Citation preview
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