The above discussion of the AR(1) process generalizes directly to the AR(p) process where p≥1 Definition A stochastic process iXt, t T) is said to be a autoregressive of order p (AR(P)) if it satisfies the stochastic difference equation, Xt=q1Xt-1+2Xt-2+…+φpXt-p+ where 1, 2,,Op are constants and ut is a white-noise process Definition: A stochastic process IXt, tE T is said to be a moving average process of order q(MA(q)) if it can be expressed in the form X +614-1+62-2+…+6q where 01, 02,.,q are constants and ut is a white-noise process That is, the white-noise process is used to build the process iXt, tET) being a linear combination of the last q ut-iS Definition: A stochastic process (Xt, t E T) is said to be an autoregressive moving average process of order p, q(ARMA(p, q)) if it can be expressed in the form Xt=@1Xt-1+2X-2+…+nXt-p+t+61-1+62u-2+…+6t-q, where 1, 2,,p, 01,B2, ,q are constants and ut is a white-noise process Definition A stochastic process Yt, t E TI is said to be an fractionally autoregres- sive integrated moving average process of order p, d, q(arFIma(p, d, q )The above discussion of the AR(1) process generalizes directly to the AR(p) process where p ≥ 1. Definition: A stochastic process {Xt , t ∈ T } is said to be a autoregressive of order p (AR(p)) if it satisfies the stochastic difference equation, Xt = φ1Xt−1 + φ2Xt−2 + ... + φpXt−p + ut , where φ1, φ2, ..., φp are constants and ut is a white-noise process. Definition: A stochastic process {Xt , t ∈ T } is said to be a moving average process of order q (MA(q)) if it can be expressed in the form Xt = ut + θ1ut−1 + θ2ut−2 + ... + θqut−q, where θ1, θ2, ..., θq are constants and ut is a white-noise process. That is, the white-noise process is used to build the process {Xt , t ∈ T }, being a linear combination of the last q ut−is. Definition: A stochastic process {Xt , t ∈ T } is said to be an autoregressive moving average process of order p, q (ARMA(p, q)) if it can be expressed in the form Xt = φ1Xt−1 + φ2Xt−2 + ... + φpXt−p + ut + θ1ut−1 + θ2ut−2 + ... + θqut−q, where φ1, φ2, ..., φp, θ1, θ2, ..., θq are constants and ut is a white-noise process. Definition: A stochastic process {Yt , t ∈ T } is said to be an fractionally autoregres￾sive integrated moving average process of order p, d, q (ARFIMA(p, d, q)) 12