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27Serial correlation and Durbin–WatsonboundsT.W. AndersonDepartment of Economics and Department of StatisticsStanford University, Stanford, CAConsider the model y = Xβ+u,wherey is an n-vector of dependent variables,X is a matrix of n×k independent variables, and u is a n-vector of unobserveddisturbance. Let z = y−Xb,whereb is the least squares estimate of β.Thedstatistictests the hypothesis that the components of u are independent versusthe alternative that the components follow a Markov process. The Durbin–Watson bounds pertain to the distribution of the d-statistics.27.1 IntroductionA time series is composed of a sequence of observations y 1 ,...,y n ,wheretheindex i of the observation y i represents time. An important feature of a timeseries is the order of observations: y i is observed after y 1 ,...,y i−1 are observed.The correlation of successive observations is called a serial correlation.Related to each y i may be a vector of independent variables (x 1i ,...,x ki ).Many questions of time series analysis relate to the possible dependence of y ion x 1i ,...,x ki ; see, e.g., Anderson (1971).Aserialcorrelation(first-order)ofasequencey 1 ,...,y n isn∑ / ∑ny i y i−1 yi 2 .i=2This coefficient measures the correlation between y 1 ,...,y n−1 and y 2 ,...,y n .There are various modifications of this correlation coefficient such as replacingy i by y i − ȳ. Seebelowforthecircularserialcoefficient.Theterm“autocorrelation”is also used for serial correlation.I shall discuss two papers coauthored by James Durbin and Geoffrey Watsonentitled “Testing for serial correlation in least squares regression I and II,”i=1