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The Academy of Economic Studies Bucharest Doctoral School of Banking and Finance. DISSERTATION PAPER CENTRAL BANK REACTION FUNCTION MSc. Student: ANDRA SERBAN Supervisor: Prof. MOISĂ ALTĂR. Contents. Introduction Review of the concepts The optimal linear regulator problem - PowerPoint PPT Presentation
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The Academy of Economic Studies The Academy of Economic Studies BucharestBucharest
Doctoral School of Banking and FinanceDoctoral School of Banking and Finance
DISSERTATION PAPER
CENTRAL BANK REACTION FUNCTION
MSc. Student: ANDRA SERBANSupervisor: Prof. MOISĂ ALTĂR
Contents
Introduction Review of the concepts The optimal linear regulator problem Model specification Empirical estimation Conclusion
IntroductionIntroduction
The purpose of this paper was The purpose of this paper was to determine an explicit to determine an explicit instrument rule and to instrument rule and to compare it to an optimal compare it to an optimal monetary policy rule (reaction monetary policy rule (reaction function) .function) .
Review of the concepts
Reaction functionReaction function::
describes the systematic describes the systematic components of economic components of economic policy in a formal model, i.e. policy in a formal model, i.e. equation.equation.
(DJC Smant 2003)(DJC Smant 2003)
General approachGeneral approach
In this approach, reaction In this approach, reaction functions are not different functions are not different than than policy rulespolicy rules, specifying , specifying how the central bank should how the central bank should adjust its instrument(s) as a adjust its instrument(s) as a function of the state of the function of the state of the economy. economy.
Rudebusch and Svensson (1998) Rudebusch and Svensson (1998) describes 2 types of rules:describes 2 types of rules:
1.1. Instrument rules: Instrument rules:
The monetary policy instrument is The monetary policy instrument is expressesd as an explicit function of expressesd as an explicit function of available information available information
22.. Targeting rules Targeting rules
Central bank is assigned to minimize Central bank is assigned to minimize a loss function that is increasing in a loss function that is increasing in the deviation between a target the deviation between a target variable and the target level for this variable and the target level for this variable.variable.
Instrument rulesInstrument rules::
It = C + B(L)Zt-1 +UtIt = C + B(L)Zt-1 +UtC – vector of constantC – vector of constantB(L) – polynomial distributed lagB(L) – polynomial distributed lagZt-1 – the central bank information at t-1Zt-1 – the central bank information at t-1It – the central bank policy instrumentsIt – the central bank policy instrumentsUt – white noiseUt – white noise
It : - the interest rate It : - the interest rate Taylor (1993), (1999); Henderson-McKibbin (1993)Taylor (1993), (1999); Henderson-McKibbin (1993)
- the monetary aggregate - the monetary aggregate McCallum (1984), (1987); Meltzer (1984), McCallum (1984), (1987); Meltzer (1984),
(1987)(1987) - domestic credit- domestic credit
Jaffee and Russell (1976); Keeton(1979); Stiglitz Jaffee and Russell (1976); Keeton(1979); Stiglitz Weiss (1981)Weiss (1981)
Targeting rulesTargeting rules
2)(min
xxE ttit
where β- discount factor, 0<β<1 Et- expectation operator
x - targeting variable x* - target level for variable x
it - instrument
Optimal control Optimal control approachapproach
More specifically, reaction More specifically, reaction functions can be regarded as functions can be regarded as solutions to an solutions to an optimal optimal controlcontrol approach to monetary approach to monetary policy.policy.
Tinbergen (1952), Theil Tinbergen (1952), Theil (1964), Klein (1965)(1964), Klein (1965)
THE OPTIMAL LINEAR THE OPTIMAL LINEAR REGULATOR PROBLEMREGULATOR PROBLEM
)()()(
.,,
1:_
)|||(|)(:
***
0
**
0
0
2
0
2
QututCXtCXtQututYtYtuJ
givenXRuRXt
CXtYt
ButAXtXtSpaceState
utYtuJCost
t
t
t
t
mn
t
t
THE OPTIMAL LINEAR THE OPTIMAL LINEAR REGULATOR PROBLEMREGULATOR PROBLEM
If the system admits a solution (V, W, If the system admits a solution (V, W, R) so that R is semipositive defined R) so that R is semipositive defined (R0) and A+BF has the |eigenvalues|(R0) and A+BF has the |eigenvalues|<1, than the command ut=FXt stabilize <1, than the command ut=FXt stabilize the system and minimize the cost J(u).the system and minimize the cost J(u).
mxnmxmpxnnxnnxm
t
RWRVRCRARB
WWCCRtAAR
VWRtBA
VRtBBQ
,,,,
***1
**
2*
From the system we obtain the Matrix From the system we obtain the Matrix Riccati Difference equation:Riccati Difference equation:
which is solved by the DLQRRICCATI which is solved by the DLQRRICCATI (Discrete Linear Quadratic Regulator) (Discrete Linear Quadratic Regulator) algorithm in Matlabalgorithm in Matlab
RtABRtBBQRtBARtAACCRt*1**2**
1 )(
MODEL SPECIFICATIONMODEL SPECIFICATION
The model is an extension of The model is an extension of Ball(1998) for an open economy:Ball(1998) for an open economy:
tttttttt
tttttt
tttttt
ttttttt
CSRCSRiiii
IR
tiCSRCSRCSRCSR
RateExchangeal
CSRCSRYY
Phillips
CSRCSRiYYYY
IS
212112101
12101
32121*
110
121111*
0*
)(
)()(
:__Re
)()(
:
)()()()(
IS
The Loss FunctionThe Loss Function
Acording to Rudenbusch and Acording to Rudenbusch and Svensson (1998) I considered the Svensson (1998) I considered the following cost function of the following cost function of the central bank :central bank :
])()()([ 21
2*2*
0
ttttt
t iicbYYa
Empirical estimation
The data sample covers the The data sample covers the period 1996:01 – 2002:12period 1996:01 – 2002:12
All time series are based on All time series are based on monthly observationmonthly observation..
TIME SERIES USED
Symbol Semnification
PI Industrial Production Chained Index Seasonal Adjusted
PI_95 Industrial Production Index computed by taking Dec 1995 as basis
LN_PI_95 Log(PI_95)
HPTREND_PI The trend of LN_PI_95 computed using a Hodrick Prescott filter
OUTPUT_GAP The difference between LN_PI_95 and its trend
IPC Consumer Price Chained Index
IPC_F Consumer Price Chained Index US
IPC_95 Consumer Price Index computed by takingDec 1995 as basis
IPC_95_F Consumer Price Index computed by taking Dec 1995 as basis
INFL Log(IPC)
CSN Nominal Exchange Rate
CSR Real Exchange Rate = log(CSN*IPC_95_F/IPC_95)
DIF_CSR First difference of CSR
RA Lending rate for non-bank customers
DIF_RA First difference of RA
RRA Real lending rate for non-bank customers
EXPNET Net exports
DUMCENTRAT dummy variable that takes the value of 1 in March 1997 and 1/(no.of observations - 1) elsewhere.
TIME SERIES USEDTIME SERIES USED
-.4
-.3
-.2
-.1
.0
.1
.2
.3
1996 1997 1998 1999 2000 2001 2002
LN_PI_95 HPTREND_PI
-.20
-.15
-.10
-.05
.00
.05
.10
.15
.20
1996 1997 1998 1999 2000 2001 2002
OUTPUT_GAP
.00
.04
.08
.12
.16
.20
.24
.28
1996 1997 1998 1999 2000 2001 2002
INFL
-.01
.00
.01
.02
.03
.04
00:01 00:07 01:01 01:07 02:01 02:07
RRA
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
1996 1997 1998 1999 2000 2001 2002
CSR
-.4
-.3
-.2
-.1
.0
.1
.2
.3
1996 1997 1998 1999 2000 2001 2002
DIF_CSR
-.02
-.01
.00
.01
.02
.03
.04
1996 1997 1998 1999 2000 2001 2002
DIF_RA
.02
.03
.04
.05
.06
.07
.08
.09
.10
1996 1997 1998 1999 2000 2001 2002
RA
Unit Root TestsUnit Root Tests
Series Order of IntegrationLevel of
Significance
ADF PP
OUTPUT_GAP I(0) I(0) 1%
INFL I(0) I(0) 1%
CSR I(1) I(1) 1%
DIF_CSR I(0) I(0) 1%
RA I(1) I(1) 1%
DIF_RA I(0) I(0) 1%
RRA I(0) I(0) 1%
model
IS estimationIS estimationDependent Variable: OUTPUT_GAP
Method: Least Squares
Date: 07/01/03 Time: 10:45
Sample(adjusted): 1996:02 2002:12
Included observations: 83 after adjusting endpoints
Variable Coefficient Std. Error
t-Statistic Prob.
OUTPUT_GAP(-1) 0.655049 0.082858 7.905.667 0.0000
RRA(-1) -0.247160 0.124724 -1.981.654 0.0510
DIF_CSR 0.179965 0.064865 2.774.448 0.0069
R-squared 0.570380 Mean dependent var 0.000877
Adjusted R-squared 0.559639 S.D. dependent var 0.049507
S.E. of regression 0.032853 Akaike info criterion -3.958.076
Sum squared resid 0.086345 Schwarz criterion -3.870.648
Log likelihood 1.672.602 Durbin-Watson stat 2.035.637
System estimation by System estimation by WTSLSWTSLS
Equation: INFL=C(1)*INFL(-1)+C(2)*DIF_CSR(-1)+C(3)*DUMCENTRAT+Equation: INFL=C(1)*INFL(-1)+C(2)*DIF_CSR(-1)+C(3)*DUMCENTRAT+ +C(4)*OUTPUT_GAP(-1)+C(5)+C(4)*OUTPUT_GAP(-1)+C(5)
Observations: 82Observations: 82R-squaredR-squared 0.8196700.819670 Mean dependent var Mean dependent var 0.0352480.035248Adjusted R-squaredAdjusted R-squared 0.8103020.810302 S.D. dependent varS.D. dependent var 0.0365970.036597S.E. of regressionS.E. of regression 0.0159400.015940 Sum squared residSum squared resid 0.0195640.019564Durbin-Watson statDurbin-Watson stat 1.7314111.731411
Equation: DIF_CSR=C(6)*DIF_CSR(-1)+C(7)*RRA+C(8)*DUMCENTRATEquation: DIF_CSR=C(6)*DIF_CSR(-1)+C(7)*RRA+C(8)*DUMCENTRAT
Observations: 82Observations: 82R-squaredR-squared 0.6042700.604270 Mean dependent var Mean dependent var -0.003343-0.003343Adjusted R-squaredAdjusted R-squared 0.5942520.594252 S.D. dependent varS.D. dependent var 0.0572430.057243S.E. of regressionS.E. of regression 0.0364630.036463 Sum squared residSum squared resid 0.1050320.105032Durbin-Watson statDurbin-Watson stat 1.9929431.992943
Equation: DIF_RA=C(9)+C(10)*INFL(-1)+C(11)*DIF_CSR(-1)+DIF_RA(-1)*C(12Equation: DIF_RA=C(9)+C(10)*INFL(-1)+C(11)*DIF_CSR(-1)+DIF_RA(-1)*C(12))
Observations: 82Observations: 82R-squaredR-squared 0.6283090.628309 Mean dependent var Mean dependent var -0.000269-0.000269Adjusted R-squaredAdjusted R-squared 0.6140130.614013 S.D. dependent var S.D. dependent var 0.0053440.005344S.E. of regressionS.E. of regression 0.0033200.003320 Sum squared resid Sum squared resid 0.0008600.000860Durbin-Watson statDurbin-Watson stat 2.0440552.044055
Coefficient Std. Error t-Statistic Prob.
C(1) 0.573273 0.062073 9.235.474 0.0000
C(2) 0.253345 0.035775 7.081.660 0.0000
C(3) -0.117939 0.018424 -6.401.340 0.0000
C(4) 0.211775 0.050501 4.193.511 0.0000
C(5) 0.015095 0.002715 5.560.436 0.0000
C(6) 0.173626 0.076520 2.269.023 0.0242
C(7) -0.455091 0.162430 -2.801.767 0.0055
C(8) 0.466524 0.045509 1.025.123 0.0000
C(9) -0.001541 0.000601 -2.564.124 0.0110
C(10) 0.042867 0.013404 3.198.055 0.0016
C(11) 0.055321 0.006843 8.084.199 0.0000
C(12) 0.590841 0.093602 6.312.248 0.0000
Determinant residualcovariance 3.01E-12
EEstimated Modelstimated Model
ISIS OUTPUT_GAP =0.655049*OUTPUT_GAP(-1) -0.247160*RRA(-1)+OUTPUT_GAP =0.655049*OUTPUT_GAP(-1) -0.247160*RRA(-1)+
+0.179965* DIF_CSR +0.179965* DIF_CSR
PHILLIPSPHILLIPS INFL=0.638493*INFL(-1)+ 0.284684*DIF_CSR(-1)-0.099128*DUMCENTRAT INFL=0.638493*INFL(-1)+ 0.284684*DIF_CSR(-1)-0.099128*DUMCENTRAT
+0.176756*OUTPUT_GAP(-1)+0.012691 +0.176756*OUTPUT_GAP(-1)+0.012691
REAL EXCHANGE RATE EQUATIONREAL EXCHANGE RATE EQUATION DIF_CSR=0.173626*DIF_CSR(-1) -0.455091*RRA+0.466524*DUMCENTRATDIF_CSR=0.173626*DIF_CSR(-1) -0.455091*RRA+0.466524*DUMCENTRAT
IRIR DIF_RA=-0.001541+ 0.590841*DIF_RA(-1)+ 0. 042867*INFL(-1)+ DIF_RA=-0.001541+ 0.590841*DIF_RA(-1)+ 0. 042867*INFL(-1)+
+0.055321*DIF_CSR(-1) +0.055321*DIF_CSR(-1)
ButAXtXtSpaceState 1:_
1
ut:Commandt
t
i
i
t
t
tt
t
t
k
Dumcentrat
CSRCSR
YY
1
*
*
Xt: vectorState
100000
010000
001000
0.4891 0.1036 0.4175 0.3008 0.2804 0.0678
1.0749 0.2276 0.1077- 0.2794 0.6162 0.1491
0.3537 1.0749 0.0485 0.1232 0.2028 0.7041
A
00
00
00
0 0.4892-
00.0749-
00.3537-
B
0.01510.5733
*0.2472
*1
*
*
ttt
ttk
The Loss FunctionThe Loss Function
For For =0.9 and c=0.2=0.9 and c=0.2
a+b=0.8a+b=0.8
])(*2.0)()([9.0 21
2*2*
0
ttttt
t iibYYa
aa bb THE OPTIMAL RULESTHE OPTIMAL RULES
0.0.88
00 it =1,9907*(y -y*)t + 0,5732* (Πt – Π*) it =1,9907*(y -y*)t + 0,5732* (Πt – Π*)
+0,3483*(CSRt -CSRt -1 )++0,3483*(CSRt -CSRt -1 )++ + 0,1372*DUM+3,039*K+λ0,1372*DUM+3,039*K+λ
0.0.44
0.0.44
it =1,7671*(y -y*)t + 1,1673* (Πt – Π*) +0,643*(CSRt it =1,7671*(y -y*)t + 1,1673* (Πt – Π*) +0,643*(CSRt -CSRt -1 ) -0,0151*DUM+2,6545*K+3,1173*λ -CSRt -1 ) -0,0151*DUM+2,6545*K+3,1173*λ
0.0.22
0.0.66
it =1,561*(y -y*)t + 1,8097* (Πt – Π*) +0,9565*(CSRt it =1,561*(y -y*)t + 1,8097* (Πt – Π*) +0,9565*(CSRt
-CSRt -1 ) -0,1714*DUM+2,3923*K+5,0057*λ -CSRt -1 ) -0,1714*DUM+2,3923*K+5,0057*λ
00 0.0.88
it =1,1829*(y -y*)t + 3,8226* (Πt – Π*) it =1,1829*(y -y*)t + 3,8226* (Πt – Π*) +1,9006*(CSRt -CSRt -1 ) -+1,9006*(CSRt -CSRt -1 ) -0,6181*DUM+2,2098*K+9,1849*λ0,6181*DUM+2,2098*K+9,1849*λ
The evolution of the coefficients in The evolution of the coefficients in the reaction function for different the reaction function for different
weights put on inflationweights put on inflation
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
b
Output_gapcoefficient(beta)
Inflationcoefficient(1+alfa)
Real exchange ratecoefficient
Optimal Taylor rulesOptimal Taylor rules
it = rr* + it = rr* + ΠΠt + t + αα ( (ΠΠt – t – ΠΠ*) + *) + (y -y*)t(y -y*)t
Taylor (1993): Taylor (1993): αα=0.5 , =0.5 , =0.5=0.5
Taylor (1999): Taylor (1999): αα=0.5 , =0.5 , =1=1
Henderson-McKibbin (1993): Henderson-McKibbin (1993): αα=1 , =1 , =2=2
Ball (1997): Ball (1997): αα=0.82 , =0.82 , =1.13=1.13
Ball (1998): Ball (1998): αα=0.82 , =0.82 , =1.04 (open =1.04 (open economy)economy)
ConclusionConclusion
For different combination of For different combination of weights (a,b), put on deviation of weights (a,b), put on deviation of output from its natural trend and output from its natural trend and deviation of inflation from its deviation of inflation from its target, I found: target, I found:
1 ≤ 1 ≤ ≤2 and 0.5≤ α ≤2.88,≤2 and 0.5≤ α ≤2.88,
results comparable to those results comparable to those existing in the literature.existing in the literature.
ConclusionConclusion
Considering a=0.2 and b=0.6 the Considering a=0.2 and b=0.6 the optimal optimal
rulerule is is::it =it =1,5611,561*(y -y*)t + *(y -y*)t + 1,80971,8097* (Πt – Π*) * (Πt – Π*) +0,9565*(CSRt -CSRt-1 ) -+0,9565*(CSRt -CSRt-1 ) -0,1714*DUM+2,3923*K+5,0057*λ0,1714*DUM+2,3923*K+5,0057*λ
compared to thecompared to the explicit IR: explicit IR:
DIF_RA=-0.001541+ 0.590841*DIF_RA(-1)+ DIF_RA=-0.001541+ 0.590841*DIF_RA(-1)+
0. 0428670. 042867*INFL(-1)+0.055321*DIF_CSR(-1) *INFL(-1)+0.055321*DIF_CSR(-1)