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306 Serial correlation and Durbin-Watson boundsµ is the expectation of y,Ey = µ = ∑ β j φ j .We shall consider testing the hypothesisH : λ =0.The first theorem characterizes tests such that the probability of the acceptanceregion when λ = 0 does not depend on the values of β 1 ,...,β k . Thesecond theorem gives conditions for a test being uniformly most powerful whenλ>0isthealternative.These theorems are applicable to the circular serial correlation when Ψ =σ 2 I and Θ = σ 2 A defined above.The equation∑(yi − y i−1 ) 2 = ∑ (y2i + yi−12 ) ∑− 2 yt y t−1suggests that a serial correlation can be studied in terms of ∑ (y t − y t−1 ) 2which may be suitable to test that y 1 ,...,y n are independent against the alternativethat y 1 ,...,y n satisfy an autoregressive process. Durbin and Watsonprefer to studyd = ∑ (z i − z i−1 ) 2/ ∑z2i ,where z is defined below.27.5 Durbin–WatsonThe model isyn×1 = Xβn×k k×1+ un×1.We consider testing the null hypothesis that u has a Normal distribution withmean 0 and covariance σ 2 I n against the alternative that u has a Normaldistribution with mean 0 and covariance σ 2 A, a positive definite matrix. Thesample regression of y is b =(X ⊤ X) −1 X ⊤ y and the vector of residuals isz = y − Xb = {I − X(X ⊤ X) −1 X ⊤ }y= {I − X(X ⊤ X) −1 X ⊤ }(Xβ + u)= Mu,whereM = I − X(X ⊤ X) −1 X ⊤ .