上級マクロ経済学 Summer 2015 講義ノート(RBC)nabe/rbc2015.pdf上級マクロ経済学...

Summer 2015 (RBC)

by Naohito Abex8347

nabe@ier.hit-u.ac.jp2015 June

1

10 AD-AS2000Matlab Dyner

1982Kydlandand Prescott Long and PlosserRBC(Real Business Cycle)RBCRBC SolowDiamonRBC RBC RBC DSGE(Dynamic Stochastic General Equilibrium) RBCKPR(King,

Plosser, and Rebelo)Forward

1

2

RBCGeorge McCandless [2008] The ABCs of RBCs: An Introduction to Dynamic

Macroeconomic Models Harvard University Press.

RBCThomas F. Cooley ed. [1995] Frontiers of Business Cycle Research, Prince-

ton University Press.

King, Robert, Plosser, Charles and Rebelo, Sergio, [1988a], Production,growth and business cycles: I. The basic neoclassical model, Journal of Mone-tary Economics, 21, issue 2-3, p. 195-232.

King, Robert, Plosser, Charles and Rebelo, Sergio, [1988b], Production,growth and business cycles: II. New directions, Journal of Monetary Economics,21, issue 2-3, p. 309-341. JME

KPR 10 RBCFrontier Rebelo

Sergio Rebelo [2005] Real Business Cycle Models: Past, Present and Fu-ture, Scandinavian Journal of Economics, vol. 107(2), pages 217-238

Ljungqvist and Sargent [2012]

Carlo A. Favero [2001] Applied Macroeconometrics, Oxford University Press.VAR, Cowles Commission Approach, GMMCaribration

John B. Taylor and Michael Woodford Ed. [1999] Handbook of Macroeco-

nomics 1A, 1B, and 1C, North-Holland.

2

Heer, Burkhard and Alfred Maussner [2009] Dynamic General EquilibriumModeling, SpringerProjection Methods, Discretization GAUSS FORTRANMatlabGali SmetsRBC

frictionNew Keynesian Dynamic Stochastic General Equilibirum Model (NKDSGE)

Gali, J., 2008. Monetary policy, intution and the businesscycle: An intro-duction to the New Keynesian framework, Princeton University Press.NKDSGERBC Ljungqvist and Sargent [2012] ()NKDSGEGali

Fernandez-Villaverde, J., 2010, The econometrics of DSGEmodels, SERIEs.Smets, F.R., and Wouters, R., 2003, An estimated Dynamic Stochastic

General Equilibrium model for the Euro area, Journal of the European Eco-nomic Association.

Friedman, B.M. and M. Woodford ed. (2011) Handbook of Monetary EconomicsVolume 3A,B. North Holland

Mark Mink & Jan P.A.M. Jacobs & Jakob de Haan, [2012]. Measuring

coherence of output gaps with an application to the euro area, Oxford EconomicPapers, Oxford University Press, vol. 64(2), pages 217-236, April.

Jeske, Karsten & Krueger, Dirk & Mitman, Kurt, [2013]. Housing, mort-gage bailout guarantees and the macro economy, Journal of Monetary Eco-nomics, Elsevier, vol. 60(8), pages 917-935.

3 RBC

RBCDSG 1970Keynesian Phillips

3

Solow (Growth Accounting) 1/32/3 (Solow Residuals)CPU12007 20082014Solow Residuals ()RBC DSGE

?RBCGDP, Solow Residuals Solow Residuals

GDPHodric-Prescott (the

H-P filter) yt components(yct , y

gt )yct Cyclical Component

ygt Component

MinT

t=1

(yct )2 +

T1t=2

[(ygt+1 ygt

) (ygt ygt1)]2 , (1)1

Solow Residuals

4

yt = yct + ygt . (2)

16002 = 0 = 1600 83Solow Residuals4 1Cooley ed.[1995] H-P filter Cross CorrelationCooley ed.[1995]

(1) 1.72 1.59 or 1.69

(2)

(3) smooth

(4)

(5) Pro Cyclical

(6)

(7)

(Stylized Facts) RBC H-Pfilter

H-Pfilter Detrending Method

Ravn, Morten O. and Uhlig, Harald, On Adjusting the HP-Filter for theFrequency of Observations, Review of Economics and Statistics, Vol. 84, Issue2 - May 2002.

2Annual 100Monthly 14400

3Burnside[1999]H-P filter

4Eviews TSP Stata web (http://ideas.repec.org/c/boc/bocode/s447001.html)

5

annual 1006.25

Baxter and King [1999] Measuring Busibess Cycles: Approximate Band-Pass Filters for Econmic Time Series, Review of Economics and Statistics,Vol. 81, Issue 4 . Band-Pass Filter

Band-Pass FilterFigure 1Band Pass FilterHP Filter ()

Favio Canova.[1998], Detrending and Business Cycle Facts, Journal ofMonetary Economics 41, 475512.

Timothy Cogley and Japnes M. Nason [1995] Effects of the Hodrick-PrescottFilter on Trend and Difference Stationary Time Series, Journal of EconomicDynamics and Control vol 19.Lawrence J. Christiano & Terry J. Fitzgerald [2003]. The Band Pass Fil-

ter, International Economic Review, vol. 44(2), pages 435-465, 05.5

4

Stylized Facts Starting Point

(support)

Max

0

etu (ct) dt, (3)

s.t.dktdt

= f (kt) kt ct for all t, ko > 0: given. (4)

5 [2002]90 July

6

1 44 4 1

Max

t=0

tu (ct) , (5)

s.t. kt+1 = f (kt) + (1 ) kt ct for all t, ko > 0: given. (6)

u (ct, 1 lt) (7)lt 1 lt

1

6

yt = eztF (kt, lt) , (8)

zt (AR1)

zt+1 = zt + t+1 (9)

(i.i.d.)0 < < 1zt 0 i.i.d.7 AR120132014

6z

7 Romer[2001]

7

89t Et

E0

[ t=0

tu (ct, 1 lt)]

. (10)

RBC

(support)

[]

Max E0

[ t=0

tu (ct, 1 lt)]

, (11)

s.t. kt+1 = eztF (kt, lt) + (1 ) kt ct, for all t, ko > 0: given, (12)

zt+1 = zt + t+1. (13)

8Behavioral Economics Robert Shiller Alan Blinder (Thaler)[1998] ( )

9

8

Saddle Path

ct = h (kt) , (14)

Saddle PathPolicy Function jump State Variable Pre-determined Variablet t Control Variable Non-Predetermined Variable Predetermined Variable Control VariablePolicy Function RBCState Variable kt zt

10tControl Variable ct lt

ct = h1 (kt, zt) (15)

lt = h2 (kt, zt) (16)

112 Policy Function12Scarf Algorithm

ClosedForm Policy Function (1)

10 State Variables

11 Stokey and Lucas [1989], Recursive Methodsin Economic Dynamics, Harvard University Press 1960

12

9

(2)Miranda and Fackler [2002] Applied ComputationalEconomics and Finance, MIT Press.

5 (Linear-Quadratic Methods)

Blanchard and Kahn[1980] King, Plosser, and Rebelo [1988a, b]Blachard-Kahn KPRPolicy Functions Predetermined Variables Saddle PathSaddle Path Saddle Path Random Lagrange Methods

L = E0

[ t=0

tu (ct, 1 lt) + tt (eztF (kt, lt) + (1 ) kt ct kt+1)]

.

(17) tctltkt+1

Et

[ct

u (ct, 1 lt) t]

= 0, (18)

Et

[

ltu (ct, 1 lt) + tezt

ltF (kt, lt)

]= 0, (19)

Et

[t + t+1

(ezt+1

kt+1F (kt+1, lt+1) + (1 )

)]= 0. (20)

2 t t(20)tt t

ctu (ct, 1 lt) = t, (21)

10

lt

u (ct, 1 lt) = tezt lt

F (kt, lt) , (22)

t = Et

[t+1

(ezt+1

kt+1F (kt+1, lt+1)

)]. (23)

t+1 t Random Multiplier 2

lt u (ct, 1 lt)

ctu (ct, 1 lt)

= ezt

ltF (kt, lt) . (24)

1 1

limtE0

[ttkt

]= 0. (25)

path13

limtE0

[tkt

]= 0, (26)

6

Cooley ed.[1995]14

13Stokey, Lucas, and Presoctt [1989], Recursive Methods in Economic Dynamics,Harvad University Press.

14 RBC t- F-

11

(1) 0.1 0.1GMMMLSimulation

(2) (1) RBC

(3) (1)(2)

Cooley ed.[1995]15

Yt = eztAKt L1t . (27)

1

u (ct, 1 lt) =(c1t (1 lt)

)1 11 . (28)

1/

151990

12

= 116

u (ct, 1 lt) = (1 ) ln ct + ln (1 lt) . (29)

[]

Max Eo

[ t=0

t ((1 ) ln ct + ln (1 lt))]

, (30)

s.t. (1 + ) kt+1 = eztAkt l1t + (1 ) kt ct, for all t, ko > 0: given, (31)

zt+1 = zt + t+1. (32)

K C A (1 + )t

Cooleyed. [1995] 0.40

Et

[t + t+1 (1 + )1

(ezt+1

Akt+1l1t+1

kt+1+ (1 )

)]= 0. (33)

zt = 0 for all t

y

k+ (1 ) = 1 +

. (34)

(1 ) yc

=

1 l

1 l . (35)

(1 + )k

y= (1 ) k

y+

i

y. (36)

16 simulation Policy Function

13

i = ( + ) k, (37)

0.076 2.8% = 0.048 0.012, = 0.947, 0.987Beckerdiscretionary time 1/3 0.31(35)y/c 1.33/ (1 ) = 1.78Solow Residuals, ztGDP, K,

L

ztzt1 = (lnYt lnYt1) (lnKt ln Kt1)(1 ) (ln lt ln lt1) , (38)

zt Cooley = 0.95t = 0.007

17

0.40 0.012 0.95 0.007 0.026 0.987 1 0.64

7 ()

(1 ) eztAkt l

1t

ct=

1 lt

1 lt , (39)

(1 )ct

= t, (40)

(1 + ) kt+1 = eztAkt l1t + (1 ) kt ct, (41)

17 A A A

14

t = Et

[t+1 (1 + )

1(

ezt+1Akt+1l

1t+1

kt+1+ (1 )

)], (42)

zt+1 = zt + t+1. (43)

Policy FunctionsLinear QuadraticRandom MultiplierDynamic ProgramingValueFunction Policy FunctionsDynamicProgramingBlanchard and Kahn[1980] () PC 1 (ct, kt, lt, t, zt, t) = (c, k, l, , 0, 0)

for all t

g (y) = f (x) , x R. (44) (x, y) = (x, y)

g (y) +dg

dy(y) (y y) = f (x) + df

dx(x) (x x) . (45)

x = eln x (46)

h = ln x, (47)

j = ln y, (48)

y = f (x)

g(ej

)= f

(eh

). (49)

j,h

15

g(ej

)+ ej

dg

dy

(j j) = f (eh) + df

dx

(eh

)eh

(h h) . (50)

j,h y,x

g (y) + ydg

dy(y) (ln y ln y) = f (x) + x df

dx(x) (lnx lnx) . (51)

(45)y x

dy = ln y ln y, dx = lnx ln x, (52)g (y) = f (x)

ydg

dy(y) dy = x

df

dx(x) dx. (53)

Y = AKL1. (54)

Y = AKL

1

Y (dY ) = AKL

1dA + AK

L

1dK + (1 ) dL. (55)

dY = dA + dK + (1 ) dL. (56)

(39) (43)ztzt

dzt + dkt + (1 ) dlt dct = 11 l dlt, (57)

dct = dt, (58)

dt = Etdt+1 + Etdzt+1 + ( 1)Etdkt+1 + (1 )Etdlt+1, (59)

(1 + )k

ydkt+1 = dzt + dkt + (1 ) dlt + (1 ) k

ydkt c

ydct, (60)

dzt+1 = dzt + dt+1. (61)

16

=y

(1 + ) k, (62)

y = Akl1, (63)

y

k+ (1 ) = 1 +

, (64)

(1 ) yc

=

1 l

1 l , (65)

( + )k

y= 1 c

y, (66)

y = kl1, (67)

y

k=

1

[1 +

(1 )

], (68)

c

y= 1 ( + ) k

y, (69)

l =1

(1 ) yc[1 + 1 (1 ) yc

] , (70)Root FindingMatlab fzero18(57)-(61)

Policy Functions

dct = C1dkt + C2dzt, (71)

dlt = C3dkt + C4dzt, (72)18 (Maximum Likelihood Methods)fzero Matlab Root Finding ToolBox

17

dct Policy Function(57) dlt Policy Function (58)dt Policy Function Policy Function Policy Function Policy FunctionBlanchard and Kahn[1980] Control VariableBurnside[1999]King, Plosser, and Rebelo [1988a, b]

(1) Control Variables State Variables, Shocks

(2) State Variables, Shocks(3) Jordan(4) Policy Function(5) (1)Control Variables Policy Function

(1)(57) (58)

(1 11l 1 + 1 0

)(dctdlt

)=

( 00 1

)(dktdt

)+

(10

)dzt. (73)

(59) (60)( (1 ) 1 (1 + ) k

y 0

)Et

(dkt+1dt+1

)+

(0 1

+ (1 ) ky 0)(

dktdt

)=

(0 (1 )0 0

)Et

(dct+1dlt+1

)+

(0 0

c/y (1 ))(

dctdlt

)+(

0

)Etdzt+1 +

(01

)dzt. (74)

2

MccVt = McsXt + Mcezt, (75)

Mss0EtXt+1 + Mss1Xt = Msc0EtVt+1 + Msc1Vt + Mse0Etzt+1 + Mse1zt, (76)

Vt = (ct, lt)Xt = (kt, t)

Control Variables 2

18

MccM1cc 19

Vt = M1cc McsXt + M1cc Mcezt, (77)

Mss0EtXt+1 + Mss1Xt = Msc0Et(M1cc McsXt+1 + M

1cc Mcezt+1

)+

Msc1(M1cc McsXt + M

1cc Mcezt

)+ Mse0Etzt+1 + Mse1zt,

(78)

EtXt+1 = WXt + REtzt+1 + Qzt (79)

W = (Mss0 Msc0M1cc Mce)1 (Mss1 Msc1M1cc Mcs) , (80)R =

(Mss0 Msc0M1cc Mce

)1 (Mse0 + Msc0M1cc Mce

), (81)

Q =(Mss0 Msc0M1cc Mce

)1 (Mse1 + Msc1M1cc Mce

), (82)

Predetermined VariablesStep (3)JordanW

P1Xt+1 = P1Xt + P1RZt+1 + P1QZt, (83)

WP

PP1 = W. (84)

(xtt

)= P1Xt. (85)

Predetermined Variables xt 1Predetermined Variables State Variables

11 P

19Mcc Control Variables Mcc

19

=(

1 00 2

), (86)

1 12 12 1 11Predetermined Variables1 PredeterminedVariables indeterminacyindeterminacy1 PredeterminedVariables

W P,R,Q 1 4

W =(

W11 W12W21 W22

), R =

(RxR

), (87)

Q =(

QxQ

), P =

(P11 P12P21 P22

), (88)

P1 =(

P 11 P 12

P 21 P 22

). (89)

(W11 W12W21 W22

)=

(P111P 11 + P122P 21 P111P 12 + P122P 22

P211P 11 + P222P 21 P211P 12 + P222P 22

).

(90)(83)

Etxt+1 = 1xt +(P 11Rx + P 12R

)Etzt+1 +

(P 11Qx + P 12Q

)zt. (91)

Etxt+1 xtPolicy Function

Ett+1 = 2t +(P 21Rx + P 22R

)Etzt+1 +

(P 21Qx + P 22Q

)zt, (92)

2 1 (2)t

20

ForwardForward

t = 12 Ett+1 12[(

P 21Rx + P 22R)Etzt+1 +

(P 21Qx + P 22Q

)zt

],

(93)

t =

j=0

(j+1)2[(

P 21Rx + P 22R)Etzt+1+j +

(P 21Qx + P 22Q

)Etzt+j

].

(94) t

12 1

limj

(12

)jEtt+j+1 = 0, (95)

zt AR1

t zt

zt+1 = zt + t+1, (96)

Etzt+1 = zt, (97)

t = zt, (98)

=

j=0

(j+1)2[(

P 21Rx + P 22R) +

(P 21Qx + P 22Q

)]jzt. (99)

t, xt Xt = (kt, t)

t = (P 22

)1P 21xt +

(P 22

)1t. (100)

xt+1 = W11xt + W12t + RxEtzt+1 + Qxzt, (101)

xt+1 =(P111P 11 + P122P 21

)xt+

(P111P 12 + P122P 22

)t+RxEtzt+1+Qxzt,

(102)

21

xt+1 =(P111

[P 11 P 12 (P 22)1 P 21]) xt+(

P111P 12 + P122P 22) (

P 22)1

t + RxEtzt+1 + Qxzt. (103)

(E FG H

)(104)

(D1 D1FH1

H1GD1 H1 + H1GD1FH1)

, (105)

D = E FH1G. (106)

xt+1 =(P111P111

)xt+

(P111P 12 + P122P 22

) (P 22

)1t+RxEtzt+1+Qxzt.

(107)t = zt

xt+1 =(P111P111

)xt+

(P111P 12 + P122P 22

) (P 22

)1zt+RxEtzt+1+Qxzt,

(108)

xt+1 =(P111P111

)xt +

[(P111P 12 + P122P 22

) (P 22

)1 + Rx + Qx

]zt,

(109)

xt+1 = xxxt + xzzt. (110)

t

t = (P 22

)1P 21xt +

(P 22

)1zt, (111)

Control Variables Vt = (ct, lt)

Vt = M1cc Mcs

(I

(P 22)1 P 21)

xt+[M1cc Mcs

(0

(P 22)1 )

+ M1cc Mce

]zt,

(112)

Vt = uxxt + uzzt. (113)

22

Policy Functions State Vari-ables

8 Impulse Response Functions

Policy FunctionsPolicy Func-tionsImpulse Response Functions 1% Impulse ResponsePolicy Functions zt+1 = zt + t+1 t+1 t+1

zt Predetermined Variables

xt+1 = xxxt + xzzt. (114)

Predetermined VariablesControl VariablesPolicy Functions

Wt = uxxt + uzzt. (115)

st =(

xtzt

). (116)

st

st+1 = Mst + t+1, (117)

M =(

zz zz0

), t =

(0t

). (118)

t t+1 1 Predetermined Variables

st+j = M jst + M j1t+1, (119)

Control Variables

Vt+j = (ux,uz) st+j , (120)

2 Impulse Response Functions

23

9 Matlab Programs

Impulse-Response FunctionsMatlab 2 1 The Main Program 2 1Policy Functions 2% Matlab

[ 1] (The Main Program)

%%%%clear all;format short;%% Parameter Values%delta = 0.012; % Depeciation (annual)beta = 0.987;gam = 0.026;ehta = 0.95;alpha = 0.64;ceta = 0.40;%% Special Values for our model%%yovk = (1/ceta)*( (1+gam)/beta - (1-delta));covy = 1-(gam+delta)*(1/yovk);le = ((1-alpha)/alpha)*(1-ceta)*(1/covy)/( 1+((1-alpha)/alpha)*(1-ceta)*(1/covy));mhu = ceta*yovk*beta/(1+gam);%n=1; % The number of the predtermined variables%iter1=30; % The number of itertation for Impulse-Responses%%%% Matrices For subroutine to solve dynamic optimization problem%

24

%% MCC matrix%mcc=zeros(2,2);mcc(1,1) = 1;mcc(1,2) = 1/(1-le) -1 +ceta;mcc(2,1) = -1;%%% MSC Matrix%mcs = zeros(2,2);mcs(1,1) = ceta;mcs(2,2) = 1;%%% MCE Matrix - no stochastic elements%mce = zeros(2,1);mce(1,1) = 1;%%% MSS0 Matrix%mss0 = zeros(2,2);mss0(1,1) = -mhu*(1-ceta);mss0(1,2) = 1;mss0(2,1) = -(1+gam)/(yovk);%%% MSS1 Matrix%mss1 = zeros(2,2);mss1(1,2) = -1;mss1(2,1) = ceta+(1-delta)*(1/yovk);%%% MSC0 Matris%msc0 = zeros(2,2);msc0(1,2) = -mhu*(1-ceta);%%% MSC1 Matrix%msc1 = zeros(2,2);

25

msc1(2,1) = covy;msc1(2,2) = -1+ceta;%%% MSE0 Matrix%mse0 = zeros(2,1);mse0(1,1) = -mhu;%%% MSE1 Matrix%mse1 = zeros(2,1);mse1(2,1) = -1;%%% PAI Matrix%pai = zeros(1,1);pai(1,1)= ehta;%%%[GXX,GXZ,GUX,GUZ,M,Psi,V] = burns6(n,mcc,mcs,mce,mss0,mss1,msc0,msc1,mse0,mse1,pai);%%% Drawning the impulse and response function%%TSE=zeros(2,1); % The state and shock variables%TSE(2,1)=1;%%%T =zeros(iter1,8); % Impulse Response Matrix%T(1,1) = 1; % The first periods index%%N = M;%%for k=1:iter1

%%

26

k1 = k;%TCC = GUX*TSE(1,1) + GUZ*TSE(2,1);%TY = (1-ceta)*TCC(2,1)+ceta*TSE(1,1) + TSE(2,1);%TW = TY - TCC(2,1);%TR = TY - TSE(1,1);%T(k,:)=[k1 TCC(1,1) TCC(2,1) TSE(1,1) TSE(2,1) TY TW TR];%TSE=M*TSE;%%

end;%%% Plot the results%%%figure;%subplot(4,2,1)plot(T(:,1),T(:,2))title( (1) Consumption)xlabel(Year)%subplot(4,2,2)plot((T(:,1)),T(:,3))title( (2) Employment)xlabel(Year)%subplot(4,2,3)plot((T(:,1)),T(:,4))title( (3) Capital)xlabel(Year)%subplot(4,2,4)plot((T(:,1)),T(:,6))title( (4) GDP)xlabel(Year)%subplot(4,2,5)plot((T(:,1)),T(:,7))

27

title( (5) Wage)xlabel(Year)%subplot(4,2,6)plot((T(:,1)),T(:,8))title( (6) r)xlabel(Year)subplot(4,2,7)plot((T(:,1)),T(:,5))title( (7) shock)xlabel(Year)

%%%%%%%%%%%%%%%%% The End of the Program %%%%%%%%%%%%

[ 2]

function [GXX,GXZ,GUX,GUZ,M,Psi,V] = burns6(n,mcc,mcs,mce,mss0,mss1,msc0,msc1,mse0,mse1,pai)%% n is the number of the predetermined variable.%% Mcc*ut = Mcs*(xt,ramt)+Mce*zt,% Mss0*(xt+1, ramt+1) +Mss1*(xt, ramt) = Msc0*ut+1 + Msc1*ut+Mse0*zt+1

+ Mse1*zt,% zt+1 = Pai * zt + et.%%% Mss0 should be square.%% The outputs of this fumction are Gxx, Gxz, Gux, and Guz which are the

coefficinents of%% xt+1 = Gxx*xt + Gxz *zt,% ut = Gux*xt + Guz *zt.%% M is a transition matrix for both xt and zt.% V is a diagonal matrix which shows the stability of the system.% The number of the diagonal elements whose absolute values are% smaller than one should be the same as the number of the state variables% to get a unique solution.%%Mss0 = mss0 - msc0*inv(mcc)*mcs;Mss1 = mss1 - msc1*inv(mcc)*mcs;Mse0 = mse0 + msc0*inv(mcc)*mce;

28

Mse1 = mse1 + msc1*inv(mcc)*mce;%W = -(Mss0)\Mss1;R = (Mss0)\Mse0;Q = (Mss0)\Mse1;%% This corresponds to (xt+1,ramt+1)=W*(xt,ramt)+Q*zt+1+R*zt;%[PO,VO]=eig(W); % The eigensystem of this economy.%n1 = length(W); % The number of the endogenous variables in the reduced

model.%% Rearranging the matrices%alamb=abs(diag(VO));[lambs, lambz]=sort(alamb);V=VO(lambz,lambz);P = PO(:,lambz);%% Partitioning the matrices%P11 = P(1:n,1:n);P12 = P(1:n, n+1:n1);P21 = P(n+1:n1,1:n);P22 = P(n+1:n1,n+1:n1);%PP = inv(P);PP11 = PP(1:n,1:n);PP12 = PP(1:n, n+1:n1);PP21 = PP(n+1:n1,1:n);PP22 = PP(n+1:n1,n+1:n1);%V1 = V(1:n,1:n); % The Partition of the Jordan Matrix.V2 = V(n+1:n1, n+1:n1);%Rx = R(1:n, :);Rr = R(n+1:n1, :);%Qx = Q(1:n, :);Qr = Q(n+1:n1,:);%Phi0 = PP21*Rx + PP22*Rr;Phi1 = PP21*Qx + PP22*Qr;Phi01 = Phi0*pai + Phi1;%

29

n2 = size(mce);n3 = n2(2);%% Making a Matrix, Psi%Psi=zeros(n1-n,n3);%

for i = 1:n1-n;%for j=1:n3;

%Psi(i,j)=-(Phi01(i,j)/(1-inv(V2(i,i))*pai(j,j)));%

end;%

end;%Psi = (V2)\Psi;%GUX0 = [eye(n);-(PP22)\PP21];GUZ0 = [zeros(n,n3);(PP22)\Psi];%% Outputs, The Coefficients for the Policy Functions.%GXX = P11*V1*inv(P11);GXZ = (P11*V1*PP12 + P12*V2*PP22)*inv(PP22)*Psi+Qx+Rx*pai;GUX = inv(mcc)*mcs*GUX0;GUZ = inv(mcc)*mcs*GUZ0+inv(mcc)*mce;M = [ GXX GXZ; zeros(n3,n) pai];%%%%%%%%%%%% The End of the Program %%%%%%%%%%%%%

10

Policy Functions

dct = 0.6013dkt + 0.4305dzt, (121)

dlt = 0.2291dkt + 0.6481dzt, (122)

dkt+1 = 0.9427dkt + 0.1362dzt. (123)

zt

30

ztAR1 1% % 2

zt 1zt ztAR1zt 2ztLucas Impulse Response Functions

1% Policy Functions0.43% 0.65% 1% (1 ) = 0.6% 0.389% 1% 1.389% 1%

(Predetermined) Policy Functions

zt Policy Functions dzt zt

31

1

Tclblel1

CycJiccllBehcIVioroheUSEconomyDeviclionsfromTrendofKeyVclriclbles19541991ll

CrossCorrelationofOutputwith

J 54 32 1Y 1Y2 r3r45

Outputeomponent

GNP l2

ConsumptlOneXpenditures

02 16 38 63 85 10 85 63 38 16 02

0 1 1

l rJ

hU 3

0 0

25 42 57 72 82 83 67 46 22

22 40 55 68 78 77 64 47 27

24 37 49 65 75 78 6l 38 11

CONS

CNDS

CD

Investmellt

INV

INVF

INVN

INVR

ChINV

Govemmentpurehases

GOVT

Exports and imports

EXP

IMP

4 4 3 0

2 4 5

4 0 h 3 5

0 0 4 3

79 91 76 50 22

82 90 8 60 35

57 79 88 83 60

74 63 39 11 14

53 67 5l 27 04

01 04 08 11 16

0 J O 1 00

5 h J 7 J

nX 3 5 5 2

J 4 0 h l

0 5 5 7 1 5 0

4 00 0 J

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4 4 1

2 3 1 7 J

OO 5 5 0 7

03 204

4 9U

4

10 15 37 50 54 54 52

45 62 72 71 52 28 04

553 48 42 29

488 11 19 31

Laborinputbasedonhouseholdsurvey

HSHOURS 159 06 09 HSAVGHRS 063 04 16

1 5

30 53 74 86 82 69 52 32

34 48 63 62 52 37 23 09

HSEMPLMT 10 04 23 46

GNPHSHOURS O90 06 14 20 30

9 L15 40

33 41 19 00 18 25

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Laborinputbasedon

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ESHOURS

ESAVGHRS

ESMPLMT

GNPESHOURS

Averagehourlyeamings

basedonestabJishment

0 00 1 J

hU 4 4 7 1 0 L O

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4 0 5 1 1 1 J

3842

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FrontiersofBusinessCycle ResearchPIEditedbvThomasFCooley PublishedbyPrincetonUniversltyPress

41 tntrodutionSome FatSabout EOnOmiFJutuatjons 169

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dU

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000 0 0

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987 6 5

0 0

0

0 0

0

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0

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4 3

19471952195719621967197219771982198719921997

FJGURE41USrealGDP19471999

TABLE41 ReeSSions jn the United States sineWorldWarIl

Yearandquarter Numberofquartersuntil ChangeinrealGDP

OfpeakinrealGDP troughinrealGDP peaktotrough

19484

19S32

19573

19601

19703

19734

19801

19813

19902

AdvancedMacroeconomicsEditedbvDavidRomer

1Consumption 2Employment 0

0

0

0

0

0

0

0

0

0

0 0 5 10 15 20 25 30

Year

3Caplal

0 5 10 15 20 25 30

Year

4GDP

1

0

0

0

0

0

1

1

1

0

0

0 0 5 10 15 20 25 30

Year

5Wage

0 5 10 15 20 25 30

Year

6r

1

1

0

0

0

0

0

0

0

0 0 5 10 15 20 25 30

Year

7shock

0 5 10 15 20 25 30

Year

1

0

0

0

0 0 5 10 15 20 25 30

Year

0 02

0

0.02

0.040:01

2:01

4:01

6:01

8:01

0:01

2:01

4:01

6:01

8:01

0:01

2:01

4:01

6:01

8:01

DetrendedGDP

HPFilter

0.08

0.06

0.04

0.02

0

0.02

0.041980:01

1982:01

1984:01

1986:01

1988:01

1990:01

1992:01

1994:01

1996:01

1998:01

2000:01

2002:01

2004:01

2006:01

2008:01

DetrendedGDP

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