Spot Volatility Estimation for High-Frequency Data

Transcription

1 Spot Volatility Estimation for High-Frequency Data Jianqing Fan Princeton University Yazhen Wang University of Connecticut Astract The availaility of high-frequency intraday data allows us to accurately estimate stock volatility. This paper employs a ivariate diffusion to model the price and volatility of an asset and investigates kernel type estimators of spot volatility ased on highfrequency return data. We estalish oth pointwise and gloal asymptotic distriutions for the estimators. Jianqing Fan is Frederick Moore 18 Professor of Finance, Department of Operation Research and Financial Engineering, Princeton University, Princeton, NJ 8544 and Director, Center for Statistical Research, Chinese Academy of Science. Yazhen Wang is professor, Department of Statistics, University of Connecticut, Storrs, CT Fan s research was partially supported y the NSF grant DMS and Chinese NSF grant , and Wang s research was partially supported y the NSF grant DMS The authors thank Per Mykland for helpful comments and suggestions.

2 ey words: Asymptotic normality, CIR model, constant elasticity of diffusion, extreme distriution, kernel estimator, long memory, stock price. 1

3 1 Introduction Volatilities of asset returns are pivotal for many issues in financial economics. For example, market participants need to estimate volatility for the purpose of hedging, option pricing, risk analysis and portfolio management. With advance of computer technology, data availaility is ecoming less and less a prolem. Nowadays it is relatively easy to otain high frequency financial data such as complete records of quotes or transaction prices for stocks. The high-frequency financial data provide an incredile experiment for understanding market microstructure and more generally for analyzing financial markets. In particular we expect to estimate volatilities etter using high-frequency returns directly. The field of high-frequency finance has evolved rapidly. Current main interests of volatility estimation are on instantaneous volatility (or spot volatility) and integrated volatility over a period of time, say, a day. Estimation methods for univariate integrated volatility include realized volatility (RV) [Andersen et. al. (23)], i-power realized variation (BPRV)[Barndorff-Nielsen and Shephard (26)], two-time scale realized volatility (TSRV)[Zhang et. al. (25)], multiple-time scale realized volatility (MSRV) [Zhang (26)], wavelet realized volatility (WRV)[Fan and Wang (27)], kernel realized volatility (RV)[Barndorff-Nielsen et. al. (24)], and Fourier realized volatility (FRV) [Mancino and Sanfelici (28)]. For the case of multiple assets, estimation approaches of multivariate integrated volatility consist of realized co-volatility for synchronized high-frequency data [Barndorff-Nielsen and Shephard (24)] and realized covolatility ased on overlap intervals and previous ticks for non-synchronized high-frequency data [Hayashi and usuoka (25) and Zhang (25)]. Wang, Yao, Li and Zou (27) has 1

4 proposed a matrix factor model to achieve dimension reduction and facilitate the estimation of integrated co-volatility in very high dimensions for non-synchronized high-frequency data. For spot volatility estimation, Foster and Nelson (1996) first showed that spot volatility can e estimated from high-frequency data y rolling and lock sampling filters. For a general class of price and volatility processes, under a numer of stringent conditions, they estalished pointwise asymptotic normality for rolling regression estimators of the spot volatility and estalish the efficiency of different weighting schemes. The conditions and results are in quite astract sense. For given examples, the conditions are hard to verify and asymptotic normality is difficult to evaluate as well. Andreou and Ghysels (22) further investigated theoretical properties of rolling-sample volatility estimator and check its finite sample performance with simulation and empirical studies. In this paper, we assume price and volatility to follow a ivariate diffusion process and investigate asymptotic ehaviors of the kernel type estimators of spot volatility for high-frequency data. Under the general ut verifiale conditions, we derive explicit expressions for their pointwise and gloal asymptotic distriutions. We show that these conditions are met y diffusion ased volatility models often used in literature. The paper is organized as follows. Section 2 presents the main results. Section 3 illustrates the common models and verifies the conditions for these models. Section 3 features key technical propositions aout strong approximation for the spot volatility estimator. 2

5 2 Estimation of spot volatility Consider d assets and let X t = (X 1t,, X dt ) T e the vector of the log prices of d assets. Assume that X t follows a continuous-time diffusion model, dx t = µ t dt + σ t dw t, t [, T ], (1) where T is a positive constant, W t is a d-dimensional Brownian motion, µ t is a drift, and σ t is a d y d matrix. We define instantaneous or spot volatility as Γ t = σ t σ t = (γ ij t ) 1 i,j d. The quadratic variation of X t has expression [X, X] t = t Γ s ds, t [, T ]. Suppose that we oserve X t at n discrete time points t i = i T/n, i = 1,, n. Our goal is to estimate Γ t = d[x, X] t dt = σ t σ t = (γ kj t ) 1 k,j d. Suppose (x) is a kernel with support on [ 1, 1]. We define the kernel type estimator ˆΓ t = 1 = 1 = 1 t+ t i =t t+ t i =t t+ t ( ) ti t (X ti X ti 1 ) (X ti X ti 1 ) ( ) s t δ X ti ( δ X ti ) ( ) s t d[x, X] s, (2) where is andwidth, δ = T/n, δ X t is the increment of X t over [t δ, t] defined y δ X t = X t X t δ, 3

6 and [X, X] t is the realized volatility given y [X, X] t = t i t δ X ti ( δ X ti ). For example, if = 1, then the estimator results in a rolling average ˆΓ t = 1 δ X ti ( δ X ti ). t i t One side kernel with support on [ 1, ] yields an estimator that uses the immediate past data, and one side exponential kernel (x) = e x 1(x ) results in an exponential smoothing in the RiskMetric [Fan et. al. (23)]. Below we will estalish asymptotic theory for ˆΓ. First we list some technical conditions. Let denote the Euclidean norm for vectors and maximum norm for matrices. A1 sup { σ s σ t, s, t [, T ], s t a} = O P (a 1/2 log a 1/2 ), sup σt 2 = O p (1), t T and for j = 1,, d, A2 sup {σ(s) σ(t i 1 )} dws j 2, i = 1,, n = O P (n 2+η ), where η > is an aritrarily small numer. The drift µ t in (1) satisfies A3 sup{ µ t µ s, t s a} = O P (a 1/2 log a 1/2 ). Bandwidth and kernel satisfy A4 n 1/2 / log n, ( ) is twice differentiale with support [ 1, 1] and 1 1 (x) dx = 1. We will show that Assumptions A1-A2 are very general and satisfied for common volatility processes in Section 3. Assumption A3 is aout the mean drift in price processes and is often 4

7 met y price models. We may select kernel and andwidth to meet Assumption A4. Now we state the two main theorems whose proofs rely on technical propositions given in Section 4. Theorem 1 Under Assumptions A1-A4, we have that n {ˆΓt Γ t } σ 2 (t) Z, where the convergence is in distriution, and Z is a random matrix whose elements are independent and have normal distriutions with mean zero and variance 2 λ() for diagonal elements and λ() for off-diagonal elements, where λ() = (x) dx. Proof. It is a direct consequence of Propositions 2 and 3 in Section 4. Remark 1. Theorem 1 provides pointwise asymptotic distriution for ˆΓ. The limiting distriution is normal with explicit covariance matrix. The convergence rate in Theorem 1 matches up with the orders of convergence in Mykland and Zhang (28) in terms of andwidth and sample size. Theorem 2 Suppose that Assumptions A1-A4 are satisfied and that σ t, t [, T ] is stationary. Let M n = sup n ˆΓt Γ t. t T Then (2 log n) 1/2 M n λ() d n exp( 2 e x ), 5

8 where the convergence is in distriution, and λ() is defined in Theorem 1, d n = (2 log n) 1/2 + if λ 1 () >, and otherwise d n = (2 log n) 1/2 + 1 (2 log n) 1/2 {log λ 1().5 log π.5 log log n}, λ 1 () = 2 ( 1) + 2 (1), 2 λ() λ 2 () = 1 (2 log n) 1/2 {log λ 2() log (2 π)} 1 2 λ() Proof. M n has the same asymptotic distriution as [ (x)] 2 dx. sup V n (t), t T where V n (t) is defined y (7) in Section 4. The representation for V n (t) given y Propositions 2 and 3 in Section 4 allows us to estalish the asymptotic distriution for the maximum of V n (t) y an application of Theorem A1 in Bickel and Rosenlatt (1973, section 5). Remark 2. Theorem 2 gives the gloal asymptotic distriution for ˆΓ. The extreme limiting distriution may e used to construct confidence and for Γ t over whole interval t [, T ]. 3 Common volatility models Common volatility processes in literature include geometric Ornstein-Uhleneck(OU) process, Nelson GARCH diffusion process (Nelson, 199), the CIR diffusion process (Cox, Ingersoll and Ross, 1985), and long-memory volatility process (Comte and Renault, 1998). We 6

9 show elow that Assumptions A1-A2 are satisfied for these volatility processes as well as their superpositions. Below we will examine the examples for which Assumptions A1-A2 are met. Example 1. Geometric OU model, d log σ 2 (t) = λ log σ 2 (t) dt + dw v (t), (3) where W v is a standard Brownian motion, λ is a parameter, and the initial value σ 2 () is finite and independent of W v. Example 2. Nelson GARCH diffusion model, dσ 2 (t) = λ {σ 2 (t) ξ} dt + ω σ 2 (t) dw v (λ t), (4) where W v is a standard Brownian motion, (λ, ξ, ω) are parameters, and the initial value σ 2 () is finite and independent of W v. Example 3. The CIR model, dσ 2 (t) = λ {σ 2 (t) ξ} dt + ω σ(t) dw v (λ t), (5) where W v is a standard Brownian motion, (λ, ξ, ω) are parameters, and the initial value σ 2 () is finite and independent of W v. Example 4. The Long-memory model, d log σ t = κ log σ 2 t dt + γ dw v,α (t), (6) where W v,α is a fractional Brownian motion with memory index α (1/2, 1), (κ, γ) are parameters, and the initial value σ 2 () is finite and independent of W v,α. 7

10 We first check Assumption A1 for each example. For Example 1, (3) has an explicit solution log σ 2 (t) = e λ t log σ 2 () + t e λ (t s) dw v (s). From the sample path property of Brownian motion W v, we immediately show that which implies sup log σ 2 (t) = O P (1), t sup σ 2 (t) = O P (1). t This is the second condition in Assumption A1. For the first condition of Assumption A1, note that log σ 2 (t) log σ 2 (s) = ( e λ t e λ s) log σ 2 () + = ( e λ t e λ s) { log σ 2 () + t s e λ (t u) dw v (u) s e λ (s u) dw v (u) } t e λ u dw v (u) + e λ t e λ u dw v (u). Since e λ t e λ s = O(t s), the first term in aove equation is O P (t s), and due to the increment property of Brownian motion W v, the second term is O P ( (t s) log t s 1/2 ). Since σ(t) σ(s) = σ(s) {exp (log σ(t) log σ(s)) 1}, s so the first condition in Assumption A1 is satisfied for Example 1. The equation (4) in Example 2 has solution σ 2 t = exp { β 1 t + β 2 W v (t) β 2 2 t/2 } { σ 2 + β t exp ( β 1 s β 2 W v (s) + β2 2 s/2 ) } ds. where β = λ ξ, β 1 = λ, β 2 = λ ω. Again the sample path property of W v shows that sup σt 2 = O P (1), t 8

11 which is the second condition of Assumption A1. For the first condition note that σ t σ s = exp { β 1 (t s)/2 + β 2 (W v (t) W v (s))/2 β2 2 (t s)/4 } 1 exp { β 1 s/2 + β 2 W v (s)/2 β 2 2 s/4 } ( σ 2 + β s + exp { β 1 t/2 + β 2 W v (t)/2 β 2 2 t/4 } β /2 t s (σ 2 + β exp ( β 1 u β 2 W v (u) + β2 2 u/2 ) 1/2 du). s exp ( β 1 u β 2 W v (u) + β2 2 u/2 ) ) 1/2 du exp ( β 1 u β 2 W v (u) + β 2 2 u/2 ) du Due to the property for the order of increments of Brownian motion, the first term in aove equation is O P ( (t s) log t s 1/2 ), and the second term is O P ( t s ). Thus, the first condition in Assumption A1 is satisfied. For Example 3, (5) has no explicit solution. However, it is well known that σ 2 (t) is a Gamma process with sup σ 2 (t) = O P (1), t sup σ 2 (t) = O P (1). t So the second condition of Assumption A1 is met. For the first condition we have that dσ(t) =.5 λ { σ(t) + (ξ.25 ω)/σ(t)} dt +.5 ω dw v (λ t) σ(t) σ(s) =.5 λ t s { σ(u) + (ξ.25 ω)/σ(u)} du +.5 ω [W v (λ t) W v (λ s)] The first term is O P (t s) and the second term has order (t s) log t s 1/2 in proaility. Thus, the first condition in Assumption A1 is met. The equation (6) in Example 4 has solution log σ 2 (t) = e κ t log σ 2 () + γ t e κ (t s) dw v,α (s). 9

12 The maximum of sample paths of W v,α in a ounded interval is O P (1), thus the max log σ 2 (t) = O P (1), t which implies the second condition of Assumption 1. For the first condition, we have log σ 2 (t) log σ 2 (s) = ( e κ t e κ s) log σ 2 () + γ γ s e κ (s u) dw v,α (u) = ( e κ t e κ s) { log σ 2 () + γ s t } e κ u dw v,α (u) + e κ t γ e κ (t u) dw v,α (u) t s e κ u dw v,α (u). Again the first term in aove equation is O P (t s). The second term is O P ( (t s) log t s α ), due to the increment property of fractional Brownian motion W v,α. Remark 3. If volatility processes satisfy Assumption 1, their superpositions also meet Assumption 1. This shows in particular that a two factor volatility model, which is a superposition of two geometric OU processes, satisfies Assumption 1. Now we consider Assumption 2. We have the following general result for models without leverage effect, where no leverage effect means that Brownian motion W in (1) driving price processes and Brownian motion W v (or fractional Brownian motion W v,α ) in (3)-(6) are independent. Proposition 1 Suppose that there is independence etween Brownian motion in (1) for price process and Brownian motion (or fractional Brownian motion) in (3)-(6) for volatility processes. If Assumption A1 is satisfied, then Assumption A2 is automatically met. Proof. Conditional on whole paths of σ 2 t, {σ(s) σ( )} dw s 1

13 are independent Gaussian random variales with mean zero and covariance {σ(s) σ( )} {σ(s) σ( )} ds. Hence, with proaility tending to one, the maximum of {σ(s) σ(t i 1 )} dw s 2, i = 1,, n, is ounded y 2 log n sup { σ(s) σ( ) 2 ds, i = 1,, n }, which, y Assumption 1, has order n 2 log 2 n. This gives Assumption A2. For price and volatility models with leverage effect, that is, W and W v are dependent, Assumption A2 needs to check case y case. Below we illustrate the check of Assumption 2 for the geometric OU model. Note that we have and thus log σ 2 (s) log σ 2 ( ) = ( e λ s e λ ) { log σ 2 () + {σ(s) σ( )} dw s = σ( ) + e λ s s 1 e λ u dw v (u), } e λ u dw v (u) {e log σ(s) log σ() 1} dw s, {e log σ(s) log σ() 1} dw s = + { ( )} s e log σ(s) log σ(ti 1) exp e λ s e λ u dw v (u) dw s { ( ) } s exp e λ s e λ u dw v (u) 1 dw s I i + J i. We need to show that for oth I i and J i, their maximum over i = 1,, n are of order n 1+η/2 log n. 11

14 I i is a stochastic integral over [, t i ], and its integrand is of order n 1. As i l=1 I i is a discrete martingale, and its quadratic variation [I, I] is of order of the sum of squares of the integrand of I i, which has order n 2. Hence, ( ) ( ) i P max I i 2 M 2 P max I l M 1 i n 1 i n l=1 2 M 2 1 M P ([I, I] > M 2 1 ), where the last equality is due to Nuglart inequality (Jacod and Shiryaev, 22), and M 1 = n 1 log 1/2 n and M = n 1 log n. We derive that the maximum of I i is of order n 1 log n. Also J i is a stochastic integral over [, t i ], ut its integrand is of order n 1/2 log 1/2 n. However, J i are independent. Applying BDG inequality (Jacod and Shiryaev, 22) to each J i, we otain E ( J i 2 p) C E C n p { E exp { ( exp e λ s s ( e λ u dw v (u) e λ s s ) e λ u dw v (u) p 2 1} ds ) 1} 2 p ds C n 2 p, where C is a generic constant and p > is a constant and will e chosen later. With M = n 1/(2 p) 1 log n we otain P ( ) max J i M = 1 i n i P ( J i M) i (1 C n 2 p /M 2 p ) = (1 C n 2 p /M 2 p ) n 1 C n 1 2 p /M 2 p = 1 C log 2 p n 1, For large enough p 1/η we conclude that the maximum of J i is of order n 1+η/2 log n. 4 Strong approximation for spot volatility estimator Define V n (t) = n {ˆΓt Γ t }, (7) 12

15 where Γ t = 1 t+ t i =t ( ) t ti Γ s ds. (8) We estalish the following strong approximation result for V n. Strong approximation constructed on some proaility spaces are held for versions of V n, σ, Γ on the new proaility spaces, which have identical distriutions as V n, σ, Γ, respectively. For simplicity, we use the same notations to denote their versions on the constructed proaility spaces. Proposition 2 Suppose that Assumptions A1-A4 are satisfied. Then there exist matrix processes B n (t) on some proaility spaces such that B n (t) = B n (t) = {(1+1(k = j)) 1/2 B kj n (t)} d d with B kj n (t) = B jk n (t) eing independent standard Brownian motions, and independent of (µ t, σ t, W t ), and V n (t) = σ(t) 1 t+ ( ) s t db n (s) σ(t) + O P (n 1/4+η/2 log n) t t/+1 ( = σ(t) u t ) d t/ 1 B n (u) σ(t) + O P (n 1/4+η/2 log n), where B n ( ) = 1/2 B n ( ) are the rescaled of B n, and the error order is uniformly over t [, T ]. Proof. Note that X ti X ti 1 = µ s ds + σ s dw s, Assumptions A1-A3 implies that t i µ s ds is dominated y t i σ s dw s, so the drift term µ t in (1) has no effect on asymptotic results (such as limiting distriutions and convergence orders) for the estimator ˆΓ t. Therefore, for simplicity we set µ t = in the rest of proofs. 13

16 The second equality results from change variale and rescaling property of Brownian motion. We prove the first equality only. Let δ = t i = T/n. Then ˆΓ t Γ t = 1 = t+ t i =t t+ t i =t t+ t i =t ( ) ( ti t ) ( ti ) ti σ(s) dw s σ(s) dw s Γ s ds ( ) ti t {σ(ti 1) (W ti W ) (W W ti 1 ti ti 1 ) σ() Γ (t ti 1 i t } i 1) ( ) ( ti t ) σ(t ti i 1) (W ti W ti 1 ) {σ(s) σ( )} dw s + {σ(s) σ( )} dw s {σ( ) (W ti W ti 1 )} {Γ(s) Γ( )} ds ( ti ) ti + {σ(s) σ( )} dw s {σ(s) σ( )} dw s = H 1 + H 2 + H 3 + H 4. (9) Lemmas 2-4 elow will derive the orders for H 2, H 3 and H 4. Simple algera shows H 1 = 1 = δ n t+ t i =t t+ t i =t ( ) { s t (Wti σ( ) W ) ( } Wti W ti 1 ti 1) δ Id σ( ) ( ) ti t σ( ) n 1/2 U i σ( ), (1) where I d denotes the d d identity matrix, U i = ( W ti W ti 1 ) ( Wti W ti 1 ) /δ Id. As matrix random variales U i are i.i.d., E(U i ) =, and the entries of U i are uncorrelated and have variance 2 at diagonal and 1 off diagonal, then [i t] n 1/2 U j j=1 weakly converges to B(t) = B(t) = {(1+1(k = j)) 1/2 B kj (t)} d d with B kj (t) = B jk (t) eing independent standard Brownian motions, and independent of (µ t, σ t, W t ). By MT strong 14

17 approximation (omlós, Major, and Tusnády, 1975, 1976), there exists B n (t) = B n (t) on some proaility spaces with B n (t) eing versions of B such that Cov(B n, W ) =, Then from (1) we get max 1 i n n 1/2 i U j B n (t i ) = O P (n 1/2 log n). (11) j=1 H 1 δ n t+ t i =t ( ) ti t σ( ) B n (t i ) σ( ) = δ n t+ ( ) ti t σ( ) { n 1/2 U i B n (t i ) } σ( ) t i =t = δ t+ δ n ( ) ti t i σ( ) n 1/2 U j B n (t i ) σ( ) t i =t +δ j=1 ( ) ti+1 t i σ(t i ) n 1/2 U j B n (t i ) σ(t i ) j=1 ± δ ( ) n ± δ n (t±) σ(t ± ( δ)) n 1/2 U j B n (t ± ) σ(t ± ( δ)) G 1 + G 2. j=1 (12) Because of (11) and order of in Assumption A4, G 2 is of order δ n The term in the racket of G 1 is equal to n 1/2 log n = n 1/2 log 2 n. ( ) ti t i σ( ) n 1/2 U j B n (t i ) {σ( ) σ(t i )} j=1 ( ) ti t i + {σ( ) σ(t i )} n 1/2 U j B n (t i ) σ(t i ) j=1 { ( ) ( )} ti t ti+1 t i + σ(t i ) n 1/2 U j B n (t i ) σ(t i ). (13) j=1 15

18 By Assumption A1, σ( ) σ(t i ) is of order n 1/2 log n, and Assumption A4 implies ( ) ( ) ti t ti+1 t is of order n 1/2 log n. These two results together with (11) show that each of the three terms in (13) is of order n 1 log 2 n. Sustituting aove orders for (13) into G 1 given y (12) and using the order of in Assumption A4, we derive the order for G 1 δ n n n 1 log 2 n = n 1/2 log 2 n. Using aove otained order n 1/2 log 2 n for oth G 1 and G 2 and from (12) we have H 1 = δ n = δ n = δ n = δ n t+ t i =t t+ ( ) ti t σ( ) B n (t i ) σ( ) + O(n 1/2 log 2 n) ( ) ti t σ( ) {B n (t i ) B n ( )} σ( ) + O(n 1/2 log 2 n) t i =t t+ t i =t t+ t ( ) ti t σ( ) db n (s) σ( ) + O(n 1/2 log 2 n) ( ) s t σ(s) db n (s) σ(s) + O(n 1/2 log 2 n), where the last equality is due to Lemma 1 elow. Collecting together aove result for H 1 and the orders for H 2, H 3 and H 4 given y Lemmas 2-4 elow, and using equation (9) we arrive at V n = n (ˆΓ t Γ t ) = 1 t+ ( ) s t σ(s) db n (s) σ(s) + O p (n 1/4+η/2 log n). (14) t Finally we complete the proof y using the order of in Assumption A4 and showing that σ(s) in the stochastic integral on the right hand side of (14) can replaced y σ(t) with an error of order n 1/4 log n. In deed, note that 1 t+ ( ) s t σ(s) db n (s) σ(s) = t 1 t+ ( ) s t σ(t) db n (s) σ(s) t 16

19 + 1 t+ ( ) s t [σ(s) σ(t)] db n (s) σ(s). (15) t The second stochastic integral on the right hand side of (15) has is of order n 1/4 log n, ecause its quadratic variation is equal to 1 t+ t ( ) s t 1 2 [σ(s) σ(t)] 2 ds σ 2 (s) = 2 (u)[σ(t + u ) σ(t)] 2 du σ 2 (t + u ) 1 = O P (n 1/2 log 2 n), where the second equality is from the fact that y Assumption A1, the maximum of σ(t + u ) σ(t) 2 over u [ 1, 1] is of order n 1/2 log 2 n. Similarly, σ(s) in the first stochastic integral on the right hand side of (15) can e replaced y σ(t) with a resulting error of order n 1/4 log n. Lemma 1 Suppose that Assumptions A1-A4 are satisfied. Then Proof. Define t+ t i =t t+ = t ( ) ti t σ( ) db n (s) σ( ) ( ) s t σ(s) db n (s) σ(s) + O P (n 1/2 log n). D i = = [ + + ( ) ti t σ( ) db n (s) σ( ) ( ) ( ti t s t ( s t ( s t )] σ( ) db n (s) σ( ) ) [σ( ) σ(s)] db n (s) σ( ) ( ) s t σ(s) db n (s) σ(s) ) σ(s) db n (s) [σ( ) σ(s)]. (16) B n are independent of volatility process σ, and the entries of B n are independent Brownian motions, then conditional on σ, the entries of D 1,, D n are independent normal random 17

20 variales with mean zero. We work on each entry of matrix σ db n σ. Denote y D i the maximum over all entries of D i. Since D i defined y (16) is equal to a sum of three stochastic integral with respect to Brownian motion, which have explicit quadratic variations, we have that conditional on σ, E [ D i 2 σ ] C 1 sup σ(s) 4 ti ( ) ( ) ti t s t 2 ds s T ( + C 2 σ(s) 2 + σ( ) 2) σ(s) σ( ) 2 ds, (17) where C 1 and C 2 are generic constants. By Assumption A1, we have that for s [, t i ], σ(s) σ( ) is of order n 1/2 log n uniformly over 1 i n, and Assumption A4 implies that ( ) ( ) ti t s t is of order n 1/2 log n. Hence, the right hand side of (17) is of order n 2 log 2 n, and so is the conditional variance of D i. Since the entries of all D i are independent normal random variales with conditional variances uniformly ounded y a quantity of order n 2 log 2 n. Conditional on σ, with proaility tending to one, max D i log (n d 2 ) max E [ D i 2 σ]. 1 i n 1 i n Hence we have max D i = O P (n 1 log 3/2 n), 1 i n and t+ t i =t D i = O P (n n 1 log 3/2 n) = O P (n 1/2 log 1/2 n). This completes the proof of Lemma 1. 18

21 Lemma 2 Suppose that Assumptions A1-A4 are satisfied. Then 1 Proof. Note that t+ t i =t ( ) t ti (Γ s Γ ti 1 ) ds = O P (n 1/2 log 1/2 n). 1 t+ ( ) t ti (Γ s Γ ti 1 ) ds t i =t 1 t+ ( ) t ti Γ s Γ ti 1 ds t i =t C t+ t i =t C sup{n Γ s Γ i 1 ds Γ s Γ ti 1 ds, i = 1,, n} C sup{ Γ s Γ ti 1, s [, t i ], i = 1,, n}, (18) where C is generic constant depending on kernel only. Since Γ t Γ s = σ t σ t σ s σ s (σ t σ s ) σ t + σ s (σ t σ s ) σ t σ s σ t + σ s σ t σ s, from Assumption A1, we immediately have sup{ Γ s Γ ti 1, s [, t i ], i = 1,, n} = O P (n 1/2 log 1/2 n). Comining it with (18) we prove the lemma. Lemma 3 Suppose that Assumptions A1-A4 are satisfied. Then 1 t+ t i =t ( ) t ti {σ(s) σ( )} dw s ( {σ(s) σ( )} dw s ) = O P (n 1+η log 2 n). 19

22 Proof. Note that 1 t+ t i =t C ( ) ( ti t ti ) ti {σ(s) σ( )} dw s {σ(s) σ( )} dw s ( d ) ti 2 t+ C sup t i =t j=1 ( d n ti j=1 {σ(s) σ( )} dw j s {σ(s) σ( )} dw j s Now the lemma is a consequence of Assumption A2. ) 2, i = 1,, n. Lemma 4 Suppose that Assumptions A1-A4 are satisfied. Then oth of 1 t+ t i =t ( ) ( ti t ) ti σ( ) (W ti W ti 1 ) {σ(s) σ( )} dw s, and 1 t+ t i =t ( ) t ti {σ(s) σ( )} dw s {σ( ) (W ti W ti 1 )} are equal to O P (n 1/2+η/2 log 3/2 n). Proof. Because of simplicity we need to prove the first one only. For Brownian motion W, we have sup { W ti W ti 1, i = 1,, n } = O P (n 1/2 log 1/2 n), and Assumption A2 implies sup { } {σ(s) σ( )} dw s, i = 1,, n = O p ((n 1+η/2 log n). Thus, ( ) sup (W ti t i W ti 1 ) {σ(s) σ( )} dw s, i = 1,, n = O p(n 3/2+η/2 log 3/2 n), 2

23 from which we conclude 1 t+ ( ) ( ti t ) ti σ( ) (W ti W ti 1 ) {σ(s) σ( )} dw s t i =t ( ) n (W ti t i W ti 1 ) {σ(s) σ( )} dw s = O p (n 1/2+η/2 log 3/2 n). Proposition 3 Suppose that Assumptions A1-A4 are satisfied. Then n {Γ t Γ t } = o P (1), where Γ t is defined in (8). Proof. Note that Γ t Γ s = σ t σ t σ s σ s (σ t σ s ) σ t + σ s (σ t σ s ) σ t σ s σ t + σ s σ t σ s. From Assumption A1 and A4, we immediately have sup{ Γ s Γ t, s [t, t + ]} = O P ( log ). Hence with δ = t i = T/n, we otain Γ s ds = Γ t δ + (Γ s Γ t ) ds = δ [ ] Γ t + O P ( log ). (19) On the other hand, Assumption A4 and simple calculus show that δ t+ t i =t ( ) ti t = 1 1 Plugging (19) and (2) into (8), we have (u) du + O(n 1/2 log n) = 1 + O(n 1/2 log n). (2) E (ˆΓt ) [ Γ t = 1 + O(n 1/2 log n) ] [ ] Γ t + O P ( log ) = Γ t + O P ( log ). 21

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