The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard task, in particular when the underlying process contains jumps. For this reason, several authors have recently discussed methods to automatically estimate the conditional variance, i.e. they build a consistent estimator from the original statistics, but computed at various different time scales. Their methods work in several situations, but are essentially restricted to the case of continuous paths always. The aim of this work is to present a new method to consistently estimate the conditional variance which works regardless of whether the underlying process is continuous or has jumps. We will discuss the case of power variations in detail and give insight to the heuristics behind the approach.
This paper presents the asymptotic theory for nondegenerate $U$-statistics of high frequency observations of continuous Itô semimartingales. We prove uniform convergence in probability and show a functional stable central limit theorem for the standardized version of the $U$-statistic. The limiting process in the central limit theorem turns out to be conditionally Gaussian with mean zero. Finally, we indicate potential statistical applications of our probabilistic results.
The approximation of integral type functionals is studied for discrete observations of a continuous Itô semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions with fractional smoothness. An explicit $L^2$-lower bound shows that already lower order quadrature rules, such as the trapezoidal rule and the classical Riemann estimator, are rate optimal, but only the trapezoidal rule is efficient, achieving the minimal asymptotic variance.
In this paper we study the effects of noise on the bipower variation (BPV), realized volatility (RV) and testing for co-jumps in high-frequency data under the small noise framework. We first establish asymptotic properties of the BPV in this framework. In the presence of the small noise, the RV is asymptotically biased and the additional asymptotic conditional variance term appears in its limit distribution. We also give feasible estimation methods of the asymptotic conditional variances of the RV. Second, we derive the asymptotic distribution of the test statistic proposed in Jacod and Todoro…
In the common nonparametric regression model the problem of testing for a specific parametric form of the variance function is considered. Recently Dette and Hetzler (2008) proposed a test statistic, which is based on an empirical process of pseudo residuals. The process converges weakly to a Gaussian process with a complicated covariance kernel depending on the data generating process. In the present paper we consider a standardized version of this process and propose a martingale transform to obtain asymptotically distribution free tests for the corresponding Kolmogorov-Smirnov and Cramér-vo…
The asymptotic normality of conditional least squares estimators for the offspring variance in critical branching processes with non-homogeneous immigration is established, under moment assumptions on both reproduction and immigration. The proofs use martingale techniques and weak convergence results in Skorokhod spaces.
We solve the problem of mean-variance hedging for general semimartingale models via stochastic control methods. After proving that the value process of the associated stochastic control problem has a quadratic structure, we characterize its three coefficient processes as solutions of semimartingale backward stochastic differential equations and show how they can be used to describe the optimal trading strategy for each conditional mean-variance hedging problem. For comparison with the existing literature, we provide alternative equivalent versions of the BSDEs and present a number of simple ex…
In this paper we study the asymptotic behaviour of power and multipower variations of processes $Y$:\[Y_t=\int_{-\in fty}^tg(t-s)σ_sW(\mathrm{d}s)+Z_t,\] where $g:(0,\infty)\rightarrow\mathbb{R}$ is deterministic, $σ>0$ is a random process, $W$ is the stochastic Wiener measure and $Z$ is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency $σ$. The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of $Y$ …
We consider noisy non-synchronous discrete observations of a continuous semimartingale with random volatility. Functional stable central limit theorems are established under high-frequency asymptotics in three setups: one-dimensional for the spectral estimator of integrated volatility, from two-dimensional asynchronous observations for a bivariate spectral covolatility estimator and multivariate for a local method of moments. The results demonstrate that local adaptivity and smoothing noise dilution in the Fourier domain facilitate substantial efficiency gains compared to previous approaches. …
Consider a high-dimensional linear regression problem, where the number of covariates is larger than the number of observations and the interest is in estimating the conditional variance of the response variable given the covariates. A conditional and unconditioned framework are considered, where conditioning is with respect to the covariates, which are ancillary to the parameter of interest. In recent papers, a consistent estimator was developed in the unconditional framework when the marginal distribution of the covariates is normal with known mean and variance. In the present work, a certai…
Confidence Interval Construction and Conditional Variance Estimation with Dense ReLU Networks
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of California, Santa Cruz（加利福尼亚大学圣克鲁兹分校）；Hong Kong University of Science and Technology（香港科技大学）；Washington University in St. Louis（圣路易斯华盛顿大学）
We present a new non-parametric estimator of the conditional density of the kernel type. It is based on an efficient transformation of the data by quantile transform. By use of the copula representation, it turns out to have a remarkable product form. We study its asymptotic properties and compare its bias and variance to competitors based on nonparametric regression.
In this paper we present the asymptotic analysis of the realised quadratic variation for multivariate symmetric $β$-stable Lévy processes, $β\in (0,2)$, and certain pure jump semimartingales. The main focus is on derivation of functional limit theorems for the realised quadratic variation and its spectrum. We will show that the limiting process is a matrix-valued $β$-stable Lévy process when the original process is symmetric $β$-stable, while the limit is conditionally $β$-stable in case of integrals with respect to symmetric $β$-stable motions. These asymptotic results are mostly related to t…
This article investigates nonparametric estimation of variance functions for functional data when the mean function is unknown. We obtain asymptotic results for the kernel estimator based on squared residuals. Similar to the finite dimensional case, our asymptotic result shows the smoothness of the unknown mean function has an effect on the rate of convergence. Our simulaton studies demonstrate that estimator based on residuals performs much better than that based on conditional second moment of the responses.
In this paper we examine the asymptotic theory for U-statistics and V-statistics of discontinuous Ito semimartingales that are observed at high frequency. For different types of kernel functions we show laws of large numbers and associated stable central limit theorems. In most of the cases the limiting process will be conditionally centered Gaussian. The structure of the kernel function determines whether the jump and/or the continuous part of the semimartingale contribute to the limit.
Response-adaptive allocation designs refer to a class of designs where the probability an observation is assigned to a treatment is changed throughout an experiment based on the accrued responses. Such procedures result in random treatment sample sizes. Most of the current literature considers unconditional inference procedures in the analysis of response-adaptive allocation designs. The focus of this work is inference conditional on the observed treatment sample sizes. The inverse of information is a description of the large sample variance of the parameter estimates. A simple form for the co…
Limit theorems of Azadkia-Chatterjee's conditional graph correlation
School of Statistics, University of International Business and Economics（对外经济贸易大学统计学院）；Department of Statistics, University of Washington（华盛顿大学统计学系）；Academy of Mathematics and System Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
We propose simultaneous mean-variance regression for the linear estimation and approximation of conditional mean functions. In the presence of heteroskedasticity of unknown form, our method accounts for varying dispersion in the regression outcome across the support of conditioning variables by using weights that are jointly determined with the mean regression parameters. Simultaneity generates outcome predictions that are guaranteed to improve over ordinary least-squares prediction error, with corresponding parameter standard errors that are automatically valid. Under shape misspecification o…
We first revisit the problem of estimating the spot volatility of an Itô semimartingale using a kernel estimator. We prove a Central Limit Theorem with optimal convergence rate for a general two-sided kernel. Next, we introduce a new pre-averaging/kernel estimator for spot volatility to handle the microstructure noise of ultra high-frequency observations. We prove a Central Limit Theorem for the estimation error with an optimal rate and study the optimal selection of the bandwidth and kernel functions. We show that the pre-averaging/kernel estimator's asymptotic variance is minimal for exponen…
We consider a square-integrable semimartingale and investigate the convex order relations between its discrete, continuous and predictable quadratic variation. As the main results, we show that if the semimartingale has conditionally independent increments and symmetric jump measure, then its discrete realized variance dominates its quadratic variation in increasing convex order. The results have immediate applications to the pricing of options on realized variance. For a class of models including time-changed Levy models and Sato processes with symmetric jumps our results show that options on…
In this note, we study the infinite-dimensional conditional laws of Brownian semistationary processes. Motivated by the fact that these processes are typically not semimartingales, we present sufficient conditions ensuring that a Brownian semistationary process has conditional full support, a property introduced by Guasoni, Rásonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008) pp. 491--520]. By the results of Guasoni, Rásonyi, and Schachermayer, this property has two important implications. It ensures, firstly, that the process admits no free lunches under proportional transaction costs, a…
This paper proposes confidence regions for the identified set in conditional moment inequality models using Kolmogorov-Smirnov statistics with a truncated inverse variance weighting with increasing truncation points. The new weighting differs from those proposed in the literature in two important ways. First, confidence regions based on KS tests with the weighting function I propose converge to the identified set at a faster rate than existing procedures based on bounded weight functions in a broad class of models. This provides a theoretical justification for inverse variance weighting in thi…
This paper considers a semiparametric generalized autoregressive conditional heteroskedasticity (S-GARCH) model. For this model, we first estimate the time-varying long run component for unconditional variance by the kernel estimator, and then estimate the non-time-varying parameters in GARCH-type short run component by the quasi maximum likelihood estimator (QMLE). We show that the QMLE is asymptotically normal with the parametric convergence rate. Next, we construct a Lagrange multiplier test for linear parameter constraint and a portmanteau test for model checking, and obtain their asymptot…
Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than a decade. One of the most well-known and widely studied problems is that of estimation of the quadratic variation of the continuous component of an Itô semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed in the literature when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of the truncated moments of a Lévy pr…
This paper derives the asymptotic distribution of variance weighted Kolmogorov-Smirnov statistics for conditional moment inequality models for the case of a one dimensional covariate. The asymptotic distribution depends on the data generating process only through the variance of a single random variable, leading to critical values that can be calculated analytically. By arguments in Armstrong (2011b), the resulting tests achieve the best minimax rate for local alternatives out of available approaches in a broad class of settings.
The runs test is a well-known test that is used for checking independence between elements of a sample data sequence. Some of runs tests are based on the longest run and others based on the total runs. In this paper, we consider order statistics of two runs statistics, and obtain their probability mass functions. In addition, the means and variances of the order statistics are derived using traditional conditional expectation.
The single-index model is one of the most popular semiparametric models in Econometrics. In this paper, we define a quantile regression single-index model, which includes the single-index structure for conditional mean and for conditional variance.
This article investigates discrete-time approximations of stochastic integrals driven by semimartingales with jumps via weighted bounded mean oscillation (BMO) approach. This approach enables $L_p$-estimates, $p \in (2, \infty)$, for the approximation error depending on the weight, and it allows a change of the underlying measure which leaves the error estimates unchanged. To take advantage of this approach, we propose a new approximation scheme obtained from a correction for the Riemann approximation based on tracking jumps of the underlying semimartingale. We also discuss a way to optimize t…
We consider the mean-variance hedging problem under partial Information. The underlying asset price process follows a continuous semimartingale and strategies have to be constructed when only part of the information in the market is available. We show that the initial mean variance hedging problem is equivalent to a new mean variance hedging problem with an additional correction term, which is formulated in terms of observable processes. We prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differentia…
The conditional expectation and conditional variance of Brownian motion is considered given the argmax, B(t|argmax), as well as those with additional information: B(t|close, argmax), B(t|max, argmax), B(t|close, max, argmax) where the close is the final value: B(t=1)=c and t in [0,1]. We compute the expectation and variance of a Brownian meander in time. By splicing together two Brownian meanders, the mean and variance of the constrained process are calculated. Computational results displaying both the expectation and variance in time are presented. Comparison of the simulation with theoretica…
We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölder-regularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an $N$-dimensional diffusion process defined as solution to a system of ordinary stochastic differential equ…
We show that deliberately introducing a nested simulation stage can lead to significant variance reductions when comparing two stopping times by Monte Carlo. We derive the optimal number of nested simulations and prove that the algorithm is remarkably robust to misspecifications of this number. The method is applied to several problems related to Bermudan/American options. In these applications, our method allows to substantially increase the efficiency of other variance reduction techniques, namely, Quasi-Control Variates and Multilevel Monte Carlo.
Improving Importance Sampling estimators for rare event probabilities requires sharp approximations of conditional densities. This is achieved for events E_{n}:=(f(X_{1})+...+f(X_{n}))\inA_{n} where the summands are i.i.d. and E_{n} is a large or moderate deviation event. The approximation of the conditional density of the real r.v's X_{i} 's, for 1\leqi\leqk_{n} with repect to E_{n} on long runs, when k_{n}/n\to1, is handled. The maximal value of k compatible with a given accuracy is discussed; algorithms and simulated results are presented.
Kipnis and Varadhan showed that for an additive functional, $S_n$ say, of a reversible Markov chain the condition $E(S_n^{2})/n \to κ\in (0,\infty)$ implies the convergence of the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$, given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, $E(S_n^{2}) = n\ell(n)$, where $\ell$ is a slowly varying function. It is shown by example that the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$ need not converge to the standard normal distribution in this case; and sufficient conditions for c…
This note provides a conditional Berry-Esseen bound for the sum of a martingale difference sequence $\{X_i\}_{i=1}^n$ in $\mathbb{R}^d$, $d\ge 1$, adapted to a filtration $\{\mathcal{F}_i\}_{i=1}^n$. We approximate the conditional distribution of $S=\sum_{i=1}^n X_i$ given some $σ$-field $\mathcal{F}_0\subset \mathcal{F}_1$ by that of a mean-zero normal random vector having the same conditional variance given $\mathcal{F}_0 $ as the vector $S$. Assuming that the conditional variances $\mathsf{E}[X_iX_i^{\top}\mid\mathcal{F}_{i-1}]$, $i\ge 1$, are $\mathcal{F}_0$-measurable and non-singular, an…
In a previous paper we determined one dimensional distributions of a stationary field with linear regressions and quadratic conditional variances under a linear constraint on the coefficients of the quadratic expression. In this paper we show that for stationary Markov chains with linear regressions and quadratic conditional variances the coefficients of the quadratic expression are indeed tied by a linear constraint which can take only one of the two alternative forms.
We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Levy processes.
The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Levy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.
