by Naohito Abe x8347
1982Kydland and Prescott Long and Plosser RBC(Real Business Cycle) RBC RBC SolowDiamon RBC RBC RBC DSGE(Dynamic Stochastic General Equilibrium) RBC KPR(King,
Plosser, and Rebelo) Forward
1
2
RBC George McCandless [2008] The ABCs of RBCs: An Introduction to Dynamic
Macroeconomic Models Harvard University Press.
RBC Thomas 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 RBC Frontier Rebelo
Sergio Rebelo [2005] “Real Business Cycle Models: Past, Present and Fu- ture,” Scandinavian Journal of Economics, vol. 107(2), pages 217-238

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
friction New 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. NKDSGE RBC Ljungqvist and Sargent [2012] ( )NKDSGE Gali
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 Economics Volume 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 Economic Papers, 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
3
?RBC GDP, Solow Residuals Solow Residuals
GDP Hodric-Prescott (the
H-P filter) yt components(yc
t , y g t )yc
t Cyclical Component yg
t Component
[( yg
4
λ 16002 λ = 0 λ λλ = 1600 83 Solow Residuals 4 1Cooley ed.[1995] H-P filter Cross Correlation Cooley ed.[1995]

(2)
(3) smooth
(6)
(7)
H-Pfilter Detrending Method
Ravn, Morten O. and Uhlig, Harald, “On Adjusting the HP-Filter for the Frequency of Observations,” Review of Economics and Statistics, Vol. 84, Issue 2 - 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 λ 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 Filter Figure 1Band Pass FilterHP Filter ()
Favio Canova.[1998], “Detrending and Business Cycle Facts,” Journal of Monetary Economics 41, 475—512.
Timothy Cogley and Japnes M. Nason [1995] “Effects of the Hodrick-Prescott Filter on Trend and Difference Stationary Time Series,” Journal of Economic Dynamics 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
(support)
Max
s.t. dkt
dt = f (kt) − δkt − ct for all t, ko > 0: given. (4)

5 [2002]90 July
6
Max ∞∑
βtu (ct) , (5)
s.t. kt+1 = f (kt) + (1 − δ) kt − ct for all t, ko > 0: given. (6)
ρ β

u (ct, 1 − lt) (7)
lt 1 − lt 1
6
yt = eztF (kt, lt) , (8)
zt (AR1)
zt+1 = ηzt + εt+1 (9)
ε (i.i.d.)0 < η < 1zt 0 η ε i.i.d. 7 AR1 20132014
6 z
7 Romer[2001]
7
E0
(support)
[]
] , (11)
s.t. kt+1 = eztF (kt, lt) + (1 − δ) kt − ct, for all t, ko > 0: given, (12)
zt+1 = ηzt + εt+1. (13)

8 Behavioral Economics Robert Shiller Alan Blinder (Thaler)[1998] ( )
9
8
ct = h (kt) , (14)
10t Control Variable ct lt
ct = h1 (kt, zt) (15)
lt = h2 (kt, zt) (16)
11 2 Policy Function 12 Scarf Algorithm
Closed Form Policy Function (1)
10 State Variables
11 Stokey and Lucas [1989], Recursive Methods in Economic Dynamics, Harvard University Press 1960
12
5 (Linear-Quadratic Methods)

L = E0
[ ∞∑ t=0
] .
Et
] = 0, (18)
zt ∂
)] = 0. (20)
∂
10
zt ∂
λt = βEt
[ λt+1
( ezt+1
)] . (23)
λt+1 t Random Multiplier 2
− ∂ ∂lt
= ezt ∂
1 1

lim t→∞E0
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 δ 0.1 GMMML Simulation
(2) (1) δδ RBC
(3) (1)(2)
Cooley ed.[1995] 15

Yt = eztAKθ t L1−θ
t . (27)
u (ct, 1 − lt) =
. (28)
151990
12
σ = 116
u (ct, 1 − lt) = (1 − α) ln ct + α ln (1 − lt) . (29)
γ []
Max Eo
[ ∞∑ t=0
] , (30)
s.t. (1 + γ) kt+1 = eztAkθ t l1−θ
t + (1 − δ) kt − ct, for all t, ko > 0: given, (31)
zt+1 = ηzt + εt+1. (32)
K C A (1 + γ)t θ
Cooley ed. [1995] 0.40

( ezt+1
)] = 0. (33)
zt = 0 for all t
θy
16σ σ simulation σ Policy Function
13

i = (δ + γ) k, (37)
0.076 γ 2.8% δ = 0.048 0.012θ, δ β β = 0.947, 0.987 Becker discretionary time 1/3 0.31(35)y/c 1.33 α/ (1 − α) = 1.78 Solow Residuals, ztGDP, K,
L
zt−zt−1 = (lnYt − lnYt−1)−θ (lnKt − ln Kt−1)−(1 − θ) (ln lt − ln lt−1) , (38)
zt η Cooley η = 0.95εt σε = 0.007
17
θ δ η σε γ β σ α 0.40 0.012 0.95 0.007 0.026 0.987 1 0.64
7 ()
t + (1 − δ) kt − ct, (41)
17 A A A
14
( ezt+1
Policy Functions Linear Quadratic Random MultiplierDynamic Programing Value Function Policy FunctionsDynamic Programing Blanchard and Kahn[1980] () PC 1 (ct, kt, lt, λt, zt, εt) = (c, k, l, λ, 0, 0)
for all t
g (y) = f (x) , x ∈ R. (44)
(x, y) = (x, y)
g (y) + dg
df

df
(45)y x
dy = ln y − ln y, dx = lnx − ln x, (52)
g (y) = f (x)
y dg
1−θ
1−θ dA + θAK

zt
dzt + θdkt + (1 − θ) dlt − dct = 1
1 − l dlt, (57)
−dct = dλt, (58)
dλt = Etdλt+1 + µEtdzt+1 + µ (θ − 1)Etdkt+1 + µ (1 − θ)Etdlt+1, (59)
(1 + γ) k
y dkt+1 = dzt + θdkt + (1 − θ) dlt + (1 − δ)
k
16

θy
1 + 1−α α (1 − θ) y
c
] , (70)
Policy Functions
dct = C1dkt + C2dzt, (71)
17
dct Policy Function (57) dlt Policy Function (58)dλt Policy Function Policy Function Policy Function Policy Function Blanchard 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 1
(59) (60)( −µ (1 − θ) 1 − (1 + γ) k
y 0
)( dkt
dλt
) Et
Mss0EtXt+1 + Mss1Xt = Msc0EtVt+1 + Msc1Vt + Mse0Etzt+1 + Mse1zt, (76)
Vt = (ct, lt) ′Xt = (kt, λt)
′
Control Variables 2
18
cc Mcezt, (77)
) +
) + Mse0Etzt+1 + Mse1zt,

Predetermined Variables Step (3)JordanW
P−1Xt+1 = ΛP−1Xt + P−1RZt+1 + P−1QZt, (83)
ΛWP
PΛP−1 = W. (84)
1 1Λ P
19Mcc Control Variables Mcc
19
Λ = (
W P,R,Q Λ1 4
W = (
) Etzt+1 +
Etλt+1 = Λ2λt + ( P 21Rx + P 22Rλ
) Etzt+1 +
20
2
Λ−1 2 1
lim j→∞
( Λ−1
λt zt
zt+1 = Πzt + εt+1, (96)
Etzt+1 = Πzt, (97)
λt = − ( P 22
)−1 P 21xt +

( E F G H
) (104)
−H−1GD−1 H−1 + H−1GD−1FH−1
) , (105)

(107) λt = Ψzt
(108)
λt
Vt = M−1 cc Mcs
( I
8 Impulse Response Functions
zt Predetermined Variables
xt+1 = Γxxxt + Γxzzt. (114)
Predetermined VariablesControl Variables Policy Functions
Wt = Γuxxt + Γuzzt. (115)
M = (
) . (118)
εt t+1 1 Predetermined Variables
st+j = M jst + M j−1εt+1, (119)
Control Variables
Vt+j = (Γux,Γuz) st+j , (120)
2 Impulse Response Functions
23
[ 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
% %
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
%%%%%%%%%%%%%%%%% 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
zt
30
AR1 1% θ% 2
zt 1 zt ztAR1 zt 2 zt Lucas Impulse Response Functions
1% Policy Functions0.43% 0.65% 1% (1 − θ) = 0.6% 0.389% 1% 1.389% 1%
(Predetermined ) Policy Functions
ztη Policy Functions dzt zt
31
CrossCorrelationofOutputwith
J ∫β 54 32 ∫1Y 1Y2 r3r45
Outputeomponent
ConsumptlOneXpenditures
02 16 38 63 85 10 85 63 38 16 02
0 1 1
CONS
CNDS
CD
Investmellt
INV
INVF
INVN
INVR
2 4 5

0 0 4 3
57 79 88 83 60
74 63 39 11 14
53 67 5l 27 04
01 04 08 11 16
0 J O ′ 1 0 0
5 ′ h J 7 J
n X 3 5 5 2
J 4 0 ′ h ′ l
0 5 5 7 1 5 0
4 0 0 ′ 0 ⊥ J
O O 4 0

03 204
4 9 U
553 48 42 29
488 11 19 31
Laborinputbasedonhouseholdsurvey
1 5
HSEMPLMT 10 04 23 46
GNPHSHOURS O90 06 14 20 30
9 L1∴¶5 40
33 41 19 00 18 25
0 0 4
′ h U 4 4 7 1 0 L O
7 ′ h U 1 4
0 0 4
4 0 5 1 1 1 J

0 0 9 U 4
7 ′ h 7 J
4
0 ′ h 0 0 3
J 5 J O
′ h 0 7 3
7 ′ 0 0 0
0 5 0
0 4 ∩ 1
4 0 0 7 5
5 J 4 4
1 0 4
3 J 4

WGE O757
alincomeaccounts
COMP O55 24 25 21 14 09 03 07 09 09 09 10
NoesGNPrealGNP1982CONSperSOnalconsump10neXPendiure1982SCNDSCOnSumPILOnOfnondurabesandservices1982CDOnSumPtlOnOf
durables1982SINVgrOSSPrlVatedomesicinvestmenI1982SINVFfixedinvestment1982SINVNnOnTeSidentiafixedinvestment1982INVRreSidenIia1
6xedinvesment1982SChINVhangeininventoTies1982GOVTgOVemnlenPurChasesofgoodsaJldservices1982EXPeXPOrtSOfgoodsandservices1982
IMPinlPOrtSOrgOOdsandservices1982HSHOURStOa10uTSOfworkHouseholdSurveyHSAVGHRSaVerageVeeklyhoursofworkHouseholdSurvey
HSEMPLMTmPlomentESHOURStOtahoursofworksEsablishmentSun′eyESAVGHRSaVerageWeeklyhoursofworkEstablishment
SurveyESEMPLMTmPloymentEstablishmenSurveyWAGEaVeragehourl′eanling1982EstablishmenSurveyCOMPaVerage0acompensationper
hour1982NationalIncomeAccounSTheEstablishmenSurveysampleisfor1964l199l
”FrontiersofBusinessCycle ResearchPIEditedbvThomasFCooley PublishedbyPrincetonUniversltyPress
41 tntrodu⊂tionSome Fa⊂tSabout E⊂OnOmi⊂FJu⊂tuatjons 169
U 1 S t s t O p p U O S u O £
d U
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7 6 5
0 0
TABLE41 Re⊂eSSions jn the United States sin⊂eWorldWarIl
Yearandquarter Numberofquartersuntil ChangeinrealGDP
OfpeakinrealGDP troughinrealGDP peaktotrough
0
0
0
0
0
0
0
0
0
Year
Year
Year
Year
Year
Year
1
0
0
0
Year
19 82 :0 1
19 84 :0 1
19 86 :0 1
19 88 :0 1
19 90 :0 1
19 92 :0 1
19 94 :0 1
19 96 :0 1
19 98 :0 1
20 00 :0 1
20 02 :0 1
20 04 :0 1
20 06 :0 1
20 08 :0 1