where all the roots of o(L)=0 and 0(L)=0 lie outside the unit circle, we say that Yt is an autoregressive integrated moving-average ARIMA(p, d, q) process In particular an unit root process, d= l or an ARIMA(p, 1, g) process is therefore o(L)△Y=a+b(L)et (1- LY=a+v(LEt where v(l)=o(L)e(L) and is absolutely summable Successive substitution yields Y=Y0+at+v(L)∑= 2.1.2 Trend-Stationary Another important class is the trend- stationary process(TSP). Consider the series v(L)Et where the coefficients of v(L)is absolute summable The mean of Xt is E(X,=u+at and is not constant over time, wh the variance of Xt is Var(Xt)=(1+1+v2 +.o2 and constant. Although the mean of Xt is not constant over the period, it can be forecasted perfectly whenever we know the value of t and the parameters a and d. In the sense it is stationary around the deterministic trend t and Xt can be transformed to stationarity by regressing it on time. Note that both DSP model equation (5) and the TSP model equation(6) exhibit a linear trend, but the appropriated method of eliminating the trend differs.(It can be seen that the DsP is trend nonstationary from the definition of TSP. Most economic analysis is based the variance and covariance among the vari- bles. For example, The OLS estimator from the regression Yt on Xt is the ratio of the covariance between Y and X, to variance of Xt. Thus if the variance of thewhere all the roots of φ(L) = 0 and θ(L) = 0 lie outside the unit circle, we say that Yt is an autoregressive integrated moving-average ARIMA(p, d, q) process. In particular an unit root process, d = 1 or an ARIMA(p, 1, q) process is therefore φ(L)4Yt = α + θ(L)εt or (1 − L)Yt = α + ψ(L)εt , (4) where ψ(L) = φ −1 (L)θ(L) and is absolutely summable. Successive substitution yields Yt = Y0 + αt + ψ(L) X t−1 i=0 εt−i . (5) 2.1.2 Trend-Stationary Another important class is the trend − stationary process (TSP). Consider the series Xt = µ + αt + ψ(L)εt , (6) where the coefficients of ψ(L) is absolute summable. The mean of Xt is E(Xt) = µ + αt and is not constant over time, while the variance of Xt is V ar(Xt) = (1+ψ 2 1 + ψ 2 2 + ...)σ 2 and constant. Although the mean of Xt is not constant over the period, it can be forecasted perfectly whenever we know the value of t and the parameters α and δ. In the sense it is stationary around the deterministic trend t and Xt can be transformed to stationarity by regressing it on time. Note that both DSP model equation (5) and the TSP model equation (6) exhibit a linear trend, but the appropriated method of eliminating the trend differs. (It can be seen that the DSP is trend − nonstationary from the definition of TSP.) Most economic analysis is based the variance and covariance among the vari￾ables. For example, The OLS estimator from the regression Yt on Xt is the ratio of the covariance between Yt and Xt to variance of Xt . Thus if the variance of the 5