Continuous-time Regression Models Research Articles

Efficient estimation methods for non-Gaussian regression models in continuous time

In this paper, we develop an efficient nonparametric estimation theory for continuous time regression models with non-Gaussian Levy noises in the case when the unknown functions belong to Sobolev ellipses. Using the Pinsker’s approach, we provide a sharp lower bound for the normalized asymptotic mean square accuracy. However, the main result obtained by Pinsker for the Gaussian white noise model is not correct without additional conditions for the ellipse coefficients. We find such constructive sufficient conditions under which we develop efficient estimation methods. We show that the obtained conditions hold for the ellipse coefficients of an exponential form. For exponential coefficients, the sharp lower bound is calculated in explicit form. Finally, we apply this result to signals number detection problems in multi-pass connection channels and we obtain an almost parametric convergence rate that is natural for this case, which significantly improves the rate with respect to power-form coefficients.

The BLUE in continuous-time regression models with correlated errors

In this paper, the problem of best linear unbiased estimation is investigated for continuous-time regression models. We prove several general statements concerning the explicit form of the best linear unbiased estimator (BLUE), in particular when the error process is a smooth process with one or several derivatives of the response process available for construction of the estimators. We derive the explicit form of the BLUE for many specific models including the cases of continuous autoregressive errors of order two and integrated error processes (such as integrated Brownian motion). The results are illustrated on many examples.

Continuous-Time Volatility Regression in Large Panels

This paper studies a semiparametric inference procedure for a finite-dimensional parameter in a continuous-time regression model involving high-frequency data in a large cross-section. The model concerns the relationship between a noisy dependent process and a possibly nonlinear transform of stochastic volatility over a fixed time span, with its coefficients allowed to depend on a set of firm-specific characteristics. The construction of the estimator involves two steps: the nonparametric recovery of stochastic volatility processes, followed by a parametric second stage that uses the volatility estimates. I show that the estimator admits a central limit theorem and provide a consistent estimator of the asymptotic variance based on a factor-analytic method. The finite sample performance of the inference procedure is satisfactory in a realistically calibrated Monte Carlo setting. In a novel empirical application, I study the relationship between bid-ask spread and the spot standard deviation of asset price. The slope coefficient estimate, which measures the heterogeneous level of information asymmetry, is closely related to firm characteristics such as measures of valuation uncertainty and institutional ownership.

IMPROVED MODEL SELECTION METHOD FOR AN ADAPTIVE ESTIMATION IN SEMIMARTINGALE REGRESSION MODELS

This paper considers the problem of robust adaptive efficient estimating of a periodic function in a continuous time regression model with the dependent noises given by a general square integrable semimartingale with a conditionally Gaussian distribution. An example of such noise is the non-Gaussian Ornstein-Uhlenbeck-Levy processes. An adaptive model selection procedure, based on the improved weighted least square estimates, is proposed. Under some conditions on the noise distribution, sharp oracle inequality for the robust risk has been proved and the robust efficiency of the model selection procedure has been established. The numerical analysis results are given.

Oracle inequalities for the stochastic differential equations

This paper is a survey of recent results on the adaptive robust non parametric methods for the continuous time regression model with the semi - martingale noises with jumps. The noises are modeled by the L\'evy processes, the Ornstein -- Uhlenbeck processes and semi-Markov processes. We represent the general model selection method and the sharp oracle inequalities methods which provide the robust efficient estimation in the adaptive setting. Moreover, we present the recent results on the improved model selection methods for the nonparametric estimation problems.

Large deviations of regression parameter estimator in continuous-time models with sub-Gaussian noise

A continuous-time regression model with a jointly strictly sub-Gaussian random noise is considered in the paper. Upper exponential bounds for probabilities of large deviations of the least squares estimator for the regression parameter are obtained.

Consistent estimator of nonparametric structural spurious regression model for high frequency data

We propose a new nonparametric estimator for continuous-time regression models with nonstationary error terms. While other conventional nonparametric estimators such as the Nadaraya–Watson and local linear estimators are not consistent, our estimator achieves consistency and asymptotic normality.

Adaptive estimation of continuous-time regression models using high-frequency data

We derive the asymptotic efficiency bound for regular estimates of the slope coefficient in a linear continuous-time regression model for the continuous martingale parts of two Itô semimartingales observed on a fixed time interval with asymptotically shrinking mesh of the observation grid. We further construct an estimator from high-frequency data that achieves this efficiency bound and, indeed, is adaptive to the presence of infinite-dimensional nuisance components. The estimator is formed by taking optimal weighted average of local nonparametric volatility estimates that are constructed over blocks of high-frequency observations. The asymptotic efficiency bound is derived under a Markov assumption for the bivariate process while the high-frequency estimator and its asymptotic properties are derived in a general Itô semimartingale setting. To study the asymptotic behavior of the proposed estimator, we introduce a general spatial localization procedure which extends known results on the estimation of integrated volatility functionals to more general classes of functions of volatility. Empirically relevant numerical examples illustrate that the proposed efficient estimator provides nontrivial improvement over alternatives in the extant literature.

Robust model selection for a semimartingale continuous time regression from discrete data

The paper considers the problem of estimating a periodic function in a continuous time regression model observed under a general semimartingale noise with an unknown distribution in the case when continuous observation cannot be provided and only discrete time measurements are available. Two specific types of noises are studied in detail: a non-Gaussian Ornstein–Uhlenbeck process and a time-varying linear combination of a Brownian motion and compound Poisson process. We develop new analytical tools to treat the adaptive estimation problems from discrete data. A lower bound for the frequency sampling, needed for the efficiency of the procedure constructed by discrete observations, has been found. Sharp non-asymptotic oracle inequalities for the robust quadratic risk have been derived. New convergence rates for the efficient procedures have been obtained. An example of the regression with a martingale noise exhibits that the minimax robust convergence rate may be both higher or lower as compared with the minimax rate for the “white noise” model. The results of Monte-Carlo simulations are given.

Improved estimation in a non-Gaussian parametric regression

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The vector of unknown parameters is assumed to belong to a compact set. The noise is specified by the Ornstein–Uhlenbeck process driven by the mixture of a Brownian motion and a compound Poisson process. Improved estimates for the unknown regression parameters, based on a special modification of the James–Stein procedure with smaller quadratic risk than the usual least squares estimates, are proposed. The developed estimation scheme is applied for the improved parameter estimation in the discrete time regression with the autoregressive noise depending on unknown nuisance parameters.

Efficient robust nonparametric estimation in a semimartingale regression model

Dans cette article nous considérons le problème d’estimation robuste d’une fonction périodique dans un modèle de régression en temps continu avec un bruit dépendant décrit par une semi martingale carrée intégrable de distribution inconnue. Un exemple de ce bruit est un processus d’Ornstein–Uhlenbeck non gaussien avec sauts (voir (J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167–241), (Ann. Appl. Probab. 18 (2008) 879–908)). Nous proposons une procédure adaptative de sélection de modèle basée sur les estimateurs des moindres carrés pondérés. Sous des conditions générales sur les deux premiers moments de la distribution du bruit, des inégalités d’Oracle non asymptotiques pointues pour des risques quadratiques robustes sont obtenues et l’efficacité robuste est établie. Nous avons établi aussi que dans le cas du processus d’Ornstein–Uhlenbeck non Gaussian, la borne inférieure pour le risque quadratique robuste est donnée par la limite de l’intensité du bruit quand la fréquence tend vers l’infini. Nous donnons un exemple d’un modèle de régression avec un bruit martingale où la vitesse de convergence du risque quadratique devient plus lente si l’intensité du bruit tend vers l’infini.

Asymptotic Properties of Koenker–Bassett Estimator in Regression Model with Long-Range Dependence

This article deals with asymptotic distribution of Koenker-Bassett estimator in continuous time regression model with long-range dependent noise. It is proved that the asymptotic distribution of the normed estimator coincides with asymptotic distribution of the integral of an indicator random process generated by the random noise weighted by regression function gradient. A theorem is formulated also on asymptotic normality of the last integral.

General model selection estimation of a periodic regression with a Gaussian noise

This paper considers the problem of estimating a periodic function in a continuous time regression model with an additive stationary Gaussian noise having unknown correlation function. A general model selection procedure on the basis of arbitrary projective estimates, which does not need the knowledge of the noise correlation function, is proposed. A non-asymptotic upper bound for $${\mathcal{L}_2}$$ -risk (oracle inequality) has been derived under mild conditions on the noise. For the Ornstein–Uhlenbeck noise the risk upper bound is shown to be uniform in the nuisance parameter. In the case of Gaussian white noise the constructed procedure has some advantages as compared with the procedure based on the least squares estimates (LSE). The asymptotic minimaxity of the estimates has been proved. The proposed model selection scheme is extended also to the estimation problem based on the discrete data applicably to the situation when high frequency sampling can not be provided.

GENERALIZATION OF GMM TO A CONTINUUM OF MOMENT CONDITIONS

This paper proposes a version of the generalized method of moments procedure that handles both the case where the number of moment conditions is finite and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its values in an interval. The objective function to minimize is then the norm of the moment conditions in a Hilbert space. The estimator is shown to be consistent and asymptotically normal. The optimal estimator is obtained by minimizing the norm of the moment conditions in the reproducing kernel Hilbert space associated with the covariance. We show an easy way to calculate this estimator. Finally, we study properties of a specification test using overidentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.

Continuous-time approaches to identification of continuous-time systems

Linear continuous-time plants modeled, in general, with the aid of operator polynomials, are identified with the use of continuous-time regression models of two types, which result from ‘differential’ and ‘integral’ normalizations. Estimates of the parameters of these models are obtained via a ‘finite-horizon’ processing of the regression vector that contains consecutive multiple integrals of the plant input and output signals. Effective continuous-time identification is performed with the employment of the continuous-time least-squares and instrumental variable approaches. The continuous-time algorithms and the regression vectors are eventually transformed into the discrete-time domain by utilizing numerical integrating schemes. Ultimate discrete-time realizations are examined in a simulation study.

Cox's Factoring of Regression Model Likelihoods for Continuous-Time Processes

Cox showed that the likelihood of regression models for discrete-time processes factors into a partial likelihood and a product of conditional laws for the covariates, given the history. Jacod constructed a partial likelihood for continuous-time regression models in terms of the predictable characteristics of the response process. Here we prove a factorization of the likelihood, analogous to Cox's, assuming both the response and the covariates to be semimartingales. The result is useful for counting process regression modelling and inference, and also for regression involving continuous processes and diffusions with jumps.

Marital stability throughout the child-rearing years

Although there is evidence that the number and ages of children influence marital stability, studies have not systematically tracked the risk of marital disruption throughout the child-rearing years. This study uses marital and fertility histories from the June 1985 Current Population Survey to examine this issue. Continuous-time regression models with ages and numbers of children as time-varying covariates are estimated. Net of controls for age at marriage, year of marriage, education, and marital duration, stability increases with family size up to the third child but starts to decline as family size reaches five or more children. Aging of children is disruptive until the youngest child reaches adulthood, after which marriages become much more stable. Arrival and aging of children is an important dynamic with strong implications for marital stability.