if it can be expressed as a stationary ARMA(p, q) process after fractionally- differenced” d times Xt=1X-1+2Xt-2+…+nX-p+at+61tt-1+62t-2+…+6ut-q, where 1, p2, .,op, 01, 02,, g are constants, Id<0.5 and ut is a white-noise 3.2.2 Non-Stationary Process Definition: unit root process A stochastic process Yt, tE T is said to be an autoregressive integrated moving average process of order p, q(ARIMA(p, 1, q)) if it can be expressed as a stationary ARMA(p, g) process after first-differening (1-L)Y=X Xt=1Xt-1+Xt-2+…+qnXt-p+t+61tt-1+62t-2+…+6qt-q where 1, o2,.,p, 81, 82,.q are constants and ut is a white-noise processif it can be expressed as a stationary ARMA(p, q) process after fractionally￾differenced ”d” times: (1 − L) dYt = Xt , and Xt = φ1Xt−1 + φ2Xt−2 + ... + φpXt−p + ut + θ1ut−1 + θ2ut−2 + ... + θqut−q, where φ1, φ2, ..., φp, θ1, θ2, ..., θq are constants, |d| < 0.5 and ut is a white-noise process. 3.2.2 Non-Stationary Process Definition: unit root process A stochastic process {Yt , t ∈ T } is said to be an autoregressive integrated moving average process of order p, q (ARIMA(p, 1, q)) if it can be expressed as a stationary ARMA(p, q) process after first-differening (1 − L) 1Yt = Xt , and 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. 13