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.

Testing for serial correlation in least squares regression. II.

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The fitting of time series models

Serial correlation in regression analysis. i

It is well known that the ordinary least squares estimator of Xβ in the general linear model E y = Xβ, cov y = σ2 V, can be the best linear unbiased estimator even if V is not a multiple of the identity matrix. This article presents, in a historical perspective, the development of the several conditions for the ordinary least squares estimator to be best linear unbiased. Various characterizations of these conditions, using generalized inverses and orthogonal projectors, along with several examples, are also given. In addition, a complete set of references is provided.

The Equality of the Ordinary Least Squares Estimator and the Best Linear Unbiased Estimator

This paper provides a new framework for analyzing white noise disturbances in linear systems; rather than the usual stochastic approach, noise signals are described as elements in sets, and their effect is analyzed from a worst-case perspective. The paper studies how these sets must be chosen to have adequate properties for system response in the worst-case, statistics consistent with the stochastic point of view, and simple descriptions that allow for tractable worst-case analysis. The method is demonstrated by considering its implications in two problems: rejection of white noise signals in the presence of system uncertainty and worst-case system identification.

A set-based approach for white noise modeling

In this paper the observations are considered to be normally distributed with constant variance and means consisting of linear combinations of certain trigonometric functions. The likelihood ratio criterion for testing the independence of the observations against the alternatives of circular serial correlation of a given lag is found to be a function of the circular serial correlation coefficient for residuals from the fitted Fourier series (Section 4). The exact distribution (Section 5), the moments (Section 6), and approximate distributions (Section 7) are given for the cases of greatest interest. From these results significance levels have been found (Section 3). The use of these levels is indicated (Section 2), and an example of their use is given (Section 3).

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R. L. Anderson

Papers

Distribution of the Circular Serial Correlation Coefficient for Residuals from a Fitted Fourier Series