Example In then case p we have 入1 A1-A2A1-A2 入2-A1A2-A1 2.1.1 Real roots Suppose first that all the eigenvalues of F are real and all these real eigen- values are less than one in absolute value, then the system is stable, and its dynamics are represented as a weighted average of decaying exponentially or decaying exponentially oscillating in sign Consider the following second-order difference equation Yt=0.6t-1+0.2Y-2+Wt The eigenvalues are the solutions the polynomial 入-0.6入-0.2=0 which are 0.6+√0.6)2 4(0.2)=0 入2 0.6-y(06)2-4(0.2 The dynamic multiplier for this system aY aWt =c1(0.84)2+ is geometrically decaying and is plotted as a function of j in panel(a)of Hamil- ton,p.15. Note that as j becomes larger, the pattern is dominated by the larger eigenvalues(A1), approximating a simple geometric decay at rate(A1) If the eigenvalue are all real but at least one is greater than one in absolute value, the system is explosive. If A1 denotes the eigenvalue that isExample: In then case p = 2, we have c1 = λ 2−1 1 λ1 − λ2 = λ1 λ1 − λ2 , c2 = λ 2−1 2 λ2 − λ1 = λ2 λ2 − λ1 . 2.1.1 Real Roots Suppose first that all the eigenvalues of F are real and all these real eigen￾values are less than one in absolute value, then the system is stable, and its dynamics are represented as a weighted average of decaying exponentially or decaying exponentially oscillating in sign. Example: Consider the following second-order difference equation: Yt = 0.6Yt−1 + 0.2Yt−2 + Wt . The eigenvalues are the solutions the polynomial λ 2 − 0.6λ − 0.2 = 0 which are λ1 = 0.6 + p (0.6)2 − 4(0.2) 2 = 0.84 λ2 = 0.6 − p (0.6)2 − 4(0.2) 2 = −0.24. The dynamic multiplier for this system, ∂Yt+j ∂Wt = c1λ j 1 + c2λ j 2 = c1(0.84)j + c2(−0.24)j is geometrically decaying and is plotted as a function of j in panel (a) of Hamil￾ton, p.15. Note that as j becomes larger, the pattern is dominated by the larger eigenvalues (λ1), approximating a simple geometric decay at rate (λ1). If the eigenvalue are all real but at least one is greater than one in absolute value, the system is explosive. If λ1 denotes the eigenvalue that is 9