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)