EMPIRICAL LIKELIHOOD FOR HIGH FREQUENCY DATA. 1. Introduction

Transcription

1 EMPIRICAL LIKELIHOOD FOR HIGH FREQUENCY DATA LORENZO CAMPONOVO, YUKITOSHI MATSUSHITA, AND TAISUKE OTSU Abstract. With increasing availability of high frequency financial data as a background, various volatility measures and related statistical theory are develoed in the recent literature. This aer introduces the method of emirical likelihood to conduct statistical inference on the volatility measures under high frequency data environments. We roose a modified emirical likelihood statistic that is asymtotically ivotal under the infill asymtotics, where the number of high frequency observations in a fixed time interval increases to infinity. Our emirical likelihood aroach is extended to be robust to the resence of jums and microstructure noise. We also rovide an emirical likelihood test to detect resence of jums. Furthermore, we establish Bartlett correction, a higher-order refinement, for a general nonarametric likelihood statistic. Simulation and arealdataexamleillustratetheusefulnessofouraroach.. Introduction Realized volatility and its related statistics have become standard tools to exlore the behavior of high frequency financial data and to evaluate financial theoretical models including stochastic volatility models. This increase in oularity has been roelled by recent develoments of robability and statistical theory and by the increasing availability of high frequency financial data (see, Aït-Sahalia and Jacod, 04, for a review). By emloying the asymtotic framework so-called the infill asymtotics, where the number of high frequency observations in a fixed time interval (say, a day) increases to infinity, Jacod and Protter (998) and Barndorff-Nielsen and Shehard (00) established laws of large numbers and central limit theorems for realized volatility, which were extended to more general setus and statistics by Barndorff-Nielsen et al. (006). Gonçalves and Meddahi (009) studied higher-order roerties of the realized volatility statistic and its bootstra counterart. Also various volatility estimation methods are develoed to be robust to the resence of jums (e.g., Barndorff-Nielsen, Shehard and Winkel, 006, and Andersen, Dobrev and Schaumburg, 0) and microstructure noise (e.g., Zhang, Mykland and Aït-Sahalia, 005, Barndorff-Nielsen et al., 008, andjacodet al., 009). Finally, several testing methods for the resence of jums are develoed (e.g., Barndorff-Nielsen and Shehard, 006, and Aït-Sahalia and Jacod, 009). In this aer, we introduce the method of emirical likelihood (see, Owen, 00, for a review) to conduct statistical inference on the volatility measures under high frequency

2 data environments. In articular, based on estimating equations for the volatility measures, such as the integrated volatility, modified emirical likelihood statistics are roosed and shown to be asymtotically ivotal under the infill asymtotics. Our emirical likelihood aroach is extended to be robust to the resence of jums and microstructure noise. The roosed statistics share desirable roerties of the conventional emirical likelihood, such as range reserving, transformation resecting, and data decided shae for confidence region. We also rovide an emirical likelihood test to detect resence of jums. Our emirical likelihood aroach rovides useful alternatives to the existing Wald-tye inference methods and jum tests. This is illustrated by simulation studies and a real data examle. Another distinguishing feature of (conventional) emirical likelihood is that it admits Bartlett correction, a higher-order refinement (DiCiccio, Hall and Romano, 99). However, under the infill asymtotics, emirical likelihood is not Bartlett correctable even for the constant volatility case. In order to exlore further this issue, we consider a general class of nonarametric likelihood based on Cressie and Read s (984) ower divergence family, which contains emirical likelihood, exonential tilting, and Pearson s as secial cases. In this general class of likelihood functions, we find some members that admit Bartlett correction under the constant and general non-constant volatility cases. In articular, we show that the second-order refinement to the order O(n ) can be achieved. This Bartlett correctability can be considered as a unique advantage of our nonarametric likelihood aroach. The rest of the aer is organized as follows. In Section, we consider a benchmark setu which excludes jums and microstructure noise, construct the emirical likelihood statistic, and study its first-order asymtotic roerties. In Section 3, we roose a jum robust version of the emirical likelihood statistic. Also an emirical likelihood test to detect the resence of jums is resented. In Section 4, we roose a noise robust version of the emirical likelihood statistic. In Section 5, we conduct second-order analysis for the roosed statistic and establish the Bartlett correctability results. Sections 6 and 7 resent some simulation results and real data examle, resectively. All roofs of the theorems are contained in the web aendix.. Benchmark case In this section, we resent our methodology in a benchmark setu, which excludes jums and microstructure noise. Jum and noise robust methods are considered in the following sections. Here we consider a scalar continuous time rocess (tyically a log-rice) dx t = µ t dt + t dw t, (.)

3 for t 0, where µ is a drift rocess, is a volatility rocess, and W is a standard Brownian motion. Suose we observe high frequency returns r i = X i/n X (i )/n measured over the eriod [(i )/n, i/n] for i =,...,n.althoughourmethodologycanbealiedtoother functionals of (see Remark 4 below), we focus on the integrated volatility = R 0 udu over a fixed interval [0, ] (say, a day or month). As a nonarametric measure of volatility, the integrated volatility has been drawing considerable attention from researchers who face to high frequency financial data. One oular estimator of is so-called the realized volatility ˆ = P n r i. It is known that under certain conditions on the rocess (.), ˆ is consistent for and asymtotically normal under the limit n!for increasingly finely samled returns over the fixed interval [0, ] (called the infill asymtotics) (e.g., Jacod and Protter, 998, and Barndorff- Nielsen and Shehard, 00). In this section, we emloy the following setu based on Barndorff-Nielsen et al. (006). Assumtion X. The rocess X defined on a filtered robability sace follows (.), where µ is an adated redictable locally bounded drift rocess, and is an adated cadlag volatility rocess satisfying where a, t = 0 + Z t 0 a udu + Z t 0 u dw u + Z t 0 v u dv u,,andv are adated cadlag rocesses, a is redictable and locally bounded, and V is a Brownian motion indeendent of W. This assumtion is general enough to allow for intraday seasonality, long memory, and correlation between and W (called the leverage effect). Under this assumtion, Barndorff-Nielsen et al. (006) showed that ˆ is consistent and asymtotically normal n(ˆ ) d! N(0, ), (.) ˆV as n!, where ˆV = n 3 P n r4 i. Based on this result, it is customary to construct a Wald-tye confidence interval for. Also, Gonçalves and Meddahi (009) roosed bootstra inference methods on. In this aer, we introduce the emirical likelihood aroach and roose an alternative inference method for. Based on the estimating equation P n (nr i ) =0for the realized volatility ˆ, the emirical likelihood function for can be written as ( n ) Y EL( ) = max nw i w i (nri ) =0,w i 0, w i =. (.3) 3

4 By the Lagrange multilier argument, the dual form of (.3) is written as ny EL( ) = + (nri ), nr i P where solves n n =0.Inractice,weemloythisdualreresentationto + (nri ) comute EL( ). Let R q = n P q/ n r i q. The first-order asymtotic distribution of EL( ) is obtained as follows. See our web aendix for the roof. Theorem. Suose Assumtion X holds true. As n!, T EL ( ) = 3 R R 4 Remark. Based on this theorem, the 00( { logel( )} d!. )% asymtotic confidence interval for the integrated volatility is given by CI EL = { : T EL( ) ale, }, where, is the ( )-th quantile of the distribution. Remark. It should be noted that under the infill asymtotics, the conventional emirical likelihood statistic (i.e., logel( )) doesnotconvergetothe distribution. In other words, the emirical likelihood statistic is not internally studentized. This is because the asymtotic variance of the term n P n (nr i ) does not match to the P limit of n n (nr i ) under the infill asymtotics. The correction term 3 introduced to recover the studentization. R R 4 is Remark 3. We now discuss advantages of our emirical likelihood confidence interval CIEL comared to the conventional Wald-tye confidence interval (i.e., ˆ q ± z / ˆV/nfor the ( /)-th quantile z / of N(0, )). First, CIEL may be asymmetric around the oint estimate ˆ, anditsshaeisflexiblydeterminedbythedata. Second,CIEL never contains negative values (called range reserving roerty). On the other hand, the lower endoint of the Wald confidence interval may be negative. Third, CIEL is transformation resecting (i.e., the confidence interval of f( ) is given by {f( ) : T EL ( ) ale, }). However, the Wald confidence interval is not invariant for transformations of and may yield different conclusions. Remark 4. Here we discuss the emirical likelihood method for the integrated volatility = R 0 udu. Our method can be modified for other objects related to. For examle, suose we are interested in the -th ower variation = R 0 udu for > 0. By Barndorff-Nielsen et al. (006), is consistently estimated by ˆ = µ n P +/ n r i, where µ = E z with z N(0, ). Basedontheestimatingequationforˆ,theemirical 4

5 likelihood function for can be constructed as ( Y n EL( ) = max nw i w i (µ n / r i )=0,w i 0, ) w i =. The asymtotic roerty of this statistic is established in the same manner (with a different correction term for asymtotic ivotalness). Also, in the next section, we consider emirical likelihood for multiower variation to conduct jum robust inference. 3. Jum robust inference and test for jums In this section, we roose a jum robust version of the emirical likelihood statistic and test for resence of jums. The emirical likelihood function (.3) roosed in the last section is constructed from the estimating equation for the realized volatility ˆ = P n r i. Our aroach can be generalized to other estimating equations for the integrated volatility. In articular, it is useful to consider the multiower variation (e.g., Barndorff-Nielsen, Shehard, 004, and Barndorff-Nielsen, Shehard and Winkel, 006) ˆ = r i m+ r i m, i=m for a vector =(,..., m ) of ositive numbers with + + m =. Indeed the realized volatility is a secial case of the multiower variation (with m =and =). A remarkable roerty of the multiower variation is: if s are reasonably small, then the estimator ˆ enjoys certain robustness against jums in the observed rocess. for t To be recise, consider the rocess Y t = X t + J t, (3.) 0, where X is generated by the continuous time rocess in (.) satisfying Assumtion X, and J is a jum rocess, which is assumed to be a Lévy rocess with no continuous comonent and index Z =inf a 0: [,] x a (dx) < [0, ], for the Lévy measure. The Lévy rocess is a convenient and general class of rocesses to accommodate both finite and infinite activity jums. Barndorff-Nielsen, Shehard and Winkel (006, Theorem ) showed that the limiting distribution of the multiower variation ˆ remains the same regardless of resence of the jum rocess J as far as <, ale min{,..., m }alemax{,..., m } <. (3.) Aoularchoiceof for the jum robust estimator is the triower variation (i.e., m =3 and = = 3 =/3). 5

6 Suose we observe high frequency returns r i = Y i/n Y (i )/n measured over the eriod [(i )/n, i/n] for i =,...,n. Let c = Q m l= µ l, where µ = E z with z N(0, ). Based on the estimating equation for ˆ,wedefinethejumrobustemiricallikelihood function for as ( Y n gel( ) = max nw i w i (n r i m+ r i m c )=0,w i 0, Let R = ˆ and R 4 = n P n i=m r i d = my my µ l (m ) l= ) w i =. m+ r i m.alsodefinetheconstant l= mx µ l + k= l= ky Y m µ k l=m k+ my k µ l l= µ l + l+k. (3.3) The first-order asymtotic roerty of the jum robust emirical likelihood statistic g EL( ) is obtained as follows. Theorem. Suose Y is generated by (3.). Assume + + m =and (3.). Then! T EL ( ) = c d R R 4 { log g EL( )} d!, as n!.thisresultdoesnotchangeevenifj =0(the case of no jum). Remark 5. This theorem says that the emirical likelihood statistic T EL ( ) has the limiting distribution that is invariant to the resence of jums. Similar to the benchmark case, we introduce the correction term c d to achieve asymtotic ivotalness. The jum robust confidence interval for is obtained in the same manner. We note that the emirical likelihood function (.3) for the benchmark case (i.e., m =and =) does not satisfy the condition in (3.). We now consider hyothesis testing for resence of jums in the observed rocess (i.e., J =0). The basic idea is to comare the two estimating equations for the realized volatility ˆ and multiower variation ˆ. More recisely, we roose the following emirical likelihood statistic ( gel J Y n =max nw i R R 4 w i ( r i m+ r i m c r i )=0,w i 0, ) w i =. (3.4) The rationale of the above moment restriction is exlained as follows. When there is no jum in the rocess (i.e., J =0), both the multiower variation ˆ (with + + m =) P and realized volatility c n r i multilied by c are consistent for c. Therefore, the moment function P n i=m ( r i m+ r i m c r i ) converges to zero and the statistic EL g J tends to be small. On the other hand, in the resence of jums, the moment function tyically diverges and so does the statistic. 6

7 Let c l =(µ l +/µ l ) Q m k= µ k. The first-order asymtotic roerty of the emirical likelihood statistic EL g J for the resence of jums is obtained as follows. Theorem 3. Suose Y = X (i.e., no jum in the rocess), where X satisfies Assumtion X. Also assume (3.). Then T J EL = d c +3c c c P m l= c { logel (c l c )+c g J }! d, as n!.ontheotherhand,ify is generated by (3.), then the statistic T J EL diverges. 4. Noise robust inference Our emirical likelihood aroach resented above can be also modified to be robust to the resence of microstructure noise. In articular, we adot the re-averaging aroach of Jacod et al. (009), and construct emirical likelihood based on block averages of the original data. In this section, let us consider the following setu. Assumtion X. Observations {Z i/n } n are generated from Z i/n = X i/n + U i/n, where {X i/n } n is drawn from the latent rocess X satisfying Assumtion X, and {U i/n } n is an i.i.d. sequence with zero mean and finite eighth moments and is indeendent of X. We are interested in the integrated volatility = R 0 udu of the latent rocess X. Itis known that due to the resence of the noise term U i/n,theconventionalrealizedvolatility based on {Z i/n } n is inconsistent for. In this setu, Jacod et al. (009) develoed a noise robust estimator for based on the so-called re-averaging aroach. A simlified version of their estimator is described as follows. First, we transform the observed data {Z i/n } n into block averages Z i/n = K P K j=0 Z (i+j)/n for i =0,,...,n K +. Second, based on the block averages, comute (half of) the return data r i =( Z (i+k)/n Zi/n )/ for i =,...,n K +. Finally, we comute the noise robust estimator as = 6 K Xn K r i 3 K ˆ, (4.) where n K = n K + and ˆ = P n (Z i/n Z (i )/n ) is the conventional realized volatility estimator by using the original data. Intuitively, comared to the original Z i/n, the variance of the noise in the block average Z i/n is reduced by a factor of /K. Thus, the volatility estimator based on the block averages are exected to be less sensitive to the resence of the noise term. The second term in (4.) is a bias correction term. Note that the conventional estimator ˆ is inconsistent for under Assumtion X. Jacod et al. (009) showed that is consistent for and asymtotically normal with the rate of n /4. 7

8 By utilizing the estimating equation for (4.), the noise robust emirical likelihood function can be constructed as ( Y nk EL( ) =max n K w i where n K X w i (g Ki )=0,w i 0, n K X w i = g Ki = 6n K 3 K r i K ˆ. Choose the block length as K = cn/ + o(n /4 ) for some c>0. Then define R q = n q/ K P nk r i q and R 4 = 4 4 3c n K X r 4 i + 4 c 3 n (3 4 ) X X n K K i+4k r i j=i+k ) (Z j/n Z (j )/n ) + Xn c 3 n ( ) (Z i/n Z (i )/n ) (Z (i+)/n Z (i+)/n ), with = 6, = 96,and = The object R 4 aeared in Jacod et al. (009, eq. (3.7)) as an estimator of the asymtotic variance of. Based on this notation, the first-order asymtotic distribution of EL( ) is obtained as follows. Theorem 4. Suose Assumtion X holds true. As n!, i= T EL ( ) = 36n/ R4 R { logel( )}! d n K K. R 4 As ointed out by Jacod et al. (009), the re-averaging estimator can be interreted as a realized kernel estimator in Barndorff-Nielsen et al. (008). Similarly, our emirical likelihood statistic T EL ( ) using block averages may be interreted as the block emirical likelihood statistic by Kitamura (997) for weakly deendent time series data. However, here data blocking is emloyed to reduce the effect of microstructure noise. In this section, we imose Assumtion X and consider the case of additive and i.i.d. noise for brevity. We conjecture that it is ossible to extend our aroach to more general setus, such as weakly deendent noise (Aït-Sahalia, Mykland and Zhang, 0), nonadditive noise (Jacod et al., 009), and endogenous time (Li, Zhang and Zheng, 03) by modifying the moment function., 5. General nonarametric likelihood and second-order asymtotics In this section, we generalize the construction of nonarametric likelihood for the integrated volatility by using the ower divergence family (Cressie and Read, 984). This family is general enough to accommodate not only the emirical likelihood considered so far, but also other existing likelihood concets. Based on this general family of nonarametric likelihood functions, we investigate second-order asymtotic roerties of the 8

9 nonarametric likelihood statistics. In articular, we show that adequate choices of tuning constants lead to Bartlett correctability of the nonarametric likelihood statistic. 5.. General nonarametric likelihood. We first consider the benchmark setu in Section. As a general family of nonarametric likelihood functions, we emloy the ower divergence family (Cressie and Read, 984) 8 P n >< ( +) {(nw i) + } if 6=, 0, L (w...,w n )= P n >: log(nw i) if =, n P n w i log(nw i ) if =0. Based on L (w...,w n ) and using the estimating equation for the realized volatility ˆ = P n r i,wesecifythelikelihoodfunctionfortheintegratedvolatility as where the weights w,,...,w,n solve min L (w...,w n ), w,...,w n `, ( ) =L (w,...,w,n ), (5.) subject to w i =, w i (nri ) =0. (5.) Note that the nonarametric likelihood function `, ( ) contains two tuning constants, and. In the literature, it is commonly assumed =. For examle, the emirical likelihood function discussed so far corresonds to = =, andpearson s corresonds to = =. Also Baggerly (998) showed that in the class of likelihood functions with =,only emirical likelihood is Bartlett correctable for the mean of i.i.d. data. On the other hand, Schennach (005, 007) considered the case of 6= and studied the exonentially tilted emirical likelihood statistic with = and = 0 from Bayesian and frequentist ersectives. In the current setu where we emloy the infill asymtotics, it is crucial to consider the general class of `, ( ) indexed by to achieve Bartlett correction. Below we will show that even if the volatility rocess constant, the emirical likelihood statistic (i.e., `, ( ) with = = ) isnotbartlett correctable under the infill asymtotics, and the constants and need to be chosen searately to achieve Bartlett correction. By the Lagrange multilier argument, the solution of (5.) is (see, Baggerly, 998) and w,i = n ( + + (nr i )), (5.3) for 6= 0and w,i = n ex( (nr i )) for =0, where and solve n ( + + (nri )) =, n ( + + (nri )) (nri ) =0, (5.4) is 9

10 for 6= 0and solve n P n ex( (nr i )) = and n P n ex( (nr i ))(nr i ) =0 for =0.Inractice,weuse(5.3)tocomutethelikelihoodfunctionin(5.). The first-order asymtotic distribution of `, ( ) is obtained as follows. Theorem 5. Suose Assumtion X holds true. For each, R, asn!, T, ( ) = 3 R `, ( )! d R 4. Note that the first-order asymtotic distribution of the statistic T, ( ) is identical to the one in Theorem for emirical likelihood. Moreover, the first-order asymtotic distribution does not deend on the tuning constants and. In the next subsection, we study second-order asymtotic roerties of the statistic T, ( ) to comare different choices of and.for the first-order asymtotics,similar modifications can be alied to T, ( ) to be robust to jums and microstructure noise. 5.. Second-order asymtotics. The first-order asymtotic theory for the nonarametric likelihood statistic T, ( ) is silent about the choice of tuning constants. In order to address this issue, we investigate the second-order asymtotic roerty of T, ( ). Following the conventional recie ut forward in DiCiccio, Hall and Romano (99) and Baggerly (998), among others, we first derive the signed root of the nonarametric likelihood statistic, and then evaluate the cumulants of the signed root. Based on these cumulants, we seek values of and at which the third and fourth cumulants vanish at sufficiently fast rates to achieve Bartlett correction. Details are rovided in the web aendix (roofs of Theorems 6 and 7). For the second-order analysis, we add the following assumtion. and Assumtion H. The rocess X follows (.) with µ =0and bounded away from zero. is indeendent of W and This assumtion is restrictive since it rules out the drift term and leverage effect. Gonçalves and Meddahi (009,. 89) imosed a similar but stronger assumtion for higher-order analysis of the bootstra inference. Although the drift term µ is asymtotically negligible at the first-order, it will aear in the higher-order terms and comlicates our second-order analysis. Ruling out the leverage effect (i.e., indeendence between and W ) also simlifies our second-order analysis since it allows to condition on the ath of to comute the cumulants of the nonarametric likelihood statistic. Relaxing Assumtion H for the second-order analysis is beyond the scoe of this aer. To simlify the exosition of our results, first we consider the simle case where the volatility is constant ( t = over t [0, ]). In this setting, the higher order roerties of the nonarametric likelihood statistic are resented in the next theorem. 0

11 Theorem 6. Suose Assumtions X and H hold true and t = over t [0, ]. Then, for = and = ± 5 3,thenonarametriclikelihoodstatisticT, ( ) is Bartlett correctable, i.e., conditionally on the ath of, Pr T, ( ) ale, ( + 3n ) = + O(n ). This theorem says that when we choose = and = ± 5,thenonarametric 3 likelihood test based on T, ( ) using the adjusted critical value, ( + 3n ) rovides arefinementtotheordero(n ) on the null rejection robability error. It should be noted that the emirical likelihood statistic (i.e., T, ( ) with = = ) isnotbartlett correctable because the fourth cumulant of the signed root does not vanish at the order of O(n 4 ) (see the roof of Theorem 6 in the web aendix). Also note that the Bartlett factor +3n does not contain any unknown object. Finally, we dro the assumtion of constant volatility and consider the general case. Although the comutations are quite cumbersome, it is ossible to estimate some tuning constants ˆ and ˆ such that the nonarametric likelihood statistic Tˆ,ˆ( ) is Bartlett correctable. The higher order roerties of the nonarametric likelihood statistic in the general case are resented in the next theorem. Theorem 7. Suose Assumtions X and H hold true. Then, for ˆ, ˆ, anda (defined in (A.), (A.5), and (A.6), resectively, in the web aendix), the nonarametric likelihood statistic Tˆ,ˆ( ) is Bartlett correctable, i.e., conditionally on the ath of, n o Pr Tˆ,ˆ( ) ale, ( + an ) = + O(n ). This theorem says that even for the general case, the nonarametric likelihood statistic Tˆ,ˆ( ) with the estimated tuning constants ˆ and ˆ using the adjusted critical value, ( + an ) rovides a refinement to the order O(n ) on the null rejection robability error. In the general case, the Bartlett factor a can be estimated by the method of moments or wild bootstra as in Gonçalves and Meddahi (009). For the one-sided test, Gonçalves and Meddahi (009) obtained second-order refinement by the bootstra to the order o(n / ). In contrast, we consider the two-sided test and show that our Bartlett correction to the nonarametric likelihood statistic can yield a refinement to the order O(n ). 6. Simulation This section conducts simulation studies in order to evaluate finite samle roerties of the emirical likelihood methods resented above.

12 6.. Simulation : Benchmark case. We adot simulation designs considered in Gonçalves and Meddahi (009). In articular, we consider the stochastic volatility model dx t = µ t dt + t dw t + dw t + q dw 3t, where W t, W t,andw 3t are indeendent standard Brownian motions. First, we consider a general case (i.e. with drift and leverage effects) to illustrate the first-order asymtotic theory in Theorem for the nonarametric likelihood statistic T, ( ). We consider two different models for the volatility rocess for t is the GARCH(,) diffusion t. The first model d t =0.035(0.636 t )dt t dw t. The second model is the two-factor diffusion model t = f( t +.5 t), where d t = tdt + dw t, d t =.386 tdt +(+0.5 t)dw t,and ( ex(x) x ale x 0 f(x) = ex(x 0 ) x0 x0 x 0 + x x>x 0 with x 0 = log(.5). We allow for drift and leverage effects by setting µ t = 0.034, = 0.576, and =0for GARCH(,) models, and µ t =0.030 and = = 0.30 for the two-factor diffusion model. We comare three methods to construct two-sided 95% confidence intervals: (i) the Wald-tye interval (Wald), (ii) emirical likelihood (EL) and (iii) nonarametric likelihood (NL) with = and = Table gives the actual coverage rates of all the intervals across 0,000 relications for five different samle sizes: n =5, 88, 48, 4, and, corresonding to.5-minute, 5- minute, half-hour, -hour, and -hour returns. The Wald-tye intervals tend to undercover for both models. The degree of undercoverage is esecially large when samling is not too frequent. The two-factor model imlies overall larger coverage distortions than the GARCH(,) model. The nonarametric likelihood intervals (including EL intervals) outerform the Wald-tye intervals in all cases. As we discussed in Remark 3, the nonarametric likelihood intervals are range reserving but the Wald-tye confidence interval may contain negative values. To illustrate this oint, we reort the frequencies of negative left endoints of the Wald-tye confidence intervals in Table. This shows that the Wald-tye intervals tend to contain negative values articularly for small samle sizes. Second, we consider two secial cases to illustrate the second-order refinements roosed in the last section: (a) a benchmark model where volatility is constant, and (b)

13 models where volatility is not constant (with no drift term and no leverage effect). Bartlett corrected nonarametric likelihood (BNL) with the Bartlett correction factor +3/n are comared with the above methods. Table 3 shows that the Bartlett corrected nonarametric likelihood intervals outerform all the other intervals even when there is stochastic volatility desite the fact that this correction does not theoretically rovide an asymtotic refinement under the non-constant volatility case. 6.. Simulation : Test for jum. In this subsection we evaluate finite samle roerties of the nonarametric likelihood tests for the resence of jums. We adot simulation designs considered in Dovonon, Gonçalves, Hounyo and Meddahi (04). In articular, we consider the two-factor diffusion model with diurnality effects. q d log S t = µ t dt + u,t t ( dw t + dw t + dw 3t )+dj t, u,t = ex( 0t)+0.5 ex( 0( t)), t = f( t +.5 t), where d t = tdt + dw t, d t =.386 tdt +(+0.5 t)dw t,and ( ex(x) x ale x 0 f(x) = ex(x 0 ) x0 x0 x 0 + x x>x 0 with x 0 =log(.5). The rocess u,t models the diurnal U-shaed attern in intraday volatility. When u,t =for t [0, ], thereturnrocessreducestothesimlecaseof no diurnally effects. J t is a finite activity jum rocess modeled as a comound Poisson rocess with constant jum intensity and random jum size distributed as N(0, jum ). Under the null hyothesis of no jums in the return rocess, we set alternative hyothesis, we set =0.058 and jum =.74. jum =0.Underthe We comare three methods to test for jums: (i) the Wald-tye test (Wald), (signed root) emirical likelihood (EL) and (iii) (signed root) nonarametric likelihood (NL) with = and = We consider five different samle sizes: n =5, 576, 88, 96, and 48 corresonding to.5-minute,.5-minute, 5-minute, 5-minute, and half-hour returns. All results are based on,000 Monte Carlo relications. Table 4 reorts the rejection frequencies of tests at the 5% nominal significance level for both cases with and without diurnally effects. (ii) The Wald-tye test tends to overreject for the both cases, the degree of which is esecially large when samling is not too We define the statistic T n = n(rv n BV n) ˆVn where RV n = P n r i, BV n = µ P n i= r i r i and ˆV n = {(µ 4 +µ 3) } n µ 3 4/3 P n i=3 r i 4/3 r i 4/3 r i 4/3. Then, the test rejects the null of no jums at significance level when T n >z where z is the 00( )% ercentile of the N(0, ) distribution. We define the signed root of nonarametric likelihood ratio statistic as NL = sgn(rv n TV n ) `/, where TV n = µ 3 /3 P n i=3 r i /3 r i /3 r i /3. Then, the test rejects the null of no jums at significance level when NL >z where z is the 00( )% ercentile of the N(0, ) distribution. 3

14 frequent. In all cases, the nonarametric likelihood tests (including EL tests) shows better erformance in the null rejection frequencies. The rejection frequencies varies between 7.0% (n =5)and7.7%(n =48) for Wald, while it varies between 5.0% (n=5) and 8.54% (n =48) for EL and between 4.68% (n =5)and8.83%(n =48)forNL. We also analyze the ower roerties of the roosed tests under the alternative hyothesis. We comare the calibrated owers of three tests (i.e., the rejection frequencies of these tests where the critical values are given by the Monte Carlo 95% ercentiles of these test statistics under the data generation rocess satisfying the null hyothesis). Table 5 shows that the nonarametric likelihood tests is slightly less owerful than Wald. Since the nonarametric likelihood tests have better null rejection roerties than Wald, these ower roerties characterize a tradeoff between the null rejection and ower roerties of Wald-tye and nonarametric likelihood tests Simulation 3: Noise robust test. In this subsection we evaluate finite samle roerties of the noise robust nonarametric likelihood tests. We adot simulation designs considered in Jacod et al. (009). In articular, we consider two different models for the X rocesses. The first model is the constant volatility model X t = X 0 + W t, with =0./ 5. The second model is the stochastic volatility model of Heston (993): dx t = (µ t /)dt + t db t, d t = ale( t )dt + / t dw t, where t = t, µ =0.05/5, ale =5/5, =0.04/5, =0.05/5 and Corr(B,W) = 0.5. Asforthemicrostructurenoise,weassumethatU t is i.i.d. and follows N(0, ). We comare three methods to construct two-sided 95% confidence intervals: (i) the Wald-tye interval (Wald), 3 (ii) emirical likelihood (EL) and (iii) nonarametric likelihood (NL) with = and = + 5.ForthechoiceoftheblocklengthK, weused 3 K = b n/ c so that c, following Jacod et al. (009). 6 3 Table 6 gives the actual coverage rates of all the intervals across 0,000 relications for eight different samle sizes: n =3400, 700, 7800, 4680, 560, and 780. The Wald-tye intervals tend to undercover for both models. The degree of undercoverage is esecially large when samling is not too frequent. The nonarametric likelihood intervals (including EL intervals) outerform the Wald-tye interval in all cases. 3 The 00( )% asymtotic confidence interval for the integrated volatility is given by CI W = { : T n ( ) ale, }, where T n ( ) = n/ ( ) R 4 and, is the ( )-th quantile of the distribution. 4

15 7. Real data examle We consider tick rices obtained from TickData consisting of intra-day quotes of Alcoa, American Exress, Baxter, Citigrou, Dow, Gilead, Goldman Sachs, Intel Cororation, Met, Microsoft, Nike, Pfizer, Verizon and Yahoo from January, 00 to November 5, 00, which corresonds to 47 trading days. Table 7 reorts the mean and standard deviation of daily returns based on 5-min intraday returns. We can observe common features across assets belonging to the same market segment. Negative returns are likely linked to the extraordinary events of the recent financial crisis in Furthermore, Table 8 reorts the ercentage of days identified with jums for the eriod under investigation. As in the Monte Carlo analysis resented in the revious section, we consider three methods to test for jums: (i) the Wald-tye test (Wald), (ii) (signed root) emirical likelihood (EL), and (iii) (signed root) nonarametric likelihood (NL) with = and = + 5. In line with the Monte Carlo findings, 3 we note that the Wald test tends to over detect the jums. Indeed, the ercentage of days identified with jums is always larger than %. EL and NL imlies very similar emirical findings. Using nonarametric likelihood rocedures, the ercentage of days identified with jums is always smaller than 3%. Aendix A. Tables n Wald EL NL Wald EL NL GARCH(,) diffusion Two-factor diffusion Table. Coverage robabilities of nominal 95% confidence intervals for integrated volatility with leverage and drift 5

16 95% 99% 99.9% 95% 99% 99.9% n GARCH(,) diffusion Two-factor diffusion Table. Frequencies (measured by ercentages) of negative left endoints of 95%, 99%, and 99.9% Wald confidence intervals for integrated volatility with leverage and drift n Wald EL NL BNL Constant volatility GARCH(,) diffusion Two-factor diffusion Table 3. Coverage robabilities of nominal 95% confidence intervals for integrated volatility with no drift and no leverage 6

17 n Wald EL NL Wald EL NL without diurnal effects with diurnal effects Table 4. Rejection frequencies of tests at 5% level n Wald EL NL Wald EL NL without diurnal effects with diurnal effects Table 5. Calibrated ower n Wald EL NL Wald EL NL constant volatility Heston Model Table 6. Coverage robabilities of nominal 95% confidence intervals for integrated volatility 7

18 Mean 0 4 SD 0 4 Alcoa American Exress Baxter Citigrou Dow Gilead Goldman Sachs Intel Cororation Met Microsoft Nike Pfizer Verizon Yahoo Table 7. Mean and standard deviation of daily returns based on 5-min intra-day returns for the eriod Wald EL NL Alcoa American Exress Baxter Citigrou Dow Gilead Goldman Sachs Intel Cororation Met Microsoft Nike Pfizer Verizon Yahoo Table 8. Percentage of days identified with jums for the eriod

19 References [] Aït-Sahalia, Y. and J. Jacod (009) Testing for jums in a discretely observed rocess, Annals of Statistics, 37,84-. [] Aït-Sahalia, Y. and J. Jacod (04) High-Frequency Financial Econometrics, PrincetonUniversity Press. [3] Aït-Sahalia, Y., Mykland, P. A. and L. Zhang (0) Ultra high frequency volatility estimation with deendent microstructure noise, Journal of Econometrics, 60, [4] Andersen, T. G., Dobrev, D. and E. Schaumburg (0) Jum-robust volatility estimation using nearest neighbor truncation, Journal of Econometrics, 69, [5] Baggerly, K. A. (998) Emirical likelihood as a goodness-of-fit measure, Biometrika, 85, [6] Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J. and N. Shehard (006) Limit theorems for biower variation in financial econometrics, Econometric Theory,, [7] Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and N. Shehard (008) Designing realized kernels to measure the ex ost variation of equity rices in the resence of noise, Econometrica, 76, [8] Barndorff-Nielsen, O. E. and N. Shehard (00) Econometric analysis of realized volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society, B,64, [9] Barndorff-Nielsen, O. E. and N. Shehard (004) Power and biower variation with stochastic volatility and jums, Journal of Financial Econometrics,, -37. [0] Barndorff-Nielsen, O. E. and N. Shehard (006) Econometrics of testing for jums in financial economics using biower variation, Journal of Financial Econometrics, 4,-30. [] Barndorff-Nielsen, O. E., Shehard, N. and M. Winkel (006) Limit theorems for multiower variation in the resence of jums, Stochastic Processes and their Alications, 6, [] Cressie, N. and T. R. C. Read (984) Multinomial goodness-of-fit tests, Journal of the Royal Statistical Society, B,46, [3] DiCiccio, T. J., Hall, P. and J. Romano (99) Emirical likelihood is Bartlett-correctable, Annals of Statistics, 9, [4] Dovonon, P., Gonçalves, S., Hounyo, U. and N. Meddahi (04) Bootstraing high-frequency jum tests, Working aer. [5] Gonçalves, S. and N. Meddahi (009) Bootstraing realized volatility, Econometrica, 77, [6] Jacod, J. and P. Protter (998) Asymtotic error distributions for the Euler method for stochastic differential equations, Annals of Probability, 6, [7] Jacod, J., Li, Y., Mykland, P. A., Podolskij, M. and M. Vetter (009) Microstructure noise in the continuous case: the re-averaging aroach, Stochastic Processes and their Alications, 9, [8] Kitamura, Y. (997) Emirical likelihood methods with weakly deendent rocess, Annals of Statistics, 5, [9] Li, Y., Zhang, Z. and X. Zheng (03) Volatility inference in the resence of both endogenous time and microstructure noise, Stochastic Processes and their Alications, 3, [0] Owen, A. B. (988) Emirical likelihood ratio confidence intervals for a single functional, Biometrika, 75, [] Owen, A. B. (00) Emirical Likelihood, Chaman and Hall/CRC. [] Schennach, S. M. (005) Bayesian exonentially tilted emirical likelihood, Biometrika, 9,

20 [3] Schennach, S. M. (007) Point estimation with exonentially tilted emirical likelihood, Annals of Statistics, 35, [4] Zhang, L., Mykland, P. A. and Y. Aït-Sahalia (005) A tale of two time scales: determining integrated volatility with noisy high-frequency data, Journal of the American Statistical Association, 00, Faculty of Mathematics and Statistics, University of St.Gallen, Bodanstrasse 6, 9000 St.Gallen, Switzerland. address: lorenzo.camonovo@unisg.ch Graduate School of Information Science and Engineering, Tokyo Institute of Technology, --, Ookayama, Meguro-ku, Tokyo , Jaan. address: matsushita.y.ab@m.titech.ac.j Deartment of Economics, London School of Economics, Houghton Street, London, WCA AE, UK. address: t.otsu@lse.ac.uk 0

21 WEB APPENDIX TO EMPIRICAL LIKELIHOOD FOR HIGH FREQUENCY DATA Abstract. In this aendix, we resent roofs of the theorems in the aer. Aendix A. Mathematical Aendix Throughout the aendix, let R q = n q/ with z N(0, ) for q>0. P n r i q, q = R 0 q udu, andµ q = E z q A.. Proof of Theorems and 5. Theorem is a secial case of Theorem 5. Since the roof is similar, we focus on the case of, 6=, 0. From Barndorff-Nielsen et al. (006, Theorem ), Assumtion X guarantees for any q>0. This imlies R!, R 4! µ4 4, and 3 R! 3 µ4 4 R 4 R q! µq q, (A.) µ 4 4. (A.) Let g i = nr i, ḡ = n P n g i,and V = n P n g i. By (A.) and Barndorff-Nielsen et al. (006), we obtain / 3 µ / 4 4 nḡ = 3 µ 4 4 n(r )! d N(0, ), (A.3) V = R 4 R +! µ 4 4. (A.4) By these results combined with E[gi ] < for all i =,...,n,wecanalythesame argument to Owen (988) to show max aleialen + g i! 0. Thus, by exanding ( + + (nri )) =, ( + + (nri )) (nri ) =0, (A.5) n n around (, )=(0, 0), weobtain = V ḡ + O (n ), = ( +) V ḡ + O (n ). Based on these results, an exansion of `, ( ) around (, )=(0, 0) yields `, ( ) = {( + + g i ) + } = ( +) V ( nḡ) + O (n ). Therefore, the conclusion follows by (A.)-(A.4).

22 A.. Proofs of Theorems and 3. Proofs of these theorems are similar to that of Theorem 5 above. First, we show Theorem. Let g i = n r i m+ r i m c. By Barndorff-Nielsen, Shehard and Winkel (006, Theorem ), we can relace (A.3) and (A.4) with (d 4) / g i! d N(0, ), n n resectively. The remaining art is similar. g i! c 4 c, Next, we show Theorem 3. Let ḡ i = n( r i m+ r i m c r i ). Under the null hyothesis of no jum (i.e. Y = X), we can aly the limit theorems in Barndorff- Nielsen et al. (006). Thus we can relace (A.3) and (A.4) with mx ({d c [c l c ]+c } 4) / ḡ i! d N(0, ), n n ḡ i l=! (c +3c c c ) 4, resectively. The remaining art is similar. Under the alternative of the resence of P jums, Barndorff-Nielsen, Shehard and Winkel (006, Theorem ) imlies that n n ḡi converges to a non-zero constant. Therefore, the test statistic T EL J diverges as n!. A.3. Proof of Theorem 4. The roof is similar to that of Theorem 5 above. By Jacod et al. (009, Theorem 3.), we obtain ( n / X nk ) (g Ki ) = n/ ( ) n K R 4 R 4 d!. (A.6) Also, by insection of the derivations in Jacod et al. (009), we can obtain These results imly n K n K X 3 K ˆ! c Z 0 U s ds, (g Ki ) 36 K ( R 4 R )= 6 K R 6 K R! + c Z 0 U s ds. 3 K ˆ K ˆ K R By (A.6) and (A.7), a similar argument to the roof of Theorem 5 yields T EL ( ) = 36n/ n K K R4 R R 4 { P n K (g Ki )} P nk (g Ki ) + o () d!.! 0. (A.7)

23 A.4. Proof of Theorem 6. Due to indeendence between and W, the symbols such as E[ ] and O ( ) mean the conditional exectation and stochastic order given the ath of,resectively. BeforeanalyzingBartlettcorrectabilityofthenonarametriclikelihood statistic, we introduce further notation. We transform the moment function as m i = V / (nri ) with V = E[n P n (nr i ) ] and define Ā k = m k i, k = E[Āk], A k = n Āk k, for k =,,...NotethatAssumtionHimlies =0, =, A k = O (n / ), for each k =,,..., where the first equality follows from E[ri ]= R i/n (i )/n udu, thesecond equality follows by construction, and the third equality follows from Barndorff-Nielsen et al. (006, Theorem ). Based on the above notation, the nonarametric likelihood statistic is rewritten as `, ( ) =L (w,...,w,n ), where w,i = n ( + + m i ), and and solves ( + + n m i ) =, n ( + + m i ) m i =0. Exansions of these equations around + w i =0and reeated substitutions yield exansions of and as follows = ( + )A + 6 ( + )( ) 3A 3 and + ( + )A A + 8 ( + )A A ( )( + ) 3A 3 A + 6 ( )( + )A3 A 3 ( + ) 3 +( ) ( + ) 3 = A ( ) 3A + A A A A + 3 ( ) 3A A ( )A A 3 ( + )+( ) 3 3 ( )( + )( ) 4 A 4 + O (n 5/ ). 3 ( )( ) 4 A 3 + O (n ). 3

24 By inserting these formulae to an exansion of n `, ( ) around + w i =0,weobtain n `, ( ) = A + 3 ( ) 3A 3 A A + A A ( ) 3 A 3 A + 3 ( )A3 A O (n 5/ ). Let q,n = n q/ P n R i/n (i )/n R 4 =3 4,n and R = yield 3 R R 4 = 3 4,n + 4,n V / 4,n A q/. udu For the term 3 4,n R4 3 4,n V A + V A + 4,n 3 4,n ,n R R 4,exansionsaround R4 3 4,n V 3/ A A + O (n 3/ ), (A.9) 4,n where R 4 = V / A 3 4,n 3 4,n + V A 3 4,n. Consider the constant volatility case t = over t [0, ]. Inthiscase,itholds =, 4,n =, V =, 3 =, 4 =5. (A.0) Then by (A.8) and (A.9), the exansion of the nonarametric likelihood statistic n T, ( ) is written as n T, ( ) = A A A A A A ( )A3 A (9 8 )A3 A + 3 A 4 + O (n 5/ ). As in Baggerly (998), to achieve Bartlett correction, we investigate the conditions of and where the third and fourth cumulants of the signed root of the above exansion vanish at sufficiently fast rates. First, we consider the third cumulant. obtained as n T, ( ) =(S + S + S 3 ) + O (n 5/ ), where S = A, S = 3 A A 3 After some algebra, the signed root form is and S 3 = O (n 3/ ) is not dislayed since it is not used to comute the third cumulant. Based on this form, the third cumulant of S + S + S 3 is obtained as A, ale 3 (, ) = E[S 3 ]+3E[S S ] 3E[S ]E[S ]+O(n 3 ), 4 A 4 (A.8) 4

25 where by Lemma, E[S] 3 = n + O(n 3 ), E[SS ]= ( +)n + O(n 3 ), E[S]E[S ] = 3 ( +)n + O(n 3 ). Therefore, if =, thenthedominanttermofthethirdcumulantvanishesanditholds ale 3 (, )=O(n 3 ). Next, we set = and analyze the fourth cumulant. After some algebra, the signed root form of n T, ( ) with = is obtained as n T, ( ) =(T + T + T 3 ) + O (n 5/ ), where T = A, T = 3 A A + 3 A, T 3 = 6 A A A 9 A A A Then the fourth cumulant of T + T + T 3 is obtained as ale 4 (, ) = E[T 4 ]+4E[T 3 T ]+4E[T 3 T 3 ] 3(E[T ]) where by Lemma, Therefore, if A E[T T ] 4E[T 3 ]E[T ] E[T T ]E[T ] 6E[T ]E[T ] +E[T ](E[T ]) E[T ]E[T T ] E[T ]E[T T 3 ]+O(n 4 ), E[T 4 ]=3n +n 3 + O(n 4 ), E[T 3 T ]= 76 3 n 3 + O(n 4 ), 74 E[T 3 T 3 ]= n 3 + O(n 4 ), (E[T ]) = n, 4 9 E[T T ]= 6 3 n 3 + O(n 4 ), E[T 3 ]E[T ]= 4 3 n 3 + O(n 4 ), E[T T ]E[T ]= 3 n 3 + O(n 4 ), E[T ]E[T ]= 4 3 n 3 + O(n 4 ), E[T ](E[T ]) = 9 n 3 + O(n 4 ), E[T ]E[T T ]= 0 3 n 3 + O(n 4 ), 6 E[T ]E[T T 3 ]= n 3 + O(n 4 ) =0, i.e. = ± 5,thenthedominanttermofthefourthcumulantvanishesanditholds 3 ale 4, ± 5 = O(n 4 ). 3 5

26 Finally, by setting = and = ± 5 3,itholds E[T ]=n, E[T T ]= 0 3 n + O(n 3 ), E[T ]= 4 3 n + O(n 3 ), E[T T 3 ]= 5 6 n + O(n 3 ), and thus the second cumulant used to comute the Bartlett correction factor is obtained as ne[(t + T + T 3 ) ]=+3n + O(n ). A.5. Proof of Theorem 7. We now dro the assumtion of constant volatility. In the general case, the identities in (A.0) do not aly. Thus the objects such as V, 3,and 4 become unknown and need to be estimated. In this case, by (A.8) and (A.9), the exansion of the nonarametric likelihood statistic n T, ( ) is written as n T, ( ) = ca + 6 c/ c / ( ) 3 +d 3/ 6d / A c(d 3)A A where + 8 c(9 3d cd)a A + 6 c( )A3 A c/ c / (d 9)( ) 3 +8d / +9cd / 6d 3/ 4cd 3/ A 3 A + 8 c/ d / (d 3)( ) 3 9c +cd 4cd +9cf A 4 + O (n 5/ ), c = V, d =, 4,n 4,n f = First, we consider the third cumulant After some algebra, the signed root form is obtained as n T, ( ) =(S + S + S 3 ) + O (n 5/ ), where S = c/ A, S = c / ( ) 3 6d / +d 3/ A + c/ (d 3)A A, and S 3 = O (n 3/ ) is not dislayed since it is not used to comute the third cumulant. Based on this form, the third cumulant of S + S + S 3 is obtained as ale 3 (, ) = E[S 3 ]+3E[S S ] 3E[S ]E[S ]+O(n 3 ), 6

27 where by Lemma, E[S] 3 = c 3/ 3 +9d / d 3/ n + O(n 3 ), 5 E[SS ] = 5c / ( ) 3 c 3/ 3 8d / +4d 3/ n + O(n 3 ), 0 E[S]E[S ] = 5c / ( ) 3 c 3/ 3 8d / +4d 3/ n + O(n 3 ). 30 Therefore, if we set as = then it holds ale 3 (, ) = O(n 3 ). 4 5 c 5 c / d / + 8 c / d 3/, (A.) Note that under the constant volatility case, the equation (A.) reduces to =. Inthegeneralcase,however, deends on unknown objects c, d, and 3. By relacing these objects with consistent estimators, we roose the data-deendent value of : ˆ = 5ĉ 4 5 / ĉ ˆd/ + 8 ˆ 3 5 / ĉ ˆd3/, (A.) ˆ 3 where ĉ = ˆV, ˆd ˆ4,n =, ˆ ˆ4,n 3 = ˆV P 3/ n P n (nr i ) 3, ˆV = n n (nr i ),and ˆ4,n = ( ˆV + ). Since ˆ = O 3 (n / ),weneedtotaketheestimationerrorofˆ into account for the second-order analysis below. Next, we rederive the stochastic exansion of n Tˆ, ( ) with ˆ in (A.). By exanding ˆ around (ĉ, ˆd, ˆ 3 )=(c, d, 3 ),itholds where g = 8 5 c c3 5 h = 4 d)c/ 5 (9 d /. ˆ = + ga + ha 3 + O (n ), 3 c / d / c 3/ d / + 4 c / d 3/ c 3/ d 3/ 3, By using this exansion of ˆ, we can rewrite the exansion of the nonarametric likelihood statistic as n Tˆ, ( ) = ca + 6 c/ c / ( + 8 c(9 3d cd)a A + 6 c( + 8 c/ c / ((d 9)( + 8 c/ d / (d 3)( ) 3 +d 3/ 6d / A c(d 3)A A h 3 )A 3 A 3 ) 3g) 3 +8d / 6d 3/ +9cd / 4cd 3/ A 3 A ) 3 9c +cd 4cd +9cf A 4 + O (n 5/ ).(A.3) 7

28 After some algebra, the signed root form is obtained as n Tˆ, ( ) =(T + T + T 3 ) + O (n 5/ ), where and j = l = m = q = ( ) = T = c/ A, T = ja + ka A, T 3 = la A + qa A 3 + ma A + ( )A 3, c / ( ) 3 +d 3/ 6d /, k = c/ (d 3), 36 c/ (9 3d cd) 44 c/ (d 3), c / (d 5)( ) 3 +8d / d 5/ +8cd / 8cd 3/ 6c / g 3, 7 c/ ( h 3 ), 9c / f 9c / +c / d 4c / d 36 4 c/ ( ) 3 c / d 3 9c / d +6c / d. By the definition of,wecanshowthatthethirdcumulantoft + T + T 3 satisfies ale 3 (ˆ, )=O(n 3 ).Afterlengthycalculations,byusingtheexectationsinLemma,the fourth cumulant ale 4 (ˆ, ) = E[T 4 ]+4E[T 3 T ]+4E[T 3 T 3 ] 3(E[T ]) is written in the form of +6E[T T ] 4E[T 3 ]E[T ] E[T T ]E[T ] 6E[T ]E[T ] +E[T ](E[T ]) E[T ]E[T T ] E[T ]E[T T 3 ]+O(n 4 ) ale 4 (ˆ, )= ( )+ + O(n 4 ), (A.4) where and are imlicitly defined and do not deend on.although ( ), and contain unknown objects c, d, and 3,theycanbeestimatedbyĉ, ˆd, andˆ 3,resectively (denote by ˆ ( ), ˆ and ˆ ). Then if the solution exists, the ideal value ˆ is given by a solution of ˆ ˆ ( ˆ)+ˆ =0. It should be noted that in the exansion (A.3), (A.5) aears only in the term f. Therefore, the estimation error ˆ is of negligible order O (n 5/ ),anditholdsale 4 (ˆ, ˆ) =O(n 4 ), i.e., the dominant term of the fourth cumulant vanishes if we choose ˆ and ˆ as in (A.) and (A.5), resectively. 8