the objective then being to maximize C()=-(T-1)/2log(2r)-(T-1)/2]lg(2)- (yt -C-oyt-1 T-1)/2log(27)-{T-1)/2log(o2)-∑ Maximization of( 9) with respect to c and o is equivalent to minimization of >It-C-pgt-1)2=(y-xBy'(y-XB 10 which is achieved by an ordinary least square(Ols)regression of yt on a constant and its own lagged value. where X= The conditional maximum likelihood estimates of c and o are therefore given by yt-1 t=2 yt t=29t-1 9t-1 9t-1yt The conditional maximum likelihood estimator of o is found by setting (v-c-0yt-1) T-1 It is important to note if you have a sample of size T to estimate an AR(1) process by conditional MLE, you will only use T-1 observation of this samplethe objective then being to maximize L ∗ (θ) = −[(T − 1)/2] log(2π) − [(T − 1)/2] log(σ 2 ) − X T t=2  (yt − c − φyt−1) 2 2σ 2  = −[(T − 1)/2] log(2π) − [(T − 1)/2] log(σ 2 ) − X T t=2  ε 2 t 2σ 2  (9) Maximization of (9) with respect to c and φ is equivalent to minimization of X T t=2 (yt − c − φyt−1) 2 = (y − Xβ) 0 (y − Xβ), (10) which is achieved by an ordinary least square (OLS) regression of yt on a constant and its own lagged value, where y =         y2 y3 . . . yT         , X =         1 y1 1 y2 . . . . . . 1 yT −1         , and β =  c φ  . The conditional maximum likelihood estimates of c and φ are therefore given by  cˆ φˆ  =  T − 1 PT t=2 P yt−1 T t=2 yt−1 PT t=2 y 2 t−1 −1  PT t=2 P yt−1 T t=2 yt−1yt  . The conditional maximum likelihood estimator of σ 2 is found by setting ∂L ∗ ∂σ 2 = −(T − 1) 2σ 2 + X T t=2  (yt − c − φyt−1) 2 2σ 4  = 0 or σˆ 2 = X T t=2 " (yt − cˆ− φyˆ t−1) 2 T − 1 # . It is important to note if you have a sample of size T to estimate an AR(1) process by conditional MLE, you will only use T − 1 observation of this sample. 7