International macroe.. - Free

4.5. CALIBRATING THE LUCAS MODEL 135Notice that in Markov chain analysis the Þrst subscript denotes the statethat you condition on. For concreteness, consider a Markov chain with twopossible states, λ 1 and λ 2 , with transition matrix," #p11 pP =12,p 21 p 22where the rows of P sum to 1.Property 2 The transition matrix over k steps isP k = PP ···P | {z }kProof. For the two state process, deÞnep (2)ij = P[X t+2 = λ j |X t = λ i ]= P[X t+2 = λ j ∩ X t+1 = λ 1 |X t = λ i ]+P[X t+1 = λ j ∩ X t+1 = λ 2 |X t = λ i ]=2XP[X t+1 = λ j ∩ X t+1 = λ k |X t = λ i ]k=1= P[X t+1 = λ j ∩ X t+1 = λ k ∩ X t = λ i ]P(X t = λ i )Now by property 1, the numerator in last equality can be decomposed as,P[X t+2 = λ j |X t+1 = λ k ]P[X t+1 = λ k |X t = λ i ]P[X t = λ i ] (4.67)Substituting (4.67) into (4.66) gives,(4.66)p (2)ij ==2XP[X t+1 = λ j |X t+1 = λ k ]P[X t+1 = λ k |X t = λ i ]k=12Xk=1p kj p ikwhich is seen to be the ij−th element of the matrix PP. The extension toany arbitrary number of steps forward is straightforward.⇐(99)