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  • Robust Estimation and Inference for Jumps in Noisy

    High Frequency Data: A Local-to-Continuity Theory

    for the Pre-averaging Method

    Jia Li

    Department of Economics

    Duke University

    This Version: September 15, 2012

    Abstract

    We develop an asymptotic theory for the pre-averaging estimator when asset price

    jumps are weakly identified, here modeled as local to zero. The theory unifies the con-

    ventional asymptotic theory for continuous and discontinuous semimartingales as two

    polar cases with a continuum of local asymptotics, and explains the breakdown of the

    conventional procedures under weak identification. We propose simple bias-corrected

    estimators for jump power variations, and construct robust confidence sets with valid

    asymptotic size in a uniform sense. The method is also robust to microstructure noise.

    Keywords: Confidence set; high frequency data; jump power variation; market

    microstructure noise; pre-averaging; semimartingale; uniformity.

    JEL Codes: C22.

    This paper is a revised version of part of my Ph.D. dissertation at the department of economics,Princeton University. I am very grateful to my advisors Yacine At-Sahalia, Ulrich Mller and MarkWatson, as well as Jean Jacod for their guidance. I am also grateful for comments from Tim Bollerslev,Valentina Corradi, Nour Maddahi, Andrew Patton, George Tauchen and Viktor Todorov on various versionsof this paper. Comments from three referees and the co-editor have vastly improved the paper. The workis partially supported by NSF Grant SES-1227448. All errors are mine.Durham, NC 27708. E-mail: jl410@duke.edu.

    1

  • 1 Introduction.

    This note proposes a robust method for the estimation and inference of power variations

    of asset price jumps. We model the asset price as a continuous-time semimartingale and

    the pth power variation of its jumps (henceforth the jump power variation) over some time

    interval [0, T ] is defined as

    0sT |Js|p, where Js is the jump at time s. The jumppower variation is a pathwise analogue of the absolute moment of jumps. It naturally serves

    as a measure for the jump risk, and can be used for estimating parameters governing the

    jump process (At-Sahalia (2004), Todorov and Bollerslev (2010)), as well as constructing

    nonparametric specification tests related to jumps (At-Sahalia and Jacod (2011)). Dis-

    entangling the jump power variation, or functionals of the jump component in general, is

    nontrivial because jumps are convoluted with the drift and the diffusive parts of the price

    process. This task is further confounded by the presence of microstructure noise (Andersen

    et al. (2006)). To the best of our knowledge, the pre-averaging method proposed by Jacod

    et al. (2010), hereafter denoted JPV, is the only method available in the current literature

    for the estimation and inference of jump characteristics that is robust to noise. However,

    the asymptotic theory of JPV does not provide a satisfactory finite-sample approximation

    in the presence of jumps, as documented by At-Sahalia et al. (2012), henceforth AJL.

    In this note, we examine the asymptotic properties of the pre-averaging estimator when

    jumps are weakly identified, or small, here modeled as local to zero. As hinted in the title,

    we label this local asymptotic setting as local-to-continuity. While the standard theory of

    JPV describes very distinct asymptotic behaviors of the pre-averaging estimator depending

    on whether jumps are present or not, our results provide a continuum of local asymptotics

    which bridges their results as two polar cases. Our theory explains the breakdown of the

    standard method when jumps are weakly identified. Constructively, we propose a simple

    bias correction for the pre-averaging estimator; we also propose robust confidence sets (CS)

    for jump power variations, which have valid asymptotic coverage uniformly against possibly

    small jumps. The results are nonparametric in nature, are valid for almost unrestricted

    semimartingales, and are robust to microstructure noise. Monte Carlo evidence strongly

    supports our theoretical findings.

    Our contribution is twofold. Firstly, our local asymptotic theory for the pre-averaging

    estimator is novel. Secondly, to the best of our knowledge, the robust CS and the associated

    uniformity result is the first example of this kind for discretely sampled semimartingales.

    More generally, we believe that the local-to-continuity approach can be extended to many

    2

  • other applications for studying asset price jumps based on high frequency data.

    We now discuss the related literature. We analyze the estimators of JPV and AJL

    under the local-to-continuity setting and propose a robustification for these estimators. The

    inference problem considered here, i.e. constructing CSs for jump power variations, is more

    general than the testing problem of AJL. Prior works on noise-robust estimation for high

    frequency data, see e.g. Barndorff-Nielsen et al. (2008) and references therein, typically

    assume away jumps or treat jumps as a nuisance, and hence have a quite different focus

    than here. Jumps are now known to be prevalent in financial data and have been actively

    studied in financial econometrics, see At-Sahalia and Jacod (2011) for a recent survey.

    The insight that local asymptotics often provides a deeper understanding of the finite-

    sample behavior of statistical procedures is now well recognized in econometrics. Main

    examples include the local-to-unity literature, see e.g. Phillips (1987), as well as the weak

    identification literature, see Staiger and Stock (1997) and Stock andWright (2000). Further-

    more, local asymptotics have been shown to play a crucial role for constructing uniformly

    valid inference procedures, see Mikusheva (2007), Andrews and Cheng (Forthcoming) and

    references therein. Our approach here is clearly inspired by the above literature, but it is

    distinct from prior works because of the nonstandard nature of the fill-in asymptotics for

    semimartingale models.

    The note is organized as follows. Section 2 presents the model and the pre-averaging

    estimator. Section 3 presents the theory. Section 4 concludes. Technical details are collected

    in the Appendix, where we present the regularity conditions and construct an estimator for

    the asymptotic variance. The web supplement of this note contains all proofs and simulation

    results.

    2 The setting.

    2.1 The underlying process.

    The underlying process is a one-dimensional It semimartingale on a filtered space

    (,F , (Ft)t0,P) with the form

    Xt = X0 +

    t0

    bs ds+

    t0

    sdWs + Jt, where (1)

    Jt =

    ( t0

    E

    (s, z) 1{|(s,z)|1} (ds, dz) +

    t0

    E

    (s, z) 1{|(s,z)|>1} (ds, dz)

    ),

    3

  • X0 is an F0-measurable random variable, W is a Brownian motion, is the stochasticvolatility process taking values in (0,) almost surely, [0, 1] is a constant, is apredictable function, is a Poisson random measure on R+ E and its compensator is(dt, dz) = dt (dz) where (E, E) is an auxiliary space and is a -finite measure, and = . In typical financial econometrics applications, the underlying process representsthe logarithm of an asset price sampled at regularly spaced discrete times in, i 0, overa fixed time interval [0, T ], with the time lag n 0 asymptotically.On the right-hand side of (1), the component X0 +

    t0bs ds+

    t0sdWs is a continuous It

    semimartingale, and Jt is a purely discontinuous process which can be completely charac-

    terized by its jumps. In the sequel, we refer to these two components as the continuous part

    and the jump part, respectively. We use the parameter to control the scale of the jumps.

    This parameter plays an important role in our asymptotic theory. To separate from other

    modeling components in (1), we introduce an auxiliary process X by setting Xt = X1t .1 In

    particular, we have

    J = X, (2)

    where for any cdlg (i.e., right-continuous with left limits) process Y, the process Y is

    defined as Yt = Yt Yt, t 0. The fixed jump process X can be thought of as thedirectionin which J deviates from zero and quantifies this deviation.

    We stress that we are interested in the jumps of the underlying process, i.e. J , rather

    than and X separately. Indeed, while J is identifiable upon observing the underlying

    process in continuous time, and X can not be identified separately because of (2). We

    hence keep the dependence of J on implicit in our notation.2 Nevertheless, introducing

    the scaling parameter is useful for considering local asymptotics under a drifting sequence

    of data generating processes. In Section 3, we derive asymptotic properties of the pre-

    averaging estimator for a drifting sequence n while keeping other coeffi cients (i.e. b, , ,

    ) fixed. When n 0, the jump process J = nX converges to zero asymptotically,capturing the idea that jumps are small in an asymptotic sense. Moreover, the rate at

    which n vanishes to zero describes how smallthe jumps are, and not surprisingly, it plays

    an important role in the limiting theorems. Generally speaking, one could think of letting

    1We note that X is the standard model for high frequency data and has been widely studied in theliterature, see e.g. Jacod (2008), At-Sahalia and Jacod (2011), Bollerslev and Todorov (2011), Todorovand Tauchen (2012) and in particular JPV.

    2One may find that it is notationally more consistent to wr