Infill Asymptotics Research Articles

Parameter estimation and singularity of laws on the path space for SDEs driven by Rosenblatt processes

In this paper, we study parameter identification for solutions to (possibly non-linear) SDEs driven by additive Rosenblatt process and singularity of the induced laws on the path space. We propose a joint estimator for the drift parameter, diffusion intensity, and Hurst index that can be computed from discrete-time observations with a bounded time horizon and we prove its strong consistency under in-fill asymptotics with a fixed time horizon. As a consequence of this strong consistency, singularity of measures generated by the solutions with different drifts is shown. This results in the invalidity of a Girsanov-type theorem for Rosenblatt processes.

A Composite Likelihood-based Approach for Change-point Detection in Spatio-temporal Processes

This paper develops a unified and computationally efficient method for change-point estimation along the time dimension in a non-stationary spatio-temporal process. By modeling a non-stationary spatio-temporal process as a piecewise stationary spatio-temporal process, we consider simultaneous estimation of the number and locations of change-points, and model parameters in each segment. A composite likelihood-based criterion is developed for change-point and parameter estimation. Under the framework of increasing domain asymptotics, theoretical results including consistency and distribution of the estimators are derived under mild conditions. In contrast to classical results in fixed dimensional time series that the localization error of change-point estimator is , exact recovery of true change-points is possible in the spatio-temporal setting. More surprisingly, the consistency of change-point estimation can be achieved without any penalty term in the criterion function. In addition, we further establish consistency of the change-point estimator under the infill asymptotics framework where the time domain is increasing while the spatial sampling domain is fixed. A computationally efficient pruned dynamic programming algorithm is developed for the challenging criterion optimization problem. Extensive simulation studies and an application to the U.S. precipitation data are provided to demonstrate the effectiveness and practicality of the proposed method.

An Econometric Analysis of Volatility Discovery

We investigate information processing in the stochastic process driving stock’s volatility (volatility discovery). We apply fractionally cointegration techniques to decompose the estimates of the market-specific integrated variances into an estimate of the common integrated variance of the efficient price and a transitory component. The market weights on the common integrated variance of the efficient price are the volatility discovery measures. We relate the volatility discovery measure to the price discovery framework and formally show their roles on the identification of the integrated variance of the efficient price. We establish the limiting distribution of the volatility discovery measures by resorting to both long span and in-fill asymptotics. The empirical application is in line with our theoretical results, as it reveals that trading venues incorporate new information into the stochastic volatility process in an individual manner and that the volatility discovery analysis identifies a distinct information process than that based on the price discovery analysis.

Robust estimation of integrated and spot volatility

We introduce a new method to estimate the integrated volatility (IV) and the spot volatility (SV) based on noisy high-frequency data. Our method employs the ReMeDI approach introduced by Li and Linton (2022) to estimate the moments of microstructure noise and thereby eliminate their influence, and the pre-averaging method to target the volatility parameter. The method is robust: it can be applied when the efficient price exhibits stochastic volatility and jumps, the observation times are random, and the noise process is nonstationary, autocorrelated, asymptotically vanishing and dependent on the efficient price. We derive the limit distributions for the proposed estimators under the infill asymptotics in a general setting. Our extensive simulation studies demonstrate the robustness, accuracy and computational efficiency of our estimators compared to several alternative estimators recently proposed in the literature. Empirically, we show that neglecting the complexities of noise and the random observation times generates substantial biases in volatility estimation and may yield a different intraday volatility pattern.

A Functional Data Approach for Continuous-Time Analysis Subject to Modeling Discrepancy under Infill Asymptotics

Parametric continuous-time analysis often entails derivations of continuous-time models from predefined discrete formulations. However, undetermined convergence rates of frequency-dependent parameters can result in ill-defined continuous-time limits, leading to modeling discrepancy, which impairs the reliability of fitting and forecasting. To circumvent this issue, we propose a simple solution based on functional data analysis (FDA) and truncated Taylor series expansions. It is demonstrated through a simulation study that our proposed method is superior—compared with misspecified parametric methods—in fitting and forecasting continuous-time stochastic processes, while the parametric method slightly dominates under correct specification, with comparable forecast errors to the FDA-based method. Due to its generally consistent and more robust performance against possible misspecification, the proposed FDA-based method is recommended in the presence of modeling discrepancy. Further, we apply the proposed method to predict the future return of the S&P 500, utilizing observations extracted from a latent continuous-time process, and show the practical efficacy of our approach in accurately discerning the underlying dynamics.

A smoothed p-value test when there is a nuisance parameter under the alternative

We present a new test when there is a nuisance parameter λ under the alternative hypothesis. The test exploits the p-value occupation time [PVOT], the measure of the subset of λ on which a p-value test based on a test statistic Tn(λ) rejects the null hypothesis. Key contributions are: (i) An asymptotic critical value upper bound for our test is the significance level α, making inference easy. (ii) We only require Tn(λ) to have a known or bootstrappable limit distribution, hence we do not require n-Gaussian asymptotics, allowing for weak or non-identification, boundary values, heavy tails, infill asymptotics, and so on. (iii) A test based on the transformed p-value supλ∈Λpn(λ) may be conservative and in some cases have trivial power, while the PVOT naturally controls for this by smoothing over the nuisance parameter space. Finally, (iv) the PVOT uniquely allows for bootstrap inference in the presence of nuisance parameters when some estimated parameters may not be identified.

Nonparametric calibration for stochastic reaction–diffusion equations based on discrete observations

Nonparametric estimation for semilinear SPDEs, namely stochastic reaction–diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, we derive a nonparametric estimator for the reaction function of the underlying equation. The estimate is chosen from a finite-dimensional function space based on a least squares criterion. Oracle inequalities provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection. Secondly, we show that the asymptotic properties of realized quadratic variation based estimators for the diffusivity and volatility carry over from linear SPDEs. In particular, we obtain a rate-optimal joint estimator of the two parameters. The result relies on our precise analysis of the Hölder regularity of the solution process and its nonlinear component, which may be of its own interest. Both steps of the calibration can be carried out simultaneously without prior knowledge of the parameters.

Periodic repolarization dynamics

Nonparametric estimation for semilinear SPDEs, namely stochastic reaction–diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, we derive a nonparametric estimator for the reaction function of the underlying equation. The estimate is chosen from a finite-dimensional function space based on a least squares criterion. Oracle inequalities provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection. Secondly, we show that the asymptotic properties of realized quadratic variation based estimators for the diffusivity and volatility carry over from linear SPDEs. In particular, we obtain a rate-optimal joint estimator of the two parameters. The result relies on our precise analysis of the Hölder regularity of the solution process and its nonlinear component, which may be of its own interest. Both steps of the calibration can be carried out simultaneously without prior knowledge of the parameters.

On Bivariate Time-Varying Price Staleness

Price staleness refers to the extent of zero returns in price dynamics. Bandi, Pirino, and Reno introduce two types of staleness: systematic and idiosyncratic staleness. In this study, we allow price staleness to be time-varying and study the statistical inference for idiosyncratic and common price staleness between two assets. We propose consistent estimators for both time-varying idiosyncratic and systematic price staleness and derive their asymptotic theory. Moreover, we develop a feasible nonparametric test for the simultaneous constancy of idiosyncratic and common price staleness. Our inference is based on infill asymptotics. Finally, we conduct simulation studies under various scenarios to assess the finite sample performance of the proposed approaches and provide an empirical application of the proposed theory.

Beyond Matérn: On A Class of Interpretable Confluent Hypergeometric Covariance Functions

The Matérn covariance function is a popular choice for prediction in spatial statistics and uncertainty quantification literature. A key benefit of the Matérn class is that it is possible to get precise control over the degree of mean-square differentiability of the random process. However, the Matérn class possesses exponentially decaying tails, and thus, may not be suitable for modeling polynomially decaying dependence. This problem can be remedied using polynomial covariances; however, one loses control over the degree of mean-square differentiability of corresponding processes, in that random processes with existing polynomial covariances are either infinitely mean-square differentiable or nowhere mean-square differentiable at all. We construct a new family of covariance functions called the Confluent Hypergeometric (CH) class using a scale mixture representation of the Matérn class where one obtains the benefits of both Matérn and polynomial covariances. The resultant covariance contains two parameters: one controls the degree of mean-square differentiability near the origin and the other controls the tail heaviness, independently of each other. Using a spectral representation, we derive theoretical properties of this new covariance including equivalent measures and asymptotic behavior of the maximum likelihood estimators under infill asymptotics. The improved theoretical properties of the CH class are verified via extensive simulations. Application using NASA’s Orbiting Carbon Observatory-2 satellite data confirms the advantage of the CH class over the Matérn class, especially in extrapolative settings. Supplementary materials for this article are available online.

A ReMeDI for Microstructure Noise

We introduce the Realized moMents of Disjoint Increments (ReMeDI) paradigm to measure microstructure noise (the deviation of the observed asset prices from the fundamental values caused by market imperfections). We propose consistent estimators of arbitrary moments of the microstructure noise process based on high‐frequency data, where the noise process could be serially dependent, endogenous, and nonstationary. We characterize the limit distributions of the proposed estimators and construct confidence intervals under infill asymptotics. Our simulation and empirical studies show that the ReMeDI approach is very effective to measure the scale and the serial dependence of microstructure noise. Moreover, the estimators are quite robust to model specifications, sample sizes, and data frequencies.

Space-Time Estimation and Prediction under Infill Asymptotics with Compactly Supported Covariance Functions

Space-Time Estimation and Prediction under Infill Asymptotics with Compactly Supported Covariance Functions

Testing for changes in the tail behavior of Brown–Resnick Pareto processes

We consider the class of r-Pareto processes defined on [0,1] whose max-stable counterparts are Brown–Resnick processes. The aim of this paper is to propose a test whether the extreme value index function of a r-Pareto process of this class remains constant over [0,1]. We assume that we observe several independent r-Pareto processes over equispaced grids of [0,1] but with possibly heterogeneous grid resolutions. We build a test based on the normalized approximate total variations of the paths of these processes. We provide its properties under infill asymptotics and assess its finite sample performance through simulation experiments. We discuss how to proceed for random processes that belong to the domain of attraction of Brown–Resnick r-Pareto processes and illustrate the approach with an application to wind speed data from Germany.

On Identifiability and Consistency of The Nugget in Gaussian Spatial Process Models

AbstractSpatial process models popular in geostatistics often represent the observed data as the sum of a smooth underlying process and white noise. The variation in the white noise is attributed to measurement error, or microscale variability, and is called the ‘nugget’. We formally establish results on the identifiability and consistency of the nugget in spatial models based upon the Gaussian process within the framework of in-fill asymptotics, that is the sample size increases within a sampling domain that is bounded. Our work extends results in fixed domain asymptotics for spatial models without the nugget. More specifically, we establish the identifiability of parameters in the Matérn covariogram and the consistency of their maximum likelihood estimators in the presence of discontinuities due to the nugget. We also present simulation studies to demonstrate the role of the identifiable quantities in spatial interpolation.

The Grid Bootstrap for Continuous Time Models

This article proposes the new grid bootstrap to construct confidence intervals (CI) for the persistence parameter in a class of continuous-time models. It is different from the standard grid bootstrap of Hansen in dealing with the initial condition. The asymptotic validity of the CI is discussed under the in-fill scheme. The modified grid bootstrap leads to uniform inferences on the persistence parameter. Its improvement over in-fill asymptotics is achieved by expanding the coefficient-based statistic around its in-fill asymptotic distribution that is non-pivotal and depends on the initial condition. Monte Carlo studies show that the modified grid bootstrap performs better than Hansen’s grid bootstrap. Empirical applications to the U.S. interest rates and volatilities suggest significant differences between the two bootstrap procedures when the initial condition is large.

UNIT ROOT TEST WITH HIGH-FREQUENCY DATA

Deviations of asset prices from the random walk dynamic imply the predictability of asset returns and thus have important implications for portfolio construction and risk management. This paper proposes a real-time monitoring device for such deviations using intraday high-frequency data. The proposed procedures are based on unit root tests with in-fill asymptotics but extended to take the empirical features of high-frequency financial data (particularly jumps) into consideration. We derive the limiting distributions of the tests under both the null hypothesis of a random walk with jumps and the alternative of mean reversion/explosiveness with jumps. The limiting results show that ignoring the presence of jumps could potentially lead to severe size distortions of both the standard left-sided (against mean reversion) and right-sided (against explosiveness) unit root tests. The simulation results reveal satisfactory performance of the proposed tests even with data from a relatively short time span. As an illustration, we apply the procedure to the Nasdaq composite index at the 10-minute frequency over two periods: around the peak of the dot-com bubble and during the 2015–2106 stock market sell-off. We find strong evidence of explosiveness in asset prices in late 1999 and mean reversion in late 2015. We also show that accounting for jumps when testing the random walk hypothesis on intraday data is empirically relevant and that ignoring jumps can lead to different conclusions.

A ReMeDI for Microstructure Noise

We introduce the Realized moMents of Disjoint Increments (ReMeDI) paradigm to measure microstructure noise (the deviation of the observed asset prices from the fundamental values caused by market imperfections). We propose consistent estimators of arbitrary moments of the microstructure noise process based on highfrequency data, where the noise process could be serially dependent, endogenous, and nonstationary. We characterize the limit distributions of the proposed estimators and construct confidence intervals under infill asymptotics. Our simulation and empirical studies show that the ReMeDI approach is very effective to measure the scale and the serial dependence of microstructure noise. Moreover, the estimators are quite robust to model specifications, sample sizes and data frequencies.

EFFICIENT ESTIMATION OF INTEGRATED VOLATILITY FUNCTIONALS UNDER GENERAL VOLATILITY DYNAMICS

We provide an asymptotic theory for the estimation of a general class of smooth nonlinear integrated volatility functionals. Such functionals are broadly useful for measuring financial risk and estimating economic models using high-frequency transaction data. The theory is valid under general volatility dynamics, which accommodates both Itô semimartingales (e.g., jump-diffusions) and long-memory processes (e.g., fractional Brownian motions). We establish the semiparametric efficiency bound under a nonstandard nonergodic setting with infill asymptotics, and show that the proposed estimator attains this efficiency bound. These results on efficient estimation are further extended to a setting with irregularly sampled data.

Cusum tests for changes in the Hurst exponent and volatility of fractional Brownian motion

In this letter, we construct cusum change-point tests for the Hurst exponent and the volatility of a discretely observed fractional Brownian motion. As a statistical application of the functional Breuer–Major theorems by Bégyn (2007) and Nourdin and Nualart (2019), we show under infill asymptotics consistency of the tests and weak convergence to the Kolmogorov–Smirnov law under the no-change hypothesis. The test is feasible and pivotal in the sense that it is based on a statistic and critical values which do not require knowledge of any parameter values. Consistent estimation of the break date under the alternative hypothesis is established. We demonstrate the finite-sample properties in simulations and a data example.

Robust Estimation of Integrated Volatility

We introduce a new method to estimate the integrated volatility (IV) based on noisy high-frequency data. Our method employs the ReMeDI approach introduced by Li and Linton (2021a) to estimate the moments of the microstructure noise and thereby eliminate their influence, and the pre-averaging method to target the volatility parameter. The method is robust: it can be applied when the efficient price exhibits stochastic volatility and jumps, the observation times are random and endogenous, and the noise process is nonstationary, autocorrelated and dependent on the efficient price. We derive the limit distribution for the proposed estimators under infill asymptotics in a general setting. Our simulation and empirical studies demonstrate the robustness, accuracy and computational efficiency of our estimators compared to several alternatives recently proposed in the literature.