Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity

This paper explores the asymptotic distribution of the cointegrating vector estimator in error correction models with conditionally heteroskedastic errors. Asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector, which estimates the cointegrating vector and the multivariate GARCH process jointly, are provided. The MLE of the cointegrating vector follows mixture normal, and its asymptotic distribution depends on the conditional heteroskedasticity and the kurtosis of standardized innovations. The reduced rank regression (RRR) estimator and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity, and thus the efficiency gain of the MLE emerges as the magnitude of conditional heteroskedasticity increases. The simulation results indicate that the relative power of the t-statistics based on the MLE improves significantly as the GARCH effect increases.     

Consistent likelihood-based estimation of a star-shaped distribution

We consider the classical problem of nonparametrically estimating a star-shaped distribution, i.e., a distribution function F on [0,∞) with the property that F(u)/u is nondecreasing on the set {u : F(u) < 1}. This problem is intriguing because of the fact that a well defined maximum likelihood estimator (MLE) exists, but this MLE is inconsistent. In this paper, we argue that the likelihood that is commonly used in this context is somewhat unnatural and propose another, so called ‘smoothed likelihood’. However, also the resulting MLE turns out to be inconsistent. We show that more serious smoothing of the likelihood yields consistent estimators in this model.     

A Monte Carlo Study on the Finite Sample Properties of the Gibbs Sampling Method for a Stochastic Frontier Model

In this paper we use Monte Carlo study to investigate the finite sample properties of the Bayesian estimator obtained by the Gibbs sampler and its classical counterpart (i.e. the MLE) for a stochastic frontier model. Our Monte Carlo results show that the MSE performance of the estimates of Gibbs sampling are substantially better than that of the MLE.     

A Bayesian-like estimator of the process capability index Cpmk

Pearn et al. (1992) proposed the capability index C_pmk, and investigated the statistical properties of its natural estimator for stable normal processes with constant mean μ. Chen and Hsu (1995) showed that under general conditions the asymptotic distribution of is normal if μ≠m, and is a linear combination of the normal and the folded-normal distributions if μ=m, where m is the mid-point between the upper and the lower specification limits. In this paper, we consider a new estimator for stable processes under a different (more realistic) condition on process mean, namely, P (μ≥m)=p, 0≤p≤1. We obtain the exact distribution, the expected value, and the variance of under normality assumption. We show that for P (μ≥m)=0, or 1, the new estimator is the MLE of C_pmk, which is asymptotically efficient. In addition, we show that under general conditions is consistent and is asymptotically unbiased. We also show that the asymptotic distribution of is a mixture of two normal distributions. RID="*" ID="*"  The research was partially supported by National Science Council of the Republic of China (NSC-89-2213-E-346-003).     

Almost unbiased product-type estimator

In sample survey methods the use of product estimators was suggested byMurthy [1964] andSrivastava [1966] and were found to serve good purpose provided the two variables viz. the main variable under study and the auxiliary variable have a very high negative correlation between them. The product estimators suggested by them are biased. In the present paper the author has obtained unbiased product estimators (to the first degree of approximation) with the help of the technique developed byQuenouille [1956] and has established that this new estimator is better than the other product estimator in the mean square error sense.     

Estimation of bias and standard error of an improved estimator of normal mean

This paper considers an improved estimator of normal mean which is obtained by considering a feasible version of minimum mean squared error estimator. The exact expression for the bias and the mean squared error are fairly complicated and do not provide any guidelines as how to estimate the standard error of improved estimator. As is well known that any estimator without a formula for standard error has little practical utility. We therefore derive unbiased estimators for the bias and mean squared error of the improved estimator. Incidently, they turn out to be minimum variance unbiased estimators. Further, this exercise yields a simple formula for estimating the standard error. Based on the criterion of estimated standard error, the efficiency of the improved estimator with respect to the traditional unbiased estimator (i.e., sample mean) is examined numerically. The relationship with asymptotic standard error is also studied.     

Further evidence on the robustness of the Tobit estimator to heteroskedasticity

When the error terms in a Tobit model are heteroskedastic, the MLE which assumes homoskedasticity is inconsistent. For the special case of a constant-term-only model, we investigate the size of the inconsistency. The inconsistency is greater the greater the heteroskedasticity and the greater the degree of censoring (i.e., the greater the number of limit observations). However, the inconsistency is much smaller than in the corresponding truncated-normal model considered by Hurd.     

A distribution-free least squares estimator for censored linear regression models

This paper describes a method for estimating simultaneously the parameter vector of the systematic component and the distribution function of the random component of a censored linear regression model. The estimator is obtained by minimizing the sum of the squares of the differences between the observed values of the dependent variable and the corresponding expected values of this variable according to the estimated parameter vector and distribution function. The resulting least squares parameter estimator incorporates information on the distribution of the random component of the regression model that is available from the estimation sample. Hence, it may often be more efficient than are parameter estimators that do not use such information. The results of numerical experiments with the least squares estimator tend to support this hypothesis.     

Mean squared error of ridge estimators in logistic regression

It is well known that the maximum likelihood estimator (MLE) is inadmissible when estimating the multidimensional Gaussian location parameter. We show that the verdict is much more subtle for the binary location parameter. We consider this problem in a regression framework by considering a ridge logistic regression (RR) with three alternative ways of shrinking the estimates of the event probabilities. While it is shown that all three variants reduce the mean squared error (MSE) of the MLE, there is at the same time, for every amount of shrinkage, a true value of the location parameter for which we are overshrinking, thus implying the minimaxity of the MLE in this family of estimators. Little shrinkage also always reduces the MSE of individual predictions for all three RR estimators; however, only the naive estimator that shrinks toward 1/2 retains this property for any generalized MSE (GMSE). In contrast, for the two RR estimators that shrink toward the common mean probability, there is always a GMSE for which even a minute amount of shrinkage increases the error. These theoretical results are illustrated on a numerical example. The estimators are also applied to a real data set, and practical implications of our results are discussed.     

Another look at the naive estimator in a regression model

We consider the linear regression model where only a particular linear function of the dependent variables is observed, Stahlecker and Schmidt (1987) proposed a naive least squares (LS) estimator for regression coefficients in such a case. In this note we represent their estimator as a general ridge estimator. This observation leads to a view different from the previous work and provides an easy way of obtaining many important properties of the naive LS estimator. Our approach also gives some insight into the relationship between the naive LS estimator and the generalized least squares estimator.     

有效价差的极大似然估计

有效价差是刻画金融资产交易成本的一种重要度量。本文基于Roll的价格模型,利用对数价格极差分布的近似正态特征,提出了一种有效价差的近似极大似然估计,并通过数值模拟比较了这一新的估计与以往文献中提出的Roll的协方差估计、贝叶斯估计以及High-Low估计在各种不同状况下的精度。模拟的结果表明,无论是在连续交易的理想状态还是交易不连续且价格不能被完全观测到的非理想状态下,极大似然估计和High-Low估计的精度均高于协方差和贝叶斯估计;当波动率相对较小的时候,极大似然估计的精度优于High-Low估计;另外,在非理想情形下,极大似然估计要比High-Low估计更加稳健。     

A simple estimator for nonlinear error in variable models

We propose a simple estimator for nonlinear method of moment models with measurement error of the classical type when no additional data, such as validation data or double measurements, are available. We assume that the marginal distributions of the measurement errors are Laplace (double exponential) with zero means and unknown variances and the measurement errors are independent of the latent variables and are independent of each other. Under these assumptions, we derive simple revised moment conditions in terms of the observed variables. They are used to make inference about the model parameters and the variance of the measurement error. The results of this paper show that the distributional assumption on the measurement errors can be used to point identify the parameters of interest. Our estimator is a parametric method of moments estimator that uses the revised moment conditions and hence is simple to compute. Our estimation method is particularly useful in situations where no additional data are available, which is the case in many economic data sets. Simulation study demonstrates good finite sample properties of our proposed estimator. We also examine the performance of the estimator in the case where the error distribution is misspecified.     

A Simple Improvement of the IV‐estimator for the Classical Errors‐in‐Variables Problem

Two measures of an error‐ridden variable make it possible to solve the classical errors‐in‐Variable problem by using one measure as an instrument for the other. It is well known that a second IV‐estimate can be obtained by reversing the roles of the two measures. We explore the optimal linear combination of these two estimates. In a Monte Carlo study, we show that the gain in precision is significant. The proposed estimator also compares well with full information maximum likelihood under normality. We illustrate the method by estimating the capital elasticity in the Norwegian ICT‐industry.     

A local maximum likelihood estimator for Poisson regression

A local maximum likelihood estimator based on Poisson regression is presented as well as its bias, variance and asymptotic distribution. This semiparametric estimator is intended to be an alternative to the Poisson, negative binomial and zero-inflated Poisson regression models that does not depend on regularity conditions and model specification accuracy. Some simulation results are presented. The use of the local maximum likelihood procedure is illustrated on one example from the literature. This procedure is found to perform well. This research was partially supported by Calouste Gulbenkian Foundation and PRODEP III.     

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.     

Properties of the CUE estimator and a modification with moments

In this paper, we analyze properties of the Continuous Updating Estimator (CUE) proposed by Hansen et al. (1996), which has been suggested as a solution to the finite sample bias problems of the two-step GMM estimator. We show that the estimator should be expected to perform poorly in finite samples under weak identification, in particular, the estimator is not guaranteed to have finite moments of any order. We propose the Regularized CUE (RCUE) as a solution to this problem. The RCUE solves a modification of the first-order conditions for the CUE estimator and is shown to be asymptotically equivalent to CUE under many weak moment asymptotics. Our theoretical findings are confirmed by extensive Monte Carlo studies.     

Small sample properties of the mixed regression estimator

In this paper, the small sample properties of the mixed regression estimator are examined when prior information may be biased and when the ration of the variance of the prior restriction errors to the variance of the sample errors is unknown. The mean square error of the mixed regression estimator is derived, and it is shown that the mixed regression estimator gets dominated by the ordinary least squares estimator in terms of the mean square error as the bias of prior information gets larger.     

The nonlinear limited-information maximum- likelihood estimator and the modified nonlinear two-stage least-squares estimator

Several limited-information type estimators of the nonlinear simultaneous equation model are considered and their asymptotic covariance matrices are compared. Amemiya (1974) proposed the general class of nonlinear two-stage least-squares estimators. In this paper, its two specific members are considered and, in addition, the nonlinear limited-information maximum- likelihood estimator and the modified nonlinear two-stage least-squares estimator are proposed. Both are shown to be asymptotically more efficient than the nonlinear two-stage least-squares estimator, and the second has the advantage of being computationally simple.     

On the asymptotic optimality of the LIML estimator with possibly many instruments

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new as well as old, and we relate them to results in some recent studies. We have found that the variance of the limiting distribution of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models of some cases recently called many instruments and many weak instruments.