Unknown Heteroscedasticity Research Articles

A novel Bayesian framework to address unknown heteroscedasticity for the linear regression model

A common problem that we encounter in linear regression models is that the error variances are not the same for all the observations, that is, there is an issue of heteroscedasticity. To avoid the adverse effects of this issue on the estimates obtained by the ordinary least squares, it is a usual practice to use the estimated weighted least squares (EWLS) method or to use some adaptive methods, especially when the form of heteroscedasticity is unknown. The current article proposes a novel Bayesian version of the EWLS estimator. The performance of the proposed estimator has been evaluated in terms of its efficiency using the Monte Carlo simulations. An example has also been included for the demonstration.

A semi-parametric empirical likelihood approach for conditional estimating equations under endogenous selection

The estimation and inference for conditional estimating equations models with endogenous selection, are considered. The approach takes into account possible endogenous selection which may lead to a selection bias. It can be used for a wide range of statistical models not covered by the model-based sampling theory. Endogeneity can be either part of the selection or within the covariates. It is particularly well suited for models with unknown heteroscedasticity, uncontrolled confounders and measurement errors. It will not be necessary to model the relationship between the endogenous covariates and the instrumental variables, which offers major advantages over two-stage least-squares. The approach proposed has the advantage of being based on a fixed number of constraints determined by the size of the parameter.

Estimating a spatial autoregressive model with autoregressive disturbances based on the indirect inference principle

ABSTRACT This paper proposes a new estimation procedure for the first-order spatial autoregressive (SAR) model, where the disturbance term also follows a first-order autoregression and its innovations may be heteroscedastic. The estimation procedure is based on the principle of indirect inference that matches the ordinary least squares estimator of the two SAR coefficients (one in the outcome equation and the other in the disturbance equation) with its approximate analytical expectation. The resulting estimator is shown to be consistent, asymptotically normal and robust to unknown heteroscedasticity. Monte Carlo experiments are provided to show its finite-sample performance in comparison with existing estimators that are based on the generalized method of moments. The new estimation procedure is applied to empirical studies on teenage pregnancy rates and Airbnb accommodation prices.

An adaptive weighted least squares ratio approach for estimation of heteroscedastic linear regression model in the presence of outliers

The issue of heteroscedasticity and its adverse impact on the estimation of a linear regression model has been extensively discussed in the available literature. Some adaptive estimators have also been proposed in the context of weighted least squares (WLS) to address the issue. Estimation of linear regression model becomes more challenging when the issue of heteroscedasticity is bundled together with the presence of outliers. In the present article, we use a relatively latest approach that is the least squares ratio method to estimate a linear regression model in the presence of heteroscedasticity and outliers. We propose an adaptive version of this technique while taking advantage of some existing adaptive estimators to fix the issue of unknown heteroscedasticity. A Monte Carlo evidence has been presented for the performance of the stated estimator varying degree of heteroscedasticity and number of outliers.

Indirect Inference Estimation of Spatial Autoregressions

The ordinary least squares (OLS) estimator for spatial autoregressions may be consistent as pointed out by Lee (2002), provided that each spatial unit is influenced aggregately by a significant portion of the total units. This paper presents a unified asymptotic distribution result of the properly recentered OLS estimator and proposes a new estimator that is based on the indirect inference (II) procedure. The resulting estimator can always be used regardless of the degree of aggregate influence on each spatial unit from other units and is consistent and asymptotically normal. The new estimator does not rely on distributional assumptions and is robust to unknown heteroscedasticity. Its good finite-sample performance, in comparison with existing estimators that are also robust to heteroscedasticity, is demonstrated by a Monte Carlo study.

Measurement and Structural Factors Influencing China’s Provincial Total-Factor Energy Efficiency Under Resource and Environmental Constraints

We studied the measurement and structural factors influencing China’s provincial total-factor energy efficiency (TFEE) under resource and environmental constraints, using spatial weight matrix analysis, spatial econometric model selection, a generalized spatial econometric model with unknown heteroscedasticity, and a directional distance function global Malmquist–Luenberger (GML) superefficient model. The findings of this empirical research are as follows. Resource and environmental constraints should be considered while measuring TFEE. The results obtained in such cases are more accurate reflections of the actual situation in China. Furthermore, spatial effects should be considered when analyzing the factors influencing provincial TFEE; otherwise, the estimates will be biased. The following conclusions were obtained from the results of the empirical analysis: China’s provincial TFEE continued to decline under resource and environmental constraints, and the trend is not optimistic, implying an undue reliance on coal resources, which reduce TFEE by a considerable extent. Moreover, China’s interprovincial TFEE is affected by a variety of structural factors.

A Monte Carlo comparison of GMM and QMLE estimators for short dynamic panel data models with spatial errors

ABSTRACTWe suggest a generalized spatial system GMM (SGMM) estimation for short dynamic panel data models with spatial errors and fixed effects when n is large and T is fixed (usually small). Monte Carlo studies are conducted to evaluate the finite sample properties with the quasi-maximum likelihood estimation (QMLE). The results show that, QMLE, with a proper approximation for initial observation, performs better than SGMM in general cases. However, it performs poorly when spatial dependence is large. QMLE and SGMM perform better for different parameters when there is unknown heteroscedasticity in the disturbances and the data are highly persistent. Both estimates are not sensitive to the treatment of initial values. Estimation of the spatial autoregressive parameter is generally biased when either the data are highly persistent or spatial dependence is large. Choices of spatial weights matrices and the sign of spatial dependence do affect the performance of the estimates, especially in the case of the heteroscedastic disturbance. We also give empirical guidelines for the model.

Identification and estimation of partially linear censored regression models with unknown heteroscedasticity

In this paper, we introduce a new identification and estimation strategy for partially linear regression models with a general form of unknown heteroscedasticity, that is, Y=X′β0+m(Z)+U and U=σ(X,Z)ε⁠, where ε is independent of (X,Z) and the functional forms of both m(·) and σ(·) are left unspecified. We show that in such a model, β0 and m(·) can be exactly identified while σ(·) can be identified up to scale as long as σ(X,Z) permits sufficient nonlinearity in X. A two‐stage estimation procedure motivated by the identification strategy is described and its large sample properties are formally established. Moreover, our strategy is flexible enough to allow for both fixed and random censoring in the dependent variable. Simulation results show that the proposed estimator performs reasonably well in finite samples.

Estimation of spatial panel data models with randomly missing data in the dependent variable

We suggest and compare different methods for estimating spatial autoregressive panel models with randomly missing data in the dependent variable. We start with a random effects model and then generalize the model by introducing the spatial Mundlak approach. A nonlinear least squares method is suggested and a generalized method of moments estimation is developed for the model. A two-stage least squares estimation with imputation is proposed as well. We analytically compare these estimation methods and find that the generalized nonlinear least squares, best generalized two-stage least squares with imputation, and best method of moments estimators have identical asymptotic variances. The robustness of these estimation methods against unknown heteroscedasticity is also stressed since the traditional maximum likelihood approach yields inconsistent estimates under unknown heteroscedasticity. We provide finite sample evidence through Monte Carlo experiments.

Estimation of spatial autoregressive models with randomly missing data in the dependent variable

Summary We suggest and compare different methods for estimating spatial autoregressive models with randomly missing data in the dependent variable. Aside from the traditional expectation-maximization (EM) algorithm, a nonlinear least squares method is suggested and a generalized method of moments estimation is developed for the model. A two-stage least squares estimation with imputation is proposed as well. We analytically compare these estimation methods and find that generalized nonlinear least squares, best generalized two-stage least squares with imputation and best method of moments estimators have identical asymptotic variances. These methods are less efficient than maximum likelihood estimation implemented with the EM algorithm. When unknown heteroscedasticity exists, however, EM estimation produces inconsistent estimates. Under this situation, these methods outperform EM. We provide finite sample evidence through Monte Carlo experiments.