Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression

In geo-statistics, the Durbin-Watson test is frequently employed to detect the presence of residual serial correlation from least squares regression analyses. However, the Durbin-Watson statistic is only suitable for ordered time or spatial series. If the variables comprise cross-sectional data coming from spatial random sampling, the test will be ineffectual because the value of Durbin-Watson’s statistic depends on the sequence of data points. This paper develops two new statistics for testing serial correlation of residuals from least squares regression based on spatial samples. By analogy with the new form of Moran’s index, an autocorrelation coefficient is defined with a standardized residual vector and a normalized spatial weight matrix. Then by analogy with the Durbin-Watson statistic, two types of new serial correlation indices are constructed. As a case study, the two newly presented statistics are applied to a spatial sample of 29 China’s regions. These results show that the new spatial autocorrelation models can be used to test the serial correlation of residuals from regression analysis. In practice, the new statistics can make up for the deficiencies of the Durbin-Watson test.

在地理统计学（geo-statistics）中，德宾-沃森检验（Durbin-Watson test）常被用于检测最小二乘回归分析所产生的残差序列相关性。然而，德宾-沃森统计量仅适用于有序时间序列或空间序列。若研究变量为空间随机抽样得到的横截面数据，则该检验将失效，因为德宾-沃森统计量的取值依赖于数据点的序列顺序。本文提出了两种全新的统计量，用于检验基于空间抽样样本的最小二乘回归残差序列相关性。类比莫兰指数（Moran’s index）的新形式，本文通过标准化残差向量与归一化空间权重矩阵，定义了一种自相关系数。随后，类比德宾-沃森统计量，构建了两类新型序列相关指数。作为案例研究，本文将所提出的两种新型统计量应用于包含中国29个地区的空间抽样样本。研究结果表明，新型空间自相关模型可用于检验回归分析中的残差序列相关性。在实际应用中，该新型统计量可弥补德宾-沃森检验的不足。