME2	−0.2043	−0.2366	−0.2205	−0.0928	0.0734		ME2	1.7527	1.7500	1.9673	2.3810	2.6985
			ME3	−0.1120	0.0540	0.2446	0.4362	0.7306		ME3	1.2772	1.4001	1.7261	1.4380	1.5706
	0.01%	−0.0876	ME1	−3.4503	−2.0139	−1.5554	−1.0491	−0.7357	2.2998	ME1	3.8811	2.6241	2.4551	2.5735	2.7766
			ME2	0.0437	0.0089	0.0301	0.1633	0.3410		ME2	1.6266	1.6906	1.9326	2.3896	2.7213
			ME3	0.1489	0.3076	0.4972	0.6950	0.9897		ME3	1.2367	1.3710	1.7078	1.4948	1.6735
	0.02%	−0.0461	ME1	1.4407	0.6924	0.4711	0.3438	0.3824	2.3040	ME1	2.0920	1.8468	2.0100	2.4290	2.7559
			ME2	0.5181	0.5045	0.5353	0.6804	0.8449		ME2	1.5778	1.7868	2.0340	2.5118	2.8656
			ME3	0.6677	0.8253	1.0079	1.2071	1.5051		ME3	1.3320	1.4939	1.8291	1.7399	1.9751
10-s	0.005%	−0.1416	ME1	−10.9555	−6.0669	−4.6302	−3.1722	−2.4450	2.6341	ME1	11.3935	6.4347	5.0926	3.9820	3.6161
			ME2	−0.3400	−0.3899	−0.3720	−0.2555	−0.0909		ME2	2.2864	1.8765	2.0363	2.4242	2.6968
			ME3	−0.2235	−0.1976	0.0212	0.2764	0.5716		ME3	1.4774	1.4302	1.5870	1.4764	1.5774
	0.01%	−0.1178	ME1	−8.5745	−4.8751	−3.7310	−2.5558	−1.9404	2.6363	ME1	8.9899	5.2580	4.2561	3.4911	3.2988
			ME2	−0.0858	−0.1337	−0.1165	0.0015	0.1719		ME2	2.0281	1.7825	1.9930	2.4033	2.7120
			ME3	0.0409	0.0713	0.2829	0.5387	0.8356		ME3	1.4179	1.3836	1.5621	1.5129	1.6586
	0.02%	−0.0710	ME1	−4.8863	−2.6923	−2.0864	−1.4154	−1.0327	2.6399	ME1	5.3222	3.2140	2.8523	2.7546	2.8704
			ME2	0.4241	0.3767	0.3976	0.5304	0.7014		ME2	1.8341	1.7717	2.0131	2.4689	2.8195
			ME3	0.5665	0.5917	0.8060	1.0634	1.3599		ME3	1.4587	1.4454	1.6547	1.7292	1.9392
20-s	0.005%	−0.1987	ME1	−17.7796	−9.5646	−7.3266	−5.0140	−3.9444	3.0600	ME1	18.3065	9.9612	7.7382	5.6130	4.7999
			ME2	−0.6770	−0.7315	−0.7231	−0.6051	−0.4518		ME2	3.1408	2.2301	2.2493	2.5140	2.7698
			ME3	−0.4760	−0.5490	−0.3795	−0.0762	0.2182		ME3	2.2789	1.6071	1.7431	1.5932	1.6135
	0.01%	−0.1723	ME1	−16.2877	−8.6764	−6.6429	−4.5336	−3.5434	3.0621	ME1	16.8362	9.0603	7.0596	5.1679	4.4519
			ME2	−0.4067	−0.4638	−0.4586	−0.3355	−0.1811		ME2	2.9539	2.0830	2.1464	2.4481	2.7132
			ME3	−0.2070	−0.2778	−0.1000	0.1926	0.4882		ME3	2.2062	1.5119	1.7034	1.5809	1.6494
	0.02%	−0.1200	ME1	−12.8662	−7.0624	−5.3456	−3.6371	−2.8233	3.0670	ME1	13.3676	7.4466	5.7892	4.3706	3.9009
			ME2	0.1191	0.0629	0.0728	0.2021	0.3562		ME2	2.6226	1.9304	2.0461	2.4303	2.7561
			ME3	0.3287	0.2614	0.4493	0.7287	1.0260		ME3	2.1632	1.4769	1.7526	1.7006	1.8539

Notes: ME and RMSE are the mean error and root mean-squared error, respectively, of the volatility estimates in annualized standard deviation in percentage. MD1 is the Heston model and the average true daily integrated volatility is around 40%. ME1 is the ACD-ICV method of Tse and Yang (2012). ME2 is ME1 with ${\delta }^2$ replaced by $V_D$, the integrated volatility estimated using TRV with subsampling at 3 min sampling frequency. ME3 is ME2 with sampled durations computed from BTS returns. All ACD models are fitted to diurnally transformed durations using the time-transformation function based on the number of trades as in Tse and Dong (2014).

and

Table 7. Mean error (ME) and root mean-squared error (RMSE) of daily volatility estimates of the realized kernel (RK) and Autoregressive Conditional Duration-Integrated Conditional Variance (ACD-ICV) methods for Model MD5.

			ME								RMSE
Sparsity	NSR	RK	ACD-ICV	Avg. Sampling Frequency (ACD-ICV)					RK	ACD-ICV	Avg. Sampling Frequency (ACD-ICV)
				1-min	3-min	5-min	10-min	15-min			1-min	3-min	5-min	10-min	15-min
5-s	0.005%	5.4227	ME1	−5.8297	−3.3882	−2.5730	−1.6045	−1.0188	9.9602	ME1	6.2026	3.8409	3.2360	2.8656	2.8789
			ME2	0.8058	0.7685	0.7892	0.9164	1.0902		ME2	1.9098	1.9199	2.1430	2.6091	2.9672
			ME3	0.8846	1.0464	1.2556	1.4577	1.7653		ME3	1.6636	1.8671	2.1671	2.0848	2.2923
	0.01%	5.4418	ME1	−3.4340	−1.9407	−1.4617	−0.8313	−0.3769	9.9686	ME1	3.8603	2.5721	2.4208	2.5277	2.7279
			ME2	1.0512	1.0187	1.0453	1.1758	1.3516		ME2	1.9215	2.0070	2.2572	2.7266	3.0797
			ME3	1.1471	1.3082	1.5160	1.7153	2.0254		ME3	1.7863	1.9921	2.2839	2.2406	2.4745
	0.02%	5.4794	ME1	1.4728	0.7713	0.5849	0.5818	0.7257	9.9860	ME1	2.0990	1.8775	2.0612	2.5042	2.8495
			ME2	1.5310	1.5181	1.5467	1.6891	1.8715		ME2	2.1539	2.3315	2.5607	3.0135	3.3723
			ME3	1.6721	1.8222	2.0263	2.2336	2.5459		ME3	2.1213	2.3145	2.5816	2.6222	2.8867
10-s	0.005%	5.3907	ME1	−10.9449	−6.0276	−4.5318	−2.9491	−2.1042	10.0436	ME1	11.3795	6.4015	5.0126	3.8136	3.4271
			ME2	0.6633	0.6106	0.6242	0.7470	0.9182		ME2	2.3104	1.9461	2.1377	2.5742	2.9195
			ME3	0.7705	0.7948	0.9985	1.2914	1.5981		ME3	1.6828	1.8435	2.0209	2.0229	2.2163
	0.01%	5.4121	ME1	−8.6446	−4.8146	−3.6363	−2.3242	−1.5842	10.0534	ME1	9.0607	5.2047	4.1744	3.3347	3.1353
			ME2	0.9153	0.8660	0.8891	1.0125	1.1800		ME2	2.1941	1.9921	2.2058	2.6559	3.0205
			ME3	1.0358	1.0604	1.2689	1.5548	1.8631		ME3	1.7876	1.9431	2.1333	2.1752	2.3929
	0.02%	5.4548	ME1	−4.8774	−2.6232	−1.9874	−1.1786	−0.6562	10.0727	ME1	5.3053	3.1592	2.7833	2.6525	2.7837
			ME2	1.4283	1.3848	1.4085	1.5407	1.7137		ME2	2.2591	2.2408	2.4554	2.9128	3.2801
			ME3	1.5657	1.5953	1.7983	2.0797	2.3929		ME3	2.0947	2.2393	2.4394	2.5464	2.8021
20-s	0.005%	5.3322	ME1	−17.8068	−9.5419	−7.2562	−4.8115	−3.5992	10.1532	ME1	18.3394	9.9398	7.6740	5.4435	4.5299
			ME2	0.3397	0.2821	0.2882	0.4019	0.5596		ME2	3.0755	2.1117	2.1666	2.5271	2.8374
			ME3	0.5340	0.4639	0.6157	0.9473	1.2592		ME3	2.2664	1.7445	1.9529	1.9189	2.0788
	0.01%	5.3562	ME1	−16.4046	−8.6377	−6.5612	−4.3376	−3.2101	10.1639	ME1	16.9502	9.0284	6.9857	5.0049	4.2219
			ME2	0.6035	0.5473	0.5600	0.6682	0.8417		ME2	2.9903	2.1008	2.1818	2.5643	2.9096
			ME3	0.8043	0.7385	0.8828	1.2168	1.5292		ME3	2.3152	1.8216	2.0152	2.0450	2.2400
	0.02%	5.4036	ME1	−12.8619	−7.0131	−5.2494	−3.4318	−2.4897	10.1854	ME1	13.3635	7.4026	5.6999	4.2129	3.6884
			ME2	1.1378	1.0778	1.0905	1.2102	1.3818		ME2	2.8324	2.2176	2.3262	2.7620	3.1219
			ME3	1.3435	1.2746	1.4297	1.7565	2.0717		ME3	2.5068	2.0651	2.2905	2.3864	2.6243

Notes: ME and RMSE are the mean error and root mean-squared error, respectively, of the volatility estimates in annualized standard deviation in percentage. MD5 is the Heston model with price jumps and the average true daily integrated volatility is around 40%. ME1 is the ACD-ICV method of Tse and Yang (2012). ME2 is ME1 with ${\delta }^2$ replaced by $V_D$, the integrated volatility estimated using TRV with subsampling at 3-min sampling frequency. ME3 is ME2 with sampled durations computed from BTS returns. All ACD models are fitted to diurnally transformed durations using the time-transformation function based on the number of trades as in Tse and Dong (2014).

report the ME and RMSE of the RK and the ACD-ICV estimates for MD1 and MD5, respectively. Results for other models can be found in the online supplementary material (Tables S10–S12).

Table 8. Comparison of the root mean-squared error (RMSE) of daily volatility estimates of the realized kernel (RK) and Autoregressive Conditional Duration-Integrated Conditional Variance (ACD-ICV) methods.

RK	ACD-ICV	Avg. Sampling Frequency (ACD-ICV)
		1-min	3-min	5-min	10-min	15-min
3.9198	ME1	8.1955	4.7603	3.8640	3.2027	3.0690
	ME2	1.9321	1.6430	1.7542	2.1559	2.4848
	ME3	1.4483	1.3958	1.5143	1.4755	1.6280

Notes: RMSE are the root mean-squared error of the volatility estimates (over all simulation models) in annualized standard deviation in percentage. ME1 is the ACD-ICV method of Tse and Yang (2012). ME2 is ME1 with ${\delta }^2$ replaced by $V_D$, the integrated volatility estimated using TRV with subsampling at 3-min sampling frequency. ME3 is ME2 with sampled durations computed from BTS returns. All ACD models are fitted to diurnally transformed durations using the time-transformation function based on the number of trades as in Tse and Dong (2014).

summarizes the average RMSE of Realized Kernel estimate versus the ACD-ICV estimates over all models and parameter set-ups. The RK method performs quite well for the unbiasedness property, reporting small ME for all models except MD5 (model with price jump). ME1 reports quite big absolute ME and RMSE values among all the sampling frequencies considered and it often performs worse than the RK method except for MD5.18 In contrast, the modified ACD-ICV methods, ME2 and ME3, report quite small ME and RMSE. ME3 consistently reports smaller RMSE than ME2 except for few cases in MD3 and MD5 at the 3 min and 5 min sampling frequencies. Moreover, ME2 varies more across different sampling frequencies. This demonstrates the superiority of the BTS durations over the price durations when they are fitted to the ACD model to estimate the integrated volatility. The better performance of ME3 over ME2 is mainly due to the fact that the BTS scheme performs better in yielding returns with constant volatility compared against the ACD-ICV method using the price events.

Our modified ACD-ICV method ME3 consistently produces lower RMSE over all models and parameter set-ups than the RK estimates. Its performance is robust over a wide range of sampling frequencies of up to 15 min. It also outperforms the TRV estimates with subsampling, except possibly for MD5, in which case their performances are comparable.19

6. Conclusions

We propose an easy-to-use time-transformation method to implement the BTS scheme at a prespecified average sampling frequency. Using 40 stocks from the NYSE, we perform normality test to the jump-adjusted daily and weekly BTS returns. Our results show that stock prices can be considered discrete observations from a continuous-time jump-diffusion process. The BTS scheme performs better than other sampling methods in yielding iid Gaussian returns, and it also performs better than the CTS and TTS schemes in estimating daily realized volatility using the TRV method. We also show the superiority of the BTS durations over the price durations in estimating the daily integrated volatility using the ACD-ICV method. Our modified ACD-ICV estimate, ME3, which models the high-frequency BTS durations using the ACD model, performs the best in reporting smaller RMSE values.

Finally, we note that there are other possible applications of the BTS scheme. For example, we can estimate the conditional instantaneous volatility or intraday integrated volatility (such as over 30 min intervals) by modeling the high-frequency BTS durations using the ACD model. Moreover, we can estimate the integrated quarticity ($IQ={\int }_0^1{\sigma }^4\left(\tau \right)d\tau$) by using a method similar to that implemented in this paper. That is, we first obtain transactions with approximately equal quarticity increments. The conditional instantaneous quarticity or integrated quarticity can be estimated further by modeling clustering of the corresponding durations using the ACD model. Please refer to Jacod and Rosenbaum (2013) and Andersen et al. (2014), among others, for theories and empirical applications of the asset price quarticity.