Common techniques for approximating the maximum likelihood estimator often lead to biased and inconsistent estimates when a systematic bias is introduced into the log-likelihood function. This article proposes to replace the true scores in the likelihood equations with quasi-scores that are easy-to-compute yet have an expected value of zero. This innovation would dramatically simplify the likelihood-based estimation while preserving a majority of desirable statistical properties of maximum likelihood. Consequently, this article introduces an easy-to-compute consistent estimator for models with spatial dependence and documents its computational and statistical properties. To illustrate the quality and predictability of the closed form estimator in a practical setting we use Monte Carlo simulations that address both the spatial lag and spatial error settings. The estimated variance of the estimator is elevated relative to that of maximum likelihood’s asymptotic variance and actual method’s performance. However, it is shown that inflated confidence intervals are largely inconsequential to the analysis of spatial dependence in practice. The estimator is shown to be computationally superior to the true maximum likelihood for any tested matrix sizes and levels of spatial dependence.

中文翻译：

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具有空间依赖性的线性模型的闭式一致估计器

当系统偏差被引入对数似然函数时，用于逼近最大似然估计量的常用技术通常会导致有偏差和不一致的估计。本文建议将似然方程中的真实分数替换为易于计算但期望值为零的准分数。这项创新将极大地简化基于似然的估计，同时保留大多数理想的最大似然统计特性。因此，本文介绍了一种易于计算的具有空间依赖性的模型的一致估计器，并记录了其计算和统计特性。为了说明封闭形式估计器在实际设置中的质量和可预测性，我们使用蒙特卡罗模拟来解决空间滞后和空间误差设置。估计量的估计方差相对于最大似然的渐近方差和实际方法的性能有所提高。然而，事实表明，膨胀的置信区间在很大程度上与实践中的空间依赖性分析无关。对于任何测试的矩阵大小和空间相关性水平，估计器在计算上都优于真实的最大似然。结果表明，膨胀的置信区间在很大程度上与实践中的空间依赖性分析无关。对于任何测试的矩阵大小和空间相关性水平，估计器在计算上都优于真实的最大似然。结果表明，膨胀的置信区间在很大程度上与实践中的空间依赖性分析无关。对于任何测试的矩阵大小和空间相关性水平，估计器在计算上都优于真实的最大似然。