Generalized Least Squares Estimation in Contingency Tables Analysis: Asymptotic Properties and Applications

Different sorts of bilinear models (models with bilinear interaction terms) are currently used when analyzing contingency tables: association models, correlation models... All these can be included in a general family of bilinear models: power models. In this framework, Maximum Likelihood (ML) estimation is not always possible, as explained in an introductory example. Thus, Generalized Least Squares (GLS) estimation is sometimes needed in order to estimate parameters. A subclass of power models is then considered in this paper: separable reduced-rank (SRR) models. They allow an optimal choice of weights for GLS estimation and simplifications in asymptotic studies concerning GLS estimators. Power 2 models belong to the subclass of SRR models and the asymptotic properties of GLS estimators are established. Similar results are also established for association models which are not SRR models. However, these results are more difficult to prove. Finally, 2 examples are considered to illustrate our results.     

OLS与ML：回归模型两种参数估计方法的比较研究

最小二乘法(OLS)和最大似然法(ML)是回归模型参数估计的两种最重要的方法。 但二者有着明显的差别,本文就二者之间的有关差别进行比较。     

OPTIMAL IV ESTIMATION OF SYSTEMS WITH STOCHASTIC REGRESSORS AND VAR DISTURBANCES WITH APPLICATIONS TO DYNAMIC SYSTEMS

This paper considers the general problem of Feasible Generalized Least Squares Instrumental Variables (FGLS IV) estimation using optimal instruments. First we summarize the sufficient conditions for the FGLS IV estimator to be asymptotically equivalent to an optimal GLS IV estimator. Then we specialize to stationary dynamic systems with stationary VAR errors, and use the sufficient conditions to derive new moment conditions for these models. These moment conditions produce useful IVs from the lagged endogenous variables, despite the correlation between errors and endogenous variables. This use of the information contained in the lagged endogenous variables expands the class of IV estimators under consideration and thereby potentially improves both asymptotic and small-sample efficiency of the optimal IV estimator in the class. Some Monte Carlo experiments compare the new methods with those of Hatanaka (1976). For the DGP used in the Monte Carlo experiments, asymptotic efficiency is strictly improved by the new IVs, and experimental small-sample efficiency is improved as well.     

OPTIMAL IV ESTIMATION OF SYSTEMS WITH STOCHASTIC REGRESSORS AND VAR DISTURBANCES WITH APPLICATIONS TO DYNAMIC SYSTEMS

This paper considers the general problem of Feasible Generalized Least Squares Instrumental Variables (FGLS IV) estimation using optimal instruments. First we summarize the sufficient conditions for the FGLS IV estimator to be asymptotically equivalent to an optimal GLS IV estimator. Then we specialize to stationary dynamic systems with stationary VAR errors, and use the sufficient conditions to derive new moment conditions for these models. These moment conditions produce useful IVs from the lagged endogenous variables, despite the correlation between errors and endogenous variables. This use of the information contained in the lagged endogenous variables expands the class of IV estimators under consideration and thereby potentially improves both asymptotic and small-sample efficiency of the optimal IV estimator in the class. Some Monte Carlo experiments compare the new methods with those of Hatanaka (1976). For the DGP used in the Monte Carlo experiments, asymptotic efficiency is strictly improved by the new IVs, and experimental small-sample efficiency is improved as well.     

Towards Uniformly Efficient Trend Estimation Under Weak/Strong Correlation and Non‐stationary Volatility

In this paper, we consider the deterministic trend model where the error process is allowed to be weakly or strongly correlated and subject to non‐stationary volatility. Extant estimators of the trend coefficient are analysed. We find that under heteroskedasticity, the Cochrane–Orcutt‐type estimator (with some initial condition) could be less efficient than Ordinary Least Squares (OLS) when the process is highly persistent, whereas it is asymptotically equivalent to OLS when the process is less persistent. An efficient non‐parametrically weighted Cochrane–Orcutt‐type estimator is then proposed. The efficiency is uniform over weak or strong serial correlation and non‐stationary volatility of unknown form. The feasible estimator relies on non‐parametric estimation of the volatility function, and the asymptotic theory is provided. We use the data‐dependent smoothing bandwidth that can automatically adjust for the strength of non‐stationarity in volatilities. The implementation does not require pretesting persistence of the process or specification of non‐stationary volatility. Finite‐sample evaluation via simulations and an empirical application demonstrates the good performance of proposed estimators.     

On the estimation of coefficients of a simultaneous linear explosive model of higher orders with moving average errors generating a pair of time series

In this paper, estimation of coefficients of simultaneous linear partially explosive model of higher orders with moving average errors is considered. It has been shown that the above model can be decomposed into a purely explosive model and an autoregressive model. A two stage estimation, procedure is carried out towards proposing estimators for the partially explosive model. The asymptotic properties of these estimators are also studied.     

On the asymptotic bias of OLS in dynamic regression models with autocorrelated errors

It is well-known that Ordinary Least Squares (OLS) yields inconsistent estimates if applied to a regression equation with lagged dependent variables and correlated errors. Bias expressions which appear in the literature usually assume the exogenous variables to be non-stochastic. Due to this assumption the numerical sizes of these expressions cannot be determined. Further, the analysis is mostly restricted to very simple models. In this paper the problem of calculating the asymptotic bias of OLS is generalized to stationary dynamic regression models, where the errors follow a stationary ARMA process. A general bias expression is derived and a method is introduced by which its actual size can be computed numerically.     

Amemiya's generalized least squares and tests of overidentification in simultaneous equation models with qualitative or limited dependent variables

Amemiya's generalized least squares method for the estimation of simultaneous equation modeis with qualitative or limited dependent variables is known to be efficient relative to many popular two stage estimators. This note points out that test statistics for overidentification restrictions can be obtained as by-products of Amerniya's generalized least squares procedure. Amemiya's procedure is shown to be a minimum chisquare method. The Amemiya procedure is valuable both for efficient estimation and for model evaluation of such models.     

第三类Tobit模型的半参数估计方法

本文给出第三类Tobit模型的一种新的半参数估计方法。在独立性假设下,利用主方程和选择方程中可观察受限因变量的条件生存函数所满足的关系式,构造第三类Tobit模型参数的一步联立估计量。在已知选择方程中参数一致性估计量的条件下,这种方法也可用于构造主方程模型参数 的两步估计量。本文证明了所提出的一步联立估计量和两步估计量的一致性和渐近正态性。实验模拟表明,我们提出的估计量在有限样本下具有良好表现,且一步联立估计量的有限样本表现优于或接近于Chen（1997）的估计量。     

Errors in variables: consistent adjusted least squares (cals) estimation

A single equation errors-in-variables model is considered. Exact restrictions on the parameters in the model are assumed to be available such that the model is just-identified. A Consistent Adjusted Least Squares (CALS) estimator for this model is proposed and its asymptotic distribution is given. Special cases are given as illustrations. CALS is identical to the Method of Moments (MM), and to Maximum Likelihood (ML) under the structural interpretation. Under the functional interpretation it is identical to ML in cases where the latter method is consistent.     

NONNEGATIVE ESTIMATORS FOR THE ONE-WAY RANDOM EFFECTS MODEL

The relative merits of ten estimators for the variance component of the balanced and unbalanced one-way random effects models are compared. Six of the estimators are nonnegative, two of which are obtained by modifying the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and the Weighted Least Square Estimator (WLS), and two more from the positive parts of these estimators. The Minimum Norm Quadratic Estimator (MINQE), which is nonnegative, is adjusted for reducing its bias. The nonnegative Minimum Mean Square Error Estimator (MIMSQE), the Analysis of Variance (ANOVA) and Unweighted Sums of Squares (USS) estimator are also included.     

Amemiya‘s generalized least squares and tests of overidentification in simultaneous equation models with qualitative or limited dependent variables

Amemiya's generalized least squares method for the estimation of simultaneous equation modeis with qualitative or limited dependent variables is known to be efficient relative to many popular two stage estimators. This note points out that test statistics for overidentification restrictions can be obtained as by-products of Amerniya's generalized least squares procedure. Amemiya's procedure is shown to be a minimum chisquare method. The Amemiya procedure is valuable both for efficient estimation and for model evaluation of such models.     

Finite-sample properties of limited-iinformation estimators in misspecified simultaneous equation models

The two most common limited-information estimators in Simultaneous Equation Models are the two-stage least squares and limited-information maximum likelihood estimators. As both of these estimators are complicated functions of the underlying random variables, their exact distributions are difficult to derive. Consequently, their use was first justified on the basis of large sample criteria, such as consistency and asymptotic efficiency. However, in the early 1960s the analysis of the exact distributions and moments of these estimators began, and since this time substantial progress has been made. Although these estimators are asymptotically equivalent, recent research has shown that their finite-sample properties are substantially different. However, the majority of this research has simply concentrated on a correctly specified system of equations, even though, since typically in applied studies theory provides some guidance but falls short of specifying the precise form of structural relationship, the possibilities for misspecification in simultaneous equation models are numerous. The objective of this paper is to extend the finite-sample analysis of these two estimators to include various cases of misspecification.     

Regularized robust estimation in binary regression models

In this paper, we investigate robust parameter estimation and variable selection for binary regression models with grouped data. We investigate estimation procedures based on the minimum-distance approach. In particular, we employ minimum Hellinger and minimum symmetric chi-squared distances criteria and propose regularized minimum-distance estimators. These estimators appear to possess a certain degree of automatic robustness against model misspecification and/or for potential outliers. We show that the proposed non-penalized and penalized minimum-distance estimators are efficient under the model and simultaneously have excellent robustness properties. We study their asymptotic properties such as consistency, asymptotic normality and oracle properties. Using Monte Carlo studies, we examine the small-sample and robustness properties of the proposed estimators and compare them with traditional likelihood estimators. We also study two real-data applications to illustrate our methods. The numerical studies indicate the satisfactory finite-sample performance of our procedures.     

Estimation for Partially Linear Single-index Instrumental Variables Models

In this article, we generalize the partially linear single-index models to the scenario with some endogenous covariates variables. It is well known that the estimators based on the existing methods are often inconsistent because of the endogeneity of covariates. To deal with the endogenous variables, we introduce some auxiliary instrumental variables. A three-stage estimation procedure is proposed for partially linear single-index instrumental variables models. The first stage is to obtain a linear projection of endogenous variables on a set of instrumental variables, the second stage is to estimate the link function by using local linear smoother for given constant parameters, and the last stage is to obtain the estimators of constant parameters based on the estimating equation. Asymptotic normality is established for the proposed estimators. Some simulation studies are undertaken to assess the finite sample performance of the proposed estimation procedure.     

Robust inference in an heteroscedastic measurement error model

In this paper we deal with robust inference in heteroscedastic measurement error models. Rather than the normal distribution, we postulate a Student t distribution for the observed variables. Maximum likelihood estimates are computed numerically. Consistent estimation of the asymptotic covariance matrices of the maximum likelihood and generalized least squares estimators is also discussed. Three test statistics are proposed for testing hypotheses of interest with the asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Results of simulations and an application to a real data set are also reported.     

Spurious Instrumental Variables

Spurious regression phenomenon has been recognized for a wide range of Data Generating Processes: driftless unit roots, unit roots with drift, long memory, trend and broken-trend stationarity, etc. The usual framework is Ordinary Least Squares. We show that the spurious phenomenon also occurs in Instrumental Variables estimation when using non stationary variables, whether the non stationarity component is stochastic or deterministic. Finite sample evidence supports the asymptotic results.     

Two-stage Huber estimation

In this paper we propose a new robust estimator in the context of two-stage estimation methods directed towards the correction of endogeneity problems in linear models. Our estimator is a combination of Huber estimators for each of the two stages, with scale corrections implemented using preliminary median absolute deviation estimators. In this way we obtain a two-stage estimation procedure that is an interesting compromise between concerns of simplicity of calculation, robustness and efficiency. This method compares well with other possible estimators such as two-stage least-squares (2SLS) and two-stage least-absolute-deviations (2SLAD), asymptotically and in finite samples. It is notably interesting to deal with contamination affecting more heavily the distribution tails than a few outliers and not losing as much efficiency as other popular estimators in that case, e.g. under normality. An additional originality resides in the fact that we deal with random regressors and asymmetric errors, which is not often the case in the literature on robust estimators.     

Efficiency analysis of ten estimation procedures for quantitative linear models with autocorrelated errors

Many estimation procedures for quantitative linear models with autocorrelated errors have been proposed in the literature. A number of these procedures have been compared in various ways for different sample sizes and autocorrelation parameters values and for structured or random explanatory vaiables. In this paper, we revisit three situations that were considered to some extent in previous studies, by comparing ten estimation procedures: Ordinary Least Squares (OLS), Generalized Least Squares (GLS), estimated Generalized Least Squares (six procedures), Maximum Likelihood (ML), and First Differences (FD). The six estimated GLS procedures and the ML procedure differ in the way the error autocovariance matrix is estimated. The three situations can be defined as follows: Case 1, the explanatory variable x in the simple linear regression is fixed; Case 2,x is purely random; and Case 3x is first-order autoregressive. Following a theoretical presentation, the ten estimation procedures are compared in a Monte Carlo study conducted in the time domain, where the errors are first-order autoregressive in Cases 1-3. The measure of comparison for the estimation procedures is their efficiency relative to OLS. It is evaluated as a function of the time series length and the magnitude and sign of the error autocorrelation parameter. Overall, knowledge of the model of the time series process generating the errors enhances efficiency in estimated GLS. Differences in the efficiency of estimation procedures between Case 1 and Cases 2 and 3 as well as differences in efficiency among procedures in a given situation are observed and discussed.     

Identification and Efficient Estimation of Simultaneous Equations Network Models

This article considers identification and estimation of social network models in a system of simultaneous equations. We show that, with or without row-normalization of the social adjacency matrix, the network model has different equilibrium implications, needs different identification conditions, and requires different estimation strategies. When the adjacency matrix is not row-normalized, the variation in the Bonacich centrality across nodes in a network can be used as an IV to identify social interaction effects and improve estimation efficiency. The number of such IVs depends on the number of networks. When there are many networks in the data, the proposed estimators may have an asymptotic bias due to the presence of many IVs. We propose a bias-correction procedure for the many-instrument bias. Simulation experiments show that the bias-corrected estimators perform well in finite samples. We also provide an empirical example to illustrate the proposed estimation procedure.