1 MLE of a Gaussian AR(1)Process 1.1 Evaluating the Likelihood Function Using(Scalar)Con ditional Density A stationary Gaussian AR(1) process takes the form Yt=c+oYt-1+ with Et w i.i.d. N(0, o2)and | l 1(How do you know at this stage ). For this e=(c, Consider the p.d. f of Y1, the first observations in the sample. This is a random variable with mean and variance E(Y1)= Var(ri) Since Et]oo_oo is Gaussian, Yi is also Gaussian. Hence, f1(m;0)=f1(i;c,,a2) v2r√a2/(1-a2 2 Next consider the distribution of the second observation y conditional on the serving Yi=y1. From(2) Y2=c+Y1+E2 Conditional on Yi= y1 means treating the random variable Y1 as if it were the deterministic constant y1. For this case, (3)gives Y2 as the constantc+ oy1) plus the N(0, a2)variable E2. Hence (Y2Y1=y)~N(c+om),o2), meaning that f1(y29y;e) The joint density of observations 1 and 2 is then just f2n(y2,1;6)=fy(y2lyn;)fn(y1;6)1 MLE of a Gaussian AR(1) Process 1.1 Evaluating the Likelihood Function Using (Scalar) Con￾ditional Density A stationary Gaussian AR(1) process takes the form Yt = c + φYt−1 + εt , (2) with εt ∼ i.i.d. N(0, σ 2 ) and |φ| < 1 (How do you know at this stage ?). For this case, θ = (c, φ, σ 2 ) 0 . Consider the p.d.f of Y1, the first observations in the sample. This is a random variable with mean and variance E(Y1) = µ = c 1 − φ and V ar(Y1) = σ 2 1 − φ2 . Since {εt} ∞ t=−∞ is Gaussian, Y1 is also Gaussian. Hence, fY1 (y1; θ) = fY1 (y1; c, φ, σ 2 ) = 1 √ 2π p σ 2/(1 − φ 2 ) exp  − 1 2 · {y1 − [c/(1 − φ)]} 2 σ 2/(1 − φ 2 )  . Next consider the distribution of the second observation Y2 conditional on the observing Y1 = y1. From (2), Y2 = c + φY1 + ε2. (3) Conditional on Y1 = y1 means treating the random variable Y1 as if it were the deterministic constant y1. For this case, (3) gives Y2 as the constant (c + φy1) plus the N(0, σ 2 ) variable ε2. Hence, (Y2|Y1 = y1) ∼ N((c + φy1), σ 2 ), meaning that fY2|Y1 (y2|y1; θ) = 1 √ 2πσ 2 exp  − 1 2 · (y2 − c − φy1) 2 σ 2  . The joint density of observations 1 and 2 is then just fY2,Y1 (y2, y1; θ) = fY2|Y1 (y2|y1; θ)fY1 (y1; θ). 2