Similarly, the distribution of the third observation conditional on the first two is fralY2, Y(33ly2, 91; 8) 2 form which faY21(3,y2,1;6)=fya2(yay,g1;e)fy2,1(y2,1;6) frslYa, i(y3l32, 11: 0)fraly (y2ly1: 8)fYi(91: 0) Yt-1 matter for Yt only through the value Yt-1, and the density of observation t conditional on the preceding t-l observa- given by fy 8 tY ogt-1)2 2 The likelihood of the complete sample can thus be calculated as -1=2-(0m,yx-,yx2…,:日)=/1(;)·Ⅱx=-:0).(4) The log likelihood function(denoted C(o)) is theref C(O)=lg1(0n;0)+∑og/xx-(y-;0) The log likelihood for a sample of size T from a Gaussian AR(1) process is seen to b C(6) g(2)-3log2/(1-92) {1-(c/(1-o)}2 (T-1)/2log(27)-(T-1)/21lg2)-∑Similarly, the distribution of the third observation conditional on the first two is fY3|Y2,Y1 (y3|y2, y1; θ) = 1 √ 2πσ 2 exp  − 1 2 · (y3 − c − φy2) 2 σ 2  form which fY3,Y2,Y1 (y3, y2, y1; θ) = fY3|Y2,Y1 (y3|y2, y1; θ)fY2,Y1 (y2, y1; θ) = fY3|Y2,Y1 (y3|y2, y1; θ)fY2|Y1 (y2|y1; θ)fY1 (y1; θ). In general, the value of Y1, Y2, ..., Yt−1 matter for Yt only through the value Yt−1, and the density of observation t conditional on the preceding t − 1 observa￾tions is given by fYt|Yt−1,Yt−2,...,Y1 (yt |yt−1, yt−2, ..., y1; θ) = fYt|Yt−1 (yt |yt−1; θ) = 1 √ 2πσ 2 exp  − 1 2 · (yt − c − φyt−1) 2 σ 2  . The likelihood of the complete sample can thus be calculated as fYT ,YT −1,YT −2,...,Y1 (yT , yT −1, yT −2, ..., y1; θ) = fY1 (y1; θ) · Y T t=2 fYt|Yt−1 (yt |yt−1; θ). (4) The log likelihood function (denoted L(θ)) is therefore L(θ) = log fY1 (y1; θ) + X T t=2 log fYt|Yt−1 (yt |yt−1; θ). (5) The log likelihood for a sample of size T from a Gaussian AR(1) process is seen to be L(θ) = − 1 2 log(2π) − 1 2 log[σ 2 /(1 − φ 2 )] − {y1 − [c/(1 − φ)]} 2 2σ 2/(1 − φ 2 ) −[(T − 1)/2] log(2π) − [(T − 1)/2] log(σ 2 ) − X T t=2  (yt − c − φyt−1) 2 2σ 2  . (6) 3