International macroe.. - Free

4.5. CALIBRATING THE LUCAS MODEL 127appendix). Each element of the state vector is allowed to be in eitherof one of two possible states—high and low. A ‘1’ subscript indicatesthat the variable is in the high growth state and a ‘2’ subscript indicatesthat the variable is in the low growth state. Therefore, λ = λ 1indicates high domestic money growth, λ = λ 2 indicates low domesticmoney growth. Analogous designations hold for the other variables.The 16 possible states of the world areφ 1=(λ 1 , λ ∗ 1,g 1 ,g1) ∗ φ 9=(λ 2 , λ ∗ 1,g 1 ,g1)∗φ 2=(λ 1 , λ ∗ 1,g 1 ,g2) ∗ φ 10=(λ 2 , λ ∗ 1,g 1 ,g2)∗φ 3=(λ 1 , λ ∗ 1,g 2 ,g1) ∗ φ 11=(λ 2 , λ ∗ 1,g 2 ,g1)∗φ 4=(λ 1 , λ ∗ 1,g 2 ,g2) ∗ φ 12=(λ 2 , λ ∗ 1,g 2 ,g2)∗φ 5=(λ 1 , λ ∗ 2,g 1 ,g1) ∗ φ 13=(λ 2 , λ ∗ 2,g 1 ,g1)∗φ 6=(λ 1 , λ ∗ 2,g 1 ,g2) ∗ φ 14=(λ 2 , λ ∗ 2,g 1 ,g2)∗φ 7=(λ 1 , λ ∗ 2,g 2 ,g1) ∗ φ 15=(λ 2 , λ ∗ 2,g 2 ,g1)∗φ 8=(λ 1 , λ ∗ 2,g 2 ,g2) ∗ φ 16=(λ 2 , λ ∗ 2,g 2 ,g2).∗We will denote the 16 × 16 probability transition matrix for the stateby P, wherep ij =P[˜φ t+1= φ j|˜φ t= φ i]theij−th element.The price of the domestic and foreign currency bonds are,b t = βE t [(gt+1g θ ∗(1−θ)t+1 ) 1−γ ]/λ t+1 , and b ∗ t = βE t [(gt+1g θ ∗(1−θ)t+1 ) 1−γ ]/λ ∗ t+1,under the constant relative risk aversion utility function (4.22). Sincetheir values depend on the state of the world, we say that these arestate-contingent bond prices. Next, deÞne G =[(g θ g ∗(1−θ) ) 1−γ ]/λ andG ∗ =[(g θ g ∗(1−θ) ) 1−γ ]/λ ∗ ,andletd = λ/λ ∗ be the gross rate of depreciationof the home currency. The possible values of G and G ∗ and dare given in Table 4.3,Suppose the current state is φ k. By (4.56), the spot exchange rateis given by (1 − θ)d k /θ. The domestic bond price isb k = β P 16i=1 p k,i G i , the foreign bond price is b ∗ k = β P 16i=1 p k,i G ∗ i , theexpected gross change in the nominal exchange rate is P 16i=1 p k,i d i ,andthe state-k contingent risk premium isrp k =X16i=1p k,i d i − (P 16i=1 p k,i G ∗ i )( P 16i=1 p k,i G i ) .Next, we must estimate the probability transition matrix. The Þrstquestion is whether we should use consumption data or GDP? In the