2 Autoregressive Process 2.1 The First-Order Autoregressive Process A stochastic process Yt, t E T is said to be a first order autoregressive process(AR(1)) if it can be expressed in the form Yt=c+ort where c and o are constants and Et is a white- noise process 2.1.1 Check Stationarity and Ergodicity Write the AR(1) process is lag operator form OLY+Et (1-OLY=c+Et In the case lo 1, we know from the properties of lag operator in last chapter (1-L)-1=1+oL+φ2L thi Y=(c+e)·(1+oL+φ2L2+…) (c+oLc+2L2c+…)+(et+oLet+2L2et+… (c+e+2c+…)+(et+cet-1+2et-2+…) +Et+ Et-1+Et-2+ 1 This can be viewed as an MA(oo)process with p; given by o. When I<1 this AR() is an MA(oo) with absolute summable coefficient =∑|P 1-1 < Therefore, the AR(1) process is stationary and ergodic provided that o <12 Autoregressive Process 2.1 The First-Order Autoregressive Process A stochastic process {Yt , t ∈ T } is said to be a first order autoregressive process (AR(1)) if it can be expressed in the form Yt = c + φYt−1 + εt , where c and φ are constants and εt is a white-noise process. 2.1.1 Check Stationarity and Ergodicity Write the AR(1) process is lag operator form: Yt = c + φLYt + εt , then (1 − φL)Yt = c + εt . In the case |φ| < 1, we know from the properties of lag operator in last chapter that (1 − φL) −1 = 1 + φL + φ 2L 2 + ...., thus Yt = (c + εt) · (1 + φL + φ 2L 2 + ....) = (c + φLc + φ 2L 2 c + ...) + (εt + φLεt + φ 2L 2 εt + ...) = (c + φc + φ 2 c + ...) + (εt + φεt−1 + φ 2 εt−2 + ...) = c 1 − φ + εt + φεt−1 + φ 2 εt−2 + ... This can be viewed as an MA(∞) process with ϕj given by φ j . When φ| < 1, this AR(1) is an MA(∞) with absolute summable coefficient: X∞ j=0 |ϕj | = X∞ j=0 |φ| j = 1 1 − |φ| < ∞. Therefore, the AR(1) process is stationary and ergodic provided that |φ| < 1. 8