2 Deterministic Trend and stochastic Trend Many economic and financial times series do trended upward over time(such as GNP, M2, Stock Index etc. See the plots of Hamilton, p. 436. For a long time each trending (nonstationary) economic time series has been decomposed into a deterministic trend and a stationary process. In recent years the idea of stochas- tic trend has emerged, and enriched the framework of analysis to investigate economic time series 2.1 Detrending methods 2.1.1 Differencing-Stationary One of the easiest ways to analyze those nonstationary-trending series is to make those series stationary by differencing. In our example, the random walk series with drift Yt can be transformed to a stationary series by differencing once AY=Yi-Yi-1=(1-LY=a+ut Since Ut is assumed to be a white noise process, the first difference of Yt is sta- tionary. The variance of AYt is constant over the sample period. In the I(1) Yt=Yo +at+>ut-i i=0 at is a deterministic trend while 2i=o vt-i is a stochastic trend When the nonstationary series can be transformed to the stationary series by differencing once, the series is said to be integrated of order 1 and is denoted by I(1), or in common, a unit root process. If the series needs to be differenced d times to be stationery, then the series is said to be I(d). The I(d) series(d+0) is also called a dif ferencing- stationary process (DSP). When(1-L is a stationary and invertible series that can be represented by an aRMA(p, g) model. i.e (1-1L-2L2-…-qnDP)(1-DY1=a+(1+1L+2L2+…+L)et(3) o(L)△4Y1=a+6(L)t2 Deterministic Trend and Stochastic Trend Many economic and financial times series do trended upward over time (such as GNP, M2, Stock Index etc.). See the plots of Hamilton, p.436. For a long time each trending (nonstationary) economic time series has been decomposed into a deterministic trend and a stationary process. In recent years the idea of stochas￾tic trend has emerged, and enriched the framework of analysis to investigate economic time series. 2.1 Detrending Methods 2.1.1 Differencing-Stationary One of the easiest ways to analyze those nonstationary-trending series is to make those series stationary by differencing. In our example, the random walk series with drift Yt can be transformed to a stationary series by differencing once 4Yt = Yt − Yt−1 = (1 − L)Yt = α + vt . Since vt is assumed to be a white noise process, the first difference of Yt is sta￾tionary. The variance of 4Yt is constant over the sample period. In the I(1) process, Yt = Y0 + αt + X t−1 i=0 vt−i , (2) αt is a deterministic trend while Pt−1 i=0 vt−i is a stochastic trend. When the nonstationary series can be transformed to the stationary series by differencing once, the series is said to be integrated of order 1 and is denoted by I(1), or in common, a unit root process. If the series needs to be differenced d times to be stationery, then the series is said to be I(d). The I(d) series (d 6= 0) is also called a differencing − stationary process (DSP). When (1 − L) dYt is a stationary and invertible series that can be represented by an ARMA(p, q) model, i.e. (1 − φ1L − φ2L 2 − ... − φpL p )(1 − L) dYt = α + (1 + θ1L + θ2L 2 + ... + θqL q )εt (3) or φ(L)4dYt = α + θ(L)εt , 4