Showing papers on "Semiparametric model published in 1994"

Estimation of Regression Coefficients When Some Regressors are not Always Observed

Abstract: In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score...

Empirical Likelihood and General Estimating Equations

Abstract: For some time, so-called empirical likelihoods have been used heuristically for purposes of nonparametric estimation. Owen showed that empirical likelihood ratio statistics for various parameters $\theta(F)$ of an unknown distribution $F$ have limiting chi-square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods. Our objective in this paper is twofold: first, to link estimating functions or equations and empirical likelihood; second, to develop methods of combining information about parameters. We do this by assuming that information about $F$ and $\theta$ is available in the form of unbiased estimating functions. Empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood. Efficiency results for estimates of both $\theta$ and $F$ are obtained. The methods are illustrated on several problems, and areas for future investigation are noted.

The asymptotic variance of semiparametric estimators

Abstract: This paper derives a general formula for the asymptotic variance of semiparametric estimators that accounts for the presence of nonparametric estimators of functions. The general formula is specialized to show invariance of the asymptotic variance to the type of nonparametric estimator and to obtain correction terms for estimation of densities and mean-square projections (including conditional expectations). Regularity conditions for the validity of the formula are also given, including primitive conditions for asymptotic normality when series estimators are present. New examples considered include a semiparametric panel probit estimator and a series estimator of the average derivative. Copyright 1994 by The Econometric Society.

Semiparametric Analysis of Long-Memory Time Series

Abstract: We study problems of semiparametric statistical inference connected with long-memory covariance stationary time series, having spectrum which varies regularly at the origin: There is an unknown self-similarity parameter, but elsewhere the spectrum satisfies no parametric or smoothness conditions, it need not be in $L_p$, for any $p > 1$, and in some circumstances the slowly varying factor can be of unknown form. The basic statistic of interest is the discretely averaged periodogram, based on a degenerating band of frequencies around the origin. We establish some consistency properties under mild conditions. These are applied to show consistency of new estimates of the self-similarity parameter and scale factor. We also indicate applications of our results to standard errors of least squares estimates of polynomial regression with long-memory errors, to generalized least squares estimates of this model and to estimates of a "cointegrating" relationship between long-memory time series.

Semiparametric Models for Longitudinal Data with Application to CD4 Cell Numbers in HIV Seroconverters

Abstract: The paper describes a semiparametric model for longitudinal data which is illustrated by its application to data on the time evolution of CD4 cell numbers in HIV seroconverters. The essential ingredients of the model are a parametric linear model for covariate adjustment, a nonparametric estimation of a smooth time trend, serial correlation between measurements on an individual subject, and random measurement error. A back-fitting algorithm is used in conjunction with a cross-validation prescription to fit the model. A notable feature in the application is that the onset of HIV infection is associated with a sudden drop in CD4 cells followed by a longer-term slower decay. The model is also used to estimate an individual's curve by combining his data with the population curve. Shrinkage toward the population mean trajectory is controlled in a natural way by the estimated covariance structure of the data.

Semi parametric analysis of long memory time series

Abstract: We study problems of semiparametric statistical inference connected with long-memory covariance stationary time series, having spectrum which varies regularly at the origin: There is an unknown self-similarity parameter, but elsewhere the spectrum satisfies no parametric or smoothness conditions, it need not be in $L_p$, for any $p > 1$, and in some circumstances the slowly varying factor can be of unknown form. The basic statistic of interest is the discretely averaged periodogram, based on a degenerating band of frequencies around the origin. We establish some consistency properties under mild conditions. These are applied to show consistency of new estimates of the self-similarity parameter and scale factor. We also indicate applications of our results to standard errors of least squares estimates of polynomial regression with long-memory errors, to generalized least squares estimates of this model and to estimates of a "cointegrating" relationship between long-memory time series

Chapter 41 Estimation of semiparametric models

Abstract: A semiparametric model for observational data combines a parametric form for some component of the data generating process (usually the behavioral relation between the dependent and explanatory variables) with weak nonparametric restrictions on the remainder of the model (usually the distribution of the unobservable errors). This chapter surveys some of the recent literature on semiparametric methods, emphasizing microeconometric applications using limited dependent variable models. An introductory section defines semiparametric models more precisely and reviews the techniques used to derive the large-sample properties of the corresponding estimation methods. The next section describes a number of weak restrictions on error distributions — conditional mean, conditional quantile, conditional symmetry, independence, and index restrictions — and show how they can be used to derive identifying restrictions on the distributions of observables. This general discussion is followed by a survey of a number of specific estimators proposed for particular econometric models, and the chapter concludes with a brief account of applications of these methods in practice.

Quasi-Likelihood Estimation in Semiparametric Models

Abstract: Suppose the expected value of a response variable Y may be written h(Xβ +γ(T)) where X and T are covariates, each of which may be vector-valued, β is an unknown parameter vector, γ is an unknown smooth function, and h is a known function. In this article, we outline a method for estimating the parameter β, γ of this type of semiparametric model using a quasi-likelihood function. Algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed. The generalization of this approach to the case in which Y is a multivariate response is also considered. The methodology is illustrated on two data sets and the results of a small Monte Carlo study are presented.

P Values Maximized Over a Confidence Set for the Nuisance Parameter

Abstract: For testing problems of the form H 0: v = v 0 with unknown nuisance parameter θ, various methods are used to deal with θ. The simplest approach is exemplified by the t test where the unknown variance is replaced by the sample variance and the t distribution accounts for estimation of the variance. In other problems, such as the 2 × 2 contingency table, one conditions on a sufficient statistic for 0 and proceeds as in Fisher's exact test. Because neither of these standard methods is appropriate for all situations, this article suggests a new method for handling the unknown θ. This new method is a simple modification of the formal definition of a p value that involves taking a maximum over the nuisance parameter space of a p value obtained for the case when θ is known. The suggested modification is to restrict the maximization to a confidence set for the nuisance parameter. After giving a brief justification, we give various examples to show how this new method gives improved results for 2 × 2 tabl...

Asymptotics for semiparametric econometric models via stochastic equicontinuity

Abstract: This paper provides a general framework for proving the "square root of" T-consistency and asymptotic normality of a wide variety of semiparametric estimators. The class of estimators considered consists of estimators that can be defined as the solution to a minimization problem based on a criterion function that may depend on a preliminary infinite dimensional nuisance parameter estimator. The method of proof exploits results concerning the stochastic equicontinuity of stochastic processes. The results are applied to the problem of semiparametric weighted least squares estimation of partially parametric regression models. Primitive conditions are given for "square root of" T-consistency and asymptotic normality of this estimator. Copyright 1994 by The Econometric Society.

Applied nonparametric methods

Abstract: We review different approaches to nonparametric density and regression estimation. Kernel estimators are motivated from local averaging and solving ill-posed problems. Kernel estimators are compared to k-NN estimators, orthogonal series and splines. Pointwise and uniform confidence bands are described, and the choice of smoothing parameter is discussed. Finally, the method is applied to nonparametric prediction of time series and to semiparametric estimation.

A resampling method based on pivotal estimating functions

Abstract: SUMMARY Suppose that, under a semiparametric model setting, one is interested in drawing inferences about a finite-dimensi onal parameter vector /? based on an estimating function. Generally a consistent point estimator /J for /?0, the true value for /J, can be easily obtained by finding a root of the corresponding estimating equation. To estimate the variance of ft, however, may involve complicated and subjective nonparametric functional estimates. In this paper, a general and simple resampling method for inferences about jS0 based on pivotal estimating functions is proposed. The new procedure is illustrated with the quantile and rank regression models. For both cases, our proposal can be easily and efficiently implemented with existing statistical software.

Testing a Parametric Model against a Semiparametric Alternative

Abstract: This paper describes a method for testing a parametric model of the mean of a random variable Y conditional on a vector of explanatory variables X against a semiparametric alternative The test is motivated by a conditional moment test against a parametric alternative and amounts to replacing the parametric alternative model with a semiparametric model The resulting semiparametric test is consistent against a larger set of alternatives than are parametric conditional moments tests based on finitely many moment conditions The results of Monte Carlo experiments and an application illustrate the usefulness of the new test

Efficient Semiparametric Estimation in a Stochastic Frontier Model

Abstract: This article considers the semiparametric stochastic frontier model with panel data that arises in the problem of measuring technical inefficiency in production processes. We assume a parametric form for the frontier function, which is linear in production inputs. The density of the individual firm-specific effects is considered to be unknown. We construct an efficient estimator of the slope parameters in the frontier function. We also give an estimator of the level of the frontier function and investigate its asymptotic properties. Furthermore, we provide a predictor of the individual effects that can be directly translated to firm-specific technical inefficiencies. Finally, we illustrate our methods through a real data example.

U-Processes in the Analysis of a Generalized Semiparametric Regression Estimator

Abstract: We prove -consistency and asymptotic normality of a generalized semiparametric regression estimator that includes as special cases Ichimura's semiparametric least-squares estimator for single index models, and the estimator of Klein and Spady for the binary choice regression model. Two function expansions reveal a type of U-process structure in the criterion function; then new U-process maximal inequalities are applied to establish the requisite stochastic equicontinuity condition. This method of proof avoids much of the technical detail required by more traditional methods of analysis. The general framework suggests other -consistent and asymptotically normal estimators.

Data-Driven Efficient Estimators for a Partially Linear Model

Abstract: Chen and Shiau showed that a two-stage spline smoothing method and the partial regression method lead to efficient estimators for the parametric component of a partially linear model when the smoothing parameter is a deterministic sequence tending to zero at an appropriate rate This paper is concerned with the large-sample behavior of these estimators when the smoothing parameter is chosen by the generalized cross validation (GCV) method or Mallows' $C_L$ Under mild conditions, the estimated parametric component is asymptotically normal with the usual parametric rate of convergence for both spline estimation methods As a by-product, it is shown that the "optimal rate" for the smoothing parameter, with respect to expected average squared error, is the same for the two estimation methods as it is for ordinary smoothing splines

A Semiparametric Correction for Attenuation

Abstract: A correction method is proposed for models including the generalized linear model when the covariate is measured with error. The method requires a separate validation data set that consists of the surrogate W and the true covariate X or an unbiased estimate X⊃ of X. We do not require the classical additive measurement error model in which the surrogate is unbiased for the true covariates. We first obtain an estimate of E(X|W) by using nonparametric kernel regression of X or X⊃ on W based on the validation data. Then we perform a standard analysis with the unknown X replaced by the estimate of E(X|W). The asymptotic distribution of the resulting regression parameter estimator is obtained. Generalizations to include components of X measured without error are also discussed.

Semiparametric Specification Testing of Non-nested Econometric Models

Abstract: We propose a specification test of a parametrically specified nonlinear model against a weakly specified non-nested alternative. We estimate the alternative model by using nonparametric regression (nearest neighbours). The test is based on the t-statistic of an artificial regression. MonteCarlo simulations suggest that the test has good power and size characteristics. In this paper we propose a specification test for non-nested regression models where the alternative hypothesis is nonparametric. Therefore, the regressors in the null and in the alternative are non-nested. The proposed test is based on an artificial nesting procedure in the spirit of the Davidson and MacKinnon (1981) tests. We provide a natural extension of their results in a semiparametric environment. The performance of the alternative under 'the truth of the null' forms the basis for a test of the latter. We use k-nearest neighbour (k-NN) regression to estimate that a possible misspecified parameterization will have on testing the validity of the null. The consistency of the proposed test statistic, under fixed alternatives, is derived under the type of regulatory conditions required when the alternative model is parametrically specified. Nonparametric functional estimation has been recently used in deriving various specification tests. Specification tests of nonparametric and semiparametric restrictions based on average derivatives of the regression function have been considered by Stoker (1989) and Robinson (1989). Cox et al. (1988) considered generalized spline models for testing a parametric null hypothesis against semiparametric alternatives. Eubank and Spiegelman (1990) proposed a spline based method for testing the goodness of fit of a linear model. Wooldridge (1992) derived a residual based test which allows for certain non-fixed nonparametric alternatives. A Davidson and MacKinnon (1981)type test is shown to be consistent using a "sieve" estimator for the alternatives. In the next section we discuss the proposed test. In Section 3 we investigate the small sample properties of our test by means of a small Monte Carlo. Finally we conclude. The proofs are in the Appendix.

Semiparametric Regression in Likelihood-Based Models

Abstract: A weighted likelihood is used to estimate the parameters in a semiparametric model involving two covariates and allowing an association between the covariates The development is for arbitrary but specified densities of the observations The estimators are consistent and asymptotically normal Hypothesis testing of the parametric component can be performed using a Wald test Simulations and analysis of data with Bernoulli observations demonstrate the estimators' application Speckman developed kernel estimators where the conditional density of the observations is normal with p parametric covariates Speckman's estimators and the new estimators are asymptotically equivalent, with the bias of Speckman's estimators being smaller As an example, we study the relationship between a binary response indicating the occurrence of an intraoperative cardiac complication (ICC) in vascular surgery patients and two risk factors: duration of the operation (OR) and ASA score, which is an evaluation of the patien

Bayesian Approaches to Non- and Semiparametric Density Estimation

Abstract: This paper proposes and discusses several Bayesian attempts at nonparametric and semiparametric density estimation. The main categories of these ideas are as follows: ( 1) Build a non parametric prior around a given parametric model. We look at cases where the nonparametric part of the construction is a Dirichlet process or relatives thereof. (2) Express the density as an additive expansion of orthogonal basis functions, and place priors on the coefficients. Here attention is given to a certain robust Hermite expansion around the normal distribution. Multiplicative expansions are also considered. (3) Express the unknown density as locally being of a certain parametric form, then construct suitable local likelihood functions to express information content, and place local priors on the local parameters.

Group Duration Analysis of the Proportional Hazard Model: Minimum Chi-squared Estimators and Specification Tests

Abstract: This article develops a semiparametric, minimum chi-squared estimation method of the proportional hazard model for the case when durations are grouped and covariates are categorical. The proposed estimator is easy to compute, yet asymptotically as efficient as the maximum likelihood estimator. This article also suggests simple specification tests for the proportional hazard model. If proportionality holds, then two sets of minimum chi-squared estimators, one from a further grouped data and the other from the original grouped data, will converge to the same quantity; otherwise, they will not. Therefore, a test of the equality of these two sets of estimators will offer a test for proportionality. Monte Carlo simulations demonstrate the performance of these estimators and specification tests. In addition, two real data applications illustrate the implementation of the suggested methods and the contexts in which these methods are useful.

Information and Asymptotic Efficiency in Some Generalized Proportional Hazards Models for Counting Processes

Abstract: Proportional hazards models with stochastic baseline hazards and estimators of the relative risk coefficient in these models were proposed by Prentice, Williams and Peterson and by Chang and Hsiung in medical and industrial contexts. The form of the estimating functions recommended varies according to the form of the unknown stochastic baseline hazards. This paper examines the same estimation problem in the context of large-sample theory. It is shown that the proposed estimators are regular, asymptotically normal and asymptotically efficient. Asymptotic information and representation theorems in the sense of Begun, Hall, Huang and Wellner are also provided for these models.

Advances in Random Utility Models

Abstract: In recent years, major advances have taken place in three areas of random utility modeling: (1) semiparametric estimation, (2) computational methods for multinomial probit models, and (3) computational methods for Bayesian stimation. This paper summarizes these developments and discusses their implications for practice.