Journal of Econometrics. A two-stage realized volatility approach to estimation of diffusion processes with discrete data

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1 Journal of Econometrics 5 ( Contents lists available at ScienceDirect Journal of Econometrics journal homeage: A two-stage realized volatility aroach to estimation of diffusion rocesses with discrete data Peter C.B. Phillis a,b,c,d, Jun Yu e, a Cowles Foundation, Yale University, United States b Deartment of Economics, University of Auckl, New Zeal c Deartment of Economics, University of York, United ingdom d School of Economics, Singaore Management University, Singaore e School of Economics, Singaore Management University, 9 Stamford Road, Singaore 7893, Singaore a r t i c l e i n f o a b s t r a c t Article history: Available online 5 December 8 JEL classification: C3 C E43 G3 eywords: Maximum likelihood Girsanov theorem Discrete samling Continuous record Realized volatility his aer motivates introduces a two-stage method of estimating diffusion rocesses based on discretely samled observations. In the first stage we make use of the feasible central limit theory for realized volatility, as develoed in [Jacod, J., 994. Limit of rom measures associated with the increments of a Brownian semiartingal. Working aer, Laboratoire de Probabilities, Universite Pierre et Marie Curie, Paris] [Barndorff-Nielsen, O., Shehard, N.,. Econometric analysis of realized volatility its use in estimating stochastic volatility models. Journal of the Royal Statistical Society. Series B, 64, 53 8], to rovide a regression model for estimating the arameters in the diffusion function. In the second stage, the in-fill likelihood function is derived by means of the Girsanov theorem then used to estimate the arameters in the drift function. Consistency asymtotic distribution theory for these estimates are established in various contexts. he finite samle erformance of the roosed method is comared with that of the aroximate maximum likelihood method of [Aït-Sahalia, Y.,. Maximum likelihood estimation of discretely samled diffusion: A closed-form aroximation aroach. Econometrica. 7, 3 6]. 8 Elsevier B.V. All rights reserved.. Introduction For many years, continuous time models have enjoyed a great deal of success in finance (Merton, 99 as well as wide alications in economics (e.g., Dixit (993. Corresondingly, there has been growing interest in estimating continuous systems using econometric methods with discrete data. We thank two anonymous referees for their constructive comments. We also thank Yacine Aït-Sahalia, Yongmiao Hong, Chung-ming uan, Andy Lo, Neil Shehard, seminar articiants at the First Symosium on Econometric heory Alications in aiwan, the 6 North American Winter Meeting of Econometric Society, the First Finance Summer Cam at Singaore Management University for helful discussions. Phillis gratefully acknowledges visiting suort from the School of Economics at Singaore Management University, suort from a elly Fellowshi at the University of Auckl Business School from the NSF under Grant No. SES Yu gratefully acknowledges financial suort from the Ministry of Education AcRF ier fund under Grant No. 6B43-RS. Corresonding author. addresses: eter.hillis@yale.edu (P.C.B. Phillis, yujun@smu.edu.sg (J. Yu. Many models used in finance for modelling asset rices can be written in terms of a diffusion rocess as dx t = µ(x t ; θ dt + σ (X t ; θ db t, ( where B t is a stard Brownian motion, σ (X t ; θ is a known diffusion function, µ(x t ; θ is a known drift function, θ = (θ, θ is a vector of k + k unknown arameters. Note that we isolate the vector of arameters θ in the diffusion function from θ for reasons which will be clear below. he attractions of the Ito calculus make it easy to work with rocesses generated by diffusions like ( as a result these rocesses have been used widely in finance to model asset rices, including stock rices, interest rates, exchange rates. From an econometric stoint, the estimation roblem is to estimate θ from observed data, which are tyically recorded discretely at (,,..., n ( over a certain time interval [, ], where is the samling interval is the time san of the data. For examle, if X t is recorded as the annualized interest rate observed monthly (weekly or daily, we have = / (/5 or /5. yically, can be as large as 5 for US reasury Bills, but is generally much smaller for data from swa /$ see front matter 8 Elsevier B.V. All rights reserved. doi:.6/j.jeconom.8..6

2 4 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( markets. Also, due to time-of-day effects ossibly other market microstructure frictions, it is commonly believed that intra-day data do not comletely follow diffusion models such as (. As a result, daily lower frequencies are most frequently used to estimate continuous time models. However, Barndorff-Nielsen Shehard ( Bollerslev Zhou ( recently showed how to use information from intra-day data to estimate continuous time stochastic volatility models. A large class of estimation methods is based on the likelihood function derived from the transition robability density of discrete samling then resorts to long san asymtotic theory (i.e.. Excet for a few cases, the transition robability density does not have a closed form exression hence the exact maximum likelihood (ML method based on the likelihood function for the discretely samled data is not directly available. In the financial econometrics literature, interest in obtaining estimators which aroximate or aroach ML estimators has been growing, in view of the natural attractiveness of maximum likelihood its asymtotic roerties. Several alternative methods of this tye have been develoed in recent years. See Phillis Yu (in ressa for a survey of various alternative methods the discussion of their advantages drawbacks. he main urose of the resent aer is to roose an alternative method of estimating diffusion rocesses of the form given by model ( from discrete observations to establish asymtotic roerties by resorting to both the long san (i.e. in-fill asymtotics (i.e.. he estimation rocedure involves two stes. In the first ste, we roose to use a quadratic variation tye estimator of θ. In the second ste, an aroximate in-fill likelihood function is maximized to obtain a ML estimator of θ. his method has several advantages over the existing method. First, it is not deendent on finding an aroriate auxiliary model. Second, it does not require simulations or olynomial exansions hence is straightforward to imlement. hird, it decomoses the otimization roblem into two smaller scale otimization roblems, making the aroach comutationally more attractive. Finally, exerience with the rocedure both in simulations emirical alications indicates that the method works well in finite samles. he aer is organized as follows: Section reviews the literature on the ML estimation of diffusion rocesses motivates the aroach. Section 3 introduces the new method Section 4 derives the asymtotic roerties of the estimates. Section 5 resents some Monte Carlo evidence. Section 6 discusses the case of microstructure noises Section 6 concludes. Proofs are rovided in the Aendix.. Literature review motivation.. Literature review... ransition robability density based aroaches As exlained above, a large class of estimation methods is based on the likelihood function derived from the transition robability density of the discretely samled data. Suose (X i X (i, θ is the transition robability density. he Markov roerty of model ( imlies the following log-likelihood function for the discrete samle l D (θ = n log((x i X (i, θ. ( Under regularity conditions, the resulting estimator is consistent, asymtotically normally distributed asymtotically efficient (Billingsley, 96. Unfortunately, excet for a few cases, the transition density does not have a closed form exression hence the exact ML method based on the likelihood function of the discrete samle is not a ractical rocedure. In the financial econometrics literature, interest in finding estimators that aroach ML estimators in some quantifiable sense has been growing many alternative methods have been develoed in recent years. For examle, Lo (988 suggested calculating the transition robability density by solving a artial differential equation numerically. Nowman (997 suggested an aroach which assumes that the conditional volatility remains unchanged over the unit intervals so that he can aroximate the transition density using a Gaussian method. Yu Phillis ( used the stoing time technique to develo an exact Gaussian method. Pedersen (995 Brt Santa-Clara ( advocated an aroach which calculates the transition robability density using simulation with some auxiliary oints between each air of consecutive observations introduced. his method is also closely related to the Bayesian MCMC method roosed by Elerian et al. ( Eraker (. A drawback of these simulation-based aroaches is that the corresonding comutational cost will inevitably be high. As an imortant alternative to these numerical simulated ML methods, Aït-Sahalia ( roosed to aroximate the transition robability density of diffusions using analytical exansions via Hermite olynomials. Before obtaining the closed-form exansions, a Lamerti transform is erformed on the continuous time model so that the diffusion function becomes a constant. After that one then obtains a Hermite olynomial exansion of the transition density of the transformed variable around the normal distribution. Aït-Sahalia (999 imlemented the aroximate ML method documents its good erformance. As it is tyically tedious error rone to derive the Hermite exansion by h, Aït-Sahalia ( suggested using symbolic softwares, such as MAHEMAICA. Aart from these likelihood-based aroaches, numerous alternative methods are available. We simly refer readers to the book by Prakasa Rao (999a for a review of many alternative aroaches.... Aroaches based on realized volatility in-fill likelihood When the transition robability density does not have a closed form exression but X t is observed continuously over [, ], an alternative method can be used to estimate diffusion models. We now introduce motivate the aroach. When the diffusion term is known (i.e. σ (X t ; θ = σ (X t so does not deend on any unknown arameters, one can construct the exact continuous record log-likelihood via the Girsanov theorem (e.g., (Litser Shiryaev, as follows. l IF (θ = µ(x t ; θ σ (X t dx t µ (X t ; θ dt. σ (X t Lánska (979 established the consistency asymtotic normality of the continuous record ML estimator of θ when under a certain set of regularity conditions. he assumtions of a known diffusion function the availability of a continuous time record are not realistic in financial other alications. Motivated by the fact that the drift diffusion functions are of different orders (Bi Phillis, 3, 7, we argue that there can be advantages to estimating the diffusion arameters searately from the drift arameters. For examle, when σ (X t ; θ = θ, i.e., the diffusion function is an unknown constant, a two-stage aroach can be used to estimate the model. First, θ can be estimated directly by the realized volatility function, i.e., [X ] ˆθ =, (3

3 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( Fig.. Stardized realized volatility against the number of daily observations used to calculate the realized volatility. he daily data are simulated from the Vasicek model dx t =.6(.9 X t dt +.6dB t. Fig.. Stardized realized volatility against the number of daily observations used to calculate the realized volatility. he daily data are simulated from the Vasicek model dx t =.6(.9 X t dt +.db t. where [X ] imlies that (dx t = θ dt, hence = n (X i X (i. his is because model ( t, [X] = (dx t = θ dt = θ, where [X] is the quadratic variation of X, which can be consistently estimated by [X ] as. As a result, ˆθ should be a very good estimate of θ when is small, which is tyically the case for interest rate data. Indeed, in the secial case where the drift term is zero the diffusion term is an unknown constant, the exact discrete ( model (Phillis, 97 for the data is X i X (i = θ Bi B (i so the maximum likelihood estimator is trivially ˆθ (see also Aït-Sahalia et al. (5. Although this corresondence clearly does not aly to more general secifications, it seems likely that when is small the two estimators will be close to each other. Second, the following logarithmic continuous record likelihood function of model (, l IF (θ = µ(x t ; θ σ (X t ; ˆθ dx t µ (X t ; θ σ (X t ; ˆθ dt, may be aroximated by the in-fill likelihood function l AIF (θ = n µ(x (i ; θ (X i X (i ˆθ n µ (X (i ; θ, (4 ˆθ which is in turn maximized with resect to θ. Because θ is estimated by ML θ is close to an MLE, the two-stage rocedure may be interreted as a form of rofile ML estimation. his two-stage aroach is closely related to the method roosed by Florens-Zmirou (989, where a contrast function instead of the logarithmic in-fill likelihood function was used in the second ste. o better areciate the quality of aroximation of [X ] to [X], we simulate daily data from the Vasicek model dx t =.6(.9 X t dt +.6dB t, (5 Fig. 3. Stardized realized volatility against the number of weekly observations used to calculate the realized volatility. he weekly data are simulated from the Vasicek model dx t =.6(.9 X t dt +.6dB t. lot the stardized realized volatility, [X ] /, against the number of daily observations used to calculate [X ]. Quadratic variation theory imlies that as, [X ] / a.s. σ. It is clear from Fig. that although [X ] / is quite erratic initially but quickly settles down around σ. It seems that with only 3 35 observations one can get a good estimate of σ. We then increase the volatility rate from.6 to. decrease the samling frequency from daily to weekly. Results are lotted in Figs. 3, resectively. It clear that the conclusion about the raid convergence of [X ] / is quite robust to these changes. When the diffusion term is only known u to a scalar factor, that is when dx t = µ(x t ; θ dt + θ f (X t db t, (6 the above two-stage method is easily modified. First, θ estimated by ˆθ = n can be [X ]. (7 f (X (i

4 4 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( able Simulation results under the CIR model, dx t =.3(.9 X t dt +.6 X t db t, based on relications. Mean SD st for the average stard deviation across relications, resectively. Method κ =.3 µ =.9 σ =.6 Mean SD Mean SD Mean SD / MLE Yoshida Proosed / 5 MLE Yoshida Proosed /5 MLE Yoshida Proosed /5 MLE Yoshida Proosed /5 MLE Yoshida Proosed /5 MLE Yoshida Proosed Second, the following aroximate logarithmic in-fill likelihood function can then be maximized with resect to θ (denoting the resulting estimator by ˆθ l AIF (θ = n µ(x (i ; θ ˆθ f (X (i (X i X (i n µ (X (i ; θ ˆθ f (X (i. (8 his method is alicable to many oular interest rate models, including those roosed by Vasicek (977, Cox et al. (985 (CIR hereafter, Ahn Gao (999. It is also closely related to the method roosed by Yoshida (99. In articular, instead of using the estimator in (7, Yoshida (99 used the following estimator for θ : θ = n (X i X (i. (9 f (X (i Also,Yoshida (99 suggested using an iterative rocedure to construct a better estimate of θ (denoted by θ. Under the conditions of,,, Yoshida (99 derived limiting normal distributions for n ( θ θ (ˆθ θ. Since n / = /, the diffusion arameter enjoys a faster rate of convergence. Unfortunately, requiring the diffusion function to be either a constant onown u to a scalar function limits alicability the rocedures roosed by Florens-Zmirou (989 Yoshida (99 cannot be imlemented with more general diffusion rocesses. he restriction on the diffusion term regarding arameter deendence was somewhat relaxed in Hutton Nelson (986 who based estimation on the following first order condition of the logarithmic quasi-likelihood function: µ(x t ; θ/ θ dx σ t (X t ; θ µ (X t ; θ/ θ dt =. σ (X t ; θ Although their model seems to allow for a more flexible diffusion function, it requires that the drift term share the same set of arameters as the diffusion term. his assumtion is too restrictive for ractical alications. Moreover, although this onestage estimation aroach is easy to imlement, the estimation is mainly based on the drift function hence leads to inferior finite samle roerties, as we will show below in the context of a simle examle... Motivation Our two-stage method is in line with methods roosed by Florens-Zmirou (989 Yoshida (99. hat is, in the first stage, we estimate the arameters in the diffusion functions based on the realized volatility, a quantity which consistently estimates the quadratic variation under very mild conditions. In the second ste, by assuming the diffusion function to be known, we derive aroximate the logarithmic in-fill likelihood function. o motivate the two-ste aroach, we consider two simle examles.... Examle In the first examle, we consider estimating the following CIR model dx t = κ(µ X t dt + σ X t db t, ( using the exact ML method based on the transition robability density the two-stage method discussed in Section.. he natural estimator of σ based on realized volatility is [X ] ˆσ = n. ( X (i i= Moreover, since θ = (κ, µ, the logarithmic in-fill likelihood is, n κ(µ X (i (X i X (i ˆσ X (i i= n κ (µ X (i i= ˆσ X (i. ( CIR (985 showed that the distribution of X(t + conditional on X(t is non-central chi-squared, χ [cx(t, q +, λ(t], where c = κ/(σ ( e κ, λ(t = cr(te κ, q = κµ/σ, the second third arguments are the degrees of freedom non-centrality arameters, resectively. his transition robability density is used to calculate the likelihood function to obtain the exact ML estimates. able reorts some results obtained from a Monte Carlo study where we comare three estimation methods: exact ML, Yoshida s method which estimates σ by (9, the roosed method which estimates σ by (. We vary both the samling frequencies

5 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( able Simulation results under dx t =.dt +.db t based on relications. Mean variance are calculated across relications, resectively. rue value of α =. RV ML QML Mean Variance Mean Variance Mean Variance / / / time sans. Note that the arameters the samling frequencies are all set to emirically reasonable values. In all cases, the two two-stage methods are almost identical. his is not surrising as the two methods differ only in the first stage. Moreover, the two-stage methods erform comarably with the ML method. Even in the case where very coarsely samled data ( = / are available, the two-stage method works quite well. In most cases the two-stage methods erform slightly better than the ML method. In light of Phillis Yu (5, the observed bias in the estimates of κ are the result of the near unit root roblem. he observation that the two-stage method is not dominated by ML is quite remarkable, as the data generating rocess is based on the transition robability density on which ML itself is based. An interesting side result to emerge from this simulation is that the two-stage method is able to reduce the finite samle bias variance in κ in all cases, even though the reductions are small.... Examle he model in the second examle is taken from Hutton Nelson (986 dx t = αdt + αdb t. (3 Although this model is generally not well suited to interest rate data, the feature that the drift diffusion functions share the same arameter rovides a nice framework to investigate the relative erformance of the estimation method based on the diffusion only, against that based on the drift only that based on the drift diffusion jointly. he first method is based on the realized volatility hence only uses the diffusion term to estimate the model. It is easy to show that [X ] ˆα =. he second method is based on the transition robability density given by X i X (i N(X (i + α, α. Clearly this method uses information both in the drift diffusion functions. Denote the resulting estimate by ˆα. Note that this estimator is equivalent to the MLE of the discretized model via the Euler aroximation. he third method was roosed by Hutton Nelson (986. It uses mainly information in the drift function is based on maximization of the following logarithmic quasi-likelihood function α dx t α dt. As a result, the estimate has the following analytical exression: ˆα 3 = X. able reorts results obtained from a Monte Carlo study where we comare the three estimation methods with different samling frequencies. In all cases, the two-stage method ML erform much better than QML;, most remarkably, the two-stage method erforms better than ML ( hence the Euler method. Just as in examle, the fact that the simle two-stage method outerforms ML in finite samles is surrising. Moreover, the better erformance of the first second methods clearly reflects the order difference in the drift diffusion functions. 3. A two-stage method he estimation rocedure discussed in Section. is not directly alicable to general diffusions such as model, as it requires either a constant diffusion function or searability of the scalar arameter from the reminder of the diffusion function. As a result, we have to rovide a more general two-ste rocedure to estimate a diffusion rocess in the form of model (. In articular, in the first ste we roose to estimate the arameters in the diffusion function by using the feasible central limit theorem for realized volatility derived by Jacod (994 oularized by Barndorff- Nielsen Shehard (. Assume that X t is observed at a gird of discrete times t =,,..., M ( = ( M =,..., n (=,, (M +,..., where n = M with being a fixed ositive integer, is the time san of the data, is the samling frequency, M = O(n. his articular construction allows for the nonoverlaing sub-samles ((k M +,..., km, where k =,...,, so that each sub-samle has M observations over the interval ((k, k ]. For examle, if ten years of weekly observed interest rates are available we slit the data into ten blocks, then =, = /5, M = 5, =. he total number of observations is 5 the number of observations contained in each block is 5. he first limit in Box I follows by virtue of the definition of quadratic variation, while the second limit in Box I is the central limit theorem (CL which is due to Jacod (994 Barndorff-Nielsen Shehard (, the third limit in Box I involves a finite samle correction on the asymtotic theory (Barndorff-Nielsen Shehard, 5. It is shown in the latter reference that the third limit in Box I has a better finite samle erformance than that the second limit in Box I. Although in this aer we only use the realized volatility to estimate the quadratic variation, other realized ower variations, such as realized absolute variation, can be used in the same way. For the theoretical develoment of general realized ower variations, we refer readers to the articles by Barndorff-Nielsen Shehard (3 Barndorff-Nielsen et al. (6. Based on the CL i.e., the second limit in Box I, θ can be estimated in the first stage by running a (nonlinear least squares regression of the stardized realized volatility (X (k M +i X (k M +(i (4

6 44 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( As, n = M, so that M (X (k M +i X (k M +(i [X] k (X (k M +i X (k M +(i ([X] k log ( M [X] (k, (X (k M +i X (k M +(i log([x] k where = M (X (k M +i X (k M +(i 4 3 s k [X] (k d N(,, [X] (k + s k d N(,, r s k = max k, ( M M (X (k M +i X (k M +(i fo =,...,. Box I. where ˆθ = arg min θ Q (θ, Q (θ = { (X(k M +i X (k M +(i σ ( } X (k M +(i ; θ Box II. on the stardized diffusion function ([X] k [X] (k = ( σ (X (k t ; θ dt (5 σ ( X (k M +(i ; θ (6 fo =,...,. Denote the resulting estimator of θ by ˆθ. In fact, we can write ˆθ as the extremum estimator as in Box II. A similar regression in stardized log levels of realized volatility can be run using the third limit in Box I. his aroach rovides a more general estimation rocedure than those designed to estimate models with a constant diffusion or a scalar arameter in the diffusion function. Hence this aroach substantially generalizes the method offlorens-zmirou (989 the method of Yoshida (99 to a much wider class of diffusion rocesses. Indeed, when =, the least squares regression above is equivalent to minimizing the squared difference between the terms given by Eqs. (4 (5, which yields exactly the exression of the estimator (7 when the diffusion term is known u to the scalar factor. In the second stage, the aroximate log-likelihood function is maximized with resect to θ (denoting the resulting estimator by ˆθ l AIF (θ = n µ(x (i ; θ σ (X (i ; ˆθ (X i X (i n 4. Asymtotic results µ (X (i ; θ σ (X (i ; ˆθ. (7 he asymtotic theory of a slightly different two-stage estimator in the multivariate case was obtained in Yoshida (99 for models whose diffusion term is known u to a constant (matrix factor, where both infill long san asymtotics are emloyed both for the diffusion drift arameter estimators. In this section we first derive the asymtotic theory for the same class of (scalar models but only resort to long san asymtotics for the drift arameter asymtotic theory. We then investigate the asymtotic roerties of the estimators roosed in Section 3 for model ( whose diffusion function has a general form.

7 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( Scalar arameter in the diffusion function o highlight the differences between our aroach Yoshida (99, we first assume the data are generated according to the following stochastic differential equation: dx t = µ(x t ; θ dt + θ f (X tdb t. (8 Denote θ by τ θ by τ. Both µ( ; θ f ( are timehomogeneous, B-measurable functions on D = (l, u with l < u, where B is the σ -field generated by Borel sets on D. τ is estimated by ˆτ defined by Eq. (7; θ is estimated by ˆθ, the maximizer of Eq. (8. o rove consistency of ˆτ in a diffusion rocess with a constant diffusion term (i.e. f (X t =, Florens-Zmirou (989 assumed,,. he same set of assumtions were emloyed by Yoshida (99 to deal with the diffusion rocess for more general, but still known, f (X t. In this aer, using the theory of Jacod (994 Barndorff-Nielsen Shehard (, we show that the condition of an infinite time san of data (i.e. is not needed to develo the asymtotic theory for ˆτ, thereby extending the asymtotic results of Yoshida (99 in a significant way. We list the following conditions. Assumtion. Equation [X] t τ t f (X s ds = has a unique solution at τ > t >. Assumtion. inf x J f (x >, where J is a comact subset of the range of the rocess. Assumtion 3. t µ (X s ; θ ds < t <. Remark 4.. Assumtion is an identification condition Assumtion is a bounding ositivity condition on the volatility function. Assumtion 3 ensures weak convergence of the error rocess from the Euler aroximation to the diffusion rocess (Jacod Protter, 998. heorem 4. (Asymtotics of the Diffusion Parameter Estimate. Suose Assumtions hold, ˆτ τ as. If, in addition, Assumtion 3 holds, / (ˆτ τ d τ f (X s dw s f (X s ds where W t is a Brownian motion which is indeendent of X t. Remark 4.. With a different estimate for τ, we substantially imrove the results of Yoshida (99, who derived asymtotic roerties of the diffusion estimate assuming that, by only requiring in heorem 4. that. Our result is not surrising confirms the intuition that when the samling interval goes to zero, the samle ath within a finite time san, no matter how short, can erfectly reveal the quadratic variation of the rocess (see, for examle, Merton (98 at least over that time san. o establish the asymtotic roerties of the drift arameter estimate, we follow Yoshida (99 closely. In articular, we first list the following conditions. Assumtion 4. θ Θ where the arameter sace Θ R is a comact set with θ Int(Θ. Assumtion 5. Both µ( ; θ f ( functions are twice continuously differentiable. As a result, for any comact subset J of the range of the rocess, we have the following two conditions: (i (Lischitz condition here exists a constant L so that µ(x; θ µ(y; θ + θ f (x f (y L x y, for all x y in J. (ii (Growth condition here exists a constant L so that µ(x; θ + θ f (x L + x, for all x y in J. Assumtion 6. Define the scale measure of X t by ( x µ(y; θ s(x; θ = ex τ f (y dy, c where c is a generic constant. We assume the following conditions hold u c s(x; θdx = c l s(x; θdx =, u dx = A(θ <. s(x; θτ f (x l Assumtion 7. For arbitrary, su E( X t <. t Assumtion 8. Define the following function ( µ(x, θ Y(θ ; τ θ = µ(x, θ τ f (x µ(x, θ π θ (dx assume function Y( ; τ has the unique maximum at θ = θ, where π θ is defined in Remark 4.4. Assumtion 9. For fixed θ, the derivatives l µ(x; θ / x l l f (x/ x l (l =, exist they are continuous in x. For fixed x, l µ(x; θ / θ l exist. Moreover, l µ(x; θ / x l, l f (x/ x l, l µ(x; θ / θ l C( + x C, for l =,,. Assumtion. he matrix µ(x; θ Φ = θ (τ f (x µ(x; θ π θ (dx (9 θ is ositive definite. Remark 4.3. Under Assumtion 5, these exists a solution rocess for the stochastic differential equation the solution is unique. Remark 4.4. Under Assumtion 6, the rocess X t is ergodic with an invariant robability measure that has density π θ (x = A(θs(x; θτ f (x, for x (l, u with resect to Lebesque measure on (l, u, where A(θ s(x; θ are defined in Assumtion 6. We further assume that X π θ so that X t is a stationary rocess with X t π θ.

8 46 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( heorem 4. (Asymtotics of the Drift Parameter Estimates. Let ˆθ = argmax θ Θ log l AIF (θ with l AIF (θ given by Eq. (8. Suose Assumtions hold, ˆθ θ as. If, in addition,, / (ˆθ θ d N(, Φ, where Φ is given in Eq. ( General diffusions Now we consider the general case where Florens-Zmirou (989 Yoshida (99 are not alicable. Suose data are generated from the following stochastic differential equation dx t = µ(x t ; θ dt + σ (X t; θ db t, ( where θ Θ R θ Θ R. Both µ(, θ σ ( ; θ are time-homogeneous, B-measurable functions on D = (l, u with l < u, where B is the σ -field generated by Borel sets on D. θ is estimated by regressing (4 on (5, giving the extremum estimator in Box II; θ is estimated by ˆθ, defined by Eq. (7. As in the scalar factor arameter case, we show that an infinite time san (i.e. is not needed to develo the asymtotic theory for ˆθ. Some additional assumtions are required, given the nonlinear deendence of the diffusion σ (X t ; θ on θ. Also we have to modify some earlier Assumtions listed in Section 4.. Assumtion. he equation [X] t t σ (X s ; θ ds = t t has a unique solution at θ, t >. σ (X s ; θ ds σ (X s ; θ ds = ( Assumtion. inf x J σ (x; θ >, where J is a comact subset of the range of the rocess. Assumtion 4. θ Θ, θ Θ, where arameter saces Θ R k Θ R k are comact set with θ Int(Θ θ Int(Θ. Assumtion 5. Both µ( ; θ σ ( ; θ functions are twice continuously differentiable. As a result, for any comact subset J of the range of the rocess, we have the following two conditions: (i (Lischitz condition here exists a constant L so that µ(x; θ µ(y; θ + σ (x; θ σ (y; θ L x y, for all x y in J. (ii (Growth condition here exists a constant L so that µ(x; θ + σ (x; θ L + x, for all x y in J. Assumtion 6. Define the scale measure of X t by ( x µ(y; θ s(x; θ = ex σ (y; θ dy, c where c is a generic constant. We assume the following condition holds u c s(x; θdx = c l s(x; θdx =, u dx = A(θ <. s(x; θσ (x; θ l Assumtion 8. Define the following function θ Y(θ ; θ = µ(x, θ σ (x; θ ( µ(x, θ µ(x, θ assume Y( ; θ has the unique maximum at θ = θ. π θ (dx Assumtion 9. For fixed θ, the derivatives l µ(x; θ / x l l σ (x; θ / x l (l =, exist they are continuous in x. For fixed x, l µ(x; θ / θ l l σ (x; θ / θ l exist. Moreover, l µ(x; θ / x l, l σ (x; θ / x l, l µ(x; θ / θ l, l σ (x; θ / θ l C( + x C, for l =,,. Assumtion. he matrices µ(x; θ Φ = σ (x; θ θ µ(x; θ θ π θ (dx t σ ( ( X s ; θ σ Xs ; θ θ θ ds ( are ositive definite t σ 4 ( X s ; θ ds > for all t >. heorem 4.3. (Asymtotics of the Diffusion Parameter Estimate: Suose Assumtions hold. hen, ˆθ θ as σ (ˆθ / θ d (X s ;θ σ (X s ;θ (k θ θ ds σ ( 4 X (k s ; θ ds σ (X s ;θ (k θ σ ( X s ; θ dws σ (, 4 X (k s ; θ ds where W t is a Brownian motion which is indeendent of X t. heorem 4.4 (Asymtotics of the Drift Parameter Estimate. Let ˆθ = argmax log l AIF (θ with l AIF (θ given by Eq. (7. Suose Assumtions hold, then ˆθ θ as. If, in addition,, / (ˆθ θ d N(, Φ, where Φ is given in Eq. (. 5. Monte Carlo results o examine the erformance of the roosed rocedure, we estimate the following model for short-term interest rates due to Chan et al. (99, CLS hereafter, dx t = κ(µ X t dt + σ X γ t db t, (3 with κ =.6, µ =.9, σ =.6, γ =.5. We choose γ =.5 so that the true model becomes a CIR model which enables an exact data simulation. he arameters are estimated from years of daily data (5 observations. he exeriment is relicated times to get the means stard errors for each estimate. wo estimation methods are emloyed to estimate

9 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( able 3 Simulation results under the CLS model, dx t = κ(µ X t dt + σ X γ t db t, based on samles of 5 daily observations. rue values for κ, µ, σ, γ are.6,.9,.6,.5. Mean SD st for the average stard deviation across relications, resectively. AML wo-stage method = = = = 5 = Level Log Level Log Level Log Level Log Level Log γ Mean SD σ Mean SD κ Mean SD µ Mean SD the model: the aroximate ML method of Aït-Sahalia ( the roosed two-stage method. For the two-stage method, we imlement it based on both levels log levels, resectively. o use the two-stage methods, the number of subsamles has to be chosen. here is a trade-off between a large a small. On the one h, since determines the number of observations used for the nonlinear regression, a larger will generate more variation in [X] k [X] (k across subsamles hence rovide more accurate estimation of the arameters in the diffusion function. On the other h, if is too big, M will M be too small for (X (k M +i X (k M +(i to rovide a good aroximation to [X] k [X] (k, as Figs. 3 suggested. In the Monte Carlo study, we choose various values for, namely,,, 5. hese values corresond to 5, 5, 5, 5 5 for M, resectively. he simulation results are reorted in able 3. Several interesting results emerge from this table. First, the simulation results clearly indicate a trade-off between a large a small. When is, the variance is big for the two arameters in the diffusion function, indicating that the nonlinear least regression does not lead to accurate estimation to the diffusion arameters. he erformance of the rocedure imroves substantially when. However, when is large, a further increase in fails to imrove the finite samle erformance. For examle, for the two-stage method based on log-levels, the variances of the two diffusion arameters are slightly larger when = than those when = 5. Second, the estimation of arameters in the drift function does not seem to be deendent on the diffusion arameters in any critical way. his is because the information matrix for the diffusion arameters is orthogonal to that for the drift arameters in the CLS model. hirdly, when a reasonable value for is used, the quadratic variations are well estimated. Not surrisingly, therefore, our estimates are close to the aroximate ML method of Aït-Sahalia (. Finally, we note that the two-stage method based on log levels has better finite samle erformances than that based on levels, esecially when is large ( hence M is small, consistent with the finding in Barndorff-Nielsen Shehard (5. 6. Microstructure contamination Direct alication of the two-stage method roosed above requires that X t be observed. his assumtion may be too strong for ultra high frequency data because of the resence of various market microstructure effects which contaminate X t with noise, roducing bias inconsistency in realized volatility estimates. As a consequence, our two-stage rocedure has to be modified We thank a referee for ointing this out to us. in order to roduce consistent estimates of the arameters in the diffusion function. Suose the noise is orthogonal to X t indeendent identically distributed over the grid (,,..., n, i.e. Y i = X i + e i, (4 e i e j, i j e i X i, i, (5 with zero mean E (e i = ( finite variance E ei = σ, i. In the resence of market microstructure noise such as e i, we observed Y i instead of X i. While the ure noise assumtion (5 lacks realism, as discussed in Phillis Yu (6, it is commonly adoted in the literature as it simlifies identification econometric analysis see, for examle, Bi Russell (5, Hansen Lunde (6, Barndorff-Nielsen et al. (5 Zhang et al. (5. It is easy to show that as, the M realized volatility quantity (Y (k M +i Y (k M +(i diverges, making estimation in the first stage inconsistent. o rovide a consistent estimate of the quadratic variation differential [X] k [X] (k, one can use the realized kernel estimator of Barndorff-Nielsen et al. (5, the two-scale estimator of Zhang et al. (5, or the multi-scale estimator of Zhang (6. While these rocedures converge to the quadratic variation at slower rates than realized volatility in the absence of microstructure noise, they nonetheless rovide consistent nonarametric estimators of the required quantity can be emloyed to estimate the arameters in the diffusion function using the nonlinear regression, recisely as used above. Aït-Sahalia et al. (6 generalized the two-scale rocedure to cover time deendent noise this generalization can also be adoted in the first ste of our two-stage method in the same sirit. he asymtotic roerties of the resulting estimator may be extracted along lines similar to those used here, with consequential changes in the rate of convergence. he results are obviously of interest will rovide an aroach to continuous system estimation in the resence of noise. he details will be rovided in subsequent work. 7. Concluding remarks his aer rooses a two-stage method to estimate diffusion rocesses in a general form. In the first stage the realized volatility calculated from a sequence of slit samles is regressed on the corresonding quadratic variation in order to estimate all the arameters in the diffusion function. hen, conditional on the resulting consistent estimate of the diffusion, the infill likelihood function aroximation of the diffusion rocess can be readily constructed. he resulting discrete aroximation roduces estimates of all the arameters in the drift function. Monte Carlo simulations show that the finite samle erformance of the roosed method is very satisfactory as good

10 48 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( as conventional maximum likelihood even when the discrete likelihood can be obtained. One advantage of the roosed method is that a larger scale otimization roblem is decomosed into two smaller scale otimization roblems. Although, like other extreme estimators, our method tends to over estimate the mean reversion arameter, κ, the numerical attractability of our method makes it an ideal initial estimate for the jackknife method of Phillis Yu (5 or the simulation-based methods of Phillis Yu (in ress-b reduce the finite samle bias in κ. here exist alternative two-ste rocedures to estimate diffusion rocesses. For examle, in Bi Phillis (7, nonarametric estimates of the drift diffusion functions are matched with the arametric counterarts to rovide consistent estimation of arameters in the drift in the diffusion, resectively. One can certainly use our first ste estimate (i.e. via realized volatility in lace of the diffusion estimate (i.e. via kernel functions in the rocedure of Bi Phillis (7. he aroach can be readily extended to the multi-dimensional case. Both imlementation asymtotic theory only need trivial modifications. Since the method searates estimation of the drift diffusion functions, it may be a desirable method to use when the drift but not the diffusion involves certain market microstructure features. he two-ste aroach can be also adated to deal with the following diffusion model mixed with jums: dx t = µ(x t ; θ dt + σ (X t ; θ db t + f (X t ; θ 3 dz θ 3 t, where Z θ 3 t is a Lévy rocess with arameter θ 3. For this model realized volatility cannot be used to consistently estimate θ as it does not converge to σ (X t ; θ dt. However, emirical triower variation does converge to σ (X t ; θ dt. Barndorff-Nielsen Shehard et al. (6 obtained a central limit theorem for the triower variation rocess when a diffusion rocess is mixed with finite activity jums. Since the in-fill likelihood is readily available for jum-diffusion rocesses once the diffusion term is known, we can adot the following two-ste rocedure to estimate the model: first, use triower variation to estimate θ ; then, maximize the aroximated in-fill likelihood with resect to θ θ 3. he roerties of the resulting estimator will be reorted in future work. Aendix Proof of heorem 4.. It is known that all diffusion-tye rocesses are semi-martingales (Prakasa Rao, 999b. As a result, when, [X ] [X] = τ f (X s ds, where the convergence follows from the theory of quadratic variation for semi-martingales the equality follows from Assumtion. By Assumtion 3, we have ˆτ = n i= [X ] f (X (i [X] f (X s ds = τ. his roves the first art of heorem 4.. Since X t is a semi-martingale, by Ito s lemma for semimartinagles (Prakasa Rao, 999b we have X = [X] + X s dx s. Following heorem of Barndorff-Nielsen Shehard ( we have / ([X ] [X] d τ f (X s dw s, (6 where W t is a Brownian motion which is indeendent of X t. Hence, / (ˆτ τ = / = / f (X s ds By Assumtion 3, n i= f (X (i f (X s ds. n i= n i= [X ] f (X (i [X] f (X s ds f (X s ds [X ] [X]. (7 f (X (i By Slutsky s theorem, Eqs. (6 (7 imly that / (ˆτ τ d τ f (X s dw s. (8 f (X s ds his comletes the roof of heorem 4.. Proof of heorem 4.. Obviously, the roosed drift estimator is in the class of extremum estimators. Hence, one can rove consistency by checking sufficient conditions for extremum estimation roblems. It is convenient here to check the conditions given in Newey McFadden (994,., namely, comactness, continuity, uniform convergence, identifiability. Comactness of Θ, continuity of log l AIF (θ ; ˆτ the identification condition are assured by Assumtion, Assumtion 9 Assumtion 8, resectively. he uniform convergence of log l AIF (θ to Y(θ, τ follows from Proosition, Lemma Lemma in Yoshida (99. Hence the first art of the theorem is roved. o show asymtotic normality, we follow Yoshida (99 by obtaining the weak convergence of the likelihood ratio rom field, Z,n (τ, u = l AIF (θ + / u; ˆτ/l AIF (θ ; ˆτ. Under the listed conditions, Yoshida (99 showed that log Z,n (τ, u = u / i (i n i= µ(x; θ θ τ f (X (i τ f (xdw t u Φu + ρ,n (u, (9 where ρ,n (u Φ is defined in Eq. (9. From heorem 4., we have, η >, that there exists a a ositive number c such that P( / (ˆτ τ > c < η/. Let ˆτ = τ + / M we have, ɛ > P( log Z,n (ˆτ, u log Z,n (τ, u > ɛ = P( / (ˆτ τ > c + P( su M c log Z,n (ˆτ, u log Z,n (τ, u > ɛ < η. (3

11 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( where ˆθ = arg min θ Q (θ, Q (θ = { (X(k M +i X (k M +(i σ ( } X (k M +(i ; θ Box III. ( Q θ = 3/ θ 3/ = / g { i (X (k M +i X (k M +(i σ ( } X (k M +(i ; θ g i r k { (X(k M +i X (k M +(i σ ( X (k M +(i ; θ } Box IV. r k Combining Eqs. (9 (3 roves Proosition 4 of Yoshida (99. Similarly, we can obtain Proosition 5 6 of Yoshida (99 based on ˆτ. he week convergence of the likelihood rom field follows these roositions. In articular, / (ˆθ θ d N(, Φ. his comletes the roof of heorem 4.. Proof of heorem 4.3. he argument is briefly sketched here. We consider the case where the estimate ˆθ is obtained from the extremum estimation roblem suggested by the first equation in Box II. A similar argument can be emloyed in the case where stardized log levels of realized volatility are used in the regression based on the CL result suggested by the third limit in Box I. Observe that, as, Q (θ Q (θ { σ ( } X (k s ; θ k ds σ (X (k s ; θ ds = uniformly in θ, since σ ( 4 X (k s ; θ M (X (k M +i X (k M +(i [X] k [X] (k = (k σ ( X s ; θ ds, ds, M σ ( k X (k M +(i ; θ σ (X s ; θ ds, (3 (k uniformly in θ Θ in view of the comactness of Θ the smoothness of σ (X s ; θ. Next, r k = 3 M (X (k M +i X (k M +(i 4 σ ( 4 X s ; θ (k ds, (3 as in Barndorff-Nielsen Shehard (. hus, since Q (θ is minimized for θ = θ in view of (, we have ˆθ θ by a stard extremum estimator argument. Next, by a aylor series argument under the stated smoothness ositive definiteness assumtions, we have (ˆθ / θ = [ ( ] Q ( θ Q θ θ θ 3/ θ [ ( Q θ θ θ ] [ Q 3/ θ ( θ ], (33 where θ is on the line segment connecting ˆθ to θ thus satisfies θ θ. Setting g i = ( g X (k M +(i ; θ σ ( X (k M +(i ; θ =, θ we get the equation in Box IV, in view of heorem of Barndorff-Nielsen Shehard (, / M g i { (X(k M +i X (k M +(i σ ( X (k M +(i ; θ } d (k σ ( X s ; θ σ ( X s ; θ dws (34 θ where W s is a stard Brownian motion indeendent of X t. It follows from (3, (34 Box IV that Q 3/ θ ( θ Next we have ( Q θ θ θ d σ (X s ;θ (k θ σ ( X s ; θ dws σ (. (35 4 X (k s ; θ ds g i g i r k

12 5 P.C.B. Phillis, J. Yu / Journal of Econometrics 5 ( = = g i g i (k r k (k σ (X s ;θ θ σ (X s ;θ θ σ ( 4 X (k s ; θ ds σ (X s ;θ σ (X s ;θ θ θ σ ( 4 X (k s ; θ Combining (33, (35 (36 we obtain σ (ˆθ / θ d (X s ;θ (k σ ( 4 X (k s ; θ ds σ (X s ;θ (k θ σ ( X s ; θ dws σ (, 4 X (k s ; θ ds as stated. ds ds ds. (36 θ σ (X s ;θ θ ds Proof of heorem 4.4. he roof follows similar lines to the roof of heorem 4. is therefore omitted. References Ahn, D., Gao, B., 999. A arametric nonlinear model of term structure dynamics. Review of Financial Studies, Aït-Sahalia, Y., 999. ransition densities for interest rate other nonlinear diffusions. Journal of Finance 54, Aït-Sahalia, Y.,. Maximum likelihood estimation of discretely samled diffusion: A closed-form aroximation aroach. Econometrica 7, 3 6. Aït-Sahalia, Y., Mykl, P.A., Zhang, L., 5. How often to samle a continuoustime rocess in the resence of market microstructure noise. Review of Financial Studies 8, Aït-Sahalia, Y., Mykl, P.A., Zhang, L., 6. Ultra high-frequency volatility estimation with deendent microstructure noise. Working Paer, Princeton University. Bi, F.M., Russell, J., 5. Microstructure noise, realized volatility, otimal samling. Working aer, Graduate School of Business, University of Chicago. Bi, F.M., Phillis, P.C.B., 3. Fully nonarametric estimation of scalar diffusion models. Econometrica 7, Bi, F.M., Phillis, P.C.B., 7. A simle aroach to the arametric estimation of otentially nonstationary diffusions. Journal of Econometrics 37, Barndorff-Nielsen, O., Graversen, S.E., Jacod, J., Shehard, N., 6. Limit theorems for biower variation in financial econometrics. Econometric heory, Barndorff-Nielsen, O., Hansen, P., Lunde, A., Shehard, N., 5. Designing realized kernels to measure the ex-ost variation of equity rices in the resence of noise. Working aer, Nuffield College. Barndorff-Nielsen, O., Shehard, N.,. Econometric analysis of realized volatility its use in estimating stochastic volatility models. Journal of the Royal Statistical Society. Series B 64, Barndorff-Nielsen, O., Shehard, N., 3. Realized ower variation stochastic volatility. Bernoulli 9, Barndorff-Nielsen, O., Shehard, N., 5. How accurate is the asymtotic aroximation to the distribution of realized volatility? In: Andrews, D.W.., Powell, J., Ruud, P., Stock, J. (Eds., Identification Inference for Econometric Models. Cambridge University Press. Barndorff-Nielsen, O., Shehard, N., Winkel, M., 6. Limit theorems for multiower variation in the resence of jums in financial econometrics. Stochastic Processes Alications 6, Billingsley, P., 96. Statistical Inference for Markov Processes. University of Chicago Press, Chicago, US. Bollerslev,., Zhou, H.,. Estimating stochastic volatility diffusion using conditional moments of integrated volatility. Journal of Econometrics 9, Brt, M.W., Santa-Clara, P.,. Simulated likelihood estimation of diffusions with an alication to exchange rate dynamics in incomlete markets. Journal of Financial Economics 3, 6. Chan,.C., arolyi, G.A., Longstaff, F.A., Sers, A.B., 99. An emirical comarison of alternative models of short term interest rates. Journal of Finance 47, 9 7. Cox, J., Ingersoll, J., Ross, S., 985. A theory of the term structure of interest rates. Econometrica 53, Dixit, A., 993. he Art of Smooth Pasting. Harwood Academic Publishers, Readings, U. Elerian, O., Chib, S., Shehard, N.,. Likelihood inference for discretely observed non-linear diffusions. Econometrica 69, Eraker, B.,. MCMC analysis of diffusion models with alication to finance. Journal of Business Economic Statistics 9, Florens-Zmirou, D., 989. Aroximate discrete-time schemes for statistics of diffusion rocesses. Statistics, Hansen, P., Lunde, A., 6. Realized volatility market microstructure noise. Journal of Business Economic Statistics 4, 7 6. Hutton, J.E., Nelson, P., 986. Quasi-likelihood estimation for semimartingales. Stochastic Processes their Alications, Jacod, J., 994. Limit of rom measures associated with the increments of a Brownian semiartingal. Working aer, Laboratoire de Probabilities, Universite Pierre et Marie Curie, Paris. Jacod, J., Protter, P., 998. Asymtotic error distributions for the Euler method for stochastic differential equations. Annals of Probabilities 6, Lánska, V., 979. Minimum contrast estimation in diffusion rocesses. Journal of Alied Probability 6, Litser, R.S., Shiryaev, A.N.,. Statistics of Rom Processes. Sringer-Verlag, New York. Lo, Andrew W., 988. Maximum likelihood estimation of generalized Itô rocesses with discretely samled data. Econometric heory 4, Merton, R.C., 98. On estimating the exected return on the market: An exloratory investigation. Journal of Financial Economics 8, Merton, R.C., 99. Continuous-ime Finance. Blackwell, Oxford. Newey, W.., McFadden, D., 994. Large samle estimation hyothesis testing. In: Engle, R.F., McFadden, D. (Eds., Hbook of Econometrics, vol. 4. North- Holl, Amsterdam. Nowman,.B., 997. Gaussian estimation of single-factor continuous time models of the term structure of interest rates. Journal of Finance 5, Pedersen, A., 995. A new aroach to maximum likelihood estimation for stochastic differential equations based on discrete observation. Scinavian Journal of Statistics, Phillis, P.C.B., 97. he structural estimation of a stochastic differential equation system. Econometrica 4, 4. Phillis, P.C.B., Yu, J., 5. Jackknifing bond otion rices. Review of Financial Studies 8, Phillis, P.C.B., Yu, J., 6. Maximum likelihood Gaussian estimation of continuous time models in finance. Hbook of Financial ime Series (in ress-a. Phillis, P.C.B., Yu, J., 6. Realized variance market microstructure noise Comment. Journal of Business Economic Statistics 4, 8. Phillis, P.C.B., Yu, J., 8. Simulation-based estimation of contingent-claims rices. Review of Financial Studies (in ress-b. Prakasa Rao, B.L.S., 999a. Statistical Inference for Diffusion ye Processes. Arnold, London. Prakasa Rao, B.L.S., 999b. Semimartingales their Statistical Inference. Chaman Hall, Boca Raton, Florida. Vasicek, O., 977. An equilibrium characterization of the term structure. Journal of Financial Economics 5, Yoshida, N., 99. Estimation for diffusion rocesses from discrete observation. Journal of Multivariate Analysis 4, 4. Yu, J., Phillis, P.C.B.,. A Gaussion aroach for estimating continuous time models of short term interest rates. he Econometrics Journal 4, 5. Zhang, L., 6. Efficient estimation of stochastic volatility using noise observations: A multi-scale aroach. Bernoulli, Zhang, L., Mykl, P.A., Aït-Sahalia, Y., 5. A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association,