A local stable bootstrap for power variations of pure-jump semimartingales and activity index estimation

We provide a new resampling procedure–the local stable bootstrap–that is able to mimic the dependence properties of realized power variations for pure-jump semimartingales observed at different frequencies. This allows us to propose a bootstrap estimator and inference procedure for the activity index of the underlying process, β, as well as bootstrap tests for whether it obeys a jump-diffusion or a pure-jump process, that is, of the null hypothesis H0:β=2 against the alternative H1:β<2. We establish first-order asymptotic validity of the resulting bootstrap power variations, activity index estimator, and diffusion tests for H0. Moreover, the finite sample size and power properties of the proposed diffusion tests are compared to those of benchmark tests using Monte Carlo simulations. Unlike existing procedures, our bootstrap tests are correctly sized in general settings. Finally, we illustrate the use and properties of the new bootstrap diffusion tests using high-frequency data on three FX series, the S&P 500, and the VIX.

[1]  Ulrich Hounyo.  Bootstrapping integrated covariance matrix estimators in noisy jump-diffusion models with non-synchronous trading , 2017 .

[2]  V. Todorov.  Power variation from second order differences for pure jump semimartingales , 2013 .

[3]  J. Rosínski.  Tempering stable processes , 2007 .

[4]  Tim Bollerslev,et al.  Exploiting the errors: A simple approach for improved volatility forecasting , 2016 .

[5]  Zhi Liu,et al.  MODELING HIGH-FREQUENCY FINANCIAL DATA BY PURE JUMP PROCESSES , 2012, 1206.0827.

[6]  Francis X. Diebold,et al.  Parametric and Nonparametric Measurements of Volatility: Volume 1: Tools and Techniques , 2010 .

[7]  J. Rosínski.  Series Representations of Lévy Processes from the Perspective of Point Processes , 2001 .

[8]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[9]  X. Shao,et al.  The Dependent Wild Bootstrap , 2010 .

[10]  Daniela Osterrieder,et al.  Unbalanced Regressions and the Predictive Equation , 2015 .

[11]  R. T. Varneskov.  ESTIMATING THE QUADRATIC VARIATION SPECTRUM OF NOISY ASSET PRICES USING GENERALIZED FLAT-TOP REALIZED KERNELS , 2017, Econometric Theory.

[12]  P. Carr,et al.  Stochastic Skew in Currency Options , 2004 .

[13]  Markus Pauly.  Weighted resampling of martingale difference arrays with applications , 2011 .

[14]  J. Jacod,et al.  Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data , 2009 .

[15]  Rama Cont,et al.  Nonparametric tests for pathwise properties of semimartingales , 2011, 1104.4429.

[16]  N. Shephard,et al.  How accurate is the asymptotic approximation to the distribution of realised variance , 2001 .

[17]  N. Haldrup,et al.  Space-time modeling of electricity spot prices , 2017 .

[18]  Torben G. Andersen,et al.  Parametric Inference and Dynamic State Recovery from Option Panels , 2012 .

[19]  George Tauchen,et al.  Activity Signature Functions for High-Frequency Data Analysis , 2008 .

[20]  A. Gallant,et al.  Alternative models for stock price dynamics , 2003 .

[21]  Henri Nyberg,et al.  International Sign Predictability of Stock Returns: The Role of the United States , 2016 .

[22]  Susan R. Wilson,et al.  Two guidelines for bootstrap hypothesis testing , 1991 .

[23]  George Tauchen,et al.  The Fine Structure of Equity-Index Option Dynamics , 2014 .

[24]  George Tauchen,et al.  Volatility Activity: Specification and Estimation , 2011 .

[25]  N. Shephard,et al.  LIMIT THEOREMS FOR BIPOWER VARIATION IN FINANCIAL ECONOMETRICS , 2005, Econometric Theory.

[26]  N. Shephard,et al.  Power and bipower variation with stochastic volatility and jumps , 2003 .

[27]  Nour Meddahi,et al.  BOOTSTRAPPING REALIZED VOLATILITY , 2009 .

[28]  Nour Meddahi,et al.  BOOTSTRAPPING PRE-AVERAGED REALIZED VOLATILITY UNDER MARKET MICROSTRUCTURE NOISE , 2016, Econometric Theory.

[29]  W. Wu,et al.  Nonparametric inference of discretely sampled stable Lévy processes , 2009 .

[30]  James G. MacKinnon,et al.  THE SIZE DISTORTION OF BOOTSTRAP TESTS , 1999, Econometric Theory.

[31]  Bing-Yi Jing,et al.  Estimating the Jump Activity Index Under Noisy Observations Using High-Frequency Data , 2011 .

[32]  Joel L. Horowitz,et al.  Empirically relevant critical values for hypothesis tests: A bootstrap approach , 2000 .

[33]  Mark Podolskij,et al.  Understanding Limit Theorems for Semimartingales: A Short Survey , 2009 .

[34]  Liuren Wu,et al.  Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns , 2004 .

[35]  V. Todorov.  Jump activity estimation for pure-jump semimartingales via self-normalized statistics , 2015, 1508.04216.

[36]  M. Yor,et al.  Stochastic Volatility for Lévy Processes , 2003 .

[37]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[38]  P. Saikkonen,et al.  Identification and estimation of non-Gaussian structural vector autoregressions , 2015 .

[39]  J. MacKinnon,et al.  The power of bootstrap and asymptotic tests , 2006 .

[40]  N. Shephard.  Realized power variation and stochastic volatility models , 2003 .

[41]  A. Harvey,et al.  5 Stochastic volatility , 1996 .

[42]  Jean Jacod,et al.  Is Brownian motion necessary to model high-frequency data? , 2010, 1011.2635.

[43]  George Tauchen,et al.  Volatility Jumps , 2008 .

[44]  Anders Rahbek,et al.  Modeling corporate defaults: Poisson autoregressions with exogenous covariates (PARX) , 2016 .

[45]  The fine structure of equity-index option dynamics , 2015 .

[46]  N. Shephard,et al.  Realised power variation and stochastic volatility models , 2003 .

[47]  Jean Jacod,et al.  Discretization of Processes , 2011 .

[48]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[49]  Hossein Asgharian,et al.  Effects of Macroeconomic Uncertainty upon the Stock and Bond Markets , 2015 .

[50]  Viktor Todorov,et al.  Estimation of continuous-time stochastic volatility models with jumps using high-frequency data , 2009 .

[51]  N. Kiefer,et al.  Counting Processes for Retail Default Modeling , 2015 .

[52]  George Tauchen,et al.  Realized laplace transforms for pure-jump semimartingales , 2012, 1207.5615.

[53]  Liuren Wu.  Modeling Financial Security Returns Using Levy Processes , 2006 .

[54]  D. Duffie,et al.  Transform Analysis and Asset Pricing for Affine Jump-Diffusions , 1999 .

[55]  N. Shephard,et al.  Variation, Jumps, Market Frictions and High Frequency Data in Financial Econometrics , 2005 .

[56]  P. Hansen,et al.  A Markov Chain Estimator of Multivariate Volatility from High Frequency Data , 2015 .

[57]  Dick J. C. van Dijk,et al.  Dynamic Factor Models for the Volatility Surface , 2015 .

[58]  Bootstrapping realized volatility and realized beta under a local Gaussianity assumption , 2013 .

[59]  Jean Jacod,et al.  Estimating the degree of activity of jumps in high frequency data , 2009, 0908.3095.

[60]  Jean Jacod,et al.  Asymptotic properties of realized power variations and related functionals of semimartingales , 2006, math/0604450.

[61]  Jeannette H. C. Woerner.  Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models , 2003 .

[62]  T. Bollerslev,et al.  2009-26 Tails , Fears and Risk Premia , 2009 .

[63]  George Tauchen,et al.  Limit Theorems for Power Variations of Pure-Jump Processes with Application to Activity Estimation , 2010, 1104.1064.

[64]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[65]  J. MacKinnon.  Bootstrap Hypothesis Testing , 2007 .

[66]  R. Davidson.  Bootstrapping Econometric Models , 2007 .

[67]  A. Luati,et al.  Generalised Partial Autocorrelations and the Mutual Information between Past and Future , 2015 .

[68]  D. Andrews.  Inconsistency of the Bootstrap when a Parameter is on the Boundary of the Parameter Space , 2000 .

[69]  Yacine Aït-Sahalia,et al.  Disentangling diffusion from jumps , 2004 .

[70]  P. Hansen.  A Martingale Decomposition of Discrete Markov Chains , 2015 .

[71]  P. Carr,et al.  The Finite Moment Log Stable Process and Option Pricing , 2003 .

[72]  R. Rao,et al.  Normal Approximation and Asymptotic Expansions , 1976 .

[73]  Tommaso Proietti,et al.  EuroMInd-D: A Density Estimate of Monthly Gross Domestic Product for the Euro Area , 2015 .

[74]  Jeannette H. C. Woerner.  Inference in Lévy-type stochastic volatility models , 2007, Advances in Applied Probability.

[75]  S. Johansen,et al.  Data Revisions and the Statistical Relation of Global Mean Sea-Level and Temperature , 2015 .

[76]  George Tauchen,et al.  Cross-Stock Comparisons of the Relative Contribution of Jumps to Total Price Variance , 2012 .

[77]  Zhi Liu,et al.  On the jump activity index for semimartingales , 2012 .

[78]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[79]  C. Klüppelberg,et al.  A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour , 2004, Journal of Applied Probability.