It is easy to verify by direct multiplication that L'L=V-I, with 1-20 10 10 000 Then(7)becomes C(0=(T/2)log(2)+olog oll-oy-uoLl(y-u).(8) Define theT x 1)vector y to be y L(y-A) 1-020 0 10 0 0 0 T-H 1-02(Y1-1) (Y2-1)-0(Y1-p) o(Y2-F (Y-p)-(Yr-1- 1-02 Y oY The last term in(8)can thus be written 2ty-wo-LLly-w)=2 y'y =20|(1-02)1-/(1-)2+台2 1)2It is easy to verify by direct multiplication that L 0L = V−1 , with L =         p 1 − φ 2 0 . . . 0 −φ 1 0 . . 0 0 −φ 1 0 . 0 . . . . . . . . . . . . 0 0 . . −φ 1         . Then (7) becomes L(θ) = (−T/2)log(2π) + 1 2 log |σ −2L 0L| − 1 2 (y − µ) 0σ −2L 0L(y − µ). (8) Define the (T × 1) vector y˜ to be y˜ ≡ L(y − µ) =         p 1 − φ 2 0 . . . 0 −φ 1 0 . . 0 0 −φ 1 0 . 0 . . . . . . . . . . . . 0 0 . . −φ 1                 Y1 − µ Y2 − µ Y3 − µ . . YT − µ         =         p 1 − φ 2 (Y1 − µ) (Y2 − µ) − φ(Y1 − µ) (Y3 − µ) − φ(Y2 − µ) . . (YT − µ) − φ(YT −1 − µ)         =         p 1 − φ 2 [Y1 − c/(1 − φ)] Y2 − c − φY1 Y3 − c − φY2 . . YT − c − φYT −1         . The last term in (8) can thus be written 1 2 (y − µ) 0σ −2L 0L(y − µ) =  1 2σ 2  y˜ 0y˜ =  1 2σ 2  (1 − φ 2 )[Y1 − c/(1 − φ)]2 +  1 2σ 2 X T t=2 (Yt − c − φYt−1) 2 . 5