and when they are raised to the jth power Af= R(cos(0j)+isin(0j)=R'c A2= R(cos(0j)-isin(0j)=Re The contribution of the complex conjugates to the dynamic multiplier aYi+i/an CIA+C2A2=C1R'cos(0j)+isin())+c2R'cos(0j)-isin(0j) (c1+ c2)R. cos(0j)+i(c1-c2)R. sin(0j) From Proposition 2 we know that if A1 and A2 are complex conjugates, then C1 and c2 are also complex conjugates; that is they can be written as CI=a+Bi C2 =a for some real number a and B. Therefore, the dynamic multiplier art+i/awt can further be expressed as C1+c2=[a+Bi)+(a-Bi)·R·cos(6)+i[(a+Bi)-(a-Bi)]··sin(6j) (2a)R. cos(0j)+i (2Bi)R. sin(8j) 2aR cos(8j)-2BR sin(0j) Thus, when some of the(distinct)eigenvalues are complex, then if 1. R=l, that is the complex eigenvalues have unit modulus, the multipliers are periodic sine and cosine functions of j 2. R< 1, that is the complex eigenvalues are less then one in modulus, the impulse again follows a sinusoidal pattern though its amplitude decays at the rate R3 3. R>l, that is the complex eigenvalues are greater then one in modulus, its amplitude of the sinusoids explodes at the rate R3 Example Consider the following second-order difference equation Yt=0.5Yt-1-0.8Y2+Wt The eigenvalues are the solutions the polynomial 5入+0.8and when they are raised to the jth power, λ j 1 = R j [cos(θj) + isin(θj)] = R j e iθj λ j 2 = R j [cos(θj) − isin(θj)] = R j e −iθj . The contribution of the complex conjugates to the dynamic multiplier ∂Yt+j/∂Wt : c1λ j 1 + c2λ j 2 = c1R j [cos(θj) + isin(θj)] + c2R j [cos(θj) − isin(θj)] = (c1 + c2)R j · cos(θj) + i(c1 − c2)R j · sin(θj). From Proposition 2 we know that if λ1 and λ2 are complex conjugates, then c1 and c2 are also complex conjugates; that is they can be written as c1 = α + βi c2 = α − βi, for some real number α and β. Therefore, the dynamic multiplier ∂Yt+j/∂Wt can further be expressed as c1λ j 1 + c2λ j 2 = [(α + βi) + (α − βi)] · R j · cos(θj) + i[(α + βi) − (α − βi)] · R j · sin(θj) = (2α)R j · cos(θj) + i · (2βi)R j · sin(θj) = 2αR j cos(θj) − 2βR j sin(θj). Thus, when some of the (distinct) eigenvalues are complex, then if 1. R = 1, that is the complex eigenvalues have unit modulus, the multipliers are periodic sine and cosine functions of j; 2. R < 1, that is the complex eigenvalues are less then one in modulus, the impulse again follows a sinusoidal pattern though its amplitude decays at the rate Rj ; 3. R > 1, that is the complex eigenvalues are greater then one in modulus, its amplitude of the sinusoids explodes at the rate Rj . Example: Consider the following second-order difference equation: Yt = 0.5Yt−1 − 0.8Yt−2 + Wt . The eigenvalues are the solutions the polynomial λ 2 − 0.5λ + 0.8 = 0 11