Heteroscedasticity test of high-frequency data with jumps and microstructure noise
aa r X i v : . [ ec on . E M ] O c t Heteroscedasticity test of high-frequency data with jumps andmicrostructure noise
Qiang Liu a , Zhi Liu b,c , Chuanhai Zhang d a Department of Mathematics, National University of Singapore, Singapore b Department of Mathematics, University of Macau, Macau SAR, China c UMacau Zhuhai Research Institute, Zhuhai, China d School of Finance, Zhongnan University of Economics and Law, Wuhan 430073, China
Abstract
In this paper, we are interested in testing if the volatility process is constant or not during agiven time span by using high-frequency data with the presence of jumps and microstructurenoise. Based on estimators of integrated volatility and spot volatility, we propose a nonpara-metric way to depict the discrepancy between local variation and global variation. We showthat our proposed test estimator converges to a standard normal distribution if the volatilityis constant, otherwise it diverges to infinity. Simulation studies verify the theoretical resultsand show a good finite sample performance of the test procedure. We also apply our testprocedure to do the heteroscedasticity test for some real high-frequency financial data. Weobserve that in almost half of the days tested, the assumption of constant volatility within aday is violated. And this is due to that the stock prices during opening and closing periodsare highly volatile and account for a relative large proportion of intraday variation.
JEL Classification:
C12, C14, G10
Keywords:
High-frequency data, Jumps, Market microstructure noise, Heteroscedasticity,Nonparametric test
1. Introduction
It is well known that the logarithmic price of an asset is necessarily to be modeled as asemi-martingale process under the assumption of arbitrage-free and frictionless market. Thecoefficient process driving the standard Brownian motion part, which is called the volatilityprocess, serves as a measurement of risk in finance. Due to the wide applications of volatilityin pricing of asset and derivative, portfolio selection, hedging and risk management, thereare tons of research works on estimating the volatility. Many quantities targeting to measurethe magnitude of the volatility, such as integrated volatility, spot volatility, realized Laplacetransform of volatility, are proposed (see A¨ıt-Sahalia and Jacod (2014) for their concrete def-initions and a comprehensive introduction). Besides, it is also important to investigate thedynamic structure of the volatility process. Up until now, numerous models which have beenproposed and widely applied are the ones in but not limited to Black and Scholes (1973),Vasicek (1977), Cox et al. (1985), Constantinides (1992), Duffie and Harrison (1993). Or inanother way, specific functional forms of the volatility process may be postulated before one
Preprint submitted to October 16, 2020 an do goodness-of-fit tests to verify the correctness, related references are A¨ıt-Sahalia (1996),Corradi and White (1999), Dette and von Lieres und Wilkau (2003), Dette et al. (2006), Dette and Podolskij(2008), Vetter and Dette (2012), Christensen et al. (2018), and references therein. Amongthem, one of the most basic questions have been tried to be answered is that whether thevolatility process is constant or not over a period of time, say a day . Putting forward aprocedure to answer such a question is also the purpose of this paper. In the most of theprevious literatures, the test procedures are constructed based on a continuous diffusionassumption, while we consider the underlying data generating process of the return as ageneral Itˆo semi-martingale where the jumps are involved. Besides, the presence of marketmicrostructure noise is also taken into account in our paper. From a theoretical perspective,we contribute to propose a new nonparametric heteroscedasticity test procedure by usinghigh-frequency data and further extend it to different settings incorporating the jumps andthe market microstructure noise.Our goodness-of-fit test procedure is based on the estimation of integrated volatility andspot volatility, which are well documented in existing literatures and many methods arevalid under different settings. The integrated volatility quantifies the fluctuation of the assetprice over a fixed time period, while the spot volatility measures the variation instanta-neously. We take a special case of diffusion process for an example to explain the mechanismimplicated in our test. The stochastic process is discretely observed at evenly distributedpoints on the fixed time interval [0 , , Regarding the daily pattern of the volatility process, it reaches an agreement in Andersen and Bollerslev(1997); Christensen et al. (2018); Andersen et al. (2019) that there are two distinct sources of variation formany financial asset return series. One of them is a deterministic diurnal component representing the fixeddaily pattern. The other one is a stochastic part fluctuating around the fixed one, which brings in randomnessand captures volatility clustering. Recently, Christensen et al. (2018) concluded that the re-scaled log-returnsare often close to homoscedastic within a trading day and the fixed diurnal pattern accounts for a rathersignificant fraction of intraday variation in the volatility. But they also found that important sources ofheteroscedasticity remain present in the data after annihilating the diurnal effect. Thus, the diurnal patternis not sufficient to explain daily variation of the volatility. . Theoretical results
In this section, we firstly construct our heteroscedasticity test procedure by modelingthe logarithmic price process as a continuous Itˆo semi-martingale. If a jump part of finiteactivity is further involved in the underlying data-generating process, the test procedurecan be naturally extended after thresholding the raw observed data. Finally, we incorporatethe presence of market microstructure noise into the observation procedure, namely theobserved data at a given time equals to the value of the underlying process at that time plusanother stochastic error term. We apply the pre-averaging technique before implementingthe heteroscedasticity test to eliminate the bias due to the noise. Detailed descriptionsand assumptions regarding the jumps and the market microstructure noise will be givenlater. By combining the techniques used for eliminating the effects of the jumps and themarket microstructure noise, we can also extend the test procedure to the situation withsimultaneous presence of the jumps and the market microstructure noise. Since the extensioncan be evidently seen from our previous results, we omit its detailed proof and discussion inthis paper.Throughout the paper, all the processes are defined on the time interval [0 , V i/n to be the value of the process V at the time point i/n and define ∆ ni V = V i/n − V ( i − /n for i = 1 , . . . , n . The whole test procedure is based on an infill asymptotic setting, namely n → ∞ , which gives us the high-frequency data. We use the notations → p , → d , → ds to denoteconvergence in probability, convergence in distribution and stable convergence, respectively.In general, we say F -stable convergence of a sequence X n to X defined on an extension of(Ω , F , F t , P ), if for any bounded Lipschitz function g and any bounded F -measurable Q , as n → ∞ , it holds that E [ Q g ( X n )] → E ′ [ Q g ( X )] , where E ′ stands for the expectation on an extension space. The detailed definition and moreproperties of stable convergence can be found in Jacod and Shiryayev (2003). At first, we present our methodology in a benchmark setup, which excludes jumps andmarket microstructure noise when modeling the high-frequency data. We denote X to be thelogarithmic price process of an asset, and X is set to be an one-dimensional continuous Itˆosemi-martingale defined on the filtered probability space (Ω , F , ( F t ) t ≥ , P ) with the followingform: X t = X + Z t b s ds + Z t σ s dB s , (1)where b and σ are progressively measurable processes, and B is a standard Wiener process.We also assume that the volatility process σ to be a continuous Itˆo semi-martingale on thesame filtered probability space (Ω , F , ( F t ) t ≥ , P ), and it can be represented as σ t = σ + Z t b σs ds + Z t D σs dB s + Z t D ′ σs dB ′ s , (2)where b σ , D σ and D ′ σ are adapted, c`adl`ag stochastic processes, b σ is further predictable andlocally bounded, and B ′ is another standard Wiener process independent of B . It is required4hat σ is bounded away from 0, that is, σ t > ≤ t ≤ B in X and σ accommodates the leverage effectin finance, which depicts the dependence structure between these two stochastic processes.Such continuous semi-martingale models for the log-price processes and the volatility pro-cess are widely used in vast existing high-frequency literature for volatility estimation, e.g.,Barndorff-Nielsen et al. (2008), Mykland and Zhang (2009), Jing et al. (2014), and etc.In this paper, we are interested in investigating the pattern of the volatility process.Specifically, we want to test if the volatility process is constant or not during a given timeperiod. To this end, we partition the sample space Ω into two complementary subsetsΩ c = { ω : σ t ( ω ) = σ ( ω ) , t ∈ [0 , } , Ω v = Ω \ Ω c . The null hypothesis can then be written as H : ω ∈ Ω c , while the alternative H a : ω ∈ Ω v .Our target then turns to proposing a test with a pre-set asymptotic significance level andwith power going to one to test the null hypothesis, as n → ∞ .We start demonstrating our theories with the estimation of integrated volatility IV := R σ s ds . It is well known that the most frequently used estimator of the integrated volatilityis the so-called realized volatility, which is defined as c IV n = n X i =1 (∆ ni X ) . (3)It is shown in Barndorff-Nielsen and Shephard (2007) that, under our setting, √ n ( c IV n − IV ) qR t σ s ds → d N (0 , , (4)where N (0 ,
1) denotes standard normal distribution with mean 0 and variance 1. And thecentral limit theorem result can be turned feasible when we replace the integrated quarticity R t σ s ds by its consistent estimators. Based on the estimation of the integrated volatility,the estimation of spot volatility σ τ at any given time τ can be correspondingly proposed byapplying the kernel method described in Fan and Wang (2008), Kristensen (2010), Yu et al.(2014) and Liu et al. (2018). For example, using the specific one-side uniform kernel function K ( u ) = 1 { ≤ u ≤ } , we can obtain an estimator of σ τ as b σ nτ ( k n ) = 1 k n ∆ n ⌊ τ · n ⌋ + k n X i = ⌊ τ · n ⌋ +1 (∆ ni X ) , (5)where k n is the number of intervals after the time point τ and lie closest to τ . Following thetheoretical results in aforementioned references, we conclude that under our setting, and iffurther k n → ∞ and k n /n → n → ∞ , we have p k n b σ nτ ( k n ) − σ τ p σ τ → d N (0 , , as n → ∞ . (6)5he consistency result ( b σ nτ ( k n )) → p σ τ further implies that a feasible central limit theoremcan be obtained if we replace σ τ by ( b σ nτ ( k n )) .Now, we state the first test procedure, which is based on the estimators of the integratedvolatility and the spot volatility discussed above. Theorem 1. X follows the process in (1) .1. For ω ∈ Ω , if as n → ∞ , k n → ∞ and k n /n → , then it holds that, as n → ∞ , k n n ⌊ n/k n ⌋− X j =0 ( b σ njk n /n ( k n ) − c IV n ) → p Z ( σ s − IV ) ds. (7)
2. For ω ∈ Ω c , if as n → ∞ , k n → ∞ and k n /n → , then it holds that, as n → ∞ , T n ( k n ) := r k n n ⌊ n/k n ⌋− X j =0 n(cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ c IV n (cid:1) − o → d N (0 , . (8)
3. Denote z α as the α -quantile of standard normal distribution, if as n → ∞ , k n → ∞ and k n /n → , then it holds that, as n → ∞ , ( P ( T n ( k n ) > z − α | Ω c ) → α, if P (Ω c ) > , P ( T n ( k n ) > z − α | Ω v ) → . (9)The intuition is as follows. If the volatility process is constant over [0 , √ k n ( b σ njk n /n ( k n ) − IV ) with j = 0 , ..., ⌊ n/k n ⌋− σ .Then, by martingale central limit theorem in Hall and Heyde (1980), we can obtain r k n n ⌊ n/k n ⌋− X j =0 (cid:16)(cid:0) √ k n ( b σ njk n /n ( k n ) − IV ) √ σ (cid:1) − (cid:17) → d N (0 , , as n → ∞ . After replacing IV by its estimator c IV n above, we note that the independence structurebetween the terms in the summation is broken up. But it does not affect the asymptoticconclusion, because c IV n converges to IV at a faster rate, compared with the convergencerate of ( b σ njk n /n ( k n ) − σ jk n /n ) to zero. After substituting σ with its consistent estimator c IV n , we obtain the result (8). Whether the volatility process is constant or not, we alwayshave (7), where the left hand side term is approximately the Riemann sums of the term onthe right hand side. Plugging (7) into (8), we see that if the constant volatility assumptionis violated, then the quantity T n ( k n ) in (8) will tend to infinity at a fast rate of √ nk n . Thedifferent asymptotic properties of T n ( k n ) for constant volatility and time-varying volatilitylead to conclusion (9) and enable us to do the hypothesis testing in our way.6 .2. Finite activity jump Now, we consider the setting where the underlying logarithmic price process is modeledas the combination of the continuous process X and another pure jump process J , which isrestricted to be of finite activity. That is, we have Y t = X t + J t , (10)and the process Y , instead of X in the last part, is observed at the time points in , for i = 0 , , ..., n . We write J t = P N t j =1 γ τ j , where N t is a non-explosive counting process withpossibly time varying intensity, γ τ j is the size of the jump at time τ j . These jump sizes arenot necessarily i.i.d random variables, nor independent of N .To eliminate the influence of jumps on estimating the integrated volatility, Mancini (2009)proposed a thresholding technique to discriminate the time intervals with jumps from thosewithout jumps. Such a filtering procedure can be done by using a deterministic thresholdfunction r ( x ) satisfying the following conditions: Assumption The function r ( x ) : R → R satisfies, lim x → x log( 1 x ) r ( x ) = 0 , lim x → r ( x ) =0 . It is shown that for the sample paths, with probability one, there are jumps between[( i − /n, i/n ] if (∆ ni Y ) > r (1 /n ). Since these intervals with jumps are finite, exclud-ing observed data in these intervals has no influence on the asymptotic properties of theestimator of the integrated volatility. Consequently, the thresholding versions of the estima-tors of the integrated volatility (called truncated realised volatility) and the spot volatilityare formalized as c IV n,T hr = n X i =1 (∆ ni Y ) I { (∆ ni Y ) ≤ r (1 /n ) } , b σ τ n,T hr ( k n ) = 1 k n ∆ n ⌊ τ · n ⌋ + k n X i = ⌊ τ · n ⌋ +1 (∆ ni Y ) I { (∆ ni Y ) ≤ r (1 /n ) } . (11)The same conclusions in (4) and (6) also hold if we replace c IV n and b σ τ n ( k n ) with c IV n,T hr and b σ τ n,T hr ( k n ), respectively. As a by-product, their detailed proofs are also given as weprove the following main theorem in Appendix. Theorem 2. X follows the process in (1) , and Assumption 1 hold.1. For ω ∈ Ω , if k n → ∞ and k n /n → hold, as n → ∞ , then we have, as n → ∞ , k n n ⌊ n/k n ⌋− X j =0 ( b σ n,T hrjk n /n ( k n ) − c IV n,T hr ) → p Z ( σ s − IV ) ds. (12)7 . For ω ∈ Ω c , if k n → ∞ , k n /n → and √ k n log n/ √ n → hold, as n → ∞ , then wehave, as n → ∞ , T n,T hr ( k n ) := r k n n ⌊ n/k n ⌋− X j =0 (cid:16)(cid:0) √ k n ( b σ n,T hrjk n /n ( k n ) − c IV n,T hr ) √ c IV n,T hr (cid:1) − (cid:17) → d N (0 , . (13)
3. Denote z α as the α -quantile of standard normal distribution, if k n → ∞ , k n /n → and √ k n log n/ √ n → hold, as n → ∞ , then we have, as n → ∞ , ( P ( T n,T hr ( k n ) > z − α | Ω c ) → α, if P (Ω c ) > , P ( T n,T hr ( k n ) > z − α | Ω v ) → . (14)To reduce the effect of jumps (finite activity or infinite activity) in estimating the inte-grated volatility, another alternative method is the so-called realized multi-power variationestimator (see Barndorff-Nielsen and Shephard (2004), Barndorff-Nielsen et al. (2006b) andJacod (2008)), which diminishes the effect of jumps by using the products of the consecutiveabsolute increments | ∆ ni Y | . Theoretically, both of these two estimators are rate-efficient,but the truncated realised volatility is more efficient than the realised multi-power variationestimator in the sense of having a smaller variance. Indeed, the realised multi-power varia-tion estimator is mainly biased by large jumps but is less affected by small jumps, while onthe contrary, the truncated realised volatility is problematic in removing small jumps buteliminates large jumps effectively. In Veraart (2011), the properties of these two estimatorsare analyzed and compared comprehensively, their finite sample performances are verifiedby numerous Monte Carlo studies under different models. Furthermore, a combination ofthese two estimators breeds a new estimator called truncated realized multi-power variationestimator therein, which achieves the best effect of finite sample performance, since such acombination compensates the weaknesses of these two estimators. We note that our testprocedure can be constructed accordingly by using these estimators mentioned, but we onlyconsider the truncated realised volatility version here from the perspective of both simplicityand efficiency. Remark 1.
The restriction of finite activity on the jump process J can be relaxed to someextent, for example, the case of L´evy jumps of infinite activity with finite variation. Itcan be shown that the same conclusions in above theorem also hold for this relatively relaxcondition, but we only consider finite jumps for simplicity of the proof procedure. More onrelated properties and analyses can be found in Mancini and Ren`o (2011) and Jing et al.(2014). Remark 2.
One possible choice for r ( x ) is the power function cx ω , with c being a con-stant and ω ∈ (0 , r ( x ) (may be stochastic) is considered inMancini and Ren`o (2011). Furthermore, A¨ıt-Sahalia and Jacod (2009b) point out that thevalue of c should be proportional to the “average” value of σ t , which could be consistentlyestimated by the multi-power variation estimator mentioned above. The specific setting ofthe parameters c and ω are also discussed in Veraart (2011), supported by a great deal ofsimulation studies. 8 .3. Market microstructure noise In this part, the data generating process of log-price is still modeled as the continuoussemi-martingale X , but the observation procedure is conducted with disturbance. Math-ematically, the observed data Z i/n at in for i = 0 , , ..., n are the underlying process X i/n contaminated by another market microstructure noise term ǫ i/n , that is Z i/n = X i/n + ǫ i/n . (15)For the convenience of description, we define ǫ t over the whole time span for t ∈ [0 , ǫ , we assume that there exists a transition probability Q t ( ω, dx ) from(Ω , F t ) into R . We endow the space Ω ′ = R [0 , ∞ ) with the product Borel σ -field F ′ and withthe probability Q ( ω, dω ′ ) which is the product ⊗ t ≥ Q t ( ω, · ). The process Z is called the“canonical process” on (Ω ′ , F ′ ), with the filtration F ′ = σ ( Z s : s ≤ t ). We then work in thefiltered probability space (Ω ′′ , F ′′ , F ′′ t ≥ , P ) withΩ ′′ = Ω × Ω ′ , F ′′ = F × F ′ , F t = ∩ s>t F s × F ′ s , P ′′ ( dω, dω ′ ) = P ( dω ) Q ( ω, dω ′ ) . And the following assumption is satisfied:
Assumption We have Z xQ t ( ω, dx ) = X t ( ω ) , and the process α t ( ω ) = Z x Q t ( ω, dx ) − X t ( ω ) = E [( Z t ) |F ]( ω ) − X t ( ω ) is c ` a dl ` a g(necessarily ( F t )- adapted), and the process β t ( ω ) = Z x Q t ( ω, dx ) is locally bounded. Before giving our estimators of the integrated volatility and the spot volatility, we firstlyneed to pre-average the raw increments with a function g supported on the interval [0 , Assumption The function g is continuous and piecewise differentiable with a piecewiseLipschitz derivative g ′ , g (0) = g (1) = 0 , < Z g ( s ) ds < ∞ . Denote the shorthand g ni = g ( i/p n ), then the pre-averaged increments for any process V isdefined as V njp n = p n X i =1 g ni ∆ njp n + i V, for j = 0 , · · · , ⌊ n/p n ⌋ − . c IV n,P re ( p n ) = 1 ϕ n ⌊ n/p n ⌋− X j =0 ( Z njp n ) , (16)with ϕ n = 1 p n p n X i =1 ( g ni ) . Similarly in an aforementioned way of kernel smoothing, an estimatorof the spot volatility can be obtained as b σ n,P rekp n l n /n ( p n , l n ) = np n l n ϕ n ( k +1) l n X j = kl n +1 ( Z njp n ) , for k = 0 , · · · , ⌊ n/ ( p n l n ) ⌋ − , (17)where l n is the widow width of the kernel estimation.In view of (15), we have Z njp n = X njp n + ǫ njp n . Some simple variance calculations showthat X njp n = O p ( r p n n ) and ǫ njp n = O p ( r p n ). If p n → ∞ and p n /n → ∞ hold, as n → ∞ ,then Z njp n is dominated by X njp n , and the effect of the market microstructure noise can thenbe neglected. As a consequence, we have c IV n,P re ( p n ) → p R σ s ds and b σ n,P rekp n l n /n ( p n , l n ) → p σ kp n l n /n . If further that the conditions p n /n and √ nl n /p n → r np n ( c IV n,P re ( p n ) − Z σ s ds ) → ds N (0 , Z σ s ds ) , (18) p l n ( b σ n,P rekp n l n /n ( p n , l n ) − σ kp n l n /n ) → ds N (0 , σ kp n l n ) . (19)We also give a sketch of their proofs in Appendix (Lemma 1) as a by-product of this paper.Based on these results, we can establish our test procedure as Theorem 3. X follows the process in (1) , and Assumptions 2-3 hold.1. For ω ∈ Ω , if as n → ∞ , p n → ∞ , l n → ∞ , and p n /n → ∞ hold, then we have, as n → ∞ , p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 ( b σ n,P rekp n l n /n ( p n , l n ) − c IV n,P re ( p n )) → p Z ( σ s − IV ) ds. (20)
2. For ω ∈ Ω c , if as n → ∞ , p n → ∞ , l n → ∞ , √ nl n /p n → and p n /n → hold, thenwe have, as n → ∞ , T n,P re ( p n , l n ) := r p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:16)(cid:0) √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − c IV n,P re ( p n )) √ c IV n,P re ( p n ) (cid:1) − (cid:17) → d N (0 , . (21)10 . Denote z α as the α -quantile of standard normal distribution, if as n → ∞ , p n → ∞ , l n → ∞ , √ nl n /p n → and p / n /n → hold, then we have, as n → ∞ , ( P ( T n,P re ( p n , l n ) > z − α | Ω c ) → α, if P (Ω c ) > , P ( T n,P re ( p n , l n ) > z − α | Ω v ) → . (22)On one hand, we consider constructing our estimators of the integrated volatility andthe spot volatility by using non-overlapping pre-averaged data for simplicity, instead ofthe overlapping case considered in Jacod et al. (2009). It has no harm to our theoreticalresults, but at a cost of reducing the number of pre-averaged data. On the other hand, asmentioned above, we diminish the effect of the noise by choosing p n /n → ∞ . Alternatively,we can also take p n = O p ( n / ), then X njp n and ǫ njp n are of the same order. In this case,the effect of the noise should be removed by subtracting an estimator of the variance of thenoise. Furthermore, the presence of the noise also deforms the variances of the asymptoticdistributions in (18) and (19), thus new estimators of these variances are necessarily to bereconstructed. It is viable to extend our test procedure to the setting with p n = O p ( n / ) andoverlapping pre-averaged data, but such a consideration can complicate our test procedureto an undesirable degree. We mention that such a setting may be considered as a sole workfor our future research. Readers who are interested in this setting can refer to Jacod et al.(2009) for the detailed discussion when it comes to the estimation of the integrated volatility. Remark 3.
If the presence of jump process and market microstructure noise are both con-sidered simultaneously, we can obtain similar results by combining the thresholding tech-nique and the pre-averaging method. The extension can be obviously seen from our previousderivation, thus we omit the detailed discussion here. Related papers can be referred to areJing et al. (2014) and references therein.
3. Monte Carlo study
We now conduct some Monte Carlo simulation studies to examine our test procedureand investigate the finite sample performance of our test estimator in the cases of constantvolatility and stochastic volatility. As discussed in the last theoretical section, we considerthree different scenarios where continuous semi-martingale, involvement of finite jumps andcontamination from market microstructure noise are used for modeling the log-price process.For the notations, we follow their definitions given in previous sections for the old ones, andshall specify later where new ones are used.
The latent log-price process X = ( X t ) ≤ t ≤ is generated from the following two stochasticdifferential equations, one of them considers constant volatility while the other one considersstochastic volatility. • Model 1–The constant volatility model dX t = σdW t , (23)with X = 1 and σ = 1. 11 Model 2–The Heston model with stochastic volatility dX t = σ t dW t ,dσ t = κ ( α − σ t ) dt + γσ t ( ρdW t + p − ρ dB t ) , (24)with the parameters κ = 5 , α = 0 . , γ = 5, ρ = −√ . X = 1 and σ = 1. We followthe parameter setting in Wang and Mykland (2014) to calibrate the model to real financialdata.Regarding the jump component J t = P N t j =1 γ τ j in Section 2.2, we consider the jump size γ τ j ∼ N (0 , σ κ ), the number of jumps up to time point t , N t ∼ P oisson ( λt ), which is aPoisson distribution with parameter λ . We firstly generate the process N t within t ∈ [0 , γ τ j independently. We fix σ κ = 0 . λ = 10 , , , k n = ⌊ θ √ ∆ n ⌋ with θ = 1 . ν n = 4 √ BV n · ∆ ̟n with ̟ = 0 .
499 and BV n = π n X i =2 | ∆ ni Y || ∆ ni − Y | . The quantity BV n is the realized bipower variation estimator introduced in Barndorff-Nielsen and Shephard(2004) and serves as a consistent estimator of the integrated volatility which is robust tojumps.For the market microstructure noise term ǫ t in Section 2.3, it is mixed in the observedprices at t = 0 , /n, ..., n/n , with ǫ t ∼ N (0 , η ). The noise terms are independently andidentically distributed with different strengths η = 0 . , . , .
05. Recall that p n is thenumber of increments based on the raw data used for pre-averaging, l n is the number ofnon-overlapping pre-averaged blocks used for the kernel estimation of the spot volatility.For Theorem 3, we take p n = ⌊ c ∆ − (1 / χ ) n ⌋ with c = 1 / χ = 0 .
05, and l n = ⌊ a ∆ − bn ⌋ with a = 2 and b = 0 .
17, which satisfy our theoretical requirement.For each experiment, we simulate 5000 runs of daily sample paths by using the Euler dis-cretization method. We consider different sampling frequencies with n = 23400 , , , , , Table 1 records the empirical size (based on constant volatility) and power (based on time-varying volatility) of the heteroscedasticity test when finite activity jumps are present in thelogarithmic price process. The setting λ = 0, which corresponds to the continuous semi-martingale model without jumps, is also documented for comparison. We observe desirablesize performances, meaning that the probability of type I error is acceptable, for λ = 0 , , n considered. For fixed λ , the magnitude of size approaches to corresponding nominalconfidence level as the sampling frequency increases, this is even more evident for relativelylarger λ . As for the influence of jumps, we see that more intensive jumps always worsenthe performance of size, and the extent is more obvious when the sample size n is relatively12maller. We find that almost all the values of power are 1 for all n , λ and the three nominallevels, which shows our test is quite powerful in detecting the time variation in volatilityprocess. This is inline with our theoretical analysis in Section 2.1 that our test estimatordiverges at a fast rate if the volatility process is not constant.n Size Power10% 5% 1% 10% 5% 1%780 0.0882 0.0382 0.0096 1.0000 1.0000 1.0000 λ = 0 2340 0.0920 0.0422 0.0096 1.0000 1.0000 1.00007800 0.1038 0.0536 0.0118 1.0000 1.0000 1.000023400 0.1006 0.0460 0.0100 1.0000 1.0000 1.0000780 0.1050 0.0586 0.0188 1.0000 1.0000 1.0000 λ = 10 2340 0.0918 0.0438 0.0104 1.0000 1.0000 1.00007800 0.0988 0.0538 0.0130 1.0000 1.0000 1.000023400 0.0992 0.0512 0.0090 1.0000 1.0000 1.0000780 0.1270 0.0794 0.0322 1.0000 1.0000 1.0000 λ = 20 2340 0.1024 0.0564 0.0156 1.0000 1.0000 1.00007800 0.1004 0.0526 0.0142 1.0000 1.0000 1.000023400 0.1008 0.0488 0.0094 1.0000 1.0000 1.0000780 0.3422 0.2710 0.1734 1.0000 1.0000 0.9998 λ = 50 2340 0.1496 0.0934 0.0420 1.0000 1.0000 1.00007800 0.1084 0.0556 0.0140 1.0000 1.0000 1.000023400 0.0900 0.0464 0.0078 1.0000 1.0000 1.0000780 0.8048 0.7572 0.6478 1.0000 1.0000 1.0000 λ = 100 2340 0.4574 0.3806 0.2520 1.0000 1.0000 1.00007800 0.1520 0.0968 0.0408 1.0000 1.0000 1.000023400 0.0984 0.0514 0.0132 1.0000 1.0000 1.0000 Table 1: Heteroscedasticity test with jumps.
Table 2 documents the size and power of the heteroscedasticity test in the presence ofmarket microstructure noise. The finite sample performances of both size and power aresatisfying for η = 0 . , .
01. For fixed η , as the sampling frequency increases, the sizeand power perform better in the sense of getting closer to corresponding nominal confidencelevels and 1, respectively. This phenomenon is even more distinct for relatively larger η .Regarding the effect of the market microstructure noise, a larger η deviates the values ofPower away from 1 and yields a larger type II error. Moreover, such a deterioration is evenworse for relatively smaller sample size n . We also find that all the values of size are close tocorresponding nominal confidence levels for all the different parameters n and η considered.This justifies that the pre-averaging technique works well in demolishing the disturbancefrom the market microstructure noise for the estimation of volatility (the integrated volatilityor/and the spot volatility). The results of power are more sensitive to the presence of marketmicrostructure noise because our statistics T n,P re ( p n , l n ) in Theorem 3 diverges to infinity ina relatively slow rate which depends on the parameters p n , l n , when the constant volatilityassumption is violated. 13 Size Power10% 5% 1% 10% 5% 1% η = 0 .
001 1170 0.0956 0.0510 0.0224 0.9976 0.9968 0.99364680 0.1078 0.0576 0.0222 0.9994 0.9992 0.998811700 0.1130 0.0598 0.0210 0.9996 0.9996 0.999223400 0.1034 0.0528 0.0166 1.0000 1.0000 0.9996 η = 0 .
01 1170 0.0932 0.0504 0.0204 0.9934 0.9912 0.98524680 0.1008 0.0554 0.0194 0.9994 0.9992 0.999211700 0.1108 0.0570 0.0170 0.9996 0.9994 0.999423400 0.1094 0.0548 0.0162 0.9996 0.9996 0.9996 η = 0 .
05 1170 0.0922 0.0450 0.0216 0.6918 0.6586 0.60524680 0.1114 0.0576 0.0192 0.8286 0.8072 0.765211700 0.1104 0.0568 0.0182 0.9074 0.8928 0.869823400 0.1064 0.0562 0.0146 0.9366 0.9282 0.9060
Table 2: Heteroscedasticity test with market microstructure noise.
To verify the accuracy of the normal approximations of our test statistics, namely (8)in Theorem 1, (13) in Theorem 2 and (21) in Theorem 3, we demonstrate Q-Q plots andhistograms for the finite estimates of T n ( k n ), T n,T hr ( k n ) and T n,P re ( p n , l n ) under the constantvolatility model in Figures 1–3. It is shown that all the histograms approximate standardnormal distribution closely and the Q-Q plots are almost linear, which proves the asymptoticnormality of these three quantities. -3 -2 -1 0 1 2 3 4050100150200250300350 (a) Histogram -4 -3 -2 -1 0 1 2 3 4-4-3-2-101234 (b) Q-Q PlotFigure 1: Estimates of T n ( k n ) in (8) of Theorem 1 with n = 23400. In the histograms, the red real curve isthe density of standard normal random variable. (a) Histogram -4 -3 -2 -1 0 1 2 3 4-4-3-2-1012345 (b) Q-Q PlotFigure 2: Estimates of T n,T hr ( k n ) in (13) of Theorem 2, with λ = 20 and n = 23400. In the histograms, thered real curve is the density of standard normal random variable. -4 -3 -2 -1 0 1 2 3 4 5050100150200250300350 (a) Histogram -4 -3 -2 -1 0 1 2 3 4-4-3-2-1012345 (b) Q-Q PlotFigure 3: Estimates of T n,P re ( p n , l n ) in (21) of Theorem 3, with η = 0 .
01 and n = 23400. In the histograms,the red real curve is the density of standard normal random variable. . Real data analysis In this section, we apply our proposed heteroscedasticity test statistics to high-frequencydata from the NYSE TAQ database. We use the transaction price data of the InternationalBusiness Machines (IBM) in the whole year of 2011, with a total of 252 trading days. Forvarious reasons, raw trading data contains numerous errors. Therefore, the data is notimmediately suitable for analysis and data-cleaning is an essential step when dealing withtick-by-tick data. Following the pre-filtering routine of Barndorff-Nielsen et al. (2009), wecollect all transactions from 9:30 to 16:00, delete entries with zero prices, merge multipletransactions with the same time stamp by taking the weighted average of all prices andsample every 5 seconds in calendar time. We consider 5-minute data for T n ( k n ) in Theorem1 and T n,T hr ( k n ) in Theorem 2 to avoid the influence of market microstructure noise, and5-second data for T n,P re ( p n , l n ) in Theorem 3.Recall that we set k n = ⌊ θ √ ∆ n ⌋ and p n = ⌊ c ∆ − (1 / χ ) n ⌋ for the number of raw data used forkernel smoothing in the noise-free setting and number of raw data used for pre-averaging inthe noisy setting, respectively. Figure 4 depicts the proportion of the day with time-varyingvolatility tested at significance levels of 10%, 5% and 1% as a function of θ in the frictionlesscases and a function of c in the noisy setting. Regarding other related parameters notmentioned, they remain the same as the ones in the simulation section. When we implementthe test by using 5-minute high-frequency sampling to diminish the influence from the marketmicrostructure noise, the proportions of heteroscedasticity volatility are insensitive to thechoice of θ . For the three different significance levels, similar patterns are observed foreach scenario with small deviation in the magnitude of heteroscedasticity proportion forall range of θ . If we remove the jumps by the truncation method, the proportions reduceby around 10% for all the three significance levels compared to the case without removingthe jumps. This is inline with the intuition that the presence of jumps makes the priceprocess more volatile. For the scenario of considering removing the market microstructurenoise by using pre-averaged 5-second data, the heteroscedasticity proportion decreases as c increases. This verifies that the pre-averaging methodology mitigates the impact of themarket microstructure noise better for relative larger c , which corresponds to the case thatmore raw data are used for pre-averaging.In Figure 5, we demonstrate the cross-sectional average of intraday volatility curvesestimated by the spot volatility estimators b σ nτ ( k n ) in Theorem 1 for the continuous setting, b σ τ n,T hr ( k n ) in Theorem 2 for the setting with jumps, and b σ n,P rekp n l n /n ( p n , l n ) in Theorem 3 forthe noisy setting respectively. It is shown that the volatility estimates near the openingtime or the closing time are relatively larger than other time points in the middle timespan. Moreover, the estimated volatility curves are roughly with sharp decreases or increasesaround pre-scheduled macroeconomic announcements (e.g., at 10:00 or 14:00). This makesthe whole volatility curve like a reverted “J”-shape, which is also found in Christensen et al.(2018). In fact, this happens for most of the days in a year. There are also empiricalliteratures explaining the phenomenon. For example, the period covering 9:30 and 10:00is associated with market-wide news such as FOMC meetings and macroeconomic reports,which make the stock prices to be more volatile, as discussed in Lee and Mykland. (2008),Lee (2012), and etc.To quantify how does the variation of stock price during the opening and closing time16 .4 0.6 0.8 1 1.2 1.4 1.600.10.20.30.40.50.60.70.80.91 H e t e r o sc eda s t i c i t y p r opo r t i on (a) Without removing jumps andmicrostructure noise: based on T n ( k n ) in Theorem 1 by using 5-minute data. H e t e r o sc eda s t i c i t y p r opo r t i on (b) Removing jumps: based on T n,T hr ( k n ) in Theorem 2 by using5-minute data. H e t e r o sc eda s t i c i t y p r opo r t i on (c) Removing microstructurenoise: based on T n,P re ( p n , l n )in Theorem 3 by using 5-seconddata.Figure 4: Test for the constancy of daily volatility for IBM in 2011. -4 (a) Without removing jumps andmicrostructure noise: c σ nτ ( k n ) inTheorem 1 by using 5-minutedata. -4 (b) Removing jumps: c σ τ n,T hr ( k n )in Theorem 2 by using 5-minutedata. -4 (c) Removing microstructurenoise: c σ n,P rekp n l n /n ( p n , l n ) inTheorem 3 by using 5-seconddata.Figure 5: The cross-sectional average of intraday volatility curves estimated by the spot volatility estimatorsfor IBM. T n ( k n ) T n,T hr ( k n ) T n,P re ( p n , l n )10% 5% 1% 10% 5% 1% 10% 5% 1%09:30-16:00 0.6151 0.5992 0.5278 0.5119 0.4762 0.4087 0.8214 0.7897 0.730210:00-15:30 0.4325 0.3730 0.3016 0.4087 0.3492 0.2778 0.7024 0.6429 0.527810:30-15:00 0.2778 0.2302 0.1587 0.2778 0.2302 0.1587 0.5913 0.5198 0.4643 Table 3: Heteroscedasticity test for different time spans, with θ = 1 . c = 1 /
5. Conclusion
In this paper, we propose a new nonparametric way to do the heteroscedasticity test forhigh-frequency data. The test procedure is based on the estimations of integrated volatilityand spot volatility, for which a great deal of existing literatures can be found. Our testprocedure is easy to conduct and can be naturally extended to different settings, such as thecases in the presence of jumps and market microstructure noise. Our Monte Carlo simulationstudies show the good finite sample performance of the asymptotic theory. Finally, wealso apply our test procedure to do the heteroscedasticity test for some real high-frequencyfinancial data. The empirical studies indicate that the volatility is not constant in mostof days, and the opening and closing periods account for a relatively large proportion ofintraday heteroscedasticity. This paper also enlighten us on testing whether the covariancestructure between different assets is constant or not during a given time interval, as a futurework.
Acknowledgement
Qiang Liu’s work is supported by MOE-AcRF Grant of Singapore (No. R-146-000-258-114), Zhi Liu gratefully acknowledges financial support from FDCT of Macau (No.202/2017/A3) and NSFC (No. 11971507), Chuanhai Zhang’s research is supported in partby Humanity and Social Science Youth Foundation of Chinese Ministry of Education (No.18YJC790210) and in part by the Fundamental Research Funds for the Central Universities,Zhongnan University of Economics and Law (2722019PY038).
Appendix
For the following proofs, by a standard localization procedure given in Barndorff-Nielsen et al.(2006a), we can replace the local boundedness hypothesis in our setting by a boundednessone without loss of generality. We use an unified C to denote positive constants in the proofs,and it may change from line to line. Note that∆ ni X = Z i/n ( i − /n b s ds + Z i/n ( i − /n σ s dB s . It is obvious that the random variable R i/n ( i − /n b s ds is dominated by R i/n ( i − /n σ s dB s , so thedrift term b s has no effect on asymptotic properties of estimators where X is involved. Thus,setting b s ≡ roof of Theorem 1: (1) Note that k n n ⌊ n/k n ⌋− X j =0 ( b σ njk n /n ( k n ) − c IV n ) − Z ( σ s − IV ) ds = k n n ⌊ n/k n ⌋− X j =0 (cid:0) ( b σ njk n /n ( k n ) − c IV n ) − ( σ jk n /n − IV ) (cid:1) + k n n ⌊ n/k n ⌋− X j =0 ( σ jk n /n − IV ) − Z ( σ s − IV ) ds := A + A . The result A → p b σ njk n /n ( k n ) → p σ jk n /n and c IV n → p IV , whoseproofs are given below. Observing b σ njk n /n ( k n ) − σ jk n /n = 1 k n ∆ n ( j +1) k n X i = jk n +1 (cid:0) (∆ ni X ) − ( σ jk n /n ∆ ni B ) (cid:1) + 1 k n ∆ n ( j +1) k n X i = jk n +1 (cid:0) ( σ jk n /n ∆ ni B ) − σ jk n /n ∆ n (cid:1) := B + B . Since E [ | (∆ ni X ) − ( σ jk n /n ∆ ni B ) | ] ≤ C E [ | ∆ ni X − σ jk n /n ∆ ni B | ] ≤ C ( E [ | ∆ ni X − σ jk n /n ∆ ni B | ]) / ≤ C ∆ / n p k n , together with Holder’s inequality and Itˆo’s isometry, we obtain that B → p
0. Note that { (cid:0) ( σ jk n /n ∆ ni B ) − σ jk n /n ∆ n (cid:1) , F jk n /n } is a martingale difference array, thus E [( B ) ] = 1( k n ∆ n ) j +1) k n X i = jk n +1 E (cid:2)(cid:0) ( σ jk n /n ∆ ni B ) − σ jk n /n ∆ n (cid:1) (cid:12)(cid:12) F jk n /n (cid:3) = 2 σ jk n /n k n . By Chebyshev’s inequality, we obtain B → p
0, hence b σ njk n /n ( k n ) → p σ jk n /n . The proof of c IV n → p IV can be referred to Barndorff-Nielsen et al. (2006a).For A , Riemann integrability implies that A → p (2) Before the proof, we note that we have the central limit theorems √ n ( c IV n − R σ s ds ) → d N (0 , R σ s ds ) (see Barndorff-Nielsen et al. (2006a)) and √ k n ( b σ njk n /n ( k n ) − σ jk n /n ) → d N (0 , σ jk n /n ). The last conclusion can be proved by following the consistency proof in (1) ,together with the results √ k n E [ | B | ] ≤ p k n ∆ n → E [( √ k n B ) ] → σ jk n /n .Now, we are ready to give the proof of (8). Observing that E h(cid:12)(cid:12)(cid:12)r k n n ⌊ n/k n ⌋− X j =0 (cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ c IV n (cid:1) − r k n n ⌊ n/k n ⌋− X j =0 (cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ σ (cid:1) (cid:12)(cid:12)(cid:12)i r k n n ⌊ n/k n ⌋− X j =0 E h(cid:12)(cid:12)(cid:12)(cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ c IV n (cid:1) − (cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ σ (cid:1) (cid:12)(cid:12)(cid:12)i ≤ C √ k n → , where the last inequality is derived by using Holder’s inequality and the two central limit the-orems of the integrated volatility and the spot volatility given above. Chebyshev’s inequalityimplies that as n → ∞ , we have r k n n ⌊ n/k n ⌋− X j =0 (cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ c IV n (cid:1) − r k n n ⌊ n/k n ⌋− X j =0 (cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ σ (cid:1) → p . Obviously, the result (8) can be obtained by showing that as n → ∞ , A := r k n n ⌊ n/k n ⌋− P j =0 (cid:0) √ k n ( c IV n − σ ) √ σ (cid:1) → p ,A := r k n n ⌊ n/k n ⌋− P j =0 k n ( b σ njk n /n ( k n ) − σ )( c IV n − σ ) σ → p ,A := r k n n ⌊ n/k n ⌋− P j =0 (cid:16)(cid:0) √ k n ( b σ njk n /n ( k n ) − σ ) √ σ (cid:1) − (cid:17) → d N (0 , . For A , we have E [ | A | ] ≤ C r k n n , which follows from √ n ( c IV n − σ ) = O p (1) and theboundedness of σ . For A , observing that E [( σ ∆ ni B ) ] = σ /n for i = 1 , · · · , n , we have E [( A ) ] = k n n ⌊ n/k n ⌋− X j =0 E (cid:2) k n ( b σ njk n /n ( k n ) − σ ) ( c IV n − σ ) σ (cid:3) + k n n ⌊ n/k n ⌋− X X j ,j =0 ,j = j E (cid:2) k n /n ( b σ nj k n /n ( k n ) − σ ) ( b σ nj k n /n ( k n ) − σ ) σ (cid:3) ≤ C k n n → , where the last inequality is derived by plugging in the conclusions of √ n ( c IV n − R σ ds ) = O p (1) and √ k n ( b σ njk n /n ( k n ) − σ ) = O p (1). For A , denote ξ j = r k n n (cid:16)(cid:0) √ k n ( b σ njk n /n ( k n ) − σ ) √ σ (cid:1) − (cid:17) , j = 0 , · · · , ⌊ n/k n ⌋ − . Observing that E [ ξ j |F jk n /n ] = r k n n σ (cid:0) k n E [( b σ njk n /n ( k n ) − σ ) |F jk n /n ] − σ (cid:1) r k n n σ (cid:0) k n ( 3 σ k n + k n ( k n − σ k n − σ ) − σ (cid:1) ≡ , and it is obvious that ξ j is F ( j +1) k n /n -measurable, so that { ξ j , F jk n /n } is a martingale differ-ence array. And ⌊ n/k n ⌋− X j =0 E [( ξ j ) |F jk n /n ] = k n n ⌊ n/k n ⌋− X j =0 (cid:0) E (cid:2)(cid:0) √ k n ( b σ njk n /n ( k n ) − σ ) √ σ (cid:1) (cid:12)(cid:12) F jk n /n (cid:3) − (cid:1) = k n n ⌊ n/k n ⌋− X j =0 (cid:16) n σ k n (cid:0) ( j +1) k n X i = jk n +1 E (cid:2)(cid:0) ( σ ∆ ni B ) − σ /n (cid:1) (cid:12)(cid:12) F jk n /n (cid:3) + 3 ( j +1) k n X i ,i jkn +1 i = i ( σ ∆ ni B ) − σ /n (cid:1) ( σ ∆ ni B ) − σ /n (cid:1) (cid:1) − (cid:17) = k n n ⌊ n/k n ⌋− X j =0 (cid:16) n σ k n (cid:0) k n (60 σ /n ) + 3 k n ( k n − σ /n ) (cid:1) − (cid:17) = k n n ⌊ n/k n ⌋− X j =0 ( 12 k n + 2) → , as n → ∞ . According to the central limit theorem for martingale process stated in Hall and Heyde(1980), we can get A → d N (0 , (3) The first claim is a direct consequence of (2) , while the second claim follows from (1) and the Portmanteau lemma. (cid:3)
Proof of Theorem 2: (1)
Under our setting, the conditions for Theorem 1 in Mancini(2009) are satisfied, so that if n is large enough, for P-almost all ω , we have I { (∆ ni Y ) ≤ r (1 /n ) } ( ω ) = I { ∆ ni N =0 } ( ω ). Note that the stochastic integral R i/n ( i − /n σ s dB s is a time changed Brownian mo-tion (Revuz and Yor (2001), Theorems 1.9 and 1.10), and by the L´evy’s law for the modulusof continuity of Brownian motion’s paths (Karatzas and Shreve (1991), Theorem 9.25), wehave sup i ∈{ , ··· ,n } | R i/n ( i − /n σ s dB s | p n/n ≤ C. Together with that the total number of jumps N < C , we have n log n ( c IV n,T hr − c IV n ) = n log n (cid:0) n X i =1 (∆ ni Y ) I { (∆ ni Y ) ≤ r (1 /n ) } − n X i =1 (∆ ni X ) (cid:1) = n log n (cid:0) n X i =1 (cid:0) ∆ ni Y ) I { ∆ ni N =0 } − n X i =1 (∆ ni X ) (cid:1) = n log n (cid:0) n X i =1 (cid:0) ∆ ni X ) I { ∆ ni N =0 } − n X i =1 (∆ ni X ) (cid:1) n log n n X i =1 (cid:0) ∆ ni X ) I { ∆ ni N> } < C, and similarly n log n ( b σ n,T hrjk n /n ( k n ) − b σ njk n /n ( k n )) < C. Obviously, we also have the corresponding versions of central limit theorems for the thresh-olding estimators, namely √ n ( c IV n,T hr − R σ s ds ) → d N (0 , R σ s ds ) and √ k n ( b σ n,T hrjk n /n ( k n ) − σ jk n /n ) → d N (0 , σ jk n /n ).According to (1) of Theorem 1, the result (12) can be proved by showing( b σ n,T hrjk n /n ( k n ) − c IV n,T hr ) − ( b σ njk n /n ( k n ) − c IV n ) → p , which naturally follows from the results mentioned above and Chebyshev’s inequality. (2) According to the proof of (2) of Theorem 1, to obtain (13), we only need to prove r k n n ⌊ n/k n ⌋− X j =0 (cid:16)(cid:0) √ k n ( b σ n,T hrjk n /n ( k n ) − c IV n,T hr ) √ c IV n,T hr (cid:1) − (cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ c IV n (cid:1) (cid:17) → p . And from the proof of (1), we have E [ (cid:12)(cid:12)(cid:12)r k n n ⌊ n/k n ⌋− X j =0 (cid:16)(cid:0) √ k n ( b σ n,T hrjk n /n ( k n ) − c IV n,T hr ) √ c IV n,T hr (cid:1) − (cid:0) √ k n ( b σ njk n /n ( k n ) − c IV n ) √ c IV n (cid:1) (cid:17)(cid:12)(cid:12)(cid:12) ] ≤ C √ k n log n √ n → . Then, Chebyshev’s inequality gives us the desired result. (3)
The conclusion is evident from the previous proofs. (cid:3)
Lemma 1.
If as n → ∞ , l n → and n/p n → , then c IV n,P re ( p n ) → p Z σ s ds, (25) b σ n,P rekp n l n /n ( p n , l n ) → p σ kp n l n /n , (26) and if further n /p n → and l n n /p n → , then r np n ( c IV n,P re ( p n ) − Z σ s ds ) → ds N (0 , Z σ s ds ) , (27) p l n ( b σ n,P rekp n l n /n ( p n , l n ) − σ kp n l n /n ) → ds N (0 , σ kp n l n ) . (28)22 roof: Since the asymptotic normality of the estimators c IV n,P re ( p n ) and b σ n,P rekp n l n /n ( p n , l n )implies their consistent convergence to R σ s ds and σ kp n l n /n , we only present the proof pro-cedures of (27) and (28), and the relative relaxed conditions on the parameters for theconsistency results can be easily seen from the following proof.For the proof of (27), observing that r np n ( c IV n,P re ( p n ) − IV ) = r np n ( 1 ϕ n ⌊ n/p n ⌋− X j =0 ( Z njp n ) − IV )= r np n (cid:0) ϕ n ⌊ n/p n ⌋− X j =0 ( X njp n ) − ϕ n ⌊ n/p n ⌋− X j =0 ( σ jp n /n B njp n ) (cid:1) + 1 ϕ n r np n (cid:0) ⌊ n/p n ⌋− X j =0 [( σ jp n /n B njp n ) − σ jp n /n ϕ n p n /n ] (cid:1) + r np n (cid:0) ⌊ n/p n ⌋− X j =0 p n /nσ jp n /n − IV (cid:1) + r np n · O p ( np n ):= A ′ + A ′ + A ′ . The first term in A ′ converges to 0 in probability, which is deduced from (6.14) in Podolskij and Vetter(2009) and its intact proof can be found in Barndorff-Nielsen et al. (2006a), together withthe convergence n /p n →
0, we have A ′ → p
0. It’s obvious that the result (25) onlyrequires n/p n →
0. We also have A ′ → ds N (0 , R σ s ds ), which is a special case ofLemma 3 in Podolskij and Vetter (2009), by taking r = 2 , l = 0 in L n ( r, l ) without theconsideration of the microstructure noise. Now, we are left to prove A ′ → p
0. Denote γ j = r np n ϕ n (cid:0) ( X njp n ) − ( σ jp n /n B njp n ) (cid:1) , by writing A ′ = ⌊ n/p n ⌋− X j =0 ( γ j − E [ γ j |F jp n /n ]) + ⌊ n/p n ⌋− X j =0 E [ γ j |F jp n /n ] := B ′ + B ′ , equivalently, we only need to prove B ′ → p B ′ → p
0. Observing that { γ j , F jp n /n } is amartingale difference array, and E [( B ′ ) ] = ⌊ n/p n ⌋− X j =0 E [( γ j − E [ γ j |F jp n /n ]) |F jp n /n ] ≤ C ⌊ n/p n ⌋− X j =0 E [( γ j ) |F jp n /n ] , Holder’s inequality and Lemma 1 in Podolskij and Vetter (2009) yield ⌊ n/p n ⌋− P j =0 E [( γ j ) |F jp n /n ] ≤ C/n , thus Chebyshev’s inequality implies B ′ → p
0. The proof of B ′ → p p l n ( b σ n,P rekp n l n /n ( p n , l n ) − σ kp n l n /n ) = p l n ( np n l n ϕ n ( k +1) l n X j = kl n +1 ( Z njp n ) − σ kp n l n /n )= p l n ( np n l n ϕ n ( k +1) l n X j = kl n +1 [( X njp n ) − ( σ kp n l n /n B njp n ) ])+ p l n ( np n l n ϕ n ( k +1) l n X j = kl n +1 ( σ kp n l n /n B njp n ) − σ kp n l n /n )+ p l n · O p ( np n ):= A ′ + A ′ + A ′ . Obviously, we have A ′ →
0, and we only require n/p n → E [( σ kp n l n /n B njp n ) |F kp n l n /n ] = ϕ n p n /n · σ kp n l n /n , after some variancecalculations and verifications similar to the ones in the proof of A ′ above, we obtain A ′ → ds N (0 , σ kp n l n /n ). By following the proof of A ′ → p A ′ → p
0, thus wehave (28). (cid:3)
Proof of Theorem 3: (1)
According to the Proof of (1) of Theorem 1, the conclusionnaturally follows from the results c IV n,P re ( p n ) → p IV and b σ n,P rekp n l n /n ( p n , l n ) → p σ kp n l n /n . (2) Observing that E h(cid:12)(cid:12)(cid:12)r p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:0) √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − c IV n,P re ) √ c IV n,P re (cid:1) − r p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:0) √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − c IV n,P re ) √ σ (cid:1) (cid:12)(cid:12)(cid:12)i ≤ r p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 E h(cid:12)(cid:12)(cid:12)(cid:0) √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − c IV n,P re ) √ c IV n,P re (cid:1) − (cid:0) √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − c IV n,P re ) √ σ (cid:1) (cid:12)(cid:12)(cid:12)i ≤ C √ l n → , the last inequality is derived by using Holder’s inequality and the conclusions of r np n ( c IV n,P re − R σ ds ) = O p (1) and √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − σ ) = O p (1). Chebyshev’s inequality impliesthat as n → ∞ , we have r p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:0) √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − c IV n,P re ) √ c IV n,P re (cid:1) r p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:0) √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − c IV n,P re ) √ σ (cid:1) → p . (29)Obviously, the result will hold if we can prove that as n → ∞ , it holds that A ′′ := r p n l n n ⌊ n/ ( p n l n ) ⌋− P k =0 (cid:0) √ l n ( c IV n,P re − σ ) √ σ (cid:1) → p ,A ′′ := r p n l n n ⌊ n/ ( p n l n ) ⌋− P k =0 l n ( b σ n,P rekp n l n /n ( p n , l n ) − σ )( c IV n,P re − σ ) σ → p ,A ′′ := r p n l n n ⌊ n/ ( p n l n ) ⌋− P k =0 (cid:16)(cid:0) √ l n ( b σ n,P rekp n l n /n − σ ) √ σ (cid:1) − (cid:17) → d N (0 , . For A ′′ , we have E [ | A ′′ | ] ≤ C r p n l n n , which follows from r np n ( c IV n − σ ) = O p (1) and theboundedness of σ , and Chebyshev’s inequality implies the convergence in probability. For A ′′ , since A ′′ = r p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 l n (cid:0) np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) − σ (cid:1)(cid:0) ϕ n ⌊ n/p n ⌋− X j =0 ( X njp n ) − σ (cid:1) σ + O p ( 1 p n + √ nl n p n ):= B ′′ + O p ( 1 p n + √ nl n p n ) , Chebyshev’s inequality implies that A ′′ − B ′′ → p
0. Observing that E [ np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) − σ ] = 0 for k = 0 , · · · , ⌊ n/ ( p n l n ) ⌋ −
1, we have E [( B ′′ ) ] = p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 E (cid:2) l n ( b σ n,P rekp n l n /n ( p n , l n ) − σ ) ( c IV n,P re − σ ) σ (cid:3) + p n l n n ⌊ n/ ( p n l n ) ⌋− X X k ,k =1 ,k = k E (cid:2) l n p n /n ( b σ n,P rek p n l n /n ( p n , l n ) − σ ) ( b σ n,P rek p n l n /n ( p n , l n ) − σ ) σ (cid:3) ≤ C p n l n n → , the last inequality is derived by plugging in the conclusions r np n ( c IV n,P re − R σ ds ) = O p (1)and √ l n ( b σ n,P rekp n l n /n ( p n , l n ) − σ ) = O p (1), Chebyshev’s inequality implies B ′′ → p
0. For A ′′ ,25ince A ′′ = r p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:16)(cid:0) √ l n (cid:0) np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) − σ (cid:1) √ σ (cid:1) − (cid:17) + O p ( np / n ):= B ′′ + O p ( np / n ) . Then, Chebyshev’s inequality implies that A ′′ − B ′′ → p
0, thus we only need to prove B ′′ → d N (0 , ξ ′ k = r p n l n n (cid:16)(cid:0) √ l n (cid:0) np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) − σ (cid:1) √ σ (cid:1) − (cid:17) , k = 0 , · · · , ⌊ n/ ( p n l n ) ⌋− . (30)By using that E [( X i ) k ] = (2 k − ϕ n p n σ /n ) k for k = 1 , , · · · , we have E [ ξ ′ k |F kp n l n /n ] = r p n l n n σ (cid:0) l n E [ (cid:0) np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) − σ (cid:1) |F kp n l n /n ] − σ (cid:1) = r p n l n n σ (cid:0) l n ( 3 σ l n + l n ( l n − σ l n − σ ) − σ (cid:1) ≡ , and it is obvious that ξ ′ k is F ( k +1) p n l n /n -measurable, so that { ξ k , F kp n l n /n } is a martingaledifference array. And ⌊ n/ ( p n l n ) ⌋− X k =0 E [( ξ ′ k ) |F kp n l n /n ]= p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:0) E (cid:2)(cid:0) √ l n (cid:0) np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) − σ (cid:1) √ σ (cid:1) (cid:12)(cid:12) F kp n l n /n (cid:3) − (cid:1) = p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:16) l n σ (cid:0) E (cid:2) ( np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) ) (cid:12)(cid:12) F kp n l n /n (cid:3) − E (cid:2) σ ( np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) ) (cid:12)(cid:12) F kp n l n /n (cid:3) + E (cid:2) σ ( np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) ) (cid:12)(cid:12) F kp n l n /n (cid:3) − E (cid:2) σ ( np n l n ϕ n ( k +1) l n X j = kl n +1 ( X njp n ) ) (cid:12)(cid:12) F kp n l n /n (cid:3) + σ (cid:1) − (cid:17) p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 (cid:16) l n (cid:0) l n + 12 l n + 44 l n + 48 l n l n − l n + 6 l n + 8 l n ) l n + 6( l n + 2 l n ) l n − (cid:1)(cid:17) = p n l n n ⌊ n/ ( p n l n ) ⌋− X k =0 ( 12 l n + 2) → . According to the central limit theorem for martingale process in Hall and Heyde (1980), theresults above implies that B ′′ → d N (0 , (3) The conclusion is obvious from the established results. (cid:3)
References
A¨ıt-Sahalia, Y., 1996. Testing continuous time models of the spot interest rate. Review ofFinancial Studies 9, 385–426.A¨ıt-Sahalia, Y., Jacod, J., 2009a. Estimating the degree of activity of jumps in high frequencydata. Annals of Statistics 37, 2202–2244.A¨ıt-Sahalia, Y., Jacod, J., 2009b. Testing for jumps in a discretely observed process. Annalsof Statistics 37, 184–222.A¨ıt-Sahalia, Y., Jacod, J., 2010. Is brownian motion necessary to model high frequencydata? Annals of Statistics 38, 3093–3128.A¨ıt-Sahalia, Y., Jacod, J., 2014. High-Frequency Financial Econometrics. Princeton Uni-versity Press.Andersen, T., Bollerslev, T., 1997. Intraday periodicity and volatility persistence in financialmarkets. Journal of Empirical Finance 4, 115–158.Andersen, T.G., Bollerslev, T., Diebold, F., Labys, P., 2003. Modeling and forecastingrealized volatility. Econometrica 71 (3), 579–625.Andersen, T.G., Thyrsgaard, M., Todorov, V., 2019. Time-varying periodicity in intradayvolatility. Journal of the American Statistical Association 114, 1–26.Barndorff-Nielsen, O., Graversen, S., Jacod, J., Podolskij, M., Shephard, N., 2006a. A centrallimit theorem for realised power and bipower variations of continuous semimartingales.
InY. Kabanov and R. Lipster (eds.), From Stochastic Analysis to Mathematical Finance,Festschrift for Albert Shiryaev.
Springer, Berlin.Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., Shephard, N., 2008. Designing realisedkernels to measure the ex-post variation of equity prices in the presence of noise. Econo-metrica 76 (6), 1481–1536.Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., Shephard, N., 2009. Realised kernels inpractice: Trades and quotes. Econometrics Journal 12 (3), C1–C32.27arndorff-Nielsen, O.E., Shephard, N., 2004. Power and bipower variation with stochasticvolatility and jumps. Journal of Financial Econometrics 2 (1), 1–37.Barndorff-Nielsen, O.E., Shephard, N., 2006. Econometrics of testing for jumps in financialeconomics using bipower variation. Journal of Financial Econometrics 4 (1), 1–30.Barndorff-Nielsen, O.E., Shephard, N., 2007. Variation, jumps and high frequency data infinancial econometrics. In