Volatility estimation with Microstructure noise

Transcription

1 Volatility estimation with Microstructure noise Eduardo Rossi University of Pavia December 2012 Rossi Microstructure noise University of Pavia / 52

2 Outline 1 Sampling Schemes 2 General price formation mechanism 3 Assumptions 4 Sparse sampling 5 MSE-based optimal sampling 6 Subsampling 7 Kernel-based estimation 8 The Effects of the Sampling Scheme 9 Comparison of Techniques Rossi Microstructure noise University of Pavia / 52

3 The RV approach, of holding h > 0 fixed, is motivated by the fact that it is undesirable, and due to the presence of market microstructure frictions indeed practically infeasible, to sample returns infinitely often (n ) over infinitesimally short time intervals (h 0). Model specific calculations and simulations by Andersen and Bollerslev (1998), Andersen, Bollerslev and Lange (1999), Andreou and Ghysels (2002), Bai, Russell and Tiao (2000), Barndorff-Nielsen and Shephard (2002c), Barucci and RenÚ (2002), Bollerslev and Zhou (2002), and Zumbach, Corsi and Trapletti (2002), among others, illustrate the effects of finite n (and h) for a variety of settings. The discrepancies in the underlying model formulation and character of the assumed frictions render a general assessment of the results difficult. Moreover, the size of the measurement errors are often computed unconditionally rather than conditional on the realization of the RV statistic. Nonetheless, it is evident that the measurement errors typically are non-trivial. This has motivated the development of alternative realized volatility measures designed to circumvent the microstructure biases. Rossi Microstructure noise University of Pavia / 52

4 Sampling Schemes Sampling Schemes Suppose that in a given day t, we partition the interval [0, 1] in n t subintervals, and define the grid of observation times as [τ 0,..., τ n ] where 0 = τ 0 <... < τ n = 1. The length of the i-th subinterval is given by δ i,n = τ i τ i 1 It is assumed that the length of each subinterval shrinks to zero as the number of intraday observations increases. The integrated variance over each of the subintervals is defined as IV i,t = τi τ i 1 σ 2 (t + u 1)du Rossi Microstructure noise University of Pavia / 52

5 Sampling Schemes Sampling Schemes Prices are observed at discrete and irregularly spaced intervals Calendar time sampling (CTS): the intervals are equidistant. For example, the prices may be sampled every 5 or 15 minutes. As the intraday data are irregularly spaced, in most cases calendar time sampled data must be constructed artificially. Previous tick method: during a five-minute interval, we may observe several prices, in which case the previous tick method takes the first observation as the sampled price. Transaction time sampling (TrTS): Prices are recorded every m-th transaction Business time sampling (BTS): where the sampling times are chosen such that IV i,t = IVt M t Tick time sampling (TTS): Prices are recorded at every price change. An important difference among these distinct sampling schemes is that the observation times in BTS are latent, whereas in CTS, TrTS, and TkTs they are observed. Rossi Microstructure noise University of Pavia / 52

6 Sampling Schemes Market microstructure noise Recorded asset prices deviate from their equilibrium values due to the presence of market microstructure frictions. The volatility of the observed prices depends on two distinct volatility components: 1 The volatility of the unobserved equilibrium prices 2 The volatility of the equally unobserved market microstructure effects. Market microstructure noise has many sources 1 the discreteness of the price (see Harris, 1990, 1991), 2 bid-ask bounce effects 3 the properties of the trading mechanism, as in Black (1976) and Amihud and Mendelson (1987). Rossi Microstructure noise University of Pavia / 52

7 General price formation mechanism General price formation mechanism Write an observed logarithmic price as p = p + η where p denotes the logarithmic equilibrium price, i.e., the price that would prevail in the absence of market microstructure frictions, and η denotes a market microstructure contamination in the observed logarithmic price as induced by price discreteness and bid-ask bounce effects. Fix a certain time period h (a trading day, for example) and assume availability of n equispaced high-frequency prices over h. Rossi Microstructure noise University of Pavia / 52

8 General price formation mechanism General price formation mechanism Calendar time sampling: The continuously-compounded returns over any intra-period interval of length is p jδ p (j 1)δ }{{} r jδ The RV (using all data points): RV (h) = n j=1 r 2 jδ = n [ j=1 = p r 2 Conditionally on the efficient returns: δ = h n jδ p(j 1)δ }{{} rjδ r jδ = r jδ + ɛ jδ jδ + ɛ jn ] 2 = n j=1 + η jδ η (j 1)δ }{{} ɛ jδ r 2 jδ + 2 E[RV (h) r ] = RV (h) + 2nE[η 2 ] n r jδ ɛ jn + j=1 n j=1 ɛ 2 jn Rossi Microstructure noise University of Pavia / 52

9 Assumptions General price formation mechanism Assumption 1: The equilibrium price process 1 The logarithmic equilibrium price process p t is a continuous stochastic volatility semimartingale. Specifically, p(t) = A(t) + M(t) t = 1,..., T where A(t) (with A(0) = 0) is a predictable drift process of finite variation and M(t) is a continuous local martingale defined as t 0 σ(s)dw s. 2 The spot volatility process σ(t) is cadlag and bounded away from zero. 3 The process t 0 σ4 (s)ds is bounded almost surely for all t <. Assumption 1 implies that the equilibrium return process evolves in time as a stochastic volatility martingale difference plus an adapted process of finite variation. Rossi Microstructure noise University of Pavia / 52

10 Assumptions Microstructure frictions Assumption 2: The microstructure frictions 1 The microstructure frictions in the price process η have mean zero and are covariance- stationary with joint density f n ( ). 2 The variance of ɛ jδ = η jδ η (j 1)δ is O(1) for all δ. 3 The η s are independent of the p s (η p ) Assumption 2 permits general dependence features for the microstructure friction components in the recorded prices. The correlation structure of the frictions can, for instance, capture first-order negative dependence in the recorded high-frequency returns as determined by bid-ask bounce effects as well as higher order dependence as induced by clustering in order flow. In general, the characteristics of the noise returns may be a function of the sampling frequency δ = h/n. Rossi Microstructure noise University of Pavia / 52

11 Assumptions Microstructure frictions: i.i.d. case The equilibrium return process r jδ is modeled as being O p( δ) over any intra-period time horizon of size δ = h/n. The contaminations ɛ jδ in the observed return process are O p (1). This result, which is a consequence of Assumptions 1(1) and 2(2), implies that longer period returns are less contaminated by noise than shorter period returns. The magnitude of the frictions does not decrease with the distance between subsequent time stamps. Under Assumption 2, Bandi, F. M., Russell, J. R. (2005). Microstructure noise, realized volatility, and optimal sampling, Unpublished paper. Graduate School of Business, University of Chicago. showed that RV (h) a.s. as n Rossi Microstructure noise University of Pavia / 52

12 Assumptions Microstructure frictions: i.i.d. case The dependence structure of the microstructure friction process can be simplified. One can modify Assumption 2 as follows: Assumption 3: The microstructure frictions 1 The microstructure frictions in the price process η has zero mean and is an independent and identically distributed random variable. 2 η p. 3 The variance of η is O(1). If the microstructure noise contaminations in the price process η are i.i.d., then the noise returns display an MA(1) structure with a negative first-order autocorrelation. The noise return moments do not depend on n or, equivalently, the sampling frequency δ = h/n. The MA(1) model, as typically justified by bid-ask bounce effects, is bound to be an approximation. Awartani, Corradi, and Distaso (2004) propose a formal test of the MA(1) market microstructure model. Rossi Microstructure noise University of Pavia / 52

13 where L denotes stable convergence in law (when multiplied by a suitable factor, the convergence is in distribution) and Z N(0, 1). Rossi Microstructure noise University of Pavia / 52 Assumptions Microstructure frictions: i.i.d. case Under Assumption 3, it was shown in Zhang, L., Mykland, P. A., Ait-Sahalia, Y. (2005). A tale of two time scales: determining integrated volatility with noisy high frequency data. Journal of the American Statistical Association 100, that n 1/2[ ] RV (h) IV (h) 2nE(η 2 d ) 2nE[η 4 ] 1/2 N(0, 1) In practical applications, even sampling at the highest available frequency (if one uses all the data,say sampled every second), the number of intraday observations is finite and the price records are discrete. This introduces a bias due to discretization, such that (conditionally on the p(t) ) all rjδ 2 i=1 t L 0 σ 2 (u)du + 2nE[η 2 ] }{{} O(n) } {{ } O(1) [ + 4nE[η 4 ] }{{} due to noise + 2t n t σ 4 (t)dt 0 }{{} due to discretization } {{ } total variance ] 1/2Z

14 Assumptions Microstructure frictions: i.i.d. case It has long been known that sampling all data points is not a good idea. The recommendation in the literature is to sample sparsely at some lower frequency (5, 10, 15, 30 minutes), by using a RV estimator. Reducing the value of n, from say 23, 400 (1 second sampling) to 78 (5 minute sampling over the same 6.5 hours), has the advantage of reducing the magnitude of the bias term 2nE[η 2 ]. Yet, one of the most basic lessons of statistics is that discarding data is, in general, not advisable. Rossi Microstructure noise University of Pavia / 52

15 Assumptions Microstructure frictions: Dependent case Assumption 4: Dependent Noise Structure The microstructure noise, η i has a zero mean, stationary, and strong mixing stochastic process, with the mixing coefficients decaying exponentially. The noise is independent of the price process. The variance of ɛ i is O(1). Under Assumption 3, Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12, Ait-Sahalia, Y., P. A. Mykland, L. Zhang (2011) Ultra high frequency volatility estimation with dependent microstructure noise Journal of Econometrics 160, Rossi Microstructure noise University of Pavia / 52

16 Assumptions Microstructure frictions: Dependent case Showed that where all rjδ 2 i=1 t L 0 σ 2 (u)du + 2nΩ } {{ } O(1) }{{} O(n) Ω = Var[ɛ i ] + 2 [ + 4nE[η 4 ] }{{} due to noise + 2t n t σ 4 (t)dt 0 }{{} due to discretization } {{ } total variance Cov[ɛ 2 1, ɛ 2 i+1] RV (computed with all data points) diverges to infinity linearly in n. i=1 ] 1/2Z The RV, scaled by (2n) 1, estimates the variance of the microstructure noise 1 p RV (h) E(η 2 ) 2n Rossi Microstructure noise University of Pavia / 52

17 Sparse sampling Solutions: sparse sampling and filtering In order to avoid substantial contaminations at high sampling frequencies, Andersen, Bollerslev, Diebold, and Ebens (2001), for example, suggest sampling at frequencies that are lower than the highest frequencies at which the data arrives. Relying on the levelling of the volatility signature plots at frequencies around 15 minutes, Andersen, Bollerslev, Diebold, and Labys (1999, 2000) suggest using 15- to 20- minute intervals in practise. If the equilibrium returns are unpredictable (A(t) = 0), the correlation structure of the observed returns must be imputed to microstructure noise. Andersen, Bollerslev, Diebold, and Ebens (2001) and Andersen, Bollerslev, Diebold, and Labys (2003), among others, filter the data using an MA(1) filter. Rossi Microstructure noise University of Pavia / 52

18 Sparse sampling Sparse sampling Sparse sampling: However, Zhang, L., Mykland, P. A., Ait-Sahalia, Y. (2005). A tale of two time scales: determining integrated volatility with noisy high frequency data. Journal of the American Statistical Association 100: , showed that this is not an adequate solution to the problem: First, define a new grid Λ (sparse), with n (sparse) sparsely equidistant sampled observation times. Λ (sparse) t is a subgrid of Λ t. Set RV (h) (sparse) = n (sparse) j=1 r 2 t,j all rjδ 2 i=1 t L 0 σ 2 (u)du + 2n (sparse) E[η 2 ] }{{} O(n) } {{ } O(1) [ + 4n (sparse) E[η 4 ] + 2t }{{} due to noise t σ 4 (t)dt n } (sparse) 0 {{ } due to discretization } {{ } total variance Although the bias is reduced when n (sparse) < n, the variance is increased due to discretization, leading to the well-known bias-variance trade-off. ] 1/2Z Rossi Microstructure noise University of Pavia / 52

19 MSE-based optimal sampling The Bias and MSE of RV The presence of market microstructure contaminations induces a bias-variance trade-off in IV estimation through RV. When p is observable, higher sampling frequencies over a fixed period of time result in more precise estimates of the IV of p. When the equilibrium price process is not observable, as is the case in the presence of microstructure frictions, frequency increases provide information about the underlying IV but, inevitably, entail accumulation of noise that affects both the bias and the variance of the estimator (Bandi and Russell, 2003, 2006, and Zhang, Mykland, and Ait-Sahalia, 2005). Rossi Microstructure noise University of Pavia / 52

20 MSE-based optimal sampling The Bias and MSE of RV Under 1 Assumptions 1 and 2, 2 absence of leverage effects (σ W ), 3 unpredictability of the equilibrium returns (A(t) = 0), Bandi and Russell (2003) provide an expression for the conditional (on the underlying volatility path) mean-squared error (MSE) of the RV estimator as a function of the sampling frequency δ (or n): RV (h) = n j=1 IV (h) = r 2 jδ = h 0 n [ ] 2 rjδ + ɛ jn j=1 σ 2 (u)du Rossi Microstructure noise University of Pavia / 52

21 MSE-based optimal sampling The Bias and MSE of RV Bias: MSE: n E[RV (h) IV (h)] = E rjδ 2 IV (h) = ne n [ɛ 2 ] j=1 E n [(RV (h) IV (h)) 2 ] = 2 h n (IQ(h) + o(1)) + π n π n = ne n [ɛ 4 ] + 2 The variance of RV (h) n (n j)e n [ɛ 2 j ɛ 2 j] + 4E n [ɛ 2 ]IV (t, h) j=1 E n [ (RV (h) E(RV (h))) 2] = 2 h n (IQ(h) + o(1)) + π n n 2 E n [ɛ 2 ] 2 the conditional MSE of RV can serve as the basis for an optimal sampling theory designed to choose n in order to balance bias and variance. Rossi Microstructure noise University of Pavia / 52

22 MSE-based optimal sampling MSE-based optimal sampling Bandi and Russell (2003) discuss evaluation of the MSE under Assumption 1 and 2 as well as in the MA(1) case. The MA(1) case is very convenient in that the moments of the noise do not depend on the sampling frequency. It is theoretically fairly arbitrary to choose a single optimal frequency solely based on bias considerations. While it is empirically sensible to focus on low frequencies for the purpose of bias reduction, the bias is only one of the components of the estimator s estimation error. At sufficiently low frequencies the bias can be negligible. However, at the same frequencies, the variability of the estimates might be substantial. Rossi Microstructure noise University of Pavia / 52

23 MSE-based optimal sampling MSE-based optimal sampling Under MA(1), the MSE simplifies substantially: E n [(RV (h) IV (h)) 2 ] = 2 h n (IQ(h) + o(1)) + nβ + n2 α + γ where the parameters are defined as [ ] 2 α = E(ɛ 2 ) [ β = 2E(ɛ 4 ) 3 E(ɛ 2 ) γ = 4E(ɛ 2 ) E(ɛ 4 ) + ] 2 [ ] 2 E(ɛ 2 ) Rossi Microstructure noise University of Pavia / 52

24 MSE-based optimal sampling MSE-based optimal sampling If n is large, the following approximation to the optimal number of observations applies: n ( hiq(h) ) 2 E(ɛ 2 ) If the signal coming from the underlying equilibrium price process (IQ) is large relative to the noise determined by the frictions ((E(ɛ 2 )) 2), then sampling can be conducted at relatively high frequencies. 1/3 Rossi Microstructure noise University of Pavia / 52

25 MSE-based optimal sampling MSE-based optimal sampling The MSE improvements that the MSE-based optimal frequencies guarantee over the 5- or 15-minute frequency can be substantial. Not only do the optimal frequencies vary cross-sectionally, they also change over time. Using mid-quotes dating back to 1993 for three stocks with various liquidity features, namely EXXON Mobile Corporation (XOM), SBC Communications (SBC), and Merrill Lynch (MEL), Bandi and Russell (2006) show that the daily optimal frequencies have substantially increased in recent times, generally due to decreases in the magnitude of the noise moments. This effect should in turn be attributed to an overall increase in liquidity. Rossi Microstructure noise University of Pavia / 52

26 Subsampling Subsampling Zhang et al. (2005) proposed a subsampling method in order to estimate the integrated variance consistently in the presence of MN. The simple idea is to exploit the possibility of partitioning the full gird Λ t = (τ 0,..., τ nt ) into K nonoverlapping subgrids such that where Λ (k) t Λ (k) t, k = 1,..., K Λ t = K k=1 Λ (k) t Λ (j) t = φ i j Set n (k) as the number of observations in each subgrid, and define the RV (k) RV (h) (k) = n (k) j=1 r 2 t,j Rossi Microstructure noise University of Pavia / 52

27 Subsampling Two Time Scales Estimator Rossi Microstructure noise University of Pavia / 52

28 Subsampling Two Time Scales Estimator The proposal of Zhang et al. (2005) is to use the following Two Time Scales (TTS) estimator for the daily RV : 1 n n n RV (h) (ZMA) = 1 K RV (h)(k) n RV (h) n n = 1 K K j=1 n (k) t = n K + 1 K RV (h) is a consistent estimate of the second moment of the noise return, and RV (h) is the required bias-correction. Under A(t) = 0 and the MA(1) noise case, Zhang et al. (2005) show that, as K, n with K n 0 and K 2 n RV (h) (ZMA) is a consistent estimator of IV. Rossi Microstructure noise University of Pavia / 52

29 Subsampling Two Time Scales Estimator Provided that K = cn 2/3, the rate of convergence of RV (h) (ZMA) to IV is n 1/6 and the asymptotic distribution is mixed-normal with an estimable asymptotic variance. Zhang et al. (2005) showed that, under Assumption 3, n 1/6[ ] { RV (h) (ZMA) d IV (h) ξ 2 = 16 Var[ɛ 2 i ] c 2 E[(ηi 2 )] 2 }{{} due to noise + c 4 3 IQ(h) }{{} due to discretization } {{ } total variance Cov[ɛ 0, ɛ 2 i ] i=1 } 1/2N(0, 1) Rossi Microstructure noise University of Pavia / 52

30 Subsampling Two Time Scales Estimator The proportionality factor c can be selected optimally in order to minimize the limiting variance. In the case of equidistant observations { hiq(h) c = 12E[(ηi 2)]2 } 1/3 This minimization leads to an asymptotically optimal number of subsamples given by { } 1/3 hiq(h) K asy = 12E[(ηi 2 n 2/3 )]2 Both components of the factor c: E(ηi 2 ) and IQ can be readily evaluated from the data. Rossi Microstructure noise University of Pavia / 52

31 Subsampling Two Time Scales Estimator The E(ηi 2 ) can be estimated by using a (standardized) sample average of squared continuously-compounded returns sampled at the highest frequencies. Under the MA(1) market microstructure model, hence, 1 2n n j=1 r 2 t,j E[ɛ 2 i ] = 2E[η 2 i ] p E[η 2 ] as n The quarticity can be identified by employing the Barndorff-Nielsen and Shephard s quarticity estimator n RQ = n 3h j=1 r 4 t,j with continuously-compounded returns sampled at relatively low frequencies, among other methods. The 15- or 20-minute frequency have been shown to work reasonably well in practise. Rossi Microstructure noise University of Pavia / 52

32 Subsampling Two Time Scales Estimator The estimator of Zhang, Mykland, and Ait-Sahalia (2005) is effectively a two-scale-estimator relying on very high-frequency return data to identify the bias component as well as on lower frequency return data to characterize the individual realized variances prior to averaging. In Ait-Sahalia et al. (2006), a small sample refinement to the TTS estimator is proposed. The final estimator becomes ( RV (h) (ZMA) = 1 n ) RV (h) (ZMA) n Both of the estimators are derived under Assumption 3 (IID noise). Rossi Microstructure noise University of Pavia / 52

33 Subsampling Two Time Scales Estimator In order to take into account possibly dependent noise, Zhang (2006) and Ait-Sahalia et al. (2006) proposed an alternative estimator (RV (AMZ) J ) that is also based on the two time scales idea. All the results are derived under Assumption 4 (non-iid noise). Average lag J RV RV (AL) J = 1 n J (r i+j r i ) 2 J Generalization of the TSSE derived in Zhang et al. (2005) where 1 J K n j=0 RV (AMZ) = RV (AL) K K = o(n) n (K) = n K + 1 K A small sample correction is given by RV (AMZ,adj) = (1 n(k) n(k) (AL) RV n (J) J n (J) = n J + 1 J ) RV (AMZ) n (J) Rossi Microstructure noise University of Pavia / 52

34 Subsampling Two Time Scales Estimator Zhang (2006) and Ait-Sahalia et al. (2006) showed that: RV (AMZ,adj) d t 0 σ 2 (u)du + 1 n 1/6 [ ξ 2 c 2 }{{} due to noise + c 4 3 t 0 ] 1/2N(0, σ 4 (u)du 1) }{{} due to discretization } {{ } total variance Rossi Microstructure noise University of Pavia / 52

35 Kernel-based estimation Kernel-based estimation Consistently estimating the QV under the presence of MN is, in a sense, similar to the well-known autocorrelation corrections that are frequently used in the time-series literature to estimate the long run variances and covariances of stationary stochastic processes (see, for example, Andrews, 1991; Newey and West, 1987). For example, Hansen, P. R., Lunde, A. (2006). Realized variance and market microstructure noise (with discussion), Journal of Business and Economic Statistics 24: considered the following simple kernel-based estimator: H RV HL = RV (all) n (h) + 2 n h ˆγ h where ˆγ h = h=1 n n h r t,j r t,j+h n h j=1 Rossi Microstructure noise University of Pavia / 52

36 Kernel-based estimation Kernel-based estimation RV HL = RV (all) (h) + 2 H h=1 n n h r t,j r t,j+h n h If the correlation structure of the noise returns has a finite order and A(t) = 0, under appropriate conditions on H the estimator is unbiased: E n [RV HL ] = t 0 σ 2 (u)du In the MA(1) noise case the estimator takes the simpler expression j=1 RV (all) (h) + 2 n n 1 r t,j r t,j+1 n 1 Under Assumption 1, Assumption 2b, and A(t) = 0, the covariance j=1 E n [r t,j r t,j+1 ] = E[η 2 ] Rossi Microstructure noise University of Pavia / 52

37 Kernel-based estimation Kernel-based estimation [ E n 2 n n 1 ] r t,j r t,j+1 = 2nE[η 2 ] n 1 j=1 The bias of the estimator RV HL is equal to ne[ɛ 2 ] = 2nE[η 2 ] the second term 2 n n 1 n 1 j=1 r t,jr t,j+1 provides the required bias correction. The finite sample unbiasedness of Hansen and Lunde s estimator is robust to the presence of some dependence between the underlying local martingale price process (under A(t) = 0) and market microstructure noise. Rossi Microstructure noise University of Pavia / 52

38 Kernel-based estimation Kernel-based estimation Zhou (1996) was the first to consider the use of kernel methods to deal with the problem of microstructure noise in high-frequency data. For the case of independent noise, Zhou proposed RV HL with H = 1. Hansen and Lunde (2006) examined the properties of Zhou s estimator and showed that, although unbiased under Assumption 3, the estimator is not consistent. However, Hansen and Lunde (2006) advocated that, while inconsistent, Zhou s kernel method is able to uncover several properties of the microstructure noise, and concluded that the noise: 1 is correlated with the efficient price; 2 is time dependent; 3 is quite small in the DJIA stocks; 4 has properties that have changed substantially over time. Their results are robust to both CTS and TrTS. The h-th autocovariance is scaled by n/n h to compensate for the missing autocovariance terms. In this way the estimator is unbiased. Rossi Microstructure noise University of Pavia / 52

39 Kernel-based estimation Kernel-based estimation The upward scaling has the drawback that it increases the variance of the estimator. For this reason, Hansen and Lunde (2005) consider the Bartlett kernel and define the estimator: where RV HL (h) = RV (all) (h) + 2 H = but the estimator is also inconsistent. H ( h=1 [( 4n ) 2/9 ] h H + 1 ) ˆγ h Rossi Microstructure noise University of Pavia / 52

40 Kernel-based estimation Flat-top kernel-based estimation Barndorff-Nielsen, O. E., Hansen, P. H., Lunde, A., Shephard, N. (2006). Subsampling realised kernels. Econometrics Journal proposed the flat-top kernel-based estimator: RV BHLS (h) = RV (all) (h) + H ( h 1 ) k (ˆγ h + ˆγ h ) H where k(x) for x [0, 1] is a nonstochastic weight function such that k(0) = 1 and k(1) = 0. They h=1 prove that the statement that all kernel based RV estimators were inconsistent is wrong and proposed several consistent kernel-based estimators; design a kernel that has a smaller variance than the multiscale estimator; propose an estimator for data with endogenously spaced observations, such as that in databases on transactions; consider the case where the microstructure noise is endogenous. Rossi Microstructure noise University of Pavia / 52

41 Kernel-based estimation Flat-top kernel-based estimation BHLS (2006) showed that, if H = cn 2/3, then the resulting estimator is asymptotically mixed Gaussian, converging at rate n 1/6. The constant, c, can be optimally chosen as a function of the kernel k(x). For example, the value of c that minimizes the variance of the estimator is given by { } 1/3 2[k (0) 2 + k (1) 2 ] E[ɛ 2 c = i ] 2/3 1 0 k(x)2 dx IQ(h) 3 BHLS (2006) also compared three different kernels: 1 Bartlett: k(x) = 1 x 2 2nd order: k(x) = 1 2x x 2 3 Epanechnikov k(x) = 1 x 2 Rossi Microstructure noise University of Pavia / 52

42 Kernel-based estimation Kernel-based estimation Their findings are summarized as follows: the Bartlett kernel has the same asymptotic distribution as the TTSE of Zhang et al. (2005) It is more efficient than the Epanechnikov alternative, but is less efficient than the 2nd order kernel. If k (0) = 0 and k (1) = 0, then setting H = cn 1/2, the asymptotic distribution of the estimator is mixed normal with convergence rate equal to n 1/4. If k (0) = 0 and k (1) = 0, the Tukey-Hanning kernel, k(x) = [1 cos π(1 x)2 ] 2 seems to be the best option in terms of efficiency. Rossi Microstructure noise University of Pavia / 52

43 The Effects of the Sampling Scheme The Effects of the Sampling Scheme The choice of sampling scheme can have a strong influence on the statistical properties of the realized variance. The first to contribute in that direction was Oomen (2005), who examined the following sampling alternatives: 1 calendar time sampling, 2 transaction time sampling, 3 tick time sampling, 4 business time sampling. Oomen (2006) proposed a pure jump process for the high frequency-prices, which allows for the analysis of the following sampling schemes: calendar time, business time, and transaction time sampling. Rossi Microstructure noise University of Pavia / 52

44 The Effects of the Sampling Scheme The Effects of the Sampling Scheme In Oomen (2006) the observed log-price process is expressed as N(t) N(t) p(t) = p(0) + ξ j + N(t) is a Poisson process with intensity λ(t), ξ j N(µξ, σ 2 ξ ). j=1 j=1 η j = ν j + ρ 2 ν j ρ 2 ν j q+1 ν j N(0, σ 2 ν) The process N(t) counts the number of transactions up to time t. It is assumed to be independent of both ξ and ν. The equilibrium price process N(t) p(t) = p(0) + p (t) is pure jump process of finite variation with IV given by IV = σ 2 ξ h 0 j=1 ξ j ds = σ 2 ξλ(h) Rossi Microstructure noise University of Pavia / 52 η j

45 The Effects of the Sampling Scheme The Effects of the Sampling Scheme When η MA(1) η j = ν j yields a negative first-order autocorrelation of the calendar time continuouslycompounded returns since p t p t τ = = = N(t) j=n(t τ) N(t) j=n(t τ) N(t) j=n(t τ) N(t) N(t τ) ξ j + η j ξ j + j=1 N(t) j=n(t τ)+1 j=1 η j ξ j + ν N(t) ν N(t τ) Oomen (2006) provides closed-form expressions for the MSE of the RV estimator under both calendar time sampling and transaction time sampling. η j Rossi Microstructure noise University of Pavia / 52

46 The Effects of the Sampling Scheme The Effects of the Sampling Scheme The optimal sampling frequency is derived to minimize the MSE, which is influenced by the number of trades and the noise level. It was shown that, as in the case of the diffusion-based models, the RV is a biased estimator of the jump analogue of the IV when microstructure noise is present. The bias does not diverge to infinity as the sample frequency increases. Sampling schemes: transaction time sampling leads to a lower MSE of the RV than calendar time sampling. The largest gains are obtained for days with irregular trading patterns. Rossi Microstructure noise University of Pavia / 52

47 Comparison of Techniques Comparison of Techniques From McAleer and Medeiros (2008) Realized Volatility: A Review, Econometric Reviews Rossi Microstructure noise University of Pavia / 52

48 Comparison of Techniques Comparison of Techniques 1 Although there are many unbiased estimators, only four are consistent: 1 TTSE of Zhang et al.(2005) (order of convergence is n 1/6 ) and Ait-Sahalia et al. (2005) (order of convergence is n 1/4 ). 2 Barndorff-Nielsen et al. (2006a): the realized kernel consistent estimator (generalization of Hansen-Lunde (2006)). 3 The modified MA filter of Hansen et al. (2008), which is also of order n 1/4 ). 4 The alternation estimator of Large (2006). 2 It is important to note that, while not being consistent, the kernel estimators discussed in Hansen and Lunde (2006) are important tools for uncovering, if only partially, several properties of the microstructure noise. Rossi Microstructure noise University of Pavia / 52

49 Comparison of Techniques Comparison of Techniques Barndorff-Nielsen et al.(2006), Zhang et al. (2005), and AÔt-Sahalia et al. (2005) required that the number of autocovariances (or subsamples) H and the number of observations, n, to diverge to infinity as the ratio φ = H n For a given φ, the magnitude of the finite sample MSE of the estimators can be substantially different from the asymptotic approximations. Rossi Microstructure noise University of Pavia / 52

50 Comparison of Techniques Comparison of Techniques Bandi and Russell (2006): detailed study of the finite sample performance of several kernel-based and subsampling estimators (under Assumption 3) 1 For the realistic sample sizes encountered in practical applications, the asymptotic results, in general, do not provide sufficient guidance for practical implementation, as they provide unsatisfactory representations of the finite sample properties of the estimators. 2 The finite sample MSE properties of the consistent estimator RV (ZMA) of Zhang et al. (2005), and of the inconsistent Bartlett kernel estimator RV (HL,Bartlett) (Hansen and Lunde (2005)), are similar, and a significant component of their MSE is induced by the finite sample bias. 3 Asymptotic methods to select the bandwidth can be suboptimal in their case, especially for biased kernel estimators. As their finite sample bias vanishes asymptotically, asymptotic methods do not take the finite sample bias into account and have a tendency to select an excessively small number of bandwidths. A small H can lead to a large bias component in a finite sample. Rossi Microstructure noise University of Pavia / 52

51 Comparison of Techniques Comparison of Techniques 1 This bias component can be reduced by choosing H in order to minimize the estimator s finite sample MSE. 2 In the case of Zhang et al. (2005) estimator and the Bartlett kernel estimator of Hansen and Lunde (2005), the authors proposed a simple (MSE-based) rule-of-thumb to select the ratio, φ, which is given by: ( 3 φ Bartlett,HL = φ ZMA = φ RV (h)/n 2 2 IQ(h) ) 1/3 3 While the optimal finite sample MSE values of Zhang et al. s (2005) estimator and Hansen and Lunde s (2005) Bartlett kernel estimator are generally smaller than the optimal finite sample MSE value of the classical RV estimator, the gains that these useful estimators can provide over the classical realized variance estimator might be lost or dramatically reduced by suboptimally choosing the value of φ. Rossi Microstructure noise University of Pavia / 52

52 Comparison of Techniques Comparison of Techniques Bandi and Russell (2006): evaluate the finite sample behavior of the of the consistent flat-top kernel based estimators proposed by Barndorff- Nielsen et al. (2006): 1 Despite having the same distribution as the subsampling estimator of Zhang et al. (2005), the flat-top Bartlett kernel estimator appears to be preferable to the former in finite samples. 2 The use of asymptotic criteria to select the optimal value of H can be more or less satisfactory depending on the choice of kernel. 3 Due to the lack of a substantial bias term and the flatness of the variance term as a function of φ, the suboptimal bandwidth choices do not lead to extremely large losses. 4 The asymptotic approximations to the finite sample dispersion of the symmetric estimators can be imprecise. A careful assessment of the accuracy of these estimators requires a closer examination of their finite sample properties. Rossi Microstructure noise University of Pavia / 52