Abstract Multipower estimators, widespread for their robustness to the presence of jumps, are also useful for reducing the estimation error of integrated volatility powers even in the absence of jumps. Optimizing linear combinations of multipowers can indeed drastically reduce the variance with respect to traditional estimators. In the case of quarticity, we also prove that the optimal combination is a nearly efficient estimator, being arbitrarily close to the nonparametric efficiency bound as the number of consecutive returns employed diverges. We provide guidance on how to select the optimal number of consecutive returns to minimize mean square error. The implementation on U.S. stock prices corroborates our theoretical findings and further shows that our proposed quarticity estimator noticeably reduces the number of detected jumps, and improves the quality of volatility forecasts. 1 Motivation A very active literature in financial econometrics focuses on the estimation of integrated volatility powers. The most popular example is realized volatility, which is an estimator of the integrated variance and attracted enormous recent interest. The integral of squared variance, dubbed quarticity, is also quite important since it is a necessary ingredient in relevant applications, such as estimating the confidence bands for integrated variance estimates (see, e.g., Andersen, Dobrev, and Schaumburg, 2014), applying jump tests (see, e.g., Dumitru and Urga, 2012, for a review), forecasting volatility (Bollerslev, Patton, and Quaedvlieg, 2016), and computing the optimal sampling frequency in the presence of market microstructure noise (see, e.g., Bandi and Russell, 2006). The integrated third volatility power is needed, for example, to build confidence bands in the estimator developed by Reiß (2011) to obtain efficiency in the presence of microstructure noise. Integrated volatility powers are useful, among other things, to estimate the activity of processes modeled via a jump-diffusion (Todorov and Tauchen, 2011). Disentangling price volatility due to continuous movements to that due to discontinuous movements is also of fundamental importance in a number of equally relevant applications. In this field, a primary advancement has been represented by the introduction of multipower estimators, a tool introduced by Barndorff-Nielsen and Shephard (2004) to reduce the bias, due to jumps, in estimating integrated volatility powers. Successful modifications of multipower variation estimators have been proposed in Andersen, Dobrev, and Schaumburg (2012), and Corsi, Pirino, and Renò (2010), among others. Grounding on the success of this literature in estimating integrated volatility powers in the presence of price discontinuities, this paper concentrates on efficiency, and specifically on how to minimize the variance of multipower estimators. The theory is first developed in the case without jumps. This allows to study the variance minimization problem in the simplified case of continuous semimartingales. Our first contribution is to provide a general criterion to find the optimal (in the sense of achieving minimum variance) linear combination of N multipower estimators given the number m of consecutive returns employed (e.g., m = 2 for bipower variation, m = 3 for tripower variation, and so on). We present explicit solutions in relevant cases. This shows that multipower estimators are useful even in the absence of jumps, since their usage reduces the variance of the estimator. The (empirically compelling) case with jumps is then recovered by applying the same logic to truncated returns, in the spirit of Mancini (2009). Multipower estimators are indeed robust to jumps only for powers less than 2, and their asymptotic variance depends on jumps as well. Applying the truncation method we get rid of both problems. When estimating integrated variance, standard realized variance (i.e., the sum of squared returns) is an efficient estimator, achieving the nonparametric efficiency bound (Renault, Sarisoy, and Werker, 2016). For other integrated volatility powers, the variance minimization problem is not so straightforward. As discussed in Renault, Sarisoy, and Werker (2016), Mykland and Zhang (2009) provide a nearly efficient estimator, meaning that its variance can be made arbitrarily close to the nonparametric efficiency bound, while Jacod and Rosenbaum (2013) provide a fully efficient estimator. In the case of quarticity, both these estimators can be viewed as linear combinations of multipower variations, with the number of consecutive returns m growing to infinity. Our methodology allows to optimize the variance of the estimator when the number of consecutive returns m is fixed. In this respect, our second contribution is to propose a novel quarticity estimator which is nearly efficient, as that proposed by Mykland and Zhang (2009), but with a smaller variance. Minimization of the variance can be readily exploited to improve the mean square error of any multipower estimator, as well as of the “nearest neighbor” estimators of Andersen, Dobrev, and Schaumburg (2012), which do not require truncation. Indeed, the gain in variance as m gets larger comes at the cost of a deterioration of the bias. However, these two effects can be traded off to select the value of m which minimizes the mean square error, so that the loss in terms of bias is more than compensated by the gain in terms of variance. Thus, by construction, our nearly efficient estimator can only lower the mean square error with respect to the original estimator. Our theoretical contributions have then the immediate practical application of delivering superior estimates of integrated volatility powers. Our theoretical results are corroborated by simulated experiments and empirical applications, both focused on estimating quarticity. Using simulations, we show that the quarticity estimator we propose performs better than competing alternatives, including standard multipowers, nearest neighbor estimators, and the Jacod and Rosenbaum (2013) estimator. This result is robust to the presence of frictions (market microstructure noise and flat pricing) which are not dealt with by our theory but are introduced in the data generating process for simulated returns. When we apply the proposed quarticity estimator to a set of intraday prices of liquid U.S. stocks, we confirm that estimates obtained with our estimator are less variable than competing estimators, and suffer less distortion. An immediate sanity check is that estimated quarticity should be larger than squared estimated integrated variance, the relation being exact asymptotically due to Jensen’s inequality. In our sample, with our estimator this happens in 87% of the cases, a figure much larger than what obtained with standard multipowers (58% with tripower, 49% with quadpower) and with standard nearest neighbor estimators (64% with min, 67% with med). This result is consistent with much smaller estimation error due to efficiency gains. Having at disposal superior quarticity estimates has relevant empirical implications. We provide two examples. The first shows that the adoption of our quarticity estimator for standard jump tests delivers drastically less jumps than what was obtained with standard multipowers. This finding supports the recent empirical literature (see, e.g., Christensen, Oomen, and Podolskij, 2014; Bajgrowicz, Scaillet, and Treccani, 2016) claiming that the number of jumps typically detected in high frequency data is spuriously excessive, suggesting that a significant part of spuriously detected jumps could be due to inefficient quarticity estimation. In the second example, we adopt the specification proposed by Bollerslev, Patton, and Quaedvlieg (2016) to forecast volatility. In this setting, quarticity is used to correct the coefficients of the heterogeneous autoregression (HAR) model of Corsi (2009). Also in this case, we show that our quarticity estimator improves the quality of realized volatility forecasts with respect to standard quarticity measures. Both results clearly emphasize the empirical potential of our contribution. The paper is structured as follows. In Section 2, we describe the theoretical framework, define the class of multipower estimators, and provide an expression for minimum variance estimators. Section 3 studies our proposed (nearly) efficient quarticity estimator. Section 4 reports the Monte Carlo study, whereas Section 5 reports the empirical application. Section 6 concludes. The code to compute the efficient multipower estimator is available as supplementary data at online. 2 Efficient Multipowers In a filtered probability space satisfying the usual conditions (see, e.g., Protter, 2004), denote by Xt (e.g., the log-price of a financial stock or a stock index) an Itô semimartingale of the form dXt=μtdt+σtdWt+dJt, (2.1) where μt is the predictable, σt is a non-negative and a càdlàg, and Jt is a jump process. The estimation target is the integrated volatility power over a finite interval, VT(R)=∫0T(σs)Rds, for a given R > 0. The most important cases in practice are VT(2) (integrated variance) and VT(4) (quarticity). 2.1 The Case J = 0 We start with the case without jumps, in which the methodology can be outlined clearly. The empirically compelling case with jumps is treated in the following section. Define ΔiX=XiT/n−X(i−1)T/n, for i=1,…,n, and write Δn=T/n. Multipower estimators (Barndorff-Nielsen and Shephard, 2004, 2006) are defined as follows. Consider an m–valued real vector r=[r1,…,rm] with positive coefficients, and let R=∑j=1mrj. We define the multipower variation estimator MPV(r) with powers r as MPV(r)=cr·∑i=1n−m+1(∏j=1m|Δi+j−1X|rj), (2.2) where the constant cr, meant to make the estimator unbiased in small samples under the assumption of constant volatility, is defined by cr=(nT)R2−1nn−(m−1)(∏j=1m(μrj)−1), where μr=E(|u|r) with u being a standard normal. Important special cases are bipower variation, used to estimate integrated variance: MPV([1,1])=π2nn−1∑i=1n−1|ΔiX||Δi+1X| and tripower and quadpower variation, both used to estimate quarticity: MPV([4/3,4/3,4/3])=nT1μ433nn−2∑i=1n−2|ΔiX|43|Δi+1X|43|Δi+2X|43,MPV([1,1,1,1])=nT(π2)2nn−3∑i=1n−3|ΔiX||Δi+1X||Δi+2X||Δi+3X|. When Jt = 0, it has been proven, see for example, Barndorff-Nielsen et al. (2006a), that MPV([r]) is a consistent and asymptotically normally distributed estimator, as n→∞, of ∫0T(σs)Rds. Under mild assumptions on the model (2.1), the following stable central limit theorem holds: 1Δn(MPV(r)−∫0T(σs)Rds)⇒n→∞MN(0,Vr∫0T(σs)2Rds), (2.3) where MN denotes a standard mixed normal distribution, and Vr=∏j=1mμ2rj−(2m−1)∏j=1mμrj2+2∑i=1m−1∏j=1iμrj∏j=m−i+1mμrj∏j=1m−iμrj+rj+i∏j=1mμrj2, (2.4) see Theorem 3 in Barndorff-Nielsen et al. (2006a) for details. In the large empirical literature that uses multipower estimators, these are invariably implemented with equal powers: bipower variation with r=[1,1], tripower variation with r=[4/3,4/3,4/3], quadpower variation with r=[1,1,1,1], and so forth. However, minimizing Vr in formula (2.4) reveals that the variance of the estimator can be reduced considerably by changing powers. For example, while with the standard tripower variation, we have V[4/3,4/3,4/3]=13.65, the optimal choice with three powers would be V[3.5455,0.2182,0.2362]=9.70, that is a gain of roughly 30% on the asymptotic variance. This fact has also been remarked in the paper of Mancini and Calvori (2012). As we show next, combining estimators with different powers is even more efficient.1 This also implies that it makes perfect sense to use multipower estimators even in the absence of jumps. Central limit theorems of form (2.3) hold for virtually all estimators of integrated volatility powers used in the financial econometrics. Consequently, it is natural to study efficiency in the class of such estimators in the first place. Denote by ER the class of estimators in the model defined by Equation (2.1), satisfying: if ER,n∈ER, ER,n−∫0T(σs)Rds→p0, 1Δn(ER,n−∫0T(σs)Rds)⇒n→∞MN(0,VER,n∫0T(σs)2Rds), where MN denotes a standard mixed normal distribution, and VER,n is a constant. We say, that an estimator ER,n∈ER is efficient with respect to another estimator E′R,n∈ER, if VER,n≤VE′R,n. Clearly, this implies that for every fixed T and every volatility path, the variance of random variable ER,n is not larger than the variance of E′R,n. The lower bound V* for the constants VER,n among all estimators ER,n from ER is equal to the Cramer–Rao lower bound in the model with constant volatility ( σt=1), that is R2/2. Indeed, any estimator ER,n∈ER is in particular a consistent and asymptotically normal with variance VER,n in the model with σt=1. Hence, VER,n must not exceed the Cramer–Rao lower bound. If for an estimator ER,n∈ER, VER,n is equal to the Cramer–Rao lower bound, the inequality VER,n≤VE′R,n holds for any other estimator E′R,n∈ER. In that case, ER,n is efficient in the class ER, since the constants VER,n and VE′R,n are independent from the realization of the volatility process. Renault, Sarisoy, and Werker (2016) have recently established a similar but stronger result (see Example 1 in their paper): the nonparametric lower bound for all “regular” estimators of integrated volatility powers ∫0T(σs)Rds in model (2) is given by R22∫0T(σs)2Rds. (2.5) This implies that an efficient estimator in the class ER is also efficient among all regular estimators in the sense of Renault, Sarisoy, and Werker (2016). We start by investigating efficiency among multipowers only. Clearly, this does not include all possible estimators in the class ER, so that the minimum variance estimator among multipowers will not necessarily be efficient in the above sense. In the next section, we specialize on quarticity (R = 4) and show that the corresponding estimator is nearly efficient, again in the sense of Renault, Sarisoy, and Werker (2016), meaning that its variance can get arbitrary close to the nonparametric lower bound. We now define the class of estimators inside which we look for the one with minimum variance. We call this estimator MPV-efficient to highlight this is efficient only in the restricted class in which we fix the number N of multipower estimators employed in the linear combination, and the (maximum) number m of consecutive returns employed. Minimization is achieved with respect to the weights of the linear combination and to the powers to which returns are raised. We write a linear combination of N different MPV(r) estimators which have the maximum number of adjacent returns equal to m as: GMPV(N,m;σR)=∑j=1NwjMPV(r(j)), (2.6) where G stands for “generalized”, and the weights wj,j=1,…,N are such that ∑j=1Nwj=1 and for each j=1,…,N, r(j) is a vector in ℝm with non-negative components such that ∑i=1mri(j)=R. By construction, GMPV(N,m;σR) is a consistent estimator of VT(p), that is GMPV(N,m;σR)→p∫0T(σs)Rds as n→∞ with fixed N and m. Our problem is to find the efficient estimator in this class, that is the problem of finding the optimal weights wj and power vectors r(j) delivering the minimum asymptotic variance of GMPV(N,m;σR). We denote by GMPV*(N,m;σR) the MPV-efficient estimator in this class for fixed N, m, R. The following proposition provides the desired result. Proposition 2.1 The MPV-efficient estimator in the class of linear combinations of multipower estimators, for given N, m and R, is given by: GMPV*(N,m;σR)=∑j=1Nwj*MPV(r*(j)), (2.7)where the efficient ℝm-valued power vectors r*(1),…,r*(N) minimize the quantity V(N,m,R,r(1),…,r(N))=1∑i=1N∑j=1NCij−1(r(1),…,r(N)),where C−1 is the inverse of the N × N symmetric matrix C defined as: Cij(r(1),…,r(N))=(∏k=1mμrk(i)μrk(j))−1(∏k=1mμrk(i)+rk(j)+∑k=1m−1∏ℓ=1kμrℓ(i)∏ℓ=1m−kμrk+ℓ(i)+rℓ(j)∏ℓ=m−k+1mμrℓ(i)+∑k=1m−1∏ℓ=1kμrℓ(j)∏ℓ=1m−kμrk+ℓ(j)+rℓ(i)∏ℓ=m−k+1mμrℓ(j)−(2m−1)∏k=1mμrk(i)μrk(j)), (2.8)for i,j=1,…,N; and the efficient weights coefficient wj⋆, j=1,…,N, are given by: wj⋆=V˜(N,m;σR)∑i=1NCij−1(r*(1),…,r*(N)), (2.9)where V˜(N,m;σR)=V(N,m,R;r*(1),…,r*(N)). For the MPV-efficient estimator it holds: 1Δn(GMPV*(N,m;σR)−∫0T(σs)Rds)⇒n→∞MN(0,V˜(N,m;σR)∫0T(σs)2Rds), (2.10)where the above convergence is stable in law. Proof See Appendix A. □ The above proposition provides a procedure that allows to find, by numerical optimization, the minimum variance combination with fixed N, m, and R. For example, when R = 3 the MPV-efficient estimator with N = 1, m = 2 is found to be GMPV*(1,2;σ3)=MPV([0.1358,2.8642]), with V˜(1,2;σ3)=4.7947, the lower bound imposed by maximum likelihood being 4.5 in this case. Adding more estimators in a linear combination would further minimize variance. So, when N = 2 the MPV-efficient estimator is found to be GMPV*(2,2;σ3)=0.2634MPV([1.5,1.5])+0.7366MPV([3,0]), (2.11) with V˜(2,2;σ3)=4.7324; while when N = 3 it is GMPV*(3,2;σ3)=−0.1345MPV([0.6868,2.3132])+0.3639MPV([1.3020,1.6980]) +0.7706MPV([0.0016,2.9984]), (2.12) with V˜(3,2;σ3)=4.7320. It is interesting to look, always in the case R = 3, at the MPV-efficient estimator with N=3,m=3, which is found to be GMPV*(3,3;σ3)=0.1865MPV([0,1.5,1.5])+0.1865MPV([1.5,0,1.5]) (2.13) +0.6270MPV([0,3,0]), (2.14) with V˜(3,3;3)=4.6666. We can conjecture the emergence of a clear structure for the case N = m. We will exploit this structure in the definition of our efficient quarticity estimator. Tables 1 and 2 report the values of V˜(m,N;σR) when fixing m and N in the cases R = 3 and R = 4, respectively. We can see that, as m and N increase, these values become smaller and closer to the globally efficient value R2/2. In Section 3, we will show that, when R = 4, the MPV-efficient estimator with m = N can achieve the efficient value as m→∞. Table 1 Reports the values of V˜(m,N;σ4) for several values of m and N N m 1 2 3 4 5 1 10.666 2 10.050 9.600 3 9.701 9.314 9.143 4 9.478 9.119 8.989 8.965 5 9.323 9.019 8.856 8.850 8.801 N m 1 2 3 4 5 1 10.666 2 10.050 9.600 3 9.701 9.314 9.143 4 9.478 9.119 8.989 8.965 5 9.323 9.019 8.856 8.850 8.801 Note: The globally efficient value is 8. Table 1 Reports the values of V˜(m,N;σ4) for several values of m and N N m 1 2 3 4 5 1 10.666 2 10.050 9.600 3 9.701 9.314 9.143 4 9.478 9.119 8.989 8.965 5 9.323 9.019 8.856 8.850 8.801 N m 1 2 3 4 5 1 10.666 2 10.050 9.600 3 9.701 9.314 9.143 4 9.478 9.119 8.989 8.965 5 9.323 9.019 8.856 8.850 8.801 Note: The globally efficient value is 8. Table 2 Reports the values of V˜(m,N;σ3) for several values of m and N N m 1 2 3 4 5 1 4.891 2 4.795 4.732 3 4.745 4.693 4.667 4 4.714 4.667 4.666 4.641 5 4.693 4.653 4.643 4.629 4.620 N m 1 2 3 4 5 1 4.891 2 4.795 4.732 3 4.745 4.693 4.667 4 4.714 4.667 4.666 4.641 5 4.693 4.653 4.643 4.629 4.620 Note: The globally efficient value is 4.5. Table 2 Reports the values of V˜(m,N;σ3) for several values of m and N N m 1 2 3 4 5 1 4.891 2 4.795 4.732 3 4.745 4.693 4.667 4 4.714 4.667 4.666 4.641 5 4.693 4.653 4.643 4.629 4.620 N m 1 2 3 4 5 1 4.891 2 4.795 4.732 3 4.745 4.693 4.667 4 4.714 4.667 4.666 4.641 5 4.693 4.653 4.643 4.629 4.620 Note: The globally efficient value is 4.5. 2.2 The Case J≠0 The case with J≠0 makes the analysis of the efficiency of multipower estimators more complicated. Consistency of multipower estimators is indeed lost, in the presence of jumps, if max{r1,…,rm}≥2, and the asymptotic variance of the multipower estimator also contains a part influenced by jumps if max{r1,…,rm}≥1, see Barndorff-Nielsen et al. (2006a); Barndorff-Nielsen, Shephard, and Winkel (2006b); Veraart (2010); Vetter (2010); and Woerner (2006). Thus, the central limit theorem in Equation (2.3) only holds with these two constraints. However, a simple solution at hand is provided by applying multipower estimators to truncated returns. Consider a positive stochastic process ϑt, which we call a threshold (Mancini, 2009). Write ϑt=ξtΘ(Δ), with Θ(Δ) being a real function satisfying limΔ→0Θ(Δ)=0, limΔ→0Δlog1ΔΘ(Δ)=0, (2.15) and ξt being a stochastic process on [0,T] which is a.s. bounded and with a strictly positive lower bound. Using the threshold, we define truncated equally spaced returns, observed over the interval [0,T] as Δ¯iX=(ΔiX)I{|ΔiX|≤ϑ(i−1)T/n}, i=1,…,n, (2.16) where I{A} is the indicator function of the set A. The truncation is meant to annihilate returns larger than a given threshold, while leaving all the remaining returns unchanged. We define the threshold multipower variation estimator TMPV(r) (Corsi, Pirino, and Renò, 2010) as TMPV(r)=c′r·∑i=1n−m+1(∏j=1m|Δ¯i+j−1X|rj), (2.17) where the constant c′r, meant to make the estimator unbiased in small sample under the assumption of constant volatility, is now defined by c′r=(nT)R2−1nn−(m−1)−nJ(∏j=1m(μrj)−1), (2.18) where nJ is the number of terms vanishing in the sum in Equation (2.17) because of the indicator function. Then, under mild conditions for process (2.1), the following stable central limit continues to hold as n→∞: 1Δn(TMPV(r)−∫0T(σs)Rds)⇒MN(0,Vr∫0T(σs)2Rds), (2.19) see Theorem 2.3 in Corsi, Pirino, and Renò (2010) for the case with finite activity jumps, and Theorem 13.2.1 in Jacod and Protter (2012) for the general case allowing for infinite activity jumps. Using truncated returns, the results in Section 2.1 can be readily recovered. The disadvantage of the truncation is that it entails an additional parameter, that is the threshold ϑ. Moreover, truncating implied a finite sample bias due to the elimination of large, but continuous, returns. However, using a reasonably high threshold (in this paper, we always use five “local” standard deviations) will leave out very big jumps only, whose impact would be the largest, while remaining small jumps are dealt with the multipower technique. This double-sword feature smoothes the hurdle of having to select an additional threshold and makes the proposed estimators virtually immune from the bias due do the presence of the jumps, so that we can concentrate on variance reduction.2 An alternative way to avoid threshold selection has been proposed by Andersen, Dobrev, and Schaumburg (2012) and robustified in Andersen, Dobrev, and Schaumburg (2014), consisting in a “comparison” method based on the nearest neighbors. We also consider this kind of estimators in the Monte Carlo study and empirical application, namely the minRQ estimator, defined as minRQ=π3π−8n2n−1∑i=1n−1min(|ΔiX|,|Δi+1X|)4, (2.20) and the medRQ estimators, defined as medRQ=3π9π+72−523n2n−2∑i=1n−2med(|ΔiX|,|Δi+1X|,|Δi+2X|)4. (2.21) Both estimators are consistent for quarticity in the presence of jumps, allow for a central limit theorem in the same form of Equation (2.19) with Vr replaced by 18.54 for the minRQ estimator, and by 14.16 for the medRQ estimator. Thus, the nonlinear structure of these estimators implies a quite large asymptotic variance, which means that the combined use of neighbor estimators and efficient multipowers, as suggested in Section 3.1, also leads to improvement of the estimator in the mean square error sense. Moreover, also nearest neighbor estimators might be truncated. Finally, changing nearest neighbor powers could also be of help with the variance, even if this theoretical problem appears more challenging. In what follows, we cast the theory in the more realistic case in which Jt≠0 using truncated returns, and we restrict our attention to threshold multipower estimators. 3 A Nearly Efficient Quarticity Estimator In practice, two cases appear to be more relevant: the case R = 2 (integrated volatility) and the case R = 4 (integrated quarticity). In the case R = 2, the efficient multipower estimator, for every value of m and N, is simply TMPV([2]), that is the threshold realized variance proposed by Mancini (2009). The case with R = 2 is the unique value of R for which an efficient estimator can be constructed so easily. The case R = 4 (quarticity) is more intriguing and particularly important in financial applications: it is used, for example, in determining the optimal sampling frequency in the presence of market microstructure noise (Bandi and Russell, 2006); in computing jump tests; in determining the confidence intervals of realized variance; in forecasting volatility; see, for example, the discussion in Balter (2015), who provides an estimator based on the observation of the whole price path. Based on the analysis in Section 2.1, we propose to use the MPV-efficient quarticity estimator when N = m, that is: GTMPV**(m;σ4)=32m+1TMPV([4])+22m+1∑j=0m−2TMPV([2,0,…,0︸j terms,2]). (3.1) The weights in Equation (3.1) sum up to 1, so that GTMPV**(m;σ4) is a consistent estimator, as n→∞, of VT(4) for every fixed value of m. The next proposition provides the asymptotic distribution of GTMPV**(m;σ4) for a fixed m. Proposition 3.1 As n→∞, if m is fixed, 1Δn(GTMPV**(m;σ4)−∫0Tσs4ds)⇒MN(0,V**(m,σ4)∫0Tσs8ds), (3.2)where V**(m,σ4)=8+82m+1, (3.3)and the above convergence is stable in law. Proof See Appendix A. □ Proposition 3.1 shows that estimator (3.1), when defined for a fixed m, is nearly efficient in the sense of Renault, Sarisoy, and Werker (2016), since its variance converges to 8∫0Tσs8ds (the nonparametric efficiency bound) as m→∞ (see their discussion in Section 4). Our estimator is also closely related to that proposed by Jacod and Rosenbaum (2013). Proposition 3.2 implies that, when m,n→∞ jointly with m2/n→0, estimator (3.1) is fully efficient. Indeed the Jacod and Rosenbaum quarticity estimator can also be written, ignoring negligible end-effects, as a linear combination of multipower estimators (see the proof of Proposition 3.2), so that the two estimators share the same asymptotic properties, including full efficiency in the case of regular sampling times as demonstrated by Renault, Sarisoy, and Werker (2016). When m is fixed, the difference between the Jacod and Rosenbaum quarticity estimator and GTMPV**(m;σ4) consists in the weights of the linear combination. For estimator (3.1), the weights are purposely designed to deliver the minimum variance with a fixed m. Thus, estimator (3.1) has smaller variance than the Jacod and Rosenbaum (2013) estimator implemented with the same window (while sharing its asymptotic properties as m→∞ at a given rate). As we discuss below, this fact is very important in practice. Indeed, larger m also implies larger bias. When the objective is the minimization of a loss function which depends on both variance and bias, as it is customary, an intermediate value of m is optimal. Thus, minimizing the variance in the fixed m case is beneficial even if full efficiency is lost. The problem of improving efficiency when m is fixed has also been studied by Mykland and Zhang (2009), who also propose a nearly efficient block estimator for integrated volatility powers which is Uniformly Minimum Variance Unbiased (UMVU) in each block.3 This clearly improves the asymptotic variance, but does not explore (as we do here) the possibility of interaction among blocks. For this reason, our estimator has smaller variance. The asymptotic variance of the Mykland and Zhang (2009) estimator when R = 4 is indeed given, from Equation (58) in their paper, by: VMZ(m,σ4)=8+8m2+2mm2−1 (3.4) so that the relative efficiency compares favorably for GTMPV**(m;σ4) since VMZ(m,σ4)V**(m,σ4)=12m(2m+1)(m+2)(m+1)2(m−1)>1. (3.5)Figure 1 shows the asymptotic relative efficiency (ARE) of both estimators as a function of m, showing that the advantage in using the efficient multipowers estimator can be quit large for small values of m. The figure also shows the ARE of the standard multipower estimator TMPV([4/m,…,4/m︸m terms]) with all equal powers (for m = 1 it trivially coincides with the efficient estimator; for m = 2 this is bipower variation, for m = 3 tripower variation, for m = 4 quadpower variation, and so on), which shows that using equal powers for large m is definitively not the best option; and the ARE of the minRQ (compared with the case m = 2) and the medRQ estimator (compared with the case m = 3), which shows that the min–med estimators have lower relative efficiency than traditional multipowers. For both GTMPV**(m;σ4) and the block estimator, the ARE increases with m, converging to 1 when m→∞. The convergence is however faster for GTMPV**(m;σ4). Figure 1 View largeDownload slide Asymptotic relative efficiency (with respect to the case m→∞) for quarticity estimation. Several estimators are compared: the GTMPV** estimator in Equation (3.1), the Mykland and Zhang (2009) block estimator, the standard multipower estimator with all equal powers, and the minRQ and medRQ estimator. Figure 1 View largeDownload slide Asymptotic relative efficiency (with respect to the case m→∞) for quarticity estimation. Several estimators are compared: the GTMPV** estimator in Equation (3.1), the Mykland and Zhang (2009) block estimator, the standard multipower estimator with all equal powers, and the minRQ and medRQ estimator. 3.1 The Choice of the Optimal m: Bias Considerations Achieving (near) efficiency is clearly not the end of the story. Typically, the objective is the minimization of a loss function which depends also on the bias, such as the mean square error. While the variance tends to decrease with m, the bias tends to increase with m, thus originating the usual bias–variance tradeoff. A convenient expression of the asymptotic bias is provided by the next proposition, in which m is allowed to diverge at a suitable rate. The result borrows from the work of Jacod and Rosenbaum (2013). Proposition 3.2 Assume that volatility is driven by the process dσt2=μtσdt+ΛtdWt+dJtσ,where μtσ is a predictable process, Λt is a càdlàg, and Jtσ is a jump process. If n,m→∞ in such a way that m2/n→θ with θ>0, we have 1Δn(GTMPV**(m;σ4)−∫0Tσs4ds)⇒B1+B2+B3+MN(0,8∫0Tσs8ds), (3.6) where B1=−θ2(σ04+σT4), (3.7) B2=−θ6∫0TΛs2ds, (3.8) B3=−θ6∑(Δσs2)2, (3.9)and Δσs2=σs2−σs−2 are the jumps in the variance process. The (asymptotic) bias thus consists of three negative terms, the first due to the border effect, and the second and the third due to the variability of volatility. We can take advantage of the small sample approximation of the bias provided by Proposition 3.2 with the small sample approximation of the variance provided by Proposition 3.1 to get the following approximation of the mean square error: MSE≈Δn(8+82m+1)Q2+Δn3m4B2, (3.10) where Q2=∫0Tσs8ds and B=12(σ04+σT4)+16(∫0TΛs2ds+∑(Δσs2)2). In principle, this MSE could be estimated from the data as a function of m, which could then be optimized. The main problem would be to estimate the volatility of volatility term ∫0TΛs2ds+∑(Δσs2)2 with sufficiently low error. To gain feeling of what we can get, we set ∫0Tσtdt=1, we ignore the end-effect term 12(σ04+σT4) since we use the constant cr in Equation (2.18) to compensate for it, and optimize the MSE for various values of Λ¯=(∫0TΛs2ds+∑(Δσs2)2)1/2. Table 3 reports the optimal m we found for different choices of n, and shows that the optimal m would strongly depend on the volatility-of-volatility estimate. Given the notorious difficulty in estimating this parameter, we propose the following alternative approach. Table 3 Reports the values of mopt that optimize the mean square error (3.10) for different values of n and Λ¯=(∫0TΛs2ds+∑(Δσs2)2)1/2, in the case ∫0Tσt8dt=σ0=σT=1 Λ¯=0.1 Λ¯=0.5 Λ¯=1 Λ¯=2 Λ¯=3 Λ¯=5 Optimal m n = 40 56 15 9 5 4 2 n = 80 73 20 12 7 5 3 n = 400 109 38 22 13 9 6 Λ¯=0.1 Λ¯=0.5 Λ¯=1 Λ¯=2 Λ¯=3 Λ¯=5 Optimal m n = 40 56 15 9 5 4 2 n = 80 73 20 12 7 5 3 n = 400 109 38 22 13 9 6 Table 3 Reports the values of mopt that optimize the mean square error (3.10) for different values of n and Λ¯=(∫0TΛs2ds+∑(Δσs2)2)1/2, in the case ∫0Tσt8dt=σ0=σT=1 Λ¯=0.1 Λ¯=0.5 Λ¯=1 Λ¯=2 Λ¯=3 Λ¯=5 Optimal m n = 40 56 15 9 5 4 2 n = 80 73 20 12 7 5 3 n = 400 109 38 22 13 9 6 Λ¯=0.1 Λ¯=0.5 Λ¯=1 Λ¯=2 Λ¯=3 Λ¯=5 Optimal m n = 40 56 15 9 5 4 2 n = 80 73 20 12 7 5 3 n = 400 109 38 22 13 9 6 We assume to start from a quarticity estimator Q^, which might not be efficient, but which is assumed to be unbiased (e.g., any multipower or nearest neighbor estimator). Under this assumption, we can always improve (in the mean square error sense, or in the sense of any alternative loss function combining bias and variance) with respect to the estimator Q^ by choosing m* that minimizes the MSE: MSE˜=[GTMPV**(m;σ4)−Q^]2+Var(GTMPV**(m;σ4)), (3.11) where the variance of GTMPV**(m;σ4) can be estimated, using asymptotic expression (3.2), by Var(GTMPV**(m;σ4))≈(8+82m+1)TMPV([8/3,8/3,8/3]), or replacing TMPV([8/3,8/3,8/3]) by another consistent estimator of ∫0Tσs8ds.4 By construction, the estimator GTMPV**(m*;σ4) will have a smaller mean square error than the original estimator Q^. This is the technique we use in the empirical applications on jump testing and volatility forecasting, using Q^=TMPV([4]). Note that in finite samples there is another source of negative bias, due to the truncation of the largest observations. However, it is of smaller order with respect to the other two biases, hence it is not considered here. Moreover, with observed prices, another source of bias is market microstructure noise. The bias due to the microstructure noise can be eliminated by using pre-averaging approach (see, e.g., Hautsch and Podolskij, 2013). The variance of multi-power variations based on pre-averaged returns can be minimized by changing powers as in the present paper. However, since the variance of pre-averaged estimators depends (among other terms) on the moments of the noise, the lower bound for the variance of such estimators would be different from the one considered in the present paper. The determination of this bound, possibly by methods similar to the ones used by Renault, Sarisoy, and Werker (2016), is a self-contained problem outside the scope of this paper. This said, in the Monte Carlo section below we quantify the impact of market microstructure noise and flat trading explicitly. 4 Monte Carlo Simulations In this section, we perform a series of Monte Carlo experiments focusing on quarticity estimation at different frequencies. In particular, on a simulated typical trading day in the U.S. market of 6.5 h, we consider n = 40, 80, 400 which roughly corresponds to 10, 5, and 1-min returns, respectively. In order to generate a realistic price dynamics, we simulate the jump-diffusion model: dlogpt=μdt+γtσtdWp,t+dJtdlogσt2=(α−βlogσt2)dt+ηdWσ,t, (4.1) where Wp and Wσ are standard Brownian motions with corr(dWp,dWσ)=ρ, σt is a stochastic volatility factor, and γt is an intraday effect. We use the model parameters estimated by Andersen, Benzoni, and Lund (2002) on S&P500 prices: μ=0.0304,α=−0.012,β=0.0145,η=0.1153,ρ=−0.6127, where the parameters are expressed in daily units and returns are in percentage, and we use: γt,τ=10.1033(0.1271τ2−0.1260τ+0.1239), as estimated by us on S&P500 intraday returns. We discretize model (4.1) in the interval [0,1] with the Euler scheme, using a discretization step of 1/n. Instead of specifying the jump process as a compound Poisson process with random jump sizes we restrict its realizations to a fixed number of jumps of a known size. In particular, we consider the case of absence of jumps and the case of a single jump with deterministic size equal to 31/n (small jump, notice that in the simulations σt≃1) and 101/n (big jump). In order to make the Monte Carlo simulations more realistic, we further simulate additional frictions which are not considered in the theoretical framework of this paper, but are known to be present in the data. The frictions we consider are microstructure noise, in the form of distortions to the price process, and flat prices, that is prices that do not change due, for example, to liquidity reasons.5 We thus consider three possible scenarios: The price process is observed without frictions. The price process is contaminated by microstructure noise. The price process is contaminated by flat prices. In order to save space, since microstructure noise and flat pricing have the largest impact at high frequency, we study these frictions only at n = 400. To estimate quarticity we implement standard (non truncated) quadpower variations MPV([1,1,1,1]), threshold tripower and quadpower variations, that is TMPV([43,43,43]) and TMPV([1,1,1,1]), respectively; the single power TMPV([4]) estimator; and multipower estimators GTMPV**(m;σ4) for different values of m (for simplicity, we omit the σ4 in the notation). We also implement the nearest neighbor minRQ and medRQ estimators defined in Equations (2.20) and (2.21). Finally, we consider the efficient estimator QVeff(kn) of Jacod and Rosenbaum (2013), see Equation (A.4), implemented with the bias correction proposed in Jacod and Rosenbaum (2015) for different values of kn. In order to set up the threshold for truncated multipowers, we use ϑt=cϑ·σ^tn (4.2) with cϑ=5 and and σ^tn is an estimator of local standard deviation (that is over the interval 1/n) obtained as in Corsi, Pirino, and Renò (2010). The choice of cϑ=5 is meant to truncate only returns that are extremely large with respect to the estimated local standard deviation. On each replication we compute the generated quarticity value IQk and the estimated quarticity value IQ̂k according to different estimators ( k=1,…,M). We report the relative bias, Bias=1M∑k=1MIQ̂k−IQkIQk, the relative standard deviation, Std=1M∑k=1M(IQ̂k−IQkIQk−bias)2, and the relative root mean square error: RMSE=Std2+Bias2. The figures are computed with M = 10,000 replications. 4.1 Estimation Without Frictions Tables 4, 5, and 6 report the results in the case n = 40 (10 min), n = 80 (5 min), n = 400 (1 min), respectively, for the competing estimators, and in three cases: absence of jumps, presence of a single small jump, and presence of a single large jump. Table 4 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.6036 0.5998 −0.0674 3.6144 3.0225 1.9820 11.9077 9.6054 7.0378 TMPV([4]) 0.5353 0.5353 −0.0007 0.5286 0.5277 −0.0306 0.5388 0.5379 −0.0320 TMPV([43,43,43]) 0.5914 0.5898 −0.0441 0.5652 0.5520 −0.1218 0.5864 0.5747 −0.1167 TMPV([1,1,1,1]) 0.6041 0.6002 −0.0686 0.5758 0.5519 −0.1641 0.5946 0.5738 −0.1558 minRQ 0.7094 0.7092 −0.0166 1.1396 1.1123 0.2478 1.0556 1.0307 0.2278 medRQ 0.6053 0.6042 −0.0359 0.9878 0.9566 0.2461 0.9228 0.8931 0.2326 GTMPV**(4) 0.4751 0.4743 −0.0272 0.4719 0.4695 −0.0478 0.4878 0.4855 −0.0480 GTMPV**(8) 0.4432 0.4392 −0.0594 0.4405 0.4330 −0.0810 0.4546 0.4475 −0.0801 GTMPV**(12) 0.4278 0.4202 −0.0803 0.4299 0.4178 −0.1010 0.4385 0.4265 −0.1021 GTMPV**(16) 0.4220 0.4121 −0.0912 0.4253 0.4106 −0.1109 0.4326 0.4180 −0.1115 GTMPV**(20) 0.4198 0.4095 −0.0924 0.4249 0.4105 −0.1094 0.4328 0.4187 −0.1099 GTMPV**(24) 0.4213 0.4132 −0.0820 0.4259 0.4149 −0.0961 0.4344 0.4236 −0.0963 QVeff(4) 0.4785 0.4708 −0.0857 0.4727 0.4556 −0.1261 0.4894 0.4732 −0.1250 QVeff(8) 0.4777 0.4773 −0.0192 0.4636 0.4592 −0.0636 0.4799 0.4756 −0.0635 QVeff(12) 0.4557 0.4546 −0.0325 0.4460 0.4389 −0.0790 0.4566 0.4495 −0.0802 QVeff(16) 0.4376 0.4339 −0.0567 0.4310 0.4180 −0.1052 0.4387 0.4258 −0.1056 QVeff(20) 0.4268 0.4188 −0.0821 0.4256 0.4061 −0.1275 0.4319 0.4126 −0.1278 QVeff(24) 0.4266 0.4157 −0.0957 0.4267 0.4024 −0.1418 0.4325 0.4087 −0.1412 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.6036 0.5998 −0.0674 3.6144 3.0225 1.9820 11.9077 9.6054 7.0378 TMPV([4]) 0.5353 0.5353 −0.0007 0.5286 0.5277 −0.0306 0.5388 0.5379 −0.0320 TMPV([43,43,43]) 0.5914 0.5898 −0.0441 0.5652 0.5520 −0.1218 0.5864 0.5747 −0.1167 TMPV([1,1,1,1]) 0.6041 0.6002 −0.0686 0.5758 0.5519 −0.1641 0.5946 0.5738 −0.1558 minRQ 0.7094 0.7092 −0.0166 1.1396 1.1123 0.2478 1.0556 1.0307 0.2278 medRQ 0.6053 0.6042 −0.0359 0.9878 0.9566 0.2461 0.9228 0.8931 0.2326 GTMPV**(4) 0.4751 0.4743 −0.0272 0.4719 0.4695 −0.0478 0.4878 0.4855 −0.0480 GTMPV**(8) 0.4432 0.4392 −0.0594 0.4405 0.4330 −0.0810 0.4546 0.4475 −0.0801 GTMPV**(12) 0.4278 0.4202 −0.0803 0.4299 0.4178 −0.1010 0.4385 0.4265 −0.1021 GTMPV**(16) 0.4220 0.4121 −0.0912 0.4253 0.4106 −0.1109 0.4326 0.4180 −0.1115 GTMPV**(20) 0.4198 0.4095 −0.0924 0.4249 0.4105 −0.1094 0.4328 0.4187 −0.1099 GTMPV**(24) 0.4213 0.4132 −0.0820 0.4259 0.4149 −0.0961 0.4344 0.4236 −0.0963 QVeff(4) 0.4785 0.4708 −0.0857 0.4727 0.4556 −0.1261 0.4894 0.4732 −0.1250 QVeff(8) 0.4777 0.4773 −0.0192 0.4636 0.4592 −0.0636 0.4799 0.4756 −0.0635 QVeff(12) 0.4557 0.4546 −0.0325 0.4460 0.4389 −0.0790 0.4566 0.4495 −0.0802 QVeff(16) 0.4376 0.4339 −0.0567 0.4310 0.4180 −0.1052 0.4387 0.4258 −0.1056 QVeff(20) 0.4268 0.4188 −0.0821 0.4256 0.4061 −0.1275 0.4319 0.4126 −0.1278 QVeff(24) 0.4266 0.4157 −0.0957 0.4267 0.4024 −0.1418 0.4325 0.4087 −0.1412 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n=40 intraday returns generated by model ( 4.1). The threshold is set as in Equation (4.2) with cϑ=5. Table 4 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.6036 0.5998 −0.0674 3.6144 3.0225 1.9820 11.9077 9.6054 7.0378 TMPV([4]) 0.5353 0.5353 −0.0007 0.5286 0.5277 −0.0306 0.5388 0.5379 −0.0320 TMPV([43,43,43]) 0.5914 0.5898 −0.0441 0.5652 0.5520 −0.1218 0.5864 0.5747 −0.1167 TMPV([1,1,1,1]) 0.6041 0.6002 −0.0686 0.5758 0.5519 −0.1641 0.5946 0.5738 −0.1558 minRQ 0.7094 0.7092 −0.0166 1.1396 1.1123 0.2478 1.0556 1.0307 0.2278 medRQ 0.6053 0.6042 −0.0359 0.9878 0.9566 0.2461 0.9228 0.8931 0.2326 GTMPV**(4) 0.4751 0.4743 −0.0272 0.4719 0.4695 −0.0478 0.4878 0.4855 −0.0480 GTMPV**(8) 0.4432 0.4392 −0.0594 0.4405 0.4330 −0.0810 0.4546 0.4475 −0.0801 GTMPV**(12) 0.4278 0.4202 −0.0803 0.4299 0.4178 −0.1010 0.4385 0.4265 −0.1021 GTMPV**(16) 0.4220 0.4121 −0.0912 0.4253 0.4106 −0.1109 0.4326 0.4180 −0.1115 GTMPV**(20) 0.4198 0.4095 −0.0924 0.4249 0.4105 −0.1094 0.4328 0.4187 −0.1099 GTMPV**(24) 0.4213 0.4132 −0.0820 0.4259 0.4149 −0.0961 0.4344 0.4236 −0.0963 QVeff(4) 0.4785 0.4708 −0.0857 0.4727 0.4556 −0.1261 0.4894 0.4732 −0.1250 QVeff(8) 0.4777 0.4773 −0.0192 0.4636 0.4592 −0.0636 0.4799 0.4756 −0.0635 QVeff(12) 0.4557 0.4546 −0.0325 0.4460 0.4389 −0.0790 0.4566 0.4495 −0.0802 QVeff(16) 0.4376 0.4339 −0.0567 0.4310 0.4180 −0.1052 0.4387 0.4258 −0.1056 QVeff(20) 0.4268 0.4188 −0.0821 0.4256 0.4061 −0.1275 0.4319 0.4126 −0.1278 QVeff(24) 0.4266 0.4157 −0.0957 0.4267 0.4024 −0.1418 0.4325 0.4087 −0.1412 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.6036 0.5998 −0.0674 3.6144 3.0225 1.9820 11.9077 9.6054 7.0378 TMPV([4]) 0.5353 0.5353 −0.0007 0.5286 0.5277 −0.0306 0.5388 0.5379 −0.0320 TMPV([43,43,43]) 0.5914 0.5898 −0.0441 0.5652 0.5520 −0.1218 0.5864 0.5747 −0.1167 TMPV([1,1,1,1]) 0.6041 0.6002 −0.0686 0.5758 0.5519 −0.1641 0.5946 0.5738 −0.1558 minRQ 0.7094 0.7092 −0.0166 1.1396 1.1123 0.2478 1.0556 1.0307 0.2278 medRQ 0.6053 0.6042 −0.0359 0.9878 0.9566 0.2461 0.9228 0.8931 0.2326 GTMPV**(4) 0.4751 0.4743 −0.0272 0.4719 0.4695 −0.0478 0.4878 0.4855 −0.0480 GTMPV**(8) 0.4432 0.4392 −0.0594 0.4405 0.4330 −0.0810 0.4546 0.4475 −0.0801 GTMPV**(12) 0.4278 0.4202 −0.0803 0.4299 0.4178 −0.1010 0.4385 0.4265 −0.1021 GTMPV**(16) 0.4220 0.4121 −0.0912 0.4253 0.4106 −0.1109 0.4326 0.4180 −0.1115 GTMPV**(20) 0.4198 0.4095 −0.0924 0.4249 0.4105 −0.1094 0.4328 0.4187 −0.1099 GTMPV**(24) 0.4213 0.4132 −0.0820 0.4259 0.4149 −0.0961 0.4344 0.4236 −0.0963 QVeff(4) 0.4785 0.4708 −0.0857 0.4727 0.4556 −0.1261 0.4894 0.4732 −0.1250 QVeff(8) 0.4777 0.4773 −0.0192 0.4636 0.4592 −0.0636 0.4799 0.4756 −0.0635 QVeff(12) 0.4557 0.4546 −0.0325 0.4460 0.4389 −0.0790 0.4566 0.4495 −0.0802 QVeff(16) 0.4376 0.4339 −0.0567 0.4310 0.4180 −0.1052 0.4387 0.4258 −0.1056 QVeff(20) 0.4268 0.4188 −0.0821 0.4256 0.4061 −0.1275 0.4319 0.4126 −0.1278 QVeff(24) 0.4266 0.4157 −0.0957 0.4267 0.4024 −0.1418 0.4325 0.4087 −0.1412 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n=40 intraday returns generated by model ( 4.1). The threshold is set as in Equation (4.2) with cϑ=5. Table 5 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.4376 0.4362 −0.0356 2.7541 2.3016 1.5124 9.1583 7.4072 5.3860 TMPV([4]) 0.3761 0.3759 −0.0130 0.3806 0.3802 −0.0174 0.3805 0.3801 −0.0161 TMPV([43,43,43]) 0.4248 0.4240 −0.0263 0.4263 0.4209 −0.0676 0.4247 0.4202 −0.0620 TMPV([1,1,1,1]) 0.4380 0.4364 −0.0366 0.4413 0.4319 −0.0905 0.4342 0.4257 −0.0854 minRQ 0.5051 0.5049 −0.0153 0.6991 0.6877 0.1255 0.6497 0.6376 0.1246 medRQ 0.4362 0.4354 −0.0259 0.5935 0.5799 0.1265 0.5781 0.5629 0.1317 GTMPV**(5) 0.3361 0.3350 −0.0278 0.3406 0.3389 −0.0342 0.3391 0.3377 −0.0306 GTMPV**(10) 0.3208 0.3167 −0.0510 0.3221 0.3167 −0.0591 0.3217 0.3169 −0.0551 GTMPV**(15) 0.3138 0.3060 −0.0696 0.3136 0.3037 −0.0781 0.3134 0.3043 −0.0748 GTMPV**(20) 0.3091 0.2973 −0.0845 0.3100 0.2958 −0.0930 0.3090 0.2957 −0.0898 GTMPV**(30) 0.3065 0.2890 −0.1020 0.3081 0.2879 −0.1098 0.3070 0.2878 −0.1068 GTMPV**(40) 0.3072 0.2894 −0.1030 0.3078 0.2877 −0.1094 0.3070 0.2878 −0.1069 QVeff(5) 0.3427 0.3338 −0.0776 0.3461 0.3332 −0.0935 0.3460 0.3343 −0.0889 QVeff(10) 0.3419 0.3414 −0.0192 0.3411 0.3389 −0.0389 0.3414 0.3397 −0.0346 QVeff(15) 0.3349 0.3341 −0.0231 0.3315 0.3286 −0.0438 0.3320 0.3296 −0.0399 QVeff(20) 0.3261 0.3237 −0.0388 0.3215 0.3159 −0.0599 0.3219 0.3168 −0.0569 QVeff(30) 0.3117 0.3023 −0.0762 0.3126 0.2971 −0.0973 0.3111 0.2962 −0.0951 QVeff(40) 0.3095 0.2900 −0.1081 0.3118 0.2838 −0.1291 0.3102 0.2827 −0.1276 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.4376 0.4362 −0.0356 2.7541 2.3016 1.5124 9.1583 7.4072 5.3860 TMPV([4]) 0.3761 0.3759 −0.0130 0.3806 0.3802 −0.0174 0.3805 0.3801 −0.0161 TMPV([43,43,43]) 0.4248 0.4240 −0.0263 0.4263 0.4209 −0.0676 0.4247 0.4202 −0.0620 TMPV([1,1,1,1]) 0.4380 0.4364 −0.0366 0.4413 0.4319 −0.0905 0.4342 0.4257 −0.0854 minRQ 0.5051 0.5049 −0.0153 0.6991 0.6877 0.1255 0.6497 0.6376 0.1246 medRQ 0.4362 0.4354 −0.0259 0.5935 0.5799 0.1265 0.5781 0.5629 0.1317 GTMPV**(5) 0.3361 0.3350 −0.0278 0.3406 0.3389 −0.0342 0.3391 0.3377 −0.0306 GTMPV**(10) 0.3208 0.3167 −0.0510 0.3221 0.3167 −0.0591 0.3217 0.3169 −0.0551 GTMPV**(15) 0.3138 0.3060 −0.0696 0.3136 0.3037 −0.0781 0.3134 0.3043 −0.0748 GTMPV**(20) 0.3091 0.2973 −0.0845 0.3100 0.2958 −0.0930 0.3090 0.2957 −0.0898 GTMPV**(30) 0.3065 0.2890 −0.1020 0.3081 0.2879 −0.1098 0.3070 0.2878 −0.1068 GTMPV**(40) 0.3072 0.2894 −0.1030 0.3078 0.2877 −0.1094 0.3070 0.2878 −0.1069 QVeff(5) 0.3427 0.3338 −0.0776 0.3461 0.3332 −0.0935 0.3460 0.3343 −0.0889 QVeff(10) 0.3419 0.3414 −0.0192 0.3411 0.3389 −0.0389 0.3414 0.3397 −0.0346 QVeff(15) 0.3349 0.3341 −0.0231 0.3315 0.3286 −0.0438 0.3320 0.3296 −0.0399 QVeff(20) 0.3261 0.3237 −0.0388 0.3215 0.3159 −0.0599 0.3219 0.3168 −0.0569 QVeff(30) 0.3117 0.3023 −0.0762 0.3126 0.2971 −0.0973 0.3111 0.2962 −0.0951 QVeff(40) 0.3095 0.2900 −0.1081 0.3118 0.2838 −0.1291 0.3102 0.2827 −0.1276 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n=80 intraday returns generated by model ( 4.1). The threshold is set as in Equation (4.2) with cϑ=5. Table 5 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.4376 0.4362 −0.0356 2.7541 2.3016 1.5124 9.1583 7.4072 5.3860 TMPV([4]) 0.3761 0.3759 −0.0130 0.3806 0.3802 −0.0174 0.3805 0.3801 −0.0161 TMPV([43,43,43]) 0.4248 0.4240 −0.0263 0.4263 0.4209 −0.0676 0.4247 0.4202 −0.0620 TMPV([1,1,1,1]) 0.4380 0.4364 −0.0366 0.4413 0.4319 −0.0905 0.4342 0.4257 −0.0854 minRQ 0.5051 0.5049 −0.0153 0.6991 0.6877 0.1255 0.6497 0.6376 0.1246 medRQ 0.4362 0.4354 −0.0259 0.5935 0.5799 0.1265 0.5781 0.5629 0.1317 GTMPV**(5) 0.3361 0.3350 −0.0278 0.3406 0.3389 −0.0342 0.3391 0.3377 −0.0306 GTMPV**(10) 0.3208 0.3167 −0.0510 0.3221 0.3167 −0.0591 0.3217 0.3169 −0.0551 GTMPV**(15) 0.3138 0.3060 −0.0696 0.3136 0.3037 −0.0781 0.3134 0.3043 −0.0748 GTMPV**(20) 0.3091 0.2973 −0.0845 0.3100 0.2958 −0.0930 0.3090 0.2957 −0.0898 GTMPV**(30) 0.3065 0.2890 −0.1020 0.3081 0.2879 −0.1098 0.3070 0.2878 −0.1068 GTMPV**(40) 0.3072 0.2894 −0.1030 0.3078 0.2877 −0.1094 0.3070 0.2878 −0.1069 QVeff(5) 0.3427 0.3338 −0.0776 0.3461 0.3332 −0.0935 0.3460 0.3343 −0.0889 QVeff(10) 0.3419 0.3414 −0.0192 0.3411 0.3389 −0.0389 0.3414 0.3397 −0.0346 QVeff(15) 0.3349 0.3341 −0.0231 0.3315 0.3286 −0.0438 0.3320 0.3296 −0.0399 QVeff(20) 0.3261 0.3237 −0.0388 0.3215 0.3159 −0.0599 0.3219 0.3168 −0.0569 QVeff(30) 0.3117 0.3023 −0.0762 0.3126 0.2971 −0.0973 0.3111 0.2962 −0.0951 QVeff(40) 0.3095 0.2900 −0.1081 0.3118 0.2838 −0.1291 0.3102 0.2827 −0.1276 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.4376 0.4362 −0.0356 2.7541 2.3016 1.5124 9.1583 7.4072 5.3860 TMPV([4]) 0.3761 0.3759 −0.0130 0.3806 0.3802 −0.0174 0.3805 0.3801 −0.0161 TMPV([43,43,43]) 0.4248 0.4240 −0.0263 0.4263 0.4209 −0.0676 0.4247 0.4202 −0.0620 TMPV([1,1,1,1]) 0.4380 0.4364 −0.0366 0.4413 0.4319 −0.0905 0.4342 0.4257 −0.0854 minRQ 0.5051 0.5049 −0.0153 0.6991 0.6877 0.1255 0.6497 0.6376 0.1246 medRQ 0.4362 0.4354 −0.0259 0.5935 0.5799 0.1265 0.5781 0.5629 0.1317 GTMPV**(5) 0.3361 0.3350 −0.0278 0.3406 0.3389 −0.0342 0.3391 0.3377 −0.0306 GTMPV**(10) 0.3208 0.3167 −0.0510 0.3221 0.3167 −0.0591 0.3217 0.3169 −0.0551 GTMPV**(15) 0.3138 0.3060 −0.0696 0.3136 0.3037 −0.0781 0.3134 0.3043 −0.0748 GTMPV**(20) 0.3091 0.2973 −0.0845 0.3100 0.2958 −0.0930 0.3090 0.2957 −0.0898 GTMPV**(30) 0.3065 0.2890 −0.1020 0.3081 0.2879 −0.1098 0.3070 0.2878 −0.1068 GTMPV**(40) 0.3072 0.2894 −0.1030 0.3078 0.2877 −0.1094 0.3070 0.2878 −0.1069 QVeff(5) 0.3427 0.3338 −0.0776 0.3461 0.3332 −0.0935 0.3460 0.3343 −0.0889 QVeff(10) 0.3419 0.3414 −0.0192 0.3411 0.3389 −0.0389 0.3414 0.3397 −0.0346 QVeff(15) 0.3349 0.3341 −0.0231 0.3315 0.3286 −0.0438 0.3320 0.3296 −0.0399 QVeff(20) 0.3261 0.3237 −0.0388 0.3215 0.3159 −0.0599 0.3219 0.3168 −0.0569 QVeff(30) 0.3117 0.3023 −0.0762 0.3126 0.2971 −0.0973 0.3111 0.2962 −0.0951 QVeff(40) 0.3095 0.2900 −0.1081 0.3118 0.2838 −0.1291 0.3102 0.2827 −0.1276 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n=80 intraday returns generated by model ( 4.1). The threshold is set as in Equation (4.2) with cϑ=5. Table 6 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.2094 0.2092 −0.0096 1.2485 1.0169 0.7245 4.1396 3.3419 2.4431 TMPV([4]) 0.1772 0.1772 −0.0019 0.1748 0.1746 −0.0068 0.1749 0.1748 −0.0062 TMPV([43,43,43]) 0.2017 0.2016 −0.0069 0.1953 0.1948 −0.0148 0.1986 0.1981 −0.0141 TMPV([1,1,1,1]) 0.2094 0.2092 −0.0099 0.2034 0.2024 −0.0199 0.2069 0.2060 −0.0188 minRQ 0.2373 0.2373 −0.0015 0.2449 0.2437 0.0243 0.2501 0.2490 0.0236 medRQ 0.2046 0.2046 −0.0049 0.2158 0.2144 0.0243 0.2198 0.2185 0.0238 GTMPV**(20) 0.1516 0.1499 −0.0229 0.1505 0.1482 −0.0260 0.1506 0.1483 −0.0261 GTMPV**(30) 0.1501 0.1462 −0.0337 0.1493 0.1447 −0.0367 0.1493 0.1447 −0.0369 GTMPV**(40) 0.1497 0.1432 −0.0436 0.1491 0.1417 −0.0463 0.1493 0.1418 −0.0468 GTMPV**(50) 0.1504 0.1409 −0.0527 0.1499 0.1393 −0.0554 0.1499 0.1392 −0.0558 QVeff(20) 0.1582 0.1581 −0.0044 0.1558 0.1554 −0.0104 0.1568 0.1565 −0.0099 QVeff(30) 0.1569 0.1569 −0.0042 0.1547 0.1544 −0.0105 0.1549 0.1546 −0.0099 QVeff(40) 0.1549 0.1547 −0.0083 0.1534 0.1528 −0.0141 0.1537 0.1530 −0.0141 QVeff(50) 0.1534 0.1528 −0.0139 0.1522 0.1510 −0.0192 0.1521 0.1509 −0.0198 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.2094 0.2092 −0.0096 1.2485 1.0169 0.7245 4.1396 3.3419 2.4431 TMPV([4]) 0.1772 0.1772 −0.0019 0.1748 0.1746 −0.0068 0.1749 0.1748 −0.0062 TMPV([43,43,43]) 0.2017 0.2016 −0.0069 0.1953 0.1948 −0.0148 0.1986 0.1981 −0.0141 TMPV([1,1,1,1]) 0.2094 0.2092 −0.0099 0.2034 0.2024 −0.0199 0.2069 0.2060 −0.0188 minRQ 0.2373 0.2373 −0.0015 0.2449 0.2437 0.0243 0.2501 0.2490 0.0236 medRQ 0.2046 0.2046 −0.0049 0.2158 0.2144 0.0243 0.2198 0.2185 0.0238 GTMPV**(20) 0.1516 0.1499 −0.0229 0.1505 0.1482 −0.0260 0.1506 0.1483 −0.0261 GTMPV**(30) 0.1501 0.1462 −0.0337 0.1493 0.1447 −0.0367 0.1493 0.1447 −0.0369 GTMPV**(40) 0.1497 0.1432 −0.0436 0.1491 0.1417 −0.0463 0.1493 0.1418 −0.0468 GTMPV**(50) 0.1504 0.1409 −0.0527 0.1499 0.1393 −0.0554 0.1499 0.1392 −0.0558 QVeff(20) 0.1582 0.1581 −0.0044 0.1558 0.1554 −0.0104 0.1568 0.1565 −0.0099 QVeff(30) 0.1569 0.1569 −0.0042 0.1547 0.1544 −0.0105 0.1549 0.1546 −0.0099 QVeff(40) 0.1549 0.1547 −0.0083 0.1534 0.1528 −0.0141 0.1537 0.1530 −0.0141 QVeff(50) 0.1534 0.1528 −0.0139 0.1522 0.1510 −0.0192 0.1521 0.1509 −0.0198 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n=400 intraday returns generated by model ( 4.1). The threshold is set as in Equation (4.2) with cϑ=5. Table 6 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.2094 0.2092 −0.0096 1.2485 1.0169 0.7245 4.1396 3.3419 2.4431 TMPV([4]) 0.1772 0.1772 −0.0019 0.1748 0.1746 −0.0068 0.1749 0.1748 −0.0062 TMPV([43,43,43]) 0.2017 0.2016 −0.0069 0.1953 0.1948 −0.0148 0.1986 0.1981 −0.0141 TMPV([1,1,1,1]) 0.2094 0.2092 −0.0099 0.2034 0.2024 −0.0199 0.2069 0.2060 −0.0188 minRQ 0.2373 0.2373 −0.0015 0.2449 0.2437 0.0243 0.2501 0.2490 0.0236 medRQ 0.2046 0.2046 −0.0049 0.2158 0.2144 0.0243 0.2198 0.2185 0.0238 GTMPV**(20) 0.1516 0.1499 −0.0229 0.1505 0.1482 −0.0260 0.1506 0.1483 −0.0261 GTMPV**(30) 0.1501 0.1462 −0.0337 0.1493 0.1447 −0.0367 0.1493 0.1447 −0.0369 GTMPV**(40) 0.1497 0.1432 −0.0436 0.1491 0.1417 −0.0463 0.1493 0.1418 −0.0468 GTMPV**(50) 0.1504 0.1409 −0.0527 0.1499 0.1393 −0.0554 0.1499 0.1392 −0.0558 QVeff(20) 0.1582 0.1581 −0.0044 0.1558 0.1554 −0.0104 0.1568 0.1565 −0.0099 QVeff(30) 0.1569 0.1569 −0.0042 0.1547 0.1544 −0.0105 0.1549 0.1546 −0.0099 QVeff(40) 0.1549 0.1547 −0.0083 0.1534 0.1528 −0.0141 0.1537 0.1530 −0.0141 QVeff(50) 0.1534 0.1528 −0.0139 0.1522 0.1510 −0.0192 0.1521 0.1509 −0.0198 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.2094 0.2092 −0.0096 1.2485 1.0169 0.7245 4.1396 3.3419 2.4431 TMPV([4]) 0.1772 0.1772 −0.0019 0.1748 0.1746 −0.0068 0.1749 0.1748 −0.0062 TMPV([43,43,43]) 0.2017 0.2016 −0.0069 0.1953 0.1948 −0.0148 0.1986 0.1981 −0.0141 TMPV([1,1,1,1]) 0.2094 0.2092 −0.0099 0.2034 0.2024 −0.0199 0.2069 0.2060 −0.0188 minRQ 0.2373 0.2373 −0.0015 0.2449 0.2437 0.0243 0.2501 0.2490 0.0236 medRQ 0.2046 0.2046 −0.0049 0.2158 0.2144 0.0243 0.2198 0.2185 0.0238 GTMPV**(20) 0.1516 0.1499 −0.0229 0.1505 0.1482 −0.0260 0.1506 0.1483 −0.0261 GTMPV**(30) 0.1501 0.1462 −0.0337 0.1493 0.1447 −0.0367 0.1493 0.1447 −0.0369 GTMPV**(40) 0.1497 0.1432 −0.0436 0.1491 0.1417 −0.0463 0.1493 0.1418 −0.0468 GTMPV**(50) 0.1504 0.1409 −0.0527 0.1499 0.1393 −0.0554 0.1499 0.1392 −0.0558 QVeff(20) 0.1582 0.1581 −0.0044 0.1558 0.1554 −0.0104 0.1568 0.1565 −0.0099 QVeff(30) 0.1569 0.1569 −0.0042 0.1547 0.1544 −0.0105 0.1549 0.1546 −0.0099 QVeff(40) 0.1549 0.1547 −0.0083 0.1534 0.1528 −0.0141 0.1537 0.1530 −0.0141 QVeff(50) 0.1534 0.1528 −0.0139 0.1522 0.1510 −0.0192 0.1521 0.1509 −0.0198 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n=400 intraday returns generated by model ( 4.1). The threshold is set as in Equation (4.2) with cϑ=5. Generally speaking, the performance of truncated estimators is largely better than non-truncated ones. In the case with a big jump, MPV is literally shattered: the RMSE decreases with increasing frequency, but it is still roughly +400% at the 1-min frequency. Basically, non-truncated estimators are severely biased. Standard threshold multipower estimators reduce the bias considerably. For example, in the 5-min case in the presence of a small jump, the bias goes from the +151% of the standard MPV([1,1,1,1]) estimator to the −6% of the truncated TMPV([1,1,1,1]) estimator. However, as we discuss below, the RMSE of TMPV([1,1,1,1]) (and TMPV([43,43,43])) is higher than that of the GTMPV** estimator because of the efficiency loss. The estimator based on multipowers also performs better than the neighbor truncation min–med estimators. At the frequency of 5 min (the most typical in applications) the RMSE of min–med estimators is more than double than that of GTMPV** estimators, both in the case in which there are small or large jumps, and still roughly 50% higher on paths without jumps, which again reflects the efficiency loss of these estimators. Regarding the GTMPV**(m;σ4) estimators, we can see that they generally display a small negative bias, which is consistent with the theory. Using GTMPV**(m;σ4), we obtain a gain in terms of relative RMSE, with respect to the standard multipower estimators TMPV([1,1,1,1]) and TMPV([43,43,43]), of roughly 20% at 10 min (from 60% to 40%), of 13% at 5 min (from 43% to 30%), and of 5% at 1 min (from 20% to 15%). Thus, the gain is substantial and is entirely due to the smaller variance of GTMPV**(m;σ4), which more than offsets the loss in bias due to the higher value of m employed. The performance of the efficient estimator of Jacod and Rosenbaum (2013) reflects its asymptotic nature: it improves with large kn. In order to compare it with the GTMPV**(m;σ4), we fix kn=m+1 and compute the two estimators for various m. The relative RMSE for the three different frequencies used in the simulation study in the case without jumps is shown in Figure 2. It is clear that using the estimator GTMPV**(m;σ4), which has a lower variance than the Jacod and Rosenbaum estimator for fixed m, we can obtain a better result with respect to an estimator which is designed to have the lowest possible variance when m (i.e., kn−1) diverges to +∞. Again, this is important in practice, since the number of multipowers (or the window used for preliminary estimates of spot variance) is actually fixed. The performance of the two estimators tends to be similar when m is large, since the two estimators coincide (excluding finite sample bias corrections) for large m. The Jacod and Rosenbaum estimator is also better when n is large (at the 1-min frequency) for intermediate values of m. Figure 2 View largeDownload slide Relative RMSE obtained on simulations for the GTMPV**(m;σ4) and QVeff, the Jacod and Rosenbaum estimator with kn=m+1 for different values of m, at three different sampling frequencies. Figure 2 View largeDownload slide Relative RMSE obtained on simulations for the GTMPV**(m;σ4) and QVeff, the Jacod and Rosenbaum estimator with kn=m+1 for different values of m, at three different sampling frequencies. Summarizing, the Monte Carlo experiments in this section show that i) truncating returns is essential to get reasonable quarticity estimates in the presence of jumps; ii) standard multipower estimators, including the min–med estimator, suffer substantial efficiency loss with respect to the (nearly) efficient multipower estimators, which results in a deteriorated estimate in terms of the mean square error; iii) a minimum variance estimator designed for fixed m can be beneficial, in terms of mean square error, with respect to an efficient estimator designed for diverging m. 4.2 Estimation in the Presence of Microstructure Noise The first type of friction we introduce in simulated experiments is microstructure noise in the form of autocorrelated price distortion. The observed prices, Xj, are generated as follows: Xj=Xj*+εj, (4.3) where Xj* is simulated as in the previous section, and εj=ρεεj−1+εj*, εj*∼N(0,σε2). (4.4) We consider a persistent noise process ( ρε=0.5). The microstructure noise is virtually not present at moderate frequencies; hence, we do not consider the 5- and 10-min frequencies in the present subsection. For n = 400 (1-min), as in the simulation design of Podolskij and Vetter (2009), we set σε2=0.0005IVt, with IVt denoting the daily integrated variance. Results, shown in Table 7, show that microstructure noise induces a strong distortion in all estimates, in form of an upper bias of roughly +50% which is very similar across all considered estimators. Still, the observed variance of GTMPV**(m;σ4) is the lowest among competitors. Table 7 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.6615 0.3249 0.5762 2.1188 1.3929 1.5966 5.9511 4.4924 3.9030 TMPV([4]) 0.6352 0.2703 0.5748 0.6358 0.2646 0.5781 0.6251 0.2588 0.5690 TMPV([43,43,43]) 0.6546 0.3116 0.5756 0.6529 0.3144 0.5722 0.6366 0.3009 0.5610 TMPV([1,1,1,1]) 0.6612 0.3253 0.5756 0.6555 0.3329 0.5647 0.6381 0.3204 0.5518 minRQ 0.6912 0.3640 0.5876 0.7336 0.3839 0.6251 0.7388 0.3802 0.6335 medRQ 0.6705 0.3225 0.5878 0.7060 0.3354 0.6212 0.7040 0.3257 0.6242 GTMPV**(30) 0.5889 0.2298 0.5422 0.5812 0.2252 0.5358 0.5718 0.2201 0.5277 GTMPV**(40) 0.5765 0.2262 0.5303 0.5688 0.2207 0.5243 0.5596 0.2173 0.5157 GTMPV**(50) 0.5654 0.2245 0.5190 0.5567 0.2166 0.5128 0.5493 0.2152 0.5054 QVeff(30) 0.6287 0.2430 0.5798 0.6155 0.2388 0.5673 0.6076 0.2320 0.5616 QVeff(40) 0.6233 0.2393 0.5755 0.6113 0.2349 0.5644 0.6036 0.2302 0.5579 QVeff(50) 0.6141 0.2339 0.5678 0.6056 0.2342 0.5585 0.5960 0.2282 0.5506 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.6615 0.3249 0.5762 2.1188 1.3929 1.5966 5.9511 4.4924 3.9030 TMPV([4]) 0.6352 0.2703 0.5748 0.6358 0.2646 0.5781 0.6251 0.2588 0.5690 TMPV([43,43,43]) 0.6546 0.3116 0.5756 0.6529 0.3144 0.5722 0.6366 0.3009 0.5610 TMPV([1,1,1,1]) 0.6612 0.3253 0.5756 0.6555 0.3329 0.5647 0.6381 0.3204 0.5518 minRQ 0.6912 0.3640 0.5876 0.7336 0.3839 0.6251 0.7388 0.3802 0.6335 medRQ 0.6705 0.3225 0.5878 0.7060 0.3354 0.6212 0.7040 0.3257 0.6242 GTMPV**(30) 0.5889 0.2298 0.5422 0.5812 0.2252 0.5358 0.5718 0.2201 0.5277 GTMPV**(40) 0.5765 0.2262 0.5303 0.5688 0.2207 0.5243 0.5596 0.2173 0.5157 GTMPV**(50) 0.5654 0.2245 0.5190 0.5567 0.2166 0.5128 0.5493 0.2152 0.5054 QVeff(30) 0.6287 0.2430 0.5798 0.6155 0.2388 0.5673 0.6076 0.2320 0.5616 QVeff(40) 0.6233 0.2393 0.5755 0.6113 0.2349 0.5644 0.6036 0.2302 0.5579 QVeff(50) 0.6141 0.2339 0.5678 0.6056 0.2342 0.5585 0.5960 0.2282 0.5506 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n = 400 intraday returns generated by model ( 4.3), which includes microstructure noise. The threshold is set as in Equation (4.2) with cϑ=5. The autocorrelation of the noise process is ρε=0.5. Table 7 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.6615 0.3249 0.5762 2.1188 1.3929 1.5966 5.9511 4.4924 3.9030 TMPV([4]) 0.6352 0.2703 0.5748 0.6358 0.2646 0.5781 0.6251 0.2588 0.5690 TMPV([43,43,43]) 0.6546 0.3116 0.5756 0.6529 0.3144 0.5722 0.6366 0.3009 0.5610 TMPV([1,1,1,1]) 0.6612 0.3253 0.5756 0.6555 0.3329 0.5647 0.6381 0.3204 0.5518 minRQ 0.6912 0.3640 0.5876 0.7336 0.3839 0.6251 0.7388 0.3802 0.6335 medRQ 0.6705 0.3225 0.5878 0.7060 0.3354 0.6212 0.7040 0.3257 0.6242 GTMPV**(30) 0.5889 0.2298 0.5422 0.5812 0.2252 0.5358 0.5718 0.2201 0.5277 GTMPV**(40) 0.5765 0.2262 0.5303 0.5688 0.2207 0.5243 0.5596 0.2173 0.5157 GTMPV**(50) 0.5654 0.2245 0.5190 0.5567 0.2166 0.5128 0.5493 0.2152 0.5054 QVeff(30) 0.6287 0.2430 0.5798 0.6155 0.2388 0.5673 0.6076 0.2320 0.5616 QVeff(40) 0.6233 0.2393 0.5755 0.6113 0.2349 0.5644 0.6036 0.2302 0.5579 QVeff(50) 0.6141 0.2339 0.5678 0.6056 0.2342 0.5585 0.5960 0.2282 0.5506 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.6615 0.3249 0.5762 2.1188 1.3929 1.5966 5.9511 4.4924 3.9030 TMPV([4]) 0.6352 0.2703 0.5748 0.6358 0.2646 0.5781 0.6251 0.2588 0.5690 TMPV([43,43,43]) 0.6546 0.3116 0.5756 0.6529 0.3144 0.5722 0.6366 0.3009 0.5610 TMPV([1,1,1,1]) 0.6612 0.3253 0.5756 0.6555 0.3329 0.5647 0.6381 0.3204 0.5518 minRQ 0.6912 0.3640 0.5876 0.7336 0.3839 0.6251 0.7388 0.3802 0.6335 medRQ 0.6705 0.3225 0.5878 0.7060 0.3354 0.6212 0.7040 0.3257 0.6242 GTMPV**(30) 0.5889 0.2298 0.5422 0.5812 0.2252 0.5358 0.5718 0.2201 0.5277 GTMPV**(40) 0.5765 0.2262 0.5303 0.5688 0.2207 0.5243 0.5596 0.2173 0.5157 GTMPV**(50) 0.5654 0.2245 0.5190 0.5567 0.2166 0.5128 0.5493 0.2152 0.5054 QVeff(30) 0.6287 0.2430 0.5798 0.6155 0.2388 0.5673 0.6076 0.2320 0.5616 QVeff(40) 0.6233 0.2393 0.5755 0.6113 0.2349 0.5644 0.6036 0.2302 0.5579 QVeff(50) 0.6141 0.2339 0.5678 0.6056 0.2342 0.5585 0.5960 0.2282 0.5506 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n = 400 intraday returns generated by model ( 4.3), which includes microstructure noise. The threshold is set as in Equation (4.2) with cϑ=5. The autocorrelation of the noise process is ρε=0.5. Even if the magnitude of the bias due to microstructure noise could vary given a different data generating process for the price dynamics and the noise itself, these results suggest that the impact of the noise is anyway translated into an upper bias which affects the competing estimators very similarly, suggesting that it would still be beneficial to concentrate on variance reduction even in the presence of this kind of friction. 4.3 Estimation in the Presence of Flat Prices We finally consider a form of friction which we document to be present at the highest frequencies, that is flat pricing. Flat pricing consists in the observation of spurious zero returns, which might be due to lack of liquidity in the market or asymmetric information (see the discussion in Bandi, Pirino, and Renò, 2017). In our simulation setting, we assume that the generated returns are given by: ΔjX=ΔjX*·ψj, (4.5) where X* is the process generated without frictions, and ψj={0,with probability pψ1,with probability (1−pψ). (4.6) We set pψ=0.3 and, as before, we consider only the case n = 400 (1-min data). This choice is motivated by data analysis, since the phenomenon of flat prices tends to fade away, for the stocks we consider in the empirical application, at the 5-min frequency. Results are show in Table 8. Table 8 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.5742 0.1347 −0.5582 0.7554 0.6872 −0.3135 2.1987 2.1738 0.3301 TMPV([4]) 0.3895 0.2752 0.2757 0.4212 0.2868 0.3085 0.4084 0.2875 0.2900 TMPV([43,43,43]) 0.4301 0.1555 −0.4010 0.4916 0.1465 −0.4692 0.4672 0.1470 −0.4435 TMPV([1,1,1,1]) 0.5760 0.1347 −0.5600 0.6283 0.1265 −0.6154 0.6465 0.1142 −0.6363 minRQ 0.3009 0.2286 −0.1957 0.3318 0.2575 −0.2093 0.3139 0.2627 −0.1719 medRQ 0.2337 0.2183 −0.0834 0.2596 0.2531 −0.0577 0.2575 0.2527 −0.0495 GTMPV**(30) 0.1737 0.1694 −0.0384 0.1793 0.1736 −0.0449 0.1780 0.1725 −0.0442 GTMPV**(40) 0.1731 0.1652 −0.0518 0.1789 0.1694 −0.0574 0.1777 0.1682 −0.0573 GTMPV**(50) 0.1736 0.1622 −0.0618 0.1795 0.1654 −0.0696 0.1781 0.1648 −0.0677 QVeff(30) 0.1833 0.1833 −0.0018 0.1880 0.1879 −0.0050 0.1862 0.1860 −0.0096 QVeff(40) 0.1801 0.1799 −0.0092 0.1866 0.1861 −0.0148 0.1834 0.1825 −0.0178 QVeff(50) 0.1790 0.1781 −0.0181 0.1835 0.1815 −0.0271 0.1817 0.1792 −0.0301 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.5742 0.1347 −0.5582 0.7554 0.6872 −0.3135 2.1987 2.1738 0.3301 TMPV([4]) 0.3895 0.2752 0.2757 0.4212 0.2868 0.3085 0.4084 0.2875 0.2900 TMPV([43,43,43]) 0.4301 0.1555 −0.4010 0.4916 0.1465 −0.4692 0.4672 0.1470 −0.4435 TMPV([1,1,1,1]) 0.5760 0.1347 −0.5600 0.6283 0.1265 −0.6154 0.6465 0.1142 −0.6363 minRQ 0.3009 0.2286 −0.1957 0.3318 0.2575 −0.2093 0.3139 0.2627 −0.1719 medRQ 0.2337 0.2183 −0.0834 0.2596 0.2531 −0.0577 0.2575 0.2527 −0.0495 GTMPV**(30) 0.1737 0.1694 −0.0384 0.1793 0.1736 −0.0449 0.1780 0.1725 −0.0442 GTMPV**(40) 0.1731 0.1652 −0.0518 0.1789 0.1694 −0.0574 0.1777 0.1682 −0.0573 GTMPV**(50) 0.1736 0.1622 −0.0618 0.1795 0.1654 −0.0696 0.1781 0.1648 −0.0677 QVeff(30) 0.1833 0.1833 −0.0018 0.1880 0.1879 −0.0050 0.1862 0.1860 −0.0096 QVeff(40) 0.1801 0.1799 −0.0092 0.1866 0.1861 −0.0148 0.1834 0.1825 −0.0178 QVeff(50) 0.1790 0.1781 −0.0181 0.1835 0.1815 −0.0271 0.1817 0.1792 −0.0301 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n = 400 intraday returns generated by model ( 4.5), which includes flat pricing. The threshold is set as in Equation (4.2) with cϑ=5. The probability of observing a zero return is pε=0.3. Table 8 Quarticity estimators No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.5742 0.1347 −0.5582 0.7554 0.6872 −0.3135 2.1987 2.1738 0.3301 TMPV([4]) 0.3895 0.2752 0.2757 0.4212 0.2868 0.3085 0.4084 0.2875 0.2900 TMPV([43,43,43]) 0.4301 0.1555 −0.4010 0.4916 0.1465 −0.4692 0.4672 0.1470 −0.4435 TMPV([1,1,1,1]) 0.5760 0.1347 −0.5600 0.6283 0.1265 −0.6154 0.6465 0.1142 −0.6363 minRQ 0.3009 0.2286 −0.1957 0.3318 0.2575 −0.2093 0.3139 0.2627 −0.1719 medRQ 0.2337 0.2183 −0.0834 0.2596 0.2531 −0.0577 0.2575 0.2527 −0.0495 GTMPV**(30) 0.1737 0.1694 −0.0384 0.1793 0.1736 −0.0449 0.1780 0.1725 −0.0442 GTMPV**(40) 0.1731 0.1652 −0.0518 0.1789 0.1694 −0.0574 0.1777 0.1682 −0.0573 GTMPV**(50) 0.1736 0.1622 −0.0618 0.1795 0.1654 −0.0696 0.1781 0.1648 −0.0677 QVeff(30) 0.1833 0.1833 −0.0018 0.1880 0.1879 −0.0050 0.1862 0.1860 −0.0096 QVeff(40) 0.1801 0.1799 −0.0092 0.1866 0.1861 −0.0148 0.1834 0.1825 −0.0178 QVeff(50) 0.1790 0.1781 −0.0181 0.1835 0.1815 −0.0271 0.1817 0.1792 −0.0301 No jumps Small jump Big jump RMSE Std Bias RMSE Std Bias RMSE Std Bias MPV([1,1,1,1]) 0.5742 0.1347 −0.5582 0.7554 0.6872 −0.3135 2.1987 2.1738 0.3301 TMPV([4]) 0.3895 0.2752 0.2757 0.4212 0.2868 0.3085 0.4084 0.2875 0.2900 TMPV([43,43,43]) 0.4301 0.1555 −0.4010 0.4916 0.1465 −0.4692 0.4672 0.1470 −0.4435 TMPV([1,1,1,1]) 0.5760 0.1347 −0.5600 0.6283 0.1265 −0.6154 0.6465 0.1142 −0.6363 minRQ 0.3009 0.2286 −0.1957 0.3318 0.2575 −0.2093 0.3139 0.2627 −0.1719 medRQ 0.2337 0.2183 −0.0834 0.2596 0.2531 −0.0577 0.2575 0.2527 −0.0495 GTMPV**(30) 0.1737 0.1694 −0.0384 0.1793 0.1736 −0.0449 0.1780 0.1725 −0.0442 GTMPV**(40) 0.1731 0.1652 −0.0518 0.1789 0.1694 −0.0574 0.1777 0.1682 −0.0573 GTMPV**(50) 0.1736 0.1622 −0.0618 0.1795 0.1654 −0.0696 0.1781 0.1648 −0.0677 QVeff(30) 0.1833 0.1833 −0.0018 0.1880 0.1879 −0.0050 0.1862 0.1860 −0.0096 QVeff(40) 0.1801 0.1799 −0.0092 0.1866 0.1861 −0.0148 0.1834 0.1825 −0.0178 QVeff(50) 0.1790 0.1781 −0.0181 0.1835 0.1815 −0.0271 0.1817 0.1792 −0.0301 Notes: The table reports the relative RMSE, standard deviation, and bias computed with M = 10,000 replications of n = 400 intraday returns generated by model ( 4.5), which includes flat pricing. The threshold is set as in Equation (4.2) with cϑ=5. The probability of observing a zero return is pε=0.3. Even if the probability of flat trading we use in the simulated experiments is quite high with respect to the observed values (reported in Table 9), the impact on final estimates is quite small. For GTMPV**(m;σ4), the relative RMSE increases from about 15% to 17% only. This is basically due to increased estimator variance, in line with the theoretical predictions of Phillips and Yu (2009) for realized variance. In this sense, microstructure noise and flat trading have an impact which is completely different on estimators, the first kind of friction mostly affecting the estimator bias, whereas the second kind of friction mostly affecting the estimator variance. Table 9 The list of the 16 blue chip stocks used in the empirical application, their ticker and the percentage of zero-returns at 1 and 5 min Company Ticker 1-minute zero returns (%) 5-minute zero returns (%) Bank of America BAC 9.46 4.71 Citigroup Inc. C 12.51 7.77 JPMorgan Chase & Co. JPM 6.24 2.76 Wells Fargo & Company WFC 6.24 2.76 Boeing BA 5.60 2.39 Caterpillar Inc. CAT 4.45 1.90 FedEx Corporation FDX 4.93 2.05 Honeywell International Inc. HON 7.96 3.41 Hewlett-Packard Company HPQ 8.65 3.75 International Business Machines Corp. IBM 4.02 1.83 AT&T Inc. T 9.55 4.44 Texas Instruments Incorporated TXN 12.94 6.09 Kraft Foods Inc. KFT 13.33 6.32 PepsiCo, Inc. PEP 8.19 3.61 The Procter & Gamble Company PG 7.94 3.54 Time Warner Inc. TWX 10.72 4.70 Company Ticker 1-minute zero returns (%) 5-minute zero returns (%) Bank of America BAC 9.46 4.71 Citigroup Inc. C 12.51 7.77 JPMorgan Chase & Co. JPM 6.24 2.76 Wells Fargo & Company WFC 6.24 2.76 Boeing BA 5.60 2.39 Caterpillar Inc. CAT 4.45 1.90 FedEx Corporation FDX 4.93 2.05 Honeywell International Inc. HON 7.96 3.41 Hewlett-Packard Company HPQ 8.65 3.75 International Business Machines Corp. IBM 4.02 1.83 AT&T Inc. T 9.55 4.44 Texas Instruments Incorporated TXN 12.94 6.09 Kraft Foods Inc. KFT 13.33 6.32 PepsiCo, Inc. PEP 8.19 3.61 The Procter & Gamble Company PG 7.94 3.54 Time Warner Inc. TWX 10.72 4.70 Table 9 The list of the 16 blue chip stocks used in the empirical application, their ticker and the percentage of zero-returns at 1 and 5 min Company Ticker 1-minute zero returns (%) 5-minute zero returns (%) Bank of America BAC 9.46 4.71 Citigroup Inc. C 12.51 7.77 JPMorgan Chase & Co. JPM 6.24 2.76 Wells Fargo & Company WFC 6.24 2.76 Boeing BA 5.60 2.39 Caterpillar Inc. CAT 4.45 1.90 FedEx Corporation FDX 4.93 2.05 Honeywell International Inc. HON 7.96 3.41 Hewlett-Packard Company HPQ 8.65 3.75 International Business Machines Corp. IBM 4.02 1.83 AT&T Inc. T 9.55 4.44 Texas Instruments Incorporated TXN 12.94 6.09 Kraft Foods Inc. KFT 13.33 6.32 PepsiCo, Inc. PEP 8.19 3.61 The Procter & Gamble Company PG 7.94 3.54 Time Warner Inc. TWX 10.72 4.70 Company Ticker 1-minute zero returns (%) 5-minute zero returns (%) Bank of America BAC 9.46 4.71 Citigroup Inc. C 12.51 7.77 JPMorgan Chase & Co. JPM 6.24 2.76 Wells Fargo & Company WFC 6.24 2.76 Boeing BA 5.60 2.39 Caterpillar Inc. CAT 4.45 1.90 FedEx Corporation FDX 4.93 2.05 Honeywell International Inc. HON 7.96 3.41 Hewlett-Packard Company HPQ 8.65 3.75 International Business Machines Corp. IBM 4.02 1.83 AT&T Inc. T 9.55 4.44 Texas Instruments Incorporated TXN 12.94 6.09 Kraft Foods Inc. KFT 13.33 6.32 PepsiCo, Inc. PEP 8.19 3.61 The Procter & Gamble Company PG 7.94 3.54 Time Warner Inc. TWX 10.72 4.70 Interestingly, GTMPV**(m;σ4) is much more robust than competing estimators to this form of friction. Indeed, the relative RMSE standard TMPV estimators increase from roughly 20% to 45% for tripower and 60% for quadpower; for minRQ it increases from 23% to 30%; and for medRQ it increases from 20% to 23%. The gap between GTMPV**(m;σ4) and competing estimator becomes then wider in the presence of flat trading. 5 Empirical Application The purpose of the empirical application is to apply our proposed quarticity estimator to real data from the financial market, and to compare its performance with respect to competing estimators in relevant applications, namely jump testing and volatility forecasting. As in the Monte Carlo section, we restrict our attention to the GTMPV**(m*) estimator in Equation (3.1) (we omit the σ4 in the notation for simplicity). By m* we indicate the value that minimizes the MSE in Equation (3.11), using Q^=TMPV([4]). The data set we use is the collection of 16 blue chip stocks quoted on the New York Stock Exchange. The stocks are all very liquid and they are listed, together with their corresponding ticker, in Table 9. One-minute prices were recovered from the TickData One Minute Equity Data dataset, from January 3, 2007 to June 29, 2012, for a total of 1385 trading days. Our sample then lies in the middle of the credit crunch crisis, characterized by very high volatility levels and a supposedly high number of jumps. The data went through a standard filtering procedure. TickData 1-min equity data are adjusted for corporate actions such as mergers and acquisitions or symbol changes. Moreover, the underlying tick data used to build 1-min time series are first controlled for cancelled trades, or records not temporally aligned with previous/subsequent data; then filtered to identify bad ticks which are corrected using validation with third-party sources. All the measures reported here are for daily units and percentage returns. Table 9 also shows the percentage of zero returns at 1 and 5 min. We can see that the impact of flat trading at 1 min is substantial in all stocks, while it is much less impactful at 5 min. On each day in the sample, we compute the same quarticity estimators used in the Monte Carlo section. The 1-min frequency corresponds to n = 390; the 5-min frequency to n = 78, and the 10-min frequency to n = 39. 5.1 Estimating Quarticity Table 10 reports summary statistics on pooled daily quarticity estimates ( IQ̂) and the ratio IQ̂/IV̂, where we use TRV([2]) as an estimator IV̂ of integrated volatility. Asymptotically, we expect this ratio to be always greater than 1, by Jensen inequality. However, since we are using estimated quantities, the ratio could be less than one. Using the same normalization for all measures allows the comparison of different days and different stocks. We apply the estimators at three sampling frequencies: 1, 5, and 10 min. In Table 10, we exclude the Flash Crash day (May 6, 2010) since the difference between truncated and non-truncated estimators in this specific day is many orders of magnitudes away than what observed in the other days. Table 10 Summary statistics of pooled quarticity estimators IQ̂ and the ratios IQ̂/IV̂, where IV̂ is threshold realized variance ( TRV([2])) IQ̂ IQ̂/IV̂ 1-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 316.7 3.5 9314.0 1.578 1.366 1.005 570.7 MPV([1,1,1,1]) 343.5 4.0 9852.4 1.721 1.463 1.227 526.5 minRQ 512.3 4.4 19,736.3 1.895 1.530 3.477 12,383.1 medRQ 464.7 4.4 17,007.9 1.838 1.522 2.467 8600.2 TMPV([4]) 337.7 4.6 9290.8 1.769 1.582 0.810 54.3 TMPV([43,43,43]) 310.3 3.6 8677.6 1.576 1.394 0.782 81.0 TMPV([1,1,1,1]) 293.7 3.1 8601.8 1.480 1.310 0.744 99.0 GTMPV**(m*) 331.3 4.5 9219.9 1.750 1.564 0.798 54.9 (m*) (5.7) (3.0) (7.2) IQ̂ IQ̂/IV̂ 1-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 316.7 3.5 9314.0 1.578 1.366 1.005 570.7 MPV([1,1,1,1]) 343.5 4.0 9852.4 1.721 1.463 1.227 526.5 minRQ 512.3 4.4 19,736.3 1.895 1.530 3.477 12,383.1 medRQ 464.7 4.4 17,007.9 1.838 1.522 2.467 8600.2 TMPV([4]) 337.7 4.6 9290.8 1.769 1.582 0.810 54.3 TMPV([43,43,43]) 310.3 3.6 8677.6 1.576 1.394 0.782 81.0 TMPV([1,1,1,1]) 293.7 3.1 8601.8 1.480 1.310 0.744 99.0 GTMPV**(m*) 331.3 4.5 9219.9 1.750 1.564 0.798 54.9 (m*) (5.7) (3.0) (7.2) 5-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 207.3 1.9 7595.9 1.104 1.018 0.502 427.7 MPV([1,1,1,1]) 238.6 2.2 7893.6 1.203 1.081 0.765 2801.0 minRQ 315.9 2.3 10,433.0 1.275 1.114 1.119 2102.6 medRQ 249.0 2.3 7799.7 1.236 1.103 0.957 2771.3 TMPV([4]) 209.8 2.5 6418.3 1.219 1.179 0.204 4.3 TMPV([43,43,43]) 194.2 1.9 7490.8 1.075 1.033 0.265 5.7 TMPV([1,1,1,1]) 177.9 1.7 7377.2 1.015 0.975 0.269 5.8 GTMPV**(m*) 203.2 2.5 6296.7 1.203 1.160 0.201 4.6 (m*) (8.1) (5.0) (8.2) GTMPV**(5) 189.9 2.3 6228.4 1.144 1.109 0.150 5.0 GTMPV**(10) 165.1 2.0 5530.0 1.081 1.054 0.112 4.6 GTMPV**(20) 141.1 1.7 4621.9 1.005 0.990 0.070 5.5 GTMPV**(30) 125.0 1.6 3868.2 0.962 0.957 0.054 1.7 GTMPV**(35) 120.0 1.5 3634.4 0.948 0.947 0.052 7.0 QVeff(5) 191.0 2.3 6364.7 1.149 1.108 0.179 5.2 QVeff(10) 181.4 2.2 6236.2 1.129 1.096 0.144 5.0 QVeff(20) 162.3 1.9 5759.2 1.049 1.030 0.096 4.5 QVeff(30) 137.1 1.7 4539.8 0.982 0.973 0.069 5.0 QVeff(35) 128.6 1.6 4125.7 0.955 0.948 0.063 3.1 5-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 207.3 1.9 7595.9 1.104 1.018 0.502 427.7 MPV([1,1,1,1]) 238.6 2.2 7893.6 1.203 1.081 0.765 2801.0 minRQ 315.9 2.3 10,433.0 1.275 1.114 1.119 2102.6 medRQ 249.0 2.3 7799.7 1.236 1.103 0.957 2771.3 TMPV([4]) 209.8 2.5 6418.3 1.219 1.179 0.204 4.3 TMPV([43,43,43]) 194.2 1.9 7490.8 1.075 1.033 0.265 5.7 TMPV([1,1,1,1]) 177.9 1.7 7377.2 1.015 0.975 0.269 5.8 GTMPV**(m*) 203.2 2.5 6296.7 1.203 1.160 0.201 4.6 (m*) (8.1) (5.0) (8.2) GTMPV**(5) 189.9 2.3 6228.4 1.144 1.109 0.150 5.0 GTMPV**(10) 165.1 2.0 5530.0 1.081 1.054 0.112 4.6 GTMPV**(20) 141.1 1.7 4621.9 1.005 0.990 0.070 5.5 GTMPV**(30) 125.0 1.6 3868.2 0.962 0.957 0.054 1.7 GTMPV**(35) 120.0 1.5 3634.4 0.948 0.947 0.052 7.0 QVeff(5) 191.0 2.3 6364.7 1.149 1.108 0.179 5.2 QVeff(10) 181.4 2.2 6236.2 1.129 1.096 0.144 5.0 QVeff(20) 162.3 1.9 5759.2 1.049 1.030 0.096 4.5 QVeff(30) 137.1 1.7 4539.8 0.982 0.973 0.069 5.0 QVeff(35) 128.6 1.6 4125.7 0.955 0.948 0.063 3.1 10-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 134.0 1.2 5062.8 0.914 0.836 0.484 1305.1 MPV([1,1,1,1]) 154.4 1.5 5477.1 1.009 0.906 0.601 836.2 minRQ 180.6 1.6 5316.7 1.075 0.929 0.788 756.2 medRQ 159.6 1.6 5372.0 1.043 0.936 0.706 1545.6 TMPV([4]) 165.4 1.7 7657.7 1.043 0.991 0.377 9.0 TMPV([43,43,43]) 123.7 1.2 4565.5 0.891 0.839 0.363 7.3 TMPV([1,1,1,1]) 111.8 1.0 4473.8 0.830 0.783 0.348 6.7 GTMPV**(m*) 158.3 1.7 6959.0 1.036 0.985 0.369 9.3 (m*) (5.2) (5.0) (3.4) 10-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 134.0 1.2 5062.8 0.914 0.836 0.484 1305.1 MPV([1,1,1,1]) 154.4 1.5 5477.1 1.009 0.906 0.601 836.2 minRQ 180.6 1.6 5316.7 1.075 0.929 0.788 756.2 medRQ 159.6 1.6 5372.0 1.043 0.936 0.706 1545.6 TMPV([4]) 165.4 1.7 7657.7 1.043 0.991 0.377 9.0 TMPV([43,43,43]) 123.7 1.2 4565.5 0.891 0.839 0.363 7.3 TMPV([1,1,1,1]) 111.8 1.0 4473.8 0.830 0.783 0.348 6.7 GTMPV**(m*) 158.3 1.7 6959.0 1.036 0.985 0.369 9.3 (m*) (5.2) (5.0) (3.4) Note: May 6, 2010 (the Flash Crash) is excluded. Table 10 Summary statistics of pooled quarticity estimators IQ̂ and the ratios IQ̂/IV̂, where IV̂ is threshold realized variance ( TRV([2])) IQ̂ IQ̂/IV̂ 1-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 316.7 3.5 9314.0 1.578 1.366 1.005 570.7 MPV([1,1,1,1]) 343.5 4.0 9852.4 1.721 1.463 1.227 526.5 minRQ 512.3 4.4 19,736.3 1.895 1.530 3.477 12,383.1 medRQ 464.7 4.4 17,007.9 1.838 1.522 2.467 8600.2 TMPV([4]) 337.7 4.6 9290.8 1.769 1.582 0.810 54.3 TMPV([43,43,43]) 310.3 3.6 8677.6 1.576 1.394 0.782 81.0 TMPV([1,1,1,1]) 293.7 3.1 8601.8 1.480 1.310 0.744 99.0 GTMPV**(m*) 331.3 4.5 9219.9 1.750 1.564 0.798 54.9 (m*) (5.7) (3.0) (7.2) IQ̂ IQ̂/IV̂ 1-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 316.7 3.5 9314.0 1.578 1.366 1.005 570.7 MPV([1,1,1,1]) 343.5 4.0 9852.4 1.721 1.463 1.227 526.5 minRQ 512.3 4.4 19,736.3 1.895 1.530 3.477 12,383.1 medRQ 464.7 4.4 17,007.9 1.838 1.522 2.467 8600.2 TMPV([4]) 337.7 4.6 9290.8 1.769 1.582 0.810 54.3 TMPV([43,43,43]) 310.3 3.6 8677.6 1.576 1.394 0.782 81.0 TMPV([1,1,1,1]) 293.7 3.1 8601.8 1.480 1.310 0.744 99.0 GTMPV**(m*) 331.3 4.5 9219.9 1.750 1.564 0.798 54.9 (m*) (5.7) (3.0) (7.2) 5-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 207.3 1.9 7595.9 1.104 1.018 0.502 427.7 MPV([1,1,1,1]) 238.6 2.2 7893.6 1.203 1.081 0.765 2801.0 minRQ 315.9 2.3 10,433.0 1.275 1.114 1.119 2102.6 medRQ 249.0 2.3 7799.7 1.236 1.103 0.957 2771.3 TMPV([4]) 209.8 2.5 6418.3 1.219 1.179 0.204 4.3 TMPV([43,43,43]) 194.2 1.9 7490.8 1.075 1.033 0.265 5.7 TMPV([1,1,1,1]) 177.9 1.7 7377.2 1.015 0.975 0.269 5.8 GTMPV**(m*) 203.2 2.5 6296.7 1.203 1.160 0.201 4.6 (m*) (8.1) (5.0) (8.2) GTMPV**(5) 189.9 2.3 6228.4 1.144 1.109 0.150 5.0 GTMPV**(10) 165.1 2.0 5530.0 1.081 1.054 0.112 4.6 GTMPV**(20) 141.1 1.7 4621.9 1.005 0.990 0.070 5.5 GTMPV**(30) 125.0 1.6 3868.2 0.962 0.957 0.054 1.7 GTMPV**(35) 120.0 1.5 3634.4 0.948 0.947 0.052 7.0 QVeff(5) 191.0 2.3 6364.7 1.149 1.108 0.179 5.2 QVeff(10) 181.4 2.2 6236.2 1.129 1.096 0.144 5.0 QVeff(20) 162.3 1.9 5759.2 1.049 1.030 0.096 4.5 QVeff(30) 137.1 1.7 4539.8 0.982 0.973 0.069 5.0 QVeff(35) 128.6 1.6 4125.7 0.955 0.948 0.063 3.1 5-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 207.3 1.9 7595.9 1.104 1.018 0.502 427.7 MPV([1,1,1,1]) 238.6 2.2 7893.6 1.203 1.081 0.765 2801.0 minRQ 315.9 2.3 10,433.0 1.275 1.114 1.119 2102.6 medRQ 249.0 2.3 7799.7 1.236 1.103 0.957 2771.3 TMPV([4]) 209.8 2.5 6418.3 1.219 1.179 0.204 4.3 TMPV([43,43,43]) 194.2 1.9 7490.8 1.075 1.033 0.265 5.7 TMPV([1,1,1,1]) 177.9 1.7 7377.2 1.015 0.975 0.269 5.8 GTMPV**(m*) 203.2 2.5 6296.7 1.203 1.160 0.201 4.6 (m*) (8.1) (5.0) (8.2) GTMPV**(5) 189.9 2.3 6228.4 1.144 1.109 0.150 5.0 GTMPV**(10) 165.1 2.0 5530.0 1.081 1.054 0.112 4.6 GTMPV**(20) 141.1 1.7 4621.9 1.005 0.990 0.070 5.5 GTMPV**(30) 125.0 1.6 3868.2 0.962 0.957 0.054 1.7 GTMPV**(35) 120.0 1.5 3634.4 0.948 0.947 0.052 7.0 QVeff(5) 191.0 2.3 6364.7 1.149 1.108 0.179 5.2 QVeff(10) 181.4 2.2 6236.2 1.129 1.096 0.144 5.0 QVeff(20) 162.3 1.9 5759.2 1.049 1.030 0.096 4.5 QVeff(30) 137.1 1.7 4539.8 0.982 0.973 0.069 5.0 QVeff(35) 128.6 1.6 4125.7 0.955 0.948 0.063 3.1 10-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 134.0 1.2 5062.8 0.914 0.836 0.484 1305.1 MPV([1,1,1,1]) 154.4 1.5 5477.1 1.009 0.906 0.601 836.2 minRQ 180.6 1.6 5316.7 1.075 0.929 0.788 756.2 medRQ 159.6 1.6 5372.0 1.043 0.936 0.706 1545.6 TMPV([4]) 165.4 1.7 7657.7 1.043 0.991 0.377 9.0 TMPV([43,43,43]) 123.7 1.2 4565.5 0.891 0.839 0.363 7.3 TMPV([1,1,1,1]) 111.8 1.0 4473.8 0.830 0.783 0.348 6.7 GTMPV**(m*) 158.3 1.7 6959.0 1.036 0.985 0.369 9.3 (m*) (5.2) (5.0) (3.4) 10-minute frequency Mean Median Std Mean Median Std Kurtosis MPV([43,43,43]) 134.0 1.2 5062.8 0.914 0.836 0.484 1305.1 MPV([1,1,1,1]) 154.4 1.5 5477.1 1.009 0.906 0.601 836.2 minRQ 180.6 1.6 5316.7 1.075 0.929 0.788 756.2 medRQ 159.6 1.6 5372.0 1.043 0.936 0.706 1545.6 TMPV([4]) 165.4 1.7 7657.7 1.043 0.991 0.377 9.0 TMPV([43,43,43]) 123.7 1.2 4565.5 0.891 0.839 0.363 7.3 TMPV([1,1,1,1]) 111.8 1.0 4473.8 0.830 0.783 0.348 6.7 GTMPV**(m*) 158.3 1.7 6959.0 1.036 0.985 0.369 9.3 (m*) (5.2) (5.0) (3.4) Note: May 6, 2010 (the Flash Crash) is excluded. We observe a large difference between mean and median quarticity estimates at all the considered frequencies, indicating a very skewed distribution. The difference is less pronounced for the ratios. We start the discussion with the 5-min frequency. Here, we can see that GTMPV**(m*) has the lowest in-sample standard deviation, both in terms of average estimates and ratios. The median (data-driven) m* used in these computation is 5. To gain insight, we also report GTMPV**(m) at several values of m, and compare them with the Jacod–Rosembaum estimator QVeff(m). We can see that, as m increases, the in-sample standard deviation and average of both estimators decreases, with GTMPV**(m*) being less variable at all fixed frequencies. The choice of m* balances the bias and the variance. All these results are thus in line with the theory. Figure 3 shows the estimated probability density functions of the ratio IQ̂/IV̂ for pooled daily quarticity estimates, for different quarticity estimators: TMPV([43,43,43]) (labeled threshold tripower), TMPV([1,1,1,1]) (threshold quadpower), minRV, medRV, GTMPV**(m*). We use 5-min returns. The figure shows clearly the empirical potential of the theory. The GTMPV**(m*) estimator delivers a much more concentrated ratio, with considerably thinner tails, as shown by the drop in the sample kurtosis in Table 10. The estimated ratio is spuriously less then one for 54.17% of estimates using threshold quadpower, 43.92% using threshold tripower, 35.55% using minRV, and 33.37% using medRV; the violation is instead observed only in 12.95% of cases with GTMPV**(m*).6 Figure 3 View largeDownload slide Estimated probability density functions of pooled daily ratios IQ̂/IV̂, where IQ̂ are different quarticity estimators and IV̂ is threshold realized variance ( TRV([2])). Estimates are obtained at the 5-min frequency. Figure 3 View largeDownload slide Estimated probability density functions of pooled daily ratios IQ̂/IV̂, where IQ̂ are different quarticity estimators and IV̂ is threshold realized variance ( TRV([2])). Estimates are obtained at the 5-min frequency. At 1 and 10 min, results are qualitatively the same. At 1 min, where the impact of market microstructure noise and flat trading is likely to be higher, standard multipowers display a smaller variability and a smaller average than GTMPV**(m*). At 10 min, where the bias is relatively higher since the number of observations is smaller, we observe the same phenomenon. Again, the GTMPV**(m*) estimator is implemented here to balance bias and variance in an optimal way. Finally, notice that, at all frequencies, the difference in the kurtosis between non-truncated and truncated ratios is particularly pronounced, indicating that truncated estimator have much thinner tails. 5.2 Impact on Jump Testing A popular way for testing for jumps is to take the difference between realized variance and bipower variation, and standardize this with the standard deviation of the difference. The technique has been basically laid out in Barndorff-Nielsen and Shephard (2006), see also Huang and Tauchen (2005), and is largely used in empirical work. We follow this empirical strategy using the version of the test proposed by Corsi, Pirino, and Renò (2010), namely: CTz(QV)=1n1/2RV−CTMPV([1,1])RVθ˜max(QVCTMPV([1,1])2,1) (5.1) where θ˜=(π2/4+π−5). In definition (5.1), QV is a consistent quarticity estimator. Consistently with their proposed estimators, Corsi, Pirino, and Renò (2010) suggest to use standard multipowers TMPV([4/3,4/3,4/3]) or TMPV([1,1,1,1]) to estimate quarticity. In what follows, we study the sensitivity of the test to different quarticity estimators. In total we have 2392×16≈40000 tests in our sample. Table 11 reports the number of detected jumps when standard (threshold tripower, and threshold quadpower) and the GTMPV**(m*) multipower estimators are used. At all confidence intervals, at all sampling frequencies, the number of detected jumps is significantly larger than what predicted by the confidence interval, as exhaustively reported by the empirical literature. However, when using GTMPV**(m*), the percentage of detected jumps reduces drastically, reducing the number of detection of roughly a half. Table 11 The percentage of detected jumps in our sample for three confidence intervals (99%, 99.9%, and 99.99%); for three different quarticity estimators (the standard threshold tripower and quadpower variation, and GTMPV**(m*)); and for three different sampling frequencies (1, 5, and 10 min) c.i. TMPV([43,43,43]) TMPV([1,1,1,1]) GTMPV**(m*) 1-min frequency—percentage of detected jumps 99% 45.12 48.11 35.22 99.9% 31.29 34.66 20.31 99.99% 22.77 26.22 13.68 5-min frequency—percentage of detected jumps 99% 20.56 21.65 12.30 99.9% 12.84 13.56 8.23 99.99% 8.69 9.14 5.68 10-min frequency—percentage of detected jumps 99% 18.31 18.96 11.92 99.9% 10.38 10.72 6.99 99.99% 6.06 6.26 3.92 c.i. TMPV([43,43,43]) TMPV([1,1,1,1]) GTMPV**(m*) 1-min frequency—percentage of detected jumps 99% 45.12 48.11 35.22 99.9% 31.29 34.66 20.31 99.99% 22.77 26.22 13.68 5-min frequency—percentage of detected jumps 99% 20.56 21.65 12.30 99.9% 12.84 13.56 8.23 99.99% 8.69 9.14 5.68 10-min frequency—percentage of detected jumps 99% 18.31 18.96 11.92 99.9% 10.38 10.72 6.99 99.99% 6.06 6.26 3.92 Notes: Results indicate a drastic reduction of detected jumps when GTMPV**(m*) is used. Table 11 The percentage of detected jumps in our sample for three confidence intervals (99%, 99.9%, and 99.99%); for three different quarticity estimators (the standard threshold tripower and quadpower variation, and GTMPV**(m*)); and for three different sampling frequencies (1, 5, and 10 min) c.i. TMPV([43,43,43]) TMPV([1,1,1,1]) GTMPV**(m*) 1-min frequency—percentage of detected jumps 99% 45.12 48.11 35.22 99.9% 31.29 34.66 20.31 99.99% 22.77 26.22 13.68 5-min frequency—percentage of detected jumps 99% 20.56 21.65 12.30 99.9% 12.84 13.56 8.23 99.99% 8.69 9.14 5.68 10-min frequency—percentage of detected jumps 99% 18.31 18.96 11.92 99.9% 10.38 10.72 6.99 99.99% 6.06 6.26 3.92 c.i. TMPV([43,43,43]) TMPV([1,1,1,1]) GTMPV**(m*) 1-min frequency—percentage of detected jumps 99% 45.12 48.11 35.22 99.9% 31.29 34.66 20.31 99.99% 22.77 26.22 13.68 5-min frequency—percentage of detected jumps 99% 20.56 21.65 12.30 99.9% 12.84 13.56 8.23 99.99% 8.69 9.14 5.68 10-min frequency—percentage of detected jumps 99% 18.31 18.96 11.92 99.9% 10.38 10.72 6.99 99.99% 6.06 6.26 3.92 Notes: Results indicate a drastic reduction of detected jumps when GTMPV**(m*) is used. This empirical finding might help in explaining why, in the literature, it appears that “too many” jumps are detected, as documented in Christensen, Oomen, and Podolskij (2014) using ultra-high frequency data, and by Bajgrowicz, Scaillet, and Treccani (2016). We suggest, indeed, that most of the jumps are spurious artifact of inefficient quarticity measurements. Our results show also that the impact of such spurious detections can be substantially reduced by employing a (nearly) efficient quarticity estimator. 5.3 Impact on Volatility Forecasting As suggested by Bollerslev, Patton, and Quaedvlieg (2016), quarticity estimation can assist volatility forecasting. Denote by IV̂t at day t an integrated variance estimator, and define IV̂t−j|t−h=1h∑i=jhIV̂t−j. One of the most popular model for forecasting daily integrated variance is the HAR model of Corsi (2009): IV̂t=β0+β1IV̂1+β2IV̂t−1|t−5+β3IV̂t−1|t−22+ut, (5.2) where ut is a stationary error process. It is well known that the β coefficients estimated with the HAR model are affected by measurement errors in the realized volatilities. In order to account for the presence of the measurement errors, Bollerslev, Patton, and Quaedvlieg (2016) introduce the idea of “dynamic attenuation” with the HARQ model, whose specification is: IV̂t=β0+(β1+β1,QIQ̂t−11/2)IV̂1+β2IV̂t−1|t−5+β3IV̂t−1|t−22+ut, (5.3) where IQ̂t−1 is an integrated quarticity estimator at day t. We then study the sensitivity of the results obtained with model (5.3) to different quarticity estimator. The variable to be forecasted in our exercise is IV̂=TRV, that is threshold realized variance. The different quarticity estimators are those examined so far. Our exercise is fully out-of-sample. We obtain the forecast of IV̂t at time t forecast using estimation of model (5.3) on the past year till day t – 1, so that estimates are performed on a rolling window. As the loss function, we use the traditional root mean square (relative) error. Table 12 shows the RMSE of the HARQ model corresponding to different quarticity estimates, standardized by RMSE obtained with the HAR model, a value less than one meaning that the forecasting performance of the proposed specification is better than the HAR model. Table 12 The out-of-sample forecast relative RMSE of HARQ model for TRV with different quarticity estimators, standardized by the relative RMSE of the HAR model, so that a value less than 1 indicates superiority with respect to the HAR model Quarticity estimators Ticker minRQ medRQ TMPV([43,43,43]) TMPV([1,1,1,1]) TMPV([4]) GTMPV**(m*) BAC 0.8584 0.8504 0.8704 0.8767 0.7439 0.7425 C 0.7600 0.7584 0.7842 0.7853 0.7958 0.7948 JPM 0.9008 0.9156 0.9125 0.9433 0.8610 0.8619 WFC 0.9565 0.9517 0.9539 0.9651 0.9332 0.9329 BA 0.9654 0.9602 0.9519 0.9570 0.9630 0.9641 CAT 0.9348 0.9405 0.9385 0.9342 0.9328 0.9318 FDX 0.9754 0.9607 0.9608 0.9696 0.9458 0.9460 HON 0.8769 0.8721 0.8707 0.8785 0.8672 0.8681 IBM 0.9267 0.9194 0.9182 0.9167 0.9131 0.9100 HPQ 0.8918 0.8932 0.9139 0.9345 0.9263 0.9273 TXN 0.9903 0.9812 0.9686 0.9574 0.9642 0.9640 T 0.9978 0.9153 0.9245 0.9311 0.9023 0.9020 KFT 1.0094 1.0078 0.9888 0.9858 0.9799 0.9798 PEP 0.9646 0.9653 0.9497 0.9549 0.9541 0.9541 PG 0.8581 0.8559 0.8610 0.8628 0.8620 0.8627 TWX 0.9663 0.9533 0.9545 0.9503 0.9614 0.9607 Average 0.9271 0.9188 0.9201 0.9252 0.9066 0.9064 Quarticity estimators Ticker minRQ medRQ TMPV([43,43,43]) TMPV([1,1,1,1]) TMPV([4]) GTMPV**(m*) BAC 0.8584 0.8504 0.8704 0.8767 0.7439 0.7425 C 0.7600 0.7584 0.7842 0.7853 0.7958 0.7948 JPM 0.9008 0.9156 0.9125 0.9433 0.8610 0.8619 WFC 0.9565 0.9517 0.9539 0.9651 0.9332 0.9329 BA 0.9654 0.9602 0.9519 0.9570 0.9630 0.9641 CAT 0.9348 0.9405 0.9385 0.9342 0.9328 0.9318 FDX 0.9754 0.9607 0.9608 0.9696 0.9458 0.9460 HON 0.8769 0.8721 0.8707 0.8785 0.8672 0.8681 IBM 0.9267 0.9194 0.9182 0.9167 0.9131 0.9100 HPQ 0.8918 0.8932 0.9139 0.9345 0.9263 0.9273 TXN 0.9903 0.9812 0.9686 0.9574 0.9642 0.9640 T 0.9978 0.9153 0.9245 0.9311 0.9023 0.9020 KFT 1.0094 1.0078 0.9888 0.9858 0.9799 0.9798 PEP 0.9646 0.9653 0.9497 0.9549 0.9541 0.9541 PG 0.8581 0.8559 0.8610 0.8628 0.8620 0.8627 TWX 0.9663 0.9533 0.9545 0.9503 0.9614 0.9607 Average 0.9271 0.9188 0.9201 0.9252 0.9066 0.9064 Note: In bold, we indicate the estimator with the best performance for the given stock. Table 12 The out-of-sample forecast relative RMSE of HARQ model for TRV with different quarticity estimators, standardized by the relative RMSE of the HAR model, so that a value less than 1 indicates superiority with respect to the HAR model Quarticity estimators Ticker minRQ medRQ TMPV([43,43,43]) TMPV([1,1,1,1]) TMPV([4]) GTMPV**(m*) BAC 0.8584 0.8504 0.8704 0.8767 0.7439 0.7425 C 0.7600 0.7584 0.7842 0.7853 0.7958 0.7948 JPM 0.9008 0.9156 0.9125 0.9433 0.8610 0.8619 WFC 0.9565 0.9517 0.9539 0.9651 0.9332 0.9329 BA 0.9654 0.9602 0.9519 0.9570 0.9630 0.9641 CAT 0.9348 0.9405 0.9385 0.9342 0.9328 0.9318 FDX 0.9754 0.9607 0.9608 0.9696 0.9458 0.9460 HON 0.8769 0.8721 0.8707 0.8785 0.8672 0.8681 IBM 0.9267 0.9194 0.9182 0.9167 0.9131 0.9100 HPQ 0.8918 0.8932 0.9139 0.9345 0.9263 0.9273 TXN 0.9903 0.9812 0.9686 0.9574 0.9642 0.9640 T 0.9978 0.9153 0.9245 0.9311 0.9023 0.9020 KFT 1.0094 1.0078 0.9888 0.9858 0.9799 0.9798 PEP 0.9646 0.9653 0.9497 0.9549 0.9541 0.9541 PG 0.8581 0.8559 0.8610 0.8628 0.8620 0.8627 TWX 0.9663 0.9533 0.9545 0.9503 0.9614 0.9607 Average 0.9271 0.9188 0.9201 0.9252 0.9066 0.9064 Quarticity estimators Ticker minRQ medRQ TMPV([43,43,43]) TMPV([1,1,1,1]) TMPV([4]) GTMPV**(m*) BAC 0.8584 0.8504 0.8704 0.8767 0.7439 0.7425 C 0.7600 0.7584 0.7842 0.7853 0.7958 0.7948 JPM 0.9008 0.9156 0.9125 0.9433 0.8610 0.8619 WFC 0.9565 0.9517 0.9539 0.9651 0.9332 0.9329 BA 0.9654 0.9602 0.9519 0.9570 0.9630 0.9641 CAT 0.9348 0.9405 0.9385 0.9342 0.9328 0.9318 FDX 0.9754 0.9607 0.9608 0.9696 0.9458 0.9460 HON 0.8769 0.8721 0.8707 0.8785 0.8672 0.8681 IBM 0.9267 0.9194 0.9182 0.9167 0.9131 0.9100 HPQ 0.8918 0.8932 0.9139 0.9345 0.9263 0.9273 TXN 0.9903 0.9812 0.9686 0.9574 0.9642 0.9640 T 0.9978 0.9153 0.9245 0.9311 0.9023 0.9020 KFT 1.0094 1.0078 0.9888 0.9858 0.9799 0.9798 PEP 0.9646 0.9653 0.9497 0.9549 0.9541 0.9541 PG 0.8581 0.8559 0.8610 0.8628 0.8620 0.8627 TWX 0.9663 0.9533 0.9545 0.9503 0.9614 0.9607 Average 0.9271 0.9188 0.9201 0.9252 0.9066 0.9064 Note: In bold, we indicate the estimator with the best performance for the given stock. We confirm and corroborate the empirical evidence of Bollerslev, Patton, and Quaedvlieg (2016): The HARQ model provides superior forecasts with respect to the HAR model, since the ratio is generally less than one, with rare exceptions, for all stocks and all quarticity estimators. When it comes to the choice of the quarticity estimator, the estimator we propose achieves the best value of the loss function quite often, and is also the best estimator on average, indicating a clear advantage in using our estimator. The second-best estimator is TMPV([4]), which is also a nearly efficient estimator (with m = 1), and which has been indicated by Bollerslev, Patton, and Quaedvlieg (2016) as their best-performing quarticity estimator. Again, this is an indication that an estimator built to be more precise can yield substantial improvement also in terms of volatility forecasting. 6 Conclusions We provided methods to find minimum variance estimators in the class of linear combinations of multipower estimators. In particular, we propose a specific quarticity estimator which is nearly efficient (in the sense of Renault, Sarisoy, and Werker, 2016) and with a smaller variance than traditional estimators in the literature in the case in which the numbers of multipowers employed is fixed. This result can be employed to obtain, by construction, superior estimators (in the mean square error or some other loss function sense) with respect to any unbiased estimator. Based on these results, we show, on simulated data, that optimally weighted multipowers outperform benchmark estimators in terms of mean square error. With respect to the efficient Jacod and Rosenbaum (2013) quarticity estimator, we improve in the fact that the estimator we propose has a smaller variance when m, the number of multipowers employed, is fixed. We use our theoretical results to implement a data-driven selection of m. The empirical application shows that improving the efficiency of quarticity measures by means of our estimator helps in delivering substantially less jumps than what previously found with high-frequency data, and in improving the quality of realized volatility forecasts. We thus conclude that our quarticity estimator could replace existing alternatives for empirical work. Footnotes * We thank Lucio Barabesi and Federico Bandi for insightful discussions. We are also grateful to several referees and editors who provided important feedback. The first author acknowledges the financial support from the Riksbankens Jubileumsfond Grant Dnr: P13‐1024:1 and the VR Grant Dnr: 340‐2013‐5180. Matlab® code for implementing efficient multipowers is available upon request. 1 For example, a more efficient estimator defined as cn∑i=1n−2(|ΔiX|2+|Δi+1X|2+|Δi+2X|2)2 for an appropriate constant cn can be viewed as a linear combination of multipower variations with different powers. 2 An alternative to truncation could be minimizing using powers with the constraint that all of them are less than 2, so that multipowers are sufficient to eliminate jumps asymptotically. However, this would not only, from a theoretical point of view, sacrifice efficiency; but also, from an empirical point of view, compromise the performance of the estimator given the well-known small sample bias of multipower estimators (Corsi, Pirino, and Renò, 2010), which is particularly severe when small powers are next to big powers (e.g., in the [0.5,3.5] case). 3 The setting of Mykland and Zhang (2009) is without jumps. It can however be reconciled with our framework by applying their estimator to truncated returns, as we do here, instead of the original one. 5 On flat pricing, a theoretical analysis is offered by Phillips and Yu (2009) and Bandi, Pirino, and Renò (2017). 6 We are taking advantage here also of the fact that TMPV[4] delivers, by construction, a ratio greater than one, and is also used to compute the bias when optimizing m*. 4 Alternatively, the variance of GTMPV**(m;σ4) could be estimated, in small samples, using wild bootstrap. Our numerical experiments indicate that the two ways of estimating the variance are equivalent, so that we suggest to use the handy asymptotic approximation. APPENDIX A: PROOFS Proof of Proposition 2.1 By Theorem 11.2.1 in Jacod and Protter (2012), the joint asymptotic distribution of [MPV(r(1)),...,MPV(r(N))]′ is mixed normal with conditional variance ΣT=C·∫0T(σs)2Rds. By standard mathematical arguments used in portfolio choice problems (or equivalently by straightforward calculations), we deduce that the asymptotic variance of a linear combination of multipower estimators, ∑j=1NwjMPV(r(j)), is minimal for wj⋆=∑i=1N(ΣT−1)ij∑i=1N∑j=1N(ΣT−1)ij=∑i=1N(C−1)ij∑i=1N∑j=1N(C−1)ij, (A.1) where ΣT−1 is the inverse of N × N covariance matrix ΣT, and the last equality follows by reducing the common multiplicative term ∫0T(σs)2Rds. Straightforward computations show that, for a fixed set of powers, the variance ∑j=1Nwj⋆MPV(r(j)) is indeed given by V˜(N,m;σR)∫0T(σs)2Rds=1∑i=1N∑j=1N(C−1)ij∫0T(σs)2Rds. By the properties of multivariate normal distribution, GMPV*(N,m;σR)=∑j=1Nwj*MPV(r*(j)), (A.2) it holds: 1Δn(GMPV*(N,m;σR)−∫0T(σs)Rds)⇒n→∞MN(0,V˜(N,m;σR)∫0T(σs)2Rds), (A.3) where the above convergence is stable in law. □ Proof of Proposition 3.1 Without loss of generality, we can restrict to the model Xt=σWt with constant σ, and set ϑt=+∞,T=1. The proof of Equations (3.2) and (3.3) is then straightforward. □ Proof of Proposition 3.2 The Jacod and Rosenbaum (2013) estimator takes the form: QVeff(kn)=Tn(1−2kn)∑i=1n−kn+1(cin^)2. (A.4) Write: QVeff(kn)=Tn(1−2kn)∑i=1n−kn+1(nT1kn∑j=0kn−1(Δ¯i+jX)2)2=nT(1−2kn)1kn2∑i=1n−kn+1(∑j=0kn−1(Δ¯i+jX)2)2=nT(1−2kn)1kn2(∑j=0kn−1∑i=1n−kn+1(Δ¯i+jX)4+2∑j1=0kn−1∑j2=j1+1kn−1∑i=1n−kn+1(Δ¯i+j1X)2(Δ¯i+j2X)2). Now, for all j=0,…,kn−1 we have: 13nT∑i=1n−kn+1(Δ¯i+jX)4=TMPV([4])+Op(kn/n), and, for all j1=0,…,kn−1 and j2=j1+1,…,kn−1 we have nT∑i=1n−kn+1(Δ¯i+j1X)2(Δ¯i+j2X)2=TMPV([2,0,…,0︸j2−j1−1terms,2])+Op(kn/n) This proves that the QVeff estimator can be written in a generalized form of the GTMPV**(kn−1,σ4) estimator, since it is a linear combination of the kn−1 estimators TMPV([4]), TMPV([2,2]), TMPV([2,0,2]), TMPV([2,0,0,2]),…, TMPV([2,0,…,0︸kn−2terms,2]) plus end-effects which are of order kn2/n (since there are kn−1 terms which are Op(kn/n)). 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Journal of Financial Econometrics – Oxford University Press
Published: Sep 1, 2018
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