Semiparametric ARCH Models: An Estimating Function Approach

We introduce the method of estimating functions to study the class of autoregressive conditional heteroscedasticity (ARCH) models. We derive the optimal estimating functions by combining linear and quadratic estimating functions. The resultant estimators are more efficient than the quasi-maximum likelihood estimator. If the assumption of conditional normality is imposed, the estimator obtained by using the theory of estimating functions is identical to that obtained by using the maximum likelihood method in finite samples. The relative efficiencies of the estimating function (EF) approach in comparison with the quasi-maximum likelihood estimator are developed. We illustrate the EF approach using a univariate GARCH(1,1) model with conditional normal, Student-t, and gamma distributions. The efficiency benefits of the EF approach relative to the quasi-maximum likelihood approach are substantial for the gamma distribution with large skewness. Simulation analysis shows that the finite-sample properties of the estimators from the EF approach are attractive. EF estimators tend to display less bias and root mean squared error than the quasi-maximum likelihood estimator. The efficiency gains are substantial for highly nonnormal distributions. An example demonstrates that implementation of the method is straightforward.     

A note on adaptation in garch models

In the framework of the Engle-type (G)ARCH models, I demonstrate that there is a family of symmetric and asymmetric density functions for which the asymptotic efficiency of the semiparametric estimator is equal to the asymptotic efficiency of the maximum likelihood estimator. This family of densities is bimodal (except for the normal). I also chracterize the solution to the problem of minimizing the mean squared distance between the parametric score and the semiparametric score in order to search for unimodal densities for which the semiparametric estimator is likely to perform well. The LaPlace density function emerges as one of these cases.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Statistic inference for a single-index ARCH-M model

For a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.     

Nonlinear semiparametric autoregressive model with finite mixtures of scale mixtures of skew normal innovations

We propose data generating structures which can be represented as the nonlinear autoregressive models with single and finite mixtures of scale mixtures of skew normal innovations. This class of models covers symmetric/asymmetric and light/heavy-tailed distributions, so provide a useful generalization of the symmetrical nonlinear autoregressive models. As semiparametric and nonparametric curve estimation are the approaches for exploring the structure of a nonlinear time series data set, in this article the semiparametric estimator for estimating the nonlinear function of the model is investigated based on the conditional least square method and nonparametric kernel approach. Also, an Expectation–Maximization-type algorithm to perform the maximum likelihood (ML) inference of unknown parameters of the model is proposed. Furthermore, some strong and weak consistency of the semiparametric estimator in this class of models are presented. Finally, to illustrate the usefulness of the proposed model, some simulation studies and an application to real data set are considered.     

A (Semi)Parametric Functional Coefficient Logarithmic Autoregressive Conditional Duration Model

In this article, we propose a class of logarithmic autoregressive conditional duration (ACD)-type models that accommodates overdispersion, intermittent dynamics, multiple regimes, and asymmetries in financial durations. In particular, our functional coefficient logarithmic autoregressive conditional duration (FC-LACD) model relies on a smooth-transition autoregressive specification. The motivation lies on the fact that the latter yields a universal approximation if one lets the number of regimes grows without bound. After establishing sufficient conditions for strict stationarity, we address model identifiability as well as the asymptotic properties of the quasi-maximum likelihood (QML) estimator for the FC-LACD model with a fixed number of regimes. In addition, we also discuss how to consistently estimate a semiparametric variant of the FC-LACD model that takes the number of regimes to infinity. An empirical illustration indicates that our functional coefficient model is flexible enough to model IBM price durations.     

Semiparametric Density Deconvolution

Abstract.  A new semiparametric method for density deconvolution is proposed, based on a model in which only the ratio of the unconvoluted to convoluted densities is specified parametrically. Deconvolution results from reweighting the terms in a standard kernel density estimator, where the weights are defined by the parametric density ratio. We propose that in practice, the density ratio be modelled on the log-scale as a cubic spline with a fixed number of knots. Parameter estimation is based on maximization of a type of semiparametric likelihood. The resulting asymptotic properties for our deconvolution estimator mirror the convergence rates in standard density estimation without measurement error when attention is restricted to our semiparametric class of densities. Furthermore, numerical studies indicate that for practical sample sizes our weighted kernel estimator can provide better results than the classical non-parametric kernel estimator for a range of densities outside the specified semiparametric class.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Uniformly adaptive estimation for models with arma errors

A semiparametric estimator based on an unknown density isuniformly adaptive if the expected loss of the estimator converges to the asymptotic expected loss of the maximum liklihood estimator based on teh true density (MLE), and if convergence does not depend on either the parameter values or the form of the unknown density. Without uniform adaptivity, the asymptotic expected loss of the MLE need not approximate the expected loss of a semiparametric estimator for any finite sample I show that a two step semiparametric estimator is uniformly adaptive for the parameters of nonlinear regression models with autoregressive moving average errors.     

Estimation in the Cox cure model with covariates missing not at random,with application to disease screening/prediction

In an attempt to provide a statistical tool for disease screening and prediction, we propose a semiparametric approach to analysis of the Cox proportional hazards cure model in situations where the observations on the event time are subject to right censoring and some covariates are missing not at random. To facilitate the methodological development, we begin with semiparametric maximum likelihood estimation (SPMLE) assuming that the (conditional) distribution of the missing covariates is known. A variant of the EM algorithm is used to compute the estimator. We then adapt the SPMLE to a more practical situation where the distribution is unknown and there is a consistent estimator based on available information. We establish the consistency and weak convergence of the resulting pseudo-SPMLE, and identify a suitable variance estimator. The application of our inference procedure to disease screening and prediction is illustrated via empirical studies. The proposed approach is used to analyze the tuberculosis screening study data that motivated this research. Its finite-sample performance is examined by simulation.     

Semiparametric Time Series Models with Log‐concave Innovations: Maximum Likelihood Estimation and its Consistency

We study semiparametric time series models with innovations following a log‐concave distribution. We propose a general maximum likelihood framework that allows us to estimate simultaneously the parameters of the model and the density of the innovations. This framework can be easily adapted to many well‐known models, including autoregressive moving average (ARMA), generalized autoregressive conditionally heteroscedastic (GARCH), and ARMA‐GARCH models. Furthermore, we show that the estimator under our new framework is consistent in both ARMA and ARMA‐GARCH settings. We demonstrate its finite sample performance via a thorough simulation study and apply it to model the daily log‐return of the FTSE 100 index.     

Estimation Under Inequality Constraints: Semiparametric Estimation of Conditional Duration Models

This article proposes a semiparametric estimator of the parameter in a conditional duration model when there are inequality constraints on some parameters and the error distribution may be unknown. We propose to estimate the parameter by a constrained version of an unrestricted semiparametrically efficient estimator. The main requirement for applying this method is that the initial unrestricted estimator converges in distribution. Apart from this, additional regularity conditions on the data generating process or the likelihood function, are not required. Hence the method is applicable to a broad range of models where the parameter space is constrained by inequality constraints, such as the conditional duration models. In a simulation study involving conditional duration models, the overall performance of the constrained estimator was better than its competitors, in terms of mean squared error. A data example is used to illustrate the method.     

Estimation under Cox proportional hazards model with covariates missing not at random

This paper considers likelihood-based estimation under the Cox proportional hazards model in the situations where some covariate entries are missing not at random. Assuming the conditional distribution of the missing entries is known, we demonstrate the existence of the semiparametric maximum likelihood estimator of the model parameters, establish the consistency and weak convergence. By simulation, we examine the finite-sample performance of the estimation procedure, and compare the SPMLE with the one resulted from using an estimated conditional distribution of the missing entries. We analyze the data from a tuberculosis (TB) study applying the proposed approach for illustration.     

Weighted quantile regression with missing covariates using empirical likelihood

This paper proposes an empirical likelihood-based weighted (ELW) quantile regression approach for estimating the conditional quantiles when some covariates are missing at random. The proposed ELW estimator is computationally simple and achieves semiparametric efficiency if the probability of missingness is correctly specified. The limiting covariance matrix of the ELW estimator can be estimated by a resampling technique, which does not involve nonparametric density estimation or numerical derivatives. Simulation results show that the ELW method works remarkably well in finite samples. A real data example is used to illustrate the proposed ELW method.     

A Weighted Linear Estimator of Multivariate ARCH Parameters

A weighted linear estimator (WLE) of the parameters of multivariate ARCH models is proposed. The accuracy of WLE in estimating the parameters of multivariate ARCH models is compared with the widely used quasi-maximum likelihood estimator (QMLE) through simulations. Application to real data sets are also presented and forecasts of variance-covariance matrix and value-at-risk (VaR) are obtained. The weighted resampling methods are used to approximate the sampling distribution of the proposed estimator. Our study indicates that the forecasting performance of WLE is not inferior and one-day ahead risk estimates are also found better than the QMLE.     

Indirect estimation of randomized generalized autoregressive conditional heteroskedastic models

The class of generalized autoregressive conditional heteroskedastic (GARCH) models can be used to describe the volatility with less parameters than autoregressive conditional heteroskedastic (ARCH)-type models, their distributions are heavy-tailed, with time-dependent conditional variance, and are able to model clustering of volatility. Despite all these facts, the way that GARCH models are built imposes limits on the heaviness of the tails of their unconditional distribution. The class of randomized generalized autoregressive conditional heteroskedastic (R-GARCH) models includes the ARCH and GARCH models allowing the use of stable innovations. Estimation methods and empirical analysis of R-GARCH models are the focus of this work. We present the indirect inference method to estimate the R-GARCH models, some simulations and an empirical application.     

Case-cohort analysis with semiparametric transformation models

Semiparametric transformation models provide flexible regression models for survival analysis, including the Cox proportional hazards and the proportional odds models as special cases. We consider the application of semiparametric transformation models in case-cohort studies, where the covariate data are observed only on cases and on a subcohort randomly sampled from the full cohort. We first propose an approximate profile likelihood approach with full-cohort data, which amounts to the pseudo-partial likelihood approach of Zucker [2005. A pseudo-partial likelihood method for semiparametric survival regression with covariate errors. J. Amer. Statist. Assoc. 100, 1264–1277]. Simulation results show that our proposal is almost as efficient as the nonparametric maximum likelihood estimator. We then extend this approach to the case-cohort design, applying the Horvitz–Thompson weighting method to the estimating equations from the approximated profile likelihood. Two levels of weights can be utilized to achieve unbiasedness and to gain efficiency. The resulting estimator has a closed-form asymptotic covariance matrix, and is found in simulations to be substantially more efficient than the estimator based on martingale estimating equations. The extension to left-truncated data will be discussed. We illustrate the proposed method on data from a cardiovascular risk factor study conducted in Taiwan.     

Semiparametric Efficient Estimation of the Mean of a Time Series in the Presence of Conditional Heterogeneity of Unknown Form

We obtain semiparametric efficiency bounds for estimation of a location parameter in a time series model where the innovations are stationary and ergodic conditionally symmetric martingale differences but otherwise possess general dependence and distributions of unknown form. We then describe an iterative estimator that achieves this bound when the conditional density functions of the sample are known. Finally, we develop a “semi-adaptive” estimator that achieves the bound when these densities are unknown by the investigator. This estimator employs nonparametric kernel estimates of the densities. Monte Carlo results are reported.     

Semiparametric efficient estimation for the auxiliary outcome problem with the conditional mean model

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.     

A robust closed-form estimator for the GARCH(1,1) model

In this paper we extend the closed-form estimator for the generalized autoregressive conditional heteroscedastic (GARCH(1,1)) proposed by Kristensen and Linton [A closed-form estimator for the GARCH(1,1) model. Econom Theory. 2006;22:323–337] to deal with additive outliers. It has the advantage that is per se more robust that the maximum likelihood estimator (ML) often used to estimate this model, it is easy to implement and does not require the use of any numerical optimization procedure. The robustification of the closed-form estimator is done by replacing the sample autocorrelations by a robust estimator of these correlations and by estimating the volatility using robust filters. The performance of our proposal in estimating the parameters and the volatility of the GARCH(1,1) model is compared with the proposals existing in the literature via intensive Monte Carlo experiments and the results of these experiments show that our proposal outperforms the ML and quasi-maximum likelihood estimators-based procedures. Finally, we fit the robust closed-form estimator and the benchmarks to one series of financial returns and analyse their performances in estimating and forecasting the volatility and the value-at-risk.