Likelihood estimation and inference in threshold regression

This paper studies likelihood-based estimation and inference in parametric discontinuous threshold regression models with i.i.d. data. The setup allows heteroskedasticity and threshold effects in both mean and variance. By interpreting the threshold point as a “middle” boundary of the threshold variable, we find that the Bayes estimator is asymptotically efficient among all estimators in the locally asymptotically minimax sense. In particular, the Bayes estimator of the threshold point is asymptotically strictly more efficient than the left-endpoint maximum likelihood estimator and the newly proposed middle-point maximum likelihood estimator. Algorithms are developed to calculate asymptotic distributions and risk for the estimators of the threshold point. The posterior interval is proved to be an asymptotically valid confidence interval and is attractive in both length and coverage in finite samples.     

The GMLE based Buckley–James estimator with modified case–cohort data

We consider the estimation problem under the linear regression model with the modified case–cohort design. The extensions of the Buckley–James estimator (BJE) under the case–cohort designs have been studied under an additional assumption that the censoring variable and the covariate are independent. If this assumption is violated, as is the case in a typical real data set in the literature, our simulation results suggest that those extensions are not consistent and we propose a new extension. Our estimator is based on the generalized maximum likelihood estimator (GMLE) of the underlying distributions. We propose a self-consistent algorithm, which is quite different from the one for multivariate interval-censored data. We also show that under certain regularity conditions, the GMLE and the BJE are consistent and asymptotically normally distributed. Some simulation results are presented. The BJE is also applied to the real data set in the literature.     

Conditional empirical likelihood for quantile regression models

In this paper, we propose a new Bayesian quantile regression estimator using conditional empirical likelihood as the working likelihood function. We show that the proposed estimator is asymptotically efficient and the confidence interval constructed is asymptotically valid. Our estimator has low computation cost since the posterior distribution function has explicit form. The finite sample performance of the proposed estimator is evaluated through Monte Carlo studies.     

Mixing of direct, ratio, and product method estimators

In a paper by S rivenkataramana T racy [4], four methods of estimating a population total Y with the use of an auxiliary variable were introduced, given a random sample without replacement from that population. These methods were "built around the idea that estimating the population total is essentially equivalent to estimating the total corresponding to the non-sample units, since that corresponding to the sample units is known once the sample is drawn and measurements are made on it."
However, in the case of small sampling fractions the nonsample units constitute most of the population and no great improvement over the traditional estimators is to be expected. Therefore the methods are compared with the existing estimators and it turns out that they are special cases of the "mixing estimators", introduced in this paper. The latter estimators can be made asymptotically equivalent to the regression estimator and are therefore asymptotically superior to all other estimators. An exact comparison is carried out on the artificial example given in [4]. The statement in this paper that "the proposed estimators are to be preferred to the regression estimator for., superiority of performance in the case of small samples" is evidently misleading. Finally a comparison is made with other "mixing-type" estimators, that can prove very useful in practice.     

Empiric bayes estimation of hypergeometric probability

A Bayes-empiric Bayes estimator of a parameter of the hypergeometric distribution, based on orthogonal polynomials on non-negative integers, is introduced. It is shown that this estimator is asymptotically optimal; and the resulting estimator of the prior probability function is mean square consistent.     

Recursive estimation of simultaneous equation models

A recursive instrumental variable estimator is derived. For simultaneous equation estimation, the choice of the instruments is discussed. A computationally simple and asymptotically efficient recursive estimator is proposed in this context.     

Seemingly unrelated nonlinear regressions

The article considers the estimation of the parameters of a set of nonlinear regression equations when the responses are contemporaneously but not serially correlated. Conditions are set forth such that the estimator obtained is strongly consistent, asymptotically normally distributed, and asymptotically more efficient than the single-equation least squares estimator. The methods presented allow estimation of the parameters subject to nonlinear restrictions across equations. The article includes a discussion of methods to perform the computations and a Monte Carlo simulation.     

Reduced rank regression in cointegrated models

The coefficient matrix of a cointegrated first-order autoregression is estimated by reduced rank regression (RRR), depending on the larger canonical correlations and vectors of the first difference of the observed series and the lagged variables. In a suitable coordinate system the components of the least-squares (LS) estimator associated with the lagged nonstationary variables are of order 1/T, where T is the sample size, and are asymptotically functionals of a Brownian motion process; the components associated with the lagged stationary variables are of the order T^−1/2 and are asymptotically normal. The components of the RRR estimator associated with the stationary part are asymptotically the same as for the LS estimator. Some components of the RRR estimator associated with nonstationary regressors have zero error to order 1/T and the other components have a more concentrated distribution than the corresponding components of the LS estimator.     

Relative efficiency and deficiency of kernel type estimators of smooth distribution functions

Abstract  The problem is investigated whether a given kernel type estimator of a distribution function at a single point has asymptotically better performance than the empirical estimator. A representation of the relative deficiency of the empirical distribution function with respect to a kernel type estimator is established which gives a complete solution to this problem. The problem of finding optimal kernels is studied in detail.     

An efficient GMM estimator of spatial autoregressive models

In this paper, we consider GMM estimation of the regression and MRSAR models with SAR disturbances. We derive the best GMM estimator within the class of GMM estimators based on linear and quadratic moment conditions. The best GMM estimator has the merit of computational simplicity and asymptotic efficiency. It is asymptotically as efficient as the ML estimator under normality and asymptotically more efficient than the Gaussian QML estimator otherwise. Monte Carlo studies show that, with moderate-sized samples, the best GMM estimator has its biggest advantage when the disturbances are asymmetrically distributed. When the diagonal elements of the spatial weights matrix have enough variation, incorporating kurtosis of the disturbances in the moment functions will also be helpful.     

Quantile-based nonparametric inference for first-price auctions

We propose a quantile-based nonparametric approach to inference on the probability density function (PDF) of the private values in first-price sealed-bid auctions with independent private values. Our method of inference is based on a fully nonparametric kernel-based estimator of the quantiles and PDF of observable bids. Our estimator attains the optimal rate of Guerre et al. (2000), and is also asymptotically normal with an appropriate choice of the bandwidth.     

Optimal inference for instrumental variables regression with non-Gaussian errors

This paper is concerned with inference on the coefficient on the endogenous regressor in a linear instrumental variables model with a single endogenous regressor, nonrandom exogenous regressors and instruments, and i.i.d. errors whose distribution is unknown. It is shown that under mild smoothness conditions on the error distribution it is possible to develop tests which are “nearly” efficient in the sense of Andrews et al. (2006) when identification is weak and consistent and asymptotically optimal when identification is strong. In addition, an estimator is presented which can be used in the usual way to construct valid (indeed, optimal) confidence intervals when identification is strong. The estimator is of the two stage least squares variety and is asymptotically efficient under strong identification whether or not the errors are normal.     

Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood

The paper discusses the asymptotic validity of posterior inference of pseudo‐Bayesian quantile regression methods with complete or censored data when an asymmetric Laplace likelihood is used. The asymmetric Laplace likelihood has a special place in the Bayesian quantile regression framework because the usual quantile regression estimator can be derived as the maximum likelihood estimator under such a model, and this working likelihood enables highly efficient Markov chain Monte Carlo algorithms for posterior sampling. However, it seems to be under‐recognised that the stationary distribution for the resulting posterior does not provide valid posterior inference directly. We demonstrate that a simple adjustment to the covariance matrix of the posterior chain leads to asymptotically valid posterior inference. Our simulation results confirm that the posterior inference, when appropriately adjusted, is an attractive alternative to other asymptotic approximations in quantile regression, especially in the presence of censored data.     

On the estimation of multinomial logit models from relative frequency data

Standard estimators for the binomial logit model and for the multinomial logit model allow for an error arising from the use of relative frequencies instead of the true probabilities as the dependent variable. Recently Amemiya and Nold (1975) have considered the effect of the presence of an additional specification error in the binomial logit model and have proposed a modified logit estimation scheme to take the additional error variance into account. This paper extends their idea to the multinomial logit model and proposes an estimator that is consistent and asymptotically more efficient than the standard multinomial logit estimator. The paper presents a comparison of the results of applying the new estimator and existing estimators to a logit model for the choice of automobile ownership in the United States.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

非线性计量经济模型的非参数估计方法研究

本文提出使用核估计的方法构造平滑转移模型(STR)的非参数模拟最大似然估计(NPSML),给出了NPSML估计量的构造方法、渐近性质以及相应的核函数和窗宽的选择准则,并利用滑动窗宽算法对估计量的构造过程进行了改进。通过Monte Carlo实验证明,该方法是可靠的,并且当误差项存在序列相关时,此种估计量是稳健的。     

Least absolute deviations estimation for the censored regression model

This paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression (or censored ‘Tobit’) model. The proposed estimator is a generalization of least absolute deviations estimation for the standard linear model, and, unlike estimation methods based on the assumption of normally distributed error terms, the estimator is consistent and asymptotically normal for a wide class of error distributions, and is also robust to heteroscedasticity. The paper gives the regularity conditions and proofs of these large-sample results, and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances.     

Estimating and testing a quantile regression model with interactive effects

This paper proposes a quantile regression estimator for a model with interactive effects potentially correlated with covariates. We provide conditions under which the estimator is asymptotically Gaussian and we investigate the finite sample performance of the method. An approach to testing the specification against a competing fixed effects specification is introduced. The paper presents an application to study the effect of class size and composition on educational attainment. The evidence suggests that while smaller classes are beneficial for low performers, larger classes are beneficial for high performers. The fixed effects specification is rejected in favor of the interactive effects specification.     

Calibration Estimation in Survey Sampling

Calibration estimation, where the sampling weights are adjusted to make certain estimators match known population totals, is commonly used in survey sampling. The generalized regression estimator is an example of a calibration estimator. Given the functional form of the calibration adjustment term, we establish the asymptotic equivalence between the functional-form calibration estimator and an instrumental variable calibration estimator where the instrumental variable is directly determined from the functional form in the calibration equation. Variance estimation based on linearization is discussed and applied to some recently proposed calibration estimators. The results are extended to the estimator that is a solution to the calibrated estimating equation. Results from a limited simulation study are presented.     

Three-stage least-squares estimation for a system of simultaneous,nonlinear, implicit equations

The article describes a nonlinear three-stage least-squares estimator for the parameters of a system of simultaneous, nonlinear, implicit equations; the method allows the estimation of these parameters subject to nonlinear parametric restrictions across equations. The estimator is shown to be strongly consistent, asymptotically normally distributed, and more efficient than the nonlinear two-stage least-squares estimator. Some practical implications of the regularity conditions used to obtain these results are discussed from the point of view of one whose interest is in applications, Also, computing methods using readily available nonlinear regression programs are described.