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T.W. Anderson 307Consider the serial correlation of the residualsr = z⊤ Azz ⊤ z= u⊤ M ⊤ AMuu ⊤ M ⊤ Mu .The matrix M is idempotent, i.e., M m = M, and symmetric. Its latent rootsare 0 and 1 and it has rank n − k. Let the possibly nonzero roots of M ⊤ AMbe ν 1 ,...,ν n−k . There is an n × (n − k) matrixH such that H ⊤ H = I n−kand⎡⎤ν 1 0 ··· 0H ⊤ M ⊤ 0 ν 2 ··· 0AMH = ⎢ . . . . ⎥⎣ . . . . ⎦ .0 0 ··· ν n−kLet w = H ⊤ v.Thenn−k∑r = ν j wj2j=1Durbin and Watson prove that∑wj 2 ./n−kj=1λ j ≤ ν j ≤ λ j+k ,j ∈{1,...,n− k}.Definen−k∑r L = λ j wj2∑n−k∑wj 2 , r U = λ j+k wj2/n−k∑wj 2 ./n−kj=1 j=1j=1j=1Then r L ≤ r ≤ r U .The “bounds procedure” is the following. If the observed serial correlationis greater than rU ⋆ conclude that the hypothesis of no serial correlation of thedisturbances is rejected. If the observed correlation is less than rL ⋆ ,concludethat the hypothesis of no serial correlation of the disturbance is accepted.The interval (rL ⋆ ,r⋆ U ) is called “the zone of indeterminacy.” If the observedcorrelation falls in the interval (rL ⋆ ,r⋆ U ), the data are considered as not leadingto a conclusion.ReferencesAnderson, R.L. (1942). Distribution of the serial correlation coefficient. TheAnnals of Mathematical Statistics, 13:1–13.Anderson, R.L. and Anderson, T.W. (1950). Distribution of the circular serialcorrelation coefficient for residuals from a fitted Fourier series. The Annalsof Mathematical Statistics, 21:59–81.