and variance o Only the p most recent observations matter for this distribution H (yiYi )=/yp yt|yt-1,…y (yt-C-o19t OpJt-p)2 The likelihood function for the complete sample is then fr,y-11(yr,y-1,…,y1;)=fvy=1.1(yp,y-1,…,:6) Ifx=,y=(mW-,…,y-p:0), and the loglikelihood is therefore c(e= log fy 1 g(o)+ologVpl-osyp-up'V(y-up i Plog(o2y 1y-1-02 22 Maximization of this exact log likelihood of an AR(p)process must be accom- plished numerically. 2.2 Conditional maximum Likelihood estimates The log of the likelihood conditional on the first p observation assume the simple form (6)= log fY,y1,1p+1y,1(,m-1,y+1p,…,;日) T P T 2l0g(2π) (y-c-01y-1-02y-2-….-h-p)2 3log(2)、T- g(2)-∑and variance σ 2 . Only the p most recent observations matter for this distribution. Hence for t > p fYt|Yt−1,...,Y1 (yt |yt−1, ..., y1; θ) = fYt|Yt−1,..,Yt−p (yt |yt−1, .., yt−p; θ) = 1 √ 2πσ 2 exp  −(yt − c − φ1yt−1 − φ2yt−2 − ... − φpyt−p) 2 2σ 2  . The likelihood function for the complete sample is then fYT ,YT −1,...,Y1 (yT , yT −1, ..., y1; θ) = fYp,Yp−1,...,Y1 (yp, yp−1, ..., y1; θ) × Y T t=p+1 fYt|Yt−1,..,Yt−p (yt |yt−1, .., yt−p; θ), and the loglikelihood is therefore L(θ) = log fYT ,YT −1,...,Y1 (yT , yT −1, ..., y1; θ) = − p 2 log(2π) − p 2 log(σ 2 ) + 1 2 log |V−1 p | − 1 2σ 2 (yp − µp ) 0V−1 p (y − µp ) − T − p 2 log(2π) − T − p 2 log(σ 2 ) − X T t=p+1 (yt − c − φ1yt−1 − φ2yt−2 − ... − φpyt−p) 2 2σ 2 . Maximization of this exact log likelihood of an AR(p) process must be accom￾plished numerically. 2.2 Conditional maximum Likelihood estimates The log of the likelihood conditional on the first p observation assume the simple form L ∗ (θ) = log fYT ,YT −1,..,Yp+1|Yp,...,Y1 (yT , yT −1, ..yp+1|yp, ..., y1; θ) = − T − p 2 log(2π) − T − p 2 log(σ 2 ) − X T t=p+1 (yt − c − φ1yt−1 − φ2yt−2 − ... − φpyt−p) 2 2σ 2 = − T − p 2 log(2π) − T − p 2 log(σ 2 ) − X T t=p+1 ε 2 t 2σ 2 . (11) 9