increase exponentially over time and if < -l, the multiplier exhibit explosive oscillations Thus, if o< 1, the system is stable; the consequence of a given change in t will eventually die out. If o >l, the system is explosive. An interesting possibility is the borderline case, lo|= 1. In this case, the solution(3)becomes t+=Yt-1+Wt+Wi+1+W++2+.+Wi+i-1+Wi+j Here the output variables y is the sum of the historical input w. A one-unit increase in w will cause a permanent one-unit increase in y ow,=1 for]=0,1 --unit root 2 pth-Order Difference Equations Let us now generalize the dynamic system (1) by allowing the value of y at date t to depend on p of its own lags along with the current value of the input variable Y=1Yt-1+2Yt-1+…+nYt-p+Wt,t∈T Equation(5)is a linear pth-order difference equation It is often convenient to rewrite the pth-order difference equation(5) in the calar Yt as a first-order difference equation in a vector $t. Define the (p x 1) vector st byincrease exponentially over time and if φ < −1, the multiplier exhibit explosive oscillations. Thus, if |φ| < 1, the system is stable; the consequence of a given change in Wt will eventually die out. If |φ| > 1, the system is explosive. An interesting possibility is the borderline case, |φ| = 1. In this case, the solution (3) becomes Yt+j = Yt−1 + Wt + Wt+1 + Wt+2 + ... + Wt+j−1 + Wt+j . Here the output variables Y is the sum of the historical input W. A one-unit increase in W will cause a permanent one-unit increase in Y : ∂Yt+j ∂Wt = 1 for j = 0, 1, .... − −unit root. 2 pth-Order Difference Equations Let us now generalize the dynamic system (1) by allowing the value of Y at date t to depend on p of its own lags along with the current value of the input variable Wt : Yt = φ1Yt−1 + φ2Yt−1 + .... + φpYt−p + Wt , t ∈ T . (5) Equation (5) is a linear pth-order difference equation. It is often convenient to rewrite the pth-order difference equation (5) in the scalar Yt as a first-order difference equation in a vector ξt . Define the (p × 1) vector ξt by ξt ≡           Yt Yt−1 Yt−2 . . . Yt−p+1           , 3