Additional Methods for Solving DSGE Models

Transcription

1 Addiional Mehod for Solving DSGE Model Karel Meren, Cornell Univeriy Reference King, R. G., Ploer, C. I. & Rebelo, S. T. (1988), Producion, growh and buine cycle: I. he baic neoclaical model, Journal of Moneary Economic 21(2-3), Klein, P. (2000), Uing he generalized chur form o olve a mulivariae linear raional epecaion model, Journal of Economic Dynamic and Conrol 24(3), Ljunqvi, L. & Sargen, T. (2004), Recurive Macroeconomic Theory, 2nd Ed, MIT Pre. Ch

2 We have olved he imple RBC model in King e al. (1988) applying he mehod of undeermined coefficien o he linearized yem. The mehod i generalized by Uhlig 1997 and malab ofware can be found a: hp://www2.wiwi.hu-berlin.de/iniue/wpol/hml/oolki.hm Here are wo addiional mehod o find an approimae oluion. 1 Linear Syem and he Schur (QZ) Decompoiion We can wrie he linearized yem of ochaic difference equaion a AE +1 = B (1) where =, z T collec he ae variable and he conrol z. Noe ha in he imple RBC model, we had choen uch ha A wa inverible and W = A B. The QZ decompoiion doe no require A o be inverible, which mean ha addiional aic (inraemporal) equilibrium condiion can be included among he dynamic relaionhip. We are looking for a oluion of he form +1 = G + F e +1 (2) z = H Conider he comple generalized Schur/QZ decompoiion: QAZ = S i upper riangular QBZ = T i upper riangular Moreover, Q H Q = Z H Z = I where upercrip H denoe he Hermiian ranpoe. Define he generalized eigenvalue λ i a he raio of he ih diagonal elemen of T and S. If A i inverible, han λ i i ju he ih eigenvalue of W. If A i ingular, ome of i diagonal elemen are zero and he correponding λ i i reaed a infinie. I urn ou here ei a real QZ decompoiion for every ordering of λ i. Le S and T be arranged in uch a way ha he able (i.e. maller han one) generalized eigenvalue come fir and he unable 2

3 (eceeding one and infinie) come la. Correpondingly, define he auiliary variable = Z = Z Z11 Z 12 = Z 21 Z 22 u u (3) where are he able ranformed variable and u are he unable ranformed variable. Hence AZE +1 = BZ (4) Premuliplying by Q give a new yem equivalen o (1) S11 S 12 0 S 22 SE +1 u +1 E +1 u +1 = T = u T11 T 12 0 T 22 u (5) (6) where S 11 and T 22 are quare and inverible by aumpion. Hence, we can wrie u = T 22 S 22E u +1 (7) Furhermore, he generalized eigenvalue aociaed wih S 22 and T 22 are all unable by conrucion. Therefore, afer olving forward, he oluion for u will eplode unle u = 0 Given our oluion for u, we have ha E +1 = S 11 T 11 (8) where S 11 T 11 i a able mari by conrucion. Defining ẽ +1 = Z 11 ( +1 E +1 ) a he error in epecaion and under he imporan aumpion ha Z 11 i inverible we can wrie +1 u +1 = S 11 T u + Z ẽ+1 0 (9) 3

4 and afer premuliplying by Z +1 = Z S 11 T Z H + From he definiion of he ranformed variable we have ha which lead o +1 = z = Z 11 Z Z 21 u +1 Z 22 u ẽ+1 0 (10) +1 = Z 11 S 11 T 11Z 11 + ẽ +1 z = Z 21 Z 11 The final ep i o pin down he epecaional error ẽ +1, which i done by heoreical moivaion: ẽ +1 = F e +1 The oluion o he King e al. (1988) model i compued wih he QZ decompoiion in he Malab program rbcmodel2.m uing he code provided by Klein (2000) (olab.m). 4

5 2 The Opimal Linear Regulaor Conider he following linear-quadraic conrol problem: ma {z } =0 E 0 β ( T R + z T Qz + T W z + z T W T ) =0 (11).. +1 = A + Bz + e +1 (12) 0 given where upercrip T denoe he ranpoe, R and Q are ymmeric negaive emidefinie marice and e +1 i a vecor of mean zero iid hock wih E e +1 e T +1 = Ω. Aume he value funcion i given by V ( ) = T P + d where P i a ymmeric negaive emidefinie mari. Dropping he ime ubcrip, he value funcion can be wrien a T P + d = ma ( T R + z T Qz + T W z + z T W T + βe (A + Bz + e ) T P (A + Bz + e ) + d ) z The fir order condiion i z : z T Q + z T Q T + 2 T W + 2βE(A + Bz + e ) T P B = 0 Qz + W T + βeb T P (A + Bz + e ) = 0 z = ( Q + βb T P B ) ( W T + βb T P A ) Plugging he oluion for z ino (12): = ( A B ( Q + βb T P B ) ( W T + βb T P A )) + e We ill don know P, however, plugging our oluion for z ino he value funcion, we ge ha P mu olve T P + d = T R + T F T QF T W F T F T W T +βe ((A BF ) + e ) T P ((A BF ) + e ) + d = T ( R + F T QF W F F T W T + β(a BF ) T P (A BF ) ) + Eβ (e ) T P e + βd 5

6 where F = ( Q + βb T P B ) ( W T + βb T P A ). Uing he mehod of undeermined coefficien, we have ha P and d mu olve P = R + F T QF W F F T W T + β(a BF ) T P (A BF ) d = βd + Eβ (e ) T P e and afer ome ediou algebra P = R + βa T P A + ( βa T P B + W ) ( Q + B T P B ) ( βb T P A + W T ) (13) d = β r (P Ω) 1 β (14) Equaion (13) canno be olved analyically, bu i under ome regulariy condiion raighforward o olve numerically by ieraing on he mari Ricai difference equaion. Sar wih an iniial gue P 0 and hen ierae P j+1 = R + βa T P j A + ( βa T P B + W ) ( Q + B T P j B ) ( βb T P j A + W T ) unil convergence. Afer finding P we can ue (14) o olve for d, alhough i ener only he value funcion and no he policy funcion. Now why i all hi ueful? I i ueful becaue we can approimae he planner problem in our RBC model a a linear-quadraic problem uch a (11). Recall he recurive formulaion of he planning problem in he imple RBC model v(k, a) = ma k,n u(āea k 1 α N α + i, 1 N) + βe v(k, a ) a.. i = γ k (1 δ)k a = ρa + ϵ 6

7 Le = 1, T = 1, k, a T and le z = N, i. Conider a econd order Taylor approimaion of u around he deerminiic eady ae: u ū + u ( ) + u z (z z) ( ) T u ( ) (z z) T u zz (z z) (z z) T u z ( ) ( ) T u z (z z) ū u u 1 T z z ( T u + z T u zz z + z T u z + T u z z ) 1 2 (u T u z T u z ) (ut 1 u u z z) 2 u + 1 u 1 2 T z z T u zz T u z z zt u T z u zz z u z u z zt u zz z u z T R + T W z + z T W T + z T Qz The oluion o he King e al. (1988) model i compued uing he opimal linear regulaor in he Malab program rbcmodel3.m. Noe ha in he malab program, he approimaion i linear inead of loglinear. You can read abou he opimal linear regulaor in Ljunqvi & Sargen (2004). 7