We attempt a three-stage comparison of several strategies for estimating parameters and predicting data in the simultaneous autoregressive model, which is a regression model with spatial autocorrelation in the disturbance between locations as the unit of observation. These strategies differ according to the formulation of the log-likelihood function containing a parametric weight matrix. In the first stage, a chain of logical reasoning is used to obtain theoretical findings by assuming that the data generating model and the data fitting model coincide. We consider the possibility that a subset of locations may be included in neither the parameter estimation nor the data prediction. In the second stage, a series of Monte Carlo experiments are conducted to supplement the theoretical comparison by considering also a mismatch between the two models. The prevalent strategy is defined as an approach that is not based on the exact log-likelihood function, regardless of the setting. The use of this strategy indicates that the parameter estimators do not reflect the mutual connection between all the locations included in the prediction. In the third stage, an empirical comparison is made to confirm the findings from the experimental comparison by using data observed in the real world. We conclude that the reasonable choice is not the prevalent strategy, but a strategy that can be defined as an approach based on the exact log-likelihood function, depending on the setting. The reasonable strategy tailors the parameter estimators to suit the mutual connection between all the locations included in the prediction.

中文翻译：

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基于似然性的策略，用于估计同时自回归模型中的未知参数并预测缺失数据

我们尝试对同时自回归模型中的几种用于估计参数和预测数据的策略进行三阶段比较，这是一种以位置之间的干扰为观察单位的空间自相关的回归模型。根据包含参数权重矩阵的对数似然函数的公式，这些策略会有所不同。在第一阶段，通过假设数据生成模型和数据拟合模型一致，使用逻辑推理链来获得理论结论。我们考虑了在参数估计或数据预测中都不包含位置子集的可能性。在第二阶段，通过考虑两个模型之间的不匹配，进行了一系列的蒙特卡洛实验以补充理论比较。流行策略被定义为一种不基于确切对数似然函数的方法，而与设置无关。使用此策略表示参数估计量未反映预测中包含的所有位置之间的相互联系。在第三阶段，通过使用在现实世界中观察到的数据进行经验比较，以确认实验比较的结果。我们得出的结论是，合理的选择不是普遍的策略，而是一种可以定义为基于确切对数似然函数（取决于设置）的方法的策略。合理的策略可调整参数估计量，以适合预测中包含的所有位置之间的相互联系。