International macroe.. - Free

8.4. VAR ANALYSIS OF MUNDELL—FLEMING 253To summarizeF j = C j F 0 for all j ⇒ F(1) = C(1)F 0 . (8.47)Given the C j , which you get from unrestricted VAR accounting, (8.47)says you only need to determine F 0 after which the remaining F j follow.In our 3-dimensional system, F 0 is a 3 × 3 matrix with 9 uniqueelements. To identify F 0 , you need 9 pieces of information. Start with,Σ = G 0 G =E(² t ² 0 t)=F 0 E(v t v 0 t)F 0 0 = F 0 F 0 0 where G is the uniqueupper triangular Choleski decomposition of the error covariance matrixΣ. To summarizeΣ = G 0 G = F 0 F 0 0 . (8.48)Let g ij be the ijth element of G and f ij,0 be the ijth element of F 0 .Writing (8.48) out givesg11 2 = f 11,0 2 + f 12,0 2 + f 13,0 2 , (8.49)g 11 g 12 = f 11,0 f 21,0 + f 12,0 f 22,0 + f 13,0 f 23,0 , (8.50)g 11 g 13 = f 11,0 f 31,0 + f 12,0 f 32,0 + f 13,0 f 33,0 , (8.51)g12 2 +g2 22 = f 21,0 2 + f 22,0 2 + f 23,0 2 , (8.52)g 12 g 13 + g 22 g 23 = f 21,0 f 31,0 + f 22,0 f 32,0 + f 23,0 f 33,0 , (8.53)g13 2 +g23 2 + g33 2 = f31,0 2 + f32,0 2 + f33,0. 2 (8.54)G has 6 unique elements so this decomposition gives you 6 equationsin 9 unknowns. You still need three additional pieces of information.Get them from the long-run predictions of the theory.Stochastic Mundell-Fleming predicts that neither demand shocksnor monetary shocks have a long-run effect on output which we representby setting f 12 (1) = 0 and f 13 (1) = 0, where f ij (1) is the ijthelement of F(1) = P ∞j=0 F j . The model also predicts that moneyhas no long-run effect on the real exchange rate f 33 (1) = 0. SinceF(1) = C(1)F 0 , impose these three restrictions by settingf 13 (1) = 0 = c 11 (1)f 13,0 + c 12 (1)f 23,0 + c 13 (1)f 33,0 , (8.55)f 12 (1) = 0 = c 11 (1)f 12,0 + c 12 (1)f 22,0 + c 13 (1)f 32,0 , (8.56)f 33 (1) = 1 = c 31 (1)f 13,0 + c 32 (1)f 23,0 + c 33 (1)f 33,0 . (8.57)