TESTING FOR SERIAL CORRELATION IN LEAST SQUARES REGRESSION .1.
TESTING FOR SERIAL CORRELATION IN LEAST SQUARES REGRESSION .1.
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DOI：
10.1093/biomet/37.3-4.409
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发表时间：
1950-01-01
期刊：
影响因子：
2.7
通讯作者：
WATSON, GS
中科院分区：
文献类型：
--
作者：
DURBIN, J;WATSON, GS
毫无疑问，在已知最小二乘回归方法不适用的情况下，它们被大量使用。特别是，它们经常被用于分析时间序列和类似的数据，在这些数据中，连续的观测数据是序列相关的。由此产生的并发症是众所周知的，最近Cochrane&Orutt(1949)从计量经济学的角度对其进行了研究。应用最小二乘法的一个基本假设是回归模型中的误差项是独立的。当这一假设--以及其他假设--成立时，无论观测数据本身是否具有序列相关性，该过程都是有效的。检验误差是否独立的问题构成了本文及其后继者的主题。本文主要讨论测试所依据的理论，而第二篇文章详细描述了测试程序，并给出了所采用的测试标准的有效点的界限表。在这两篇论文中，我们都不会关心如果测试结果不好应该做些什么的问题。由于在任何实际情况下的误差都是未知的，所以检验必须基于计算回归的残差。因此，不能使用普通的独立性检验，因为残差必然是相关的，无论误差是不是相关的。Moran(1950)对于单一自变量回归的情况，计算了适当检验统计量的均值和方差。构造精确检验的问题只在一种特殊情况下得到了完全解决。RL&TW Anderson(1950)证明了对于短付里叶级数回归的情形，RL Anderson(1942)得到的圆形序列相关系数的分布可以用来获得有关检验准则的精确有效点。这是由于回归向量与循环序列协方差矩阵的潜在向量重合所致。奇怪的是，这正是最不需要检验的情况，因为即使在非零的情况下，最小二乘回归系数也是最好的无偏估计，此外，还可以获得至少渐近无偏的方差估计。实际上，潜在向量情况是唯一可以获得优雅解的情况。似乎不可能找到任何其他情况的确切意义点。然而，有意义点的界限是可以得到的，在第二篇论文中，这些界限将被列成表格。我们将给出的界限在两种意义上是“最好的”：第一，它们是可以达到的(用一种将在后面讨论的回归向量)，第二，当它们达到时，所采用的检验标准对于适当的替代假设是一致的最强大的。希望这些界限将以这样或那样的方式解决实践中出现的许多情况下的意义问题。对于可疑的案件，似乎没有任何完全令人满意的程序。然而，我们将指出在某些情况下可能有用的一些近似方法。
A great deal of use has undoubtedly been made of least squares regression methods in circumstances in which they are known to be inapplicable. In particular, they have often been employed for the analysis of time series and similar data in which successive observations are serially correlated. The resulting complications are well known and have recently been studied from the standpoint of the econometrician by Cochrane & Orcutt (1949). A basic assumption underlying the application of the least squares method is that the error terms in the regression model are independent. When this assumption-among others-is satisfied the procedure is valid whether or not the observations themselves are serially correlated. The problem of testing the errors for independence forms the subject of this paper and its successor. The present paper deals mainly with the theory on which the test is based, while the second paper describes the test procedures in detail and gives tables of bounds to the significance points of the test criterion adopted. We shall not be concerned in either paper with the question of what should be done if the test gives an unfavourable result. Since the errors in any practical case will be unknown the test must be based on the residuals from the calculated regression. Consequently the ordinary tests of independence cannot be used as they stand, since the residuals are necessarily correlated whether the errors are dependent or not. The mean and variance of an appropriate test statistic have been calculated by Moran (1950) for the case of regression on a single independent variable. The problem of constructing an exact test has been completely solved only in one special case. RL & TW Anderson (1950) have shown that for the case of regression on a short Fourier series the distribution of the circular serial correlation coefficient obtained by RL Anderson (1942) can be used to obtain exact significance points for the test criterion concerned. This is due to the coincidence of the regression vectors with the latent vectors of the circular serial covariance matrix. Perversely enough, this is the very case in which the test is least needed, since the least squares regression coefficients are best unbiased estimates even in the non-null case, and in addition estimates of their variance can be obtained which are at least asymptotically unbiased.The latent vector case is in fact the only one for which an elegant solution can be obtained. It does not seem possible to find exact significance points for any other case. Nevertheless, bounds to the significance points can be obtained, and in the second paper such bounds will be tabulated. The bounds we shall give are'best'in two senses: first they can be attained (with regression vectors of a type that will be discussed later), and secondly, when they are attained the test criterion adopted is uniformly most powerful against suitable alternative hypotheses. It is hoped that these bounds will settle the question of significance one way or the other in many cases arising in practice. For doubtful cases there does not seem to be any completely satisfactory procedure. We shall, however, indicate some approximate methods which may be useful in certain circumstances.