Thus, for a pth-order difference equation, the dynamic multiplier is given by fi re fi denotes the(, 1)element of F3 Example The(1, 1)elements of F is 1 and the(1, 1)elements of F2([91, 2,op 1, 1,0,0) is 1+o2. Thus D+1=o aY+2 01+2 in a pth-order syster For larger values of j, an easy way to obtain a numerical value for the dynamic multiplier aYt+i/awt in terms the eigenvalues of the matrix F. Recall that the eigenvalues of a matrix F are those numbers a for which F-A=0 (1 For example, for p=2 the eigenvalues are the solutions to 入0 10 0入 (1-入)2 2-02)-02=0 on For a general pth-order system, the determinant in(12)is a pth-order ploy- minal in A whose p solutions characterize the p eigenvalues of F. This polyno- mial turns out to take a very similar form to(13 The eigenvalues of the matrix F defines in equation(12)are the values of A that satisfy -01-1-02-2- 中p
Thus, for a pth-order difference equation, the dynamic multiplier is given by ∂Yt+j ∂Wt = f j 11, where f j 11 denotes the (1, 1) element of F j . Example: The (1, 1) elements of F 1 is φ1 and the (1, 1) elements of F 2 (= [φ1, φ2, ..., φp][φ1, 1, 0, ..., 0]0 ) is φ 2 1 + φ2. Thus, ∂Yt+1 ∂Wt = φ1; and ∂Yt+2 ∂Wt = φ 2 1 + φ2 in a pth-order system. For larger values of j, an easy way to obtain a numerical value for the dynamic multiplier ∂Yt+j/∂Wt in terms the eigenvalues of the matrix F. Recall that the eigenvalues of a matrix F are those numbers λ for which |F − λIp| = 0. (12) For example, for p = 2 the eigenvalues are the solutions to φ1 φ2 1 0 − λ 0 0 λ = 0 or (φ1 − λ) φ2 1 −λ = λ 2 − φ2λ − φ2 = 0. (13) For a general pth-order system, the determinant in (12) is a pth-order ploynominal in λ whose p solutions characterize the p eigenvalues of F. This polynomial turns out to take a very similar form to (13). Proposition: The eigenvalues of the matrix F defines in equation (12) are the values of λ that satisfy λ p − φ1λ p−1 − φ2λ p−2 − ... − φp−1λ − φp = 0. 6
2.1 General Solution of a pth-order Difference Equation with Distinct Eigenvalues Recall that if the eigenvalues of a(p x p)matrix F are distinct, there exists a nonsingular(p x p) matrix T such that F=TAT-I where T=[x1, x2, .,xp], xi, i= 1, 2, ., p are the eigenvectors of F corresponding to its eigenvalues Ai; and A is a(p x p) matrix such that h100 0A20 00 0 This enables us to characterize the dynamic multiplier (the(1, 1)elements of F2) very easily. In general, we have F=TAT-1×TAT-1×….×TAT TAT-I (15) where A00 0
2.1 General Solution of a pth-order Difference Equation with Distinct Eigenvalues Recall that if the eigenvalues of a (p × p) matrix F are distinct, there exists a nonsingular (p × p) matrix T such that F = TΛT−1 where T = [x1, x2, ..., xp], xi , i = 1, 2, ..., p are the eigenvectors of F corresponding to its eigenvalues λi ; and Λ is a (p × p) matrix such that Λ = λ1 0 0 . . . 0 0 λ2 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . λp . This enables us to characterize the dynamic multiplier (the (1,1) elements of F j ) very easily. In general, we have F j = TΛT −1 × TΛT−1 × ... × TΛT −1 (14) = TΛjT −1 , (15) where Λ j = λ j 1 0 0 . . . 0 0 λ j 2 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . λ j p . 7
Let tii denote the row i column j element of T and let to denote the row i column element of T. Equation(15) written out explicitly become A00 0 t11t1 00 000 tpa t21t22 t12 入 from which the(1, 1)element of F3 is given by f1=c1+c2 where and a1+c2+…+cp=t1t1+th21+…+tpt=1 Therefore the dynamic multiplier of a pth-order difference equation fi1=c1+c2+…+cp2 that is the dynamic multiplier is a weighted average of each of the p eigenvalues raised to the jth power The following result provides a closed-form expression for the constant c1, C2,.,p If the eigenvalues(A1, A2, ...,Ap) of the matrix F are distinct, then the magnitude Ci can be written as I=1,k≠(A2-Ak)
Let tij denote the row i column j element of T and let t ij denote the row i column j element of T−1 . Equation (15) written out explicitly become F j = t11 t12 . . . . t1p t21 t22 . . . . t2p . . . . . . . . . . . . . . . . . . . . . tp1 tp2 . . . . tpp λ j 1 0 0 . . . 0 0 λ j 2 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . λ j p t 11 t 12 . . . . t 1p t 21 t 22 . . . . t 2p . . . . . . . . . . . . . . . . . . . . . t p1 t p2 . . . . t pp = t11λ j 1 t12λ j 2 . . . . t1pλ j p t21λ j 1 t22λ j 2 . . . . t2pλ j p . . . . . . . . . . . . . . . . . . . . . tp1λ j 1 tp2λ j 2 . . . . tppλ j p t 11 t 12 . . . . t 1p t 21 t 22 . . . . t 2p . . . . . . . . . . . . . . . . . . . . . t p1 t p2 . . . . t pp from which the (1,1) element of F j is given by f j 11 = c1λ j 1 + c2λ j 2 + ... + cpλ j p where ci = t1it i1 and c1 + c2 + ... + cp = t11t 11 + t12t 21 + ... + t1pt p1 = 1. Therefore the dynamic multiplier of a pth-order difference equation is: ∂Yt+j ∂Wt = f j 11 = c1λ j 1 + c2λ j 2 + ... + cpλ j p , that is the dynamic multiplier is a weighted average of each of the p eigenvalues raised to the jth power. The following result provides a closed-form expression for the constant c1, c2, ..., cp. Proposition 2: If the eigenvalues (λ1, λ2, ..., λp) of the matrix F are distinct, then the magnitude ci can be written as ci = λ p−1 Q i p k=1, k6=i (λi − λk) . 8