a For the first-order autoregressive model defined by (2.77), verify that the errors have the covariance structure in (2.76).

b. Verify the expected values of the numerator and denominator of (2.79) under the null and alternative hypotheses when r is replaced by

e. Hint: Write z=Ae, where A (10)-(0), e~N(0, 2), and use the expression for the expected value of a quadratic form in Exercise 2.11.

c. Under the assumption that p is known, verify that the matrix 1 0 0 T 0 P 10 0 0 -p1 will diagonalize the matrix in (2.76). That is TVTT is diagonal.

d. Use the results of part

(c) to remove the autocorrelation by analyzing the transformed model y = Ty=%+ BTx+u, where u = Te and Y = (1 - B)- To see the effect of this transformation, write the resulting model in algebraic form. Fitting this model requires an estimate of p. One suggestion is to fit the model for a range of values of p and choose that value that minimizes the residual sum of squares.