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An empirical study of interest rate determination rulesKeshab Bhattarai aa Business School, University of Hull , Cottingham Road, Hull HU6 7RX, UK E-mail:Published online: 20 Mar 2008.

To cite this article: Keshab Bhattarai (2008) An empirical study of interest rate determination rules, Applied FinancialEconomics, 18:4, 327-343, DOI: 10.1080/09603100500447560

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Applied Financial Economics, 2008, 18, 327–343

An empirical study of interest rate

determination rules

Keshab Bhattarai

Business School, University of Hull, Cottingham Road, Hull HU6 7RX, UK

E-mail: k.r.bhattarai@hull.ac.uk

This paper finds empirical support for a Taylor (1993) type interest rate

determination rule. The model is solved analytically, estimated and used

for simulation, impulse response analyses and forecasting with quarterly

time series data for the UK and annual time series data for Germany,

France, Japan, the UK and the US. The results confirm that such rules

implicitly exists during the period of analysis.

I. Introduction

Changes in the interest rates have profound impactson saving and consumption behaviours of house-holds, on investment and capital accumulationdecisions of firms, and on portfolio allocation ofdomestic and foreign traders in the financial andexchange rate markets. It is generally agreed thatthese changes affect the aggregate demand andaggregate supply positions in an economy that mayoccur immediately or over a lag of up to two years(Keynes, 1936; Hicks, 1937; Phillips, 1958; Friedman,1968; Phelps, 1968; Tobin, 1969; Laidler and Parkin,1975; Kydland and Prescott, 1977; Taylor, 1987;Nickell, 1990; Taylor, 1993; Gali and Gertler, 1999).They also influence the expectations and plans ofeconomic agents about their own future and theirperceptions about welfare and redistribution ofincome and about the prospects of the economy. Asthe public fears that the policymakers may reactunpredictably even violating promises they mighthave made for dynamic time consistency and cred-ibility of policy, the process of determination ofinterest requires transparency and coordination atnational and international levels. The UK, Europeand the majority of industrial economies have tried tosolve these time inconsistency and credibility pro-blems by making their central banks independentfrom the whims of the policymakers and initiated arule-based monetary policy aimed at achieving a pre-

set inflation target. The short-term interest rates havebecome key instruments to be determined by eco-nomic realities rather than by the discretion of thepolicymakers. When the interest rate policy is basedon rules like this it is possible to trace out thepotential effects of interest rates on market rates onvarious types of financial transactions and subse-quent impacts on asset prices, expectations of house-holds and firms and the exchange rates and ultimatelythrough these prices into the aggregate demand,inflation and the rate of unemployment system-atically with minimum errors. How could macro-economic stability and higher growth rate of outputbe achieved under such policy rules is explainedsufficiently in non-technical terms in Bernanke andMishkin (1997) and MPC Bank of England (1999).Taylor (1993) uses a small-scale model which showshow the interest rate can be systematically determinedlooking at the output gaps and inflation gaps toinsure internal and external stability and to reducethe degree of fluctuations in aggregate economicactivities. Woodford (2001) has shown how even asmall Taylor-type model can be consistent to detailedoptimization in more elaborate general equilibriummodels though more detailed analysis of conse-quences of the interest rates by a central bank isoften carried out using more comprehensive econo-metric and general equilibrium models, as discussedin Altig et al. (1995) or in Holly and Weale (2000) orin HM Treasury (2002).

Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online � 2008 Taylor & Francis 327http://www.tandf.co.uk/journalsDOI: 10.1080/09603100500447560

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Given the strengths of a small-scale model inexplaining changes in the interest rate and itscontribution in reducing the fluctuations in aneconomy, this paper aims to investigate howTaylor’s (1993) model fits to the interest rate seriesof the UK and five major industrial economies overthe past three decades. It complements appliedeconomic studies on this topic that have appearedin recent years, particularly those relating to theinterdependency of real interest rates among G7economies (Cheung and Westerman, 2002; Ghazaliand Ramlee, 2003) or to G3 economies (Yamada,2002) or to the Fisher hypothesis (Berument andJelashi, 2002; Silvapulle and Hewarathna, 2002). Themodel discussed in this paper neither directly derivesthe interest rate rules using optimizing models withnon-negativity constraints on the interest rate asfound in Sugo and Teranishi (2005) nor usespredominance of the majority vote rule over theconsensus in setting policy as discussed in Gerlack-Kristen (2005). The empirical results emerging froma single equation or simultaneous equations or paneldata or VAR-cointegration models generate resultsthat are comparable to findings seen in studiesby Asimakopoulos et al. (2000), Bacchetta andBallabriga (2000), Brooks and Skinner (2000),Camarero et al. (2002), Lee (2002), Mills and Wood(2002), Castelnouvo (2003), Ferris and Galbraith(2003), Ghazali and Ramlee (2003), Valente (2003),Wetherilt (2003), Buch (2004), Staikouras (2004).Some details on the issues, methods and majorfindings of these various studies are given in theAppendix. The study reported here also focuses onthe long-run relationship of the determinants ofinterest rate and its impacts on output and pricesbased on cointegration analyses for the long-runrelationship between the interest rate and time seriesof output gap and inflation gaps in the UK and G7economies. A simultaneous equation model is used toinvestigate the interdependency among these vari-ables and a VAR impulse response model is used foranalysing the impacts of unit shocks in output,inflation and the interest rate and for forecastingfuture values of these variables using informationcontained in their time series. Determinants of theinterest rate in this paper are based analytically on thesolutions of a second-order difference equation for aninterest rate determination rule similar to that ofTaylor (1993) in Section II. A brief discussion of thedataset used for study is in Section III. The empiricalrelevance of this model is tested with quarterlymacroeconomic time series data for the UK andannual data for five major industrial economiesduring last three decades in Section IV. Conclusionsand references are in Section V.

II. A Simple Interest RateDetermination Model

A simple interest rate determination model, originallyproposed by Taylor (1993) for the Federal Reservesin the US, can be constructed using three equations.The first equation states the current output gap

(yt� y�), the actual output relative to the trendoutput, as a function of the deviation of the interestrate one period earlier from the target interest rate ofthe monetary authority (it�1� i �). This relationship is

expected to be a negative one as the higher interestrate is expected to slow down expenses by consumersand firms and generate contractionary impacts in theeconomy such as:

yt � y�t¼ �dðit�1 � i �t�1Þ d > 0 ð1Þ

where yt and y�t are actual and natural levels ofoutput, it is the actual rate of interest in period t, i �t is

the target or the natural rate of interest for themonetary authority. This is similar to the equationfor the investment-saving equilibrium relation (IScurve) in Woodford (2001), particularly when trends

and targets are treated as expectations. More thanone period lag can be assumed between the periods ofthe decisions of the interest rate and the changes inthe output, though it was found not necessary for thecurrent study.

The next equation shows how the price level inthis economy responds to the level of economic

activities, the aggregate supply. The expectationaugmented Phillips curve in terms of output isgiven by:

�t ¼ ��t þ cðyt�1 � y�t�1Þ c > 0 ð2Þ

where �t and ��t are actual and target rates ofinflation. When the output is above the trend in thelast period, it creates an upward pressure in the

labour market which raises the wage rate. An increasein the wage rate translates into higher prices andhigher rates of inflation. Again this is similar toEquation 3 in Woodford (2001) when the target

inflation rate is treated as the expected rate ofinflation. A simple interest rate rule is derived bycombining Equations 1 and 2 to show how thepolicymakers like to reduce the interest rate whenboth output and inflation rates are higher relative to

their natural rates as:

it ¼ i �t þ aðyt � y�t Þ þ bð�t � ��t Þ a > 0; b > 0 ð3Þ

If the output gap from Equation 1 and the inflationrate gap from Equation 2 are substituted in theinterest rate rule in Equation 3 it generates anautoregressive reduced form single equation of

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interest that can explain the cycles of interest rate interms of reduced form parameters as:

it ¼ i �t � adðit�1 � i �t�1Þ � bcdðit�2 � i �t�2Þ

it þ adit�1 þ bcdit�2 ¼ i �t þ adi�t�1 þ bcdi �t�2 ð4Þ

The stability or convergence properties of thesecond-order difference Equation 4 essentiallydepends upon values of the parameters a, b, c and dand two initial conditions for i0and i1. For simplicitydefine �0 ¼ ði

�t þ adi �t�1 þ bcdi �t�2Þ, and �1¼ ad and

�2¼ bcd. Then Equation 4 can be written as:

it þ �1it�1 þ �2it�2 ¼ �0 ð5Þ

The general solution to the reduced form differenceEquation 5 has complementary and particular parts.The particular solution refers to the steady state andthe complementary solution shows a dynamicadjustment towards that steady state when theinterest rate is above or below its natural rate. Itexplains the dynamics of the interest rate series. Theconvergence or divergence from the steady state orthe natural rate of interest rate depend on this part ofthe equation.

The particular or steady state solution is easy, asthe interest rate in each period equals the steady stateinterest rate which can be also considered a naturalrate of interest, i.e. it¼ itþ1¼ itþ2¼ � � � ¼ itþn. Thuswith some manipulation the steady state or thenatural rate of interest for the above model can beexpressed as:

�i¼i �t þ adi �t�1þ bcdi �t�2

1þ �1þ �2or �i¼

i �t þ adi �t�1þ bcdi �t�21þ adþ bcd

in terms of the original model parameters withflexible targets i �t�2 i

�t�1 and i �t and as

�i ¼i �t ð1þ adþ bcdÞ

1þ adþ bcdwith fixed targets i �t : ð6Þ

Any short-run disturbances from this natural rateshould ultimately return to it due to forces of demandand supply in the financial markets and is representedby a homogeneous part of the solution.

it þ �1it�1 þ �2it�2 ¼ 0 ð7Þ

Theoretically the complementary solutions ofEquation 7 can have three different cases dependingon the values of parameters �0, and �1 and �2:

(a) real and distinct root, when �21 � 4�2 > 0,guarantees convergence to the steady state;

(b) real and equal roots case, �21 � 4�2 ¼ 0,generates repeated cycles; and

(c) complex roots case with �21 � 4�2 < 0 gives acyclical pattern which may converge or

diverge from the steady state rate dependingthe absolute values of parameters.

The general solutions of the model in these threedifferent cases are:

it ¼ A1�t1 þ A2�

t2 þ

�i ð8Þ

where A1 and A2 are arbitrary constants and �t1 and�t2 are the characteristic roots.

In case (a) the value of �t1 ¼ ð��1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�21 � 4�2

q=2Þ and

�t2 ¼ ð��1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�21 � 4�2

q=2Þ.

Threrfore the general solution Equation 8 can bewritten as:

it ¼ A1

��1 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�21 � 4�2

q

2

0@

1A

t

þ A2

��1 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�21 � 4�2

q

2

0@

1A

t

þ �i ð9Þ

More specifically using all the parameters of themodel this turns to be

it ¼ A1

�adþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiadð Þ2�4bcd

q

2

0@

1A

t

þ A2

ad�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiadð Þ2�4bcd

q

2

0@

1A

t

þ �i ð10Þ

The definite solution requires values of constantterms A1 and A2 which can be obtained using the twoinitial conditions, i0 and i1. Values of a, b, c and dparameters can be obtained from an econometricestimation. Literature suggests that interest rate,determined objectively in this manner, can be usedto achieve price stability and real growth in theeconomy (Fisher, 1977; Hanson, 1980; Barro andGordon, 1983a, b; Sargent, 1986; Mankiw, 1987;Driffil, 1988; Goodhart, 1989; Ball and Romer, 1990;Alesina and Summers, 1993; Dornbush and Fisher,1993; Nordhaus, 1995; Lockwood et al., 1998;Vickers, 1999; Nelson, 2000; Corsetti and Pesenti,2001; Benigno, 2002; Ellis and Price, 2003).

III. Dataset

Interest rates have changed significantly over theyears as shown in Figs 1a–c. In general they were lowand helped to generate an unprecedented rate ofeconomic growth in major industrial economies till

Interest rate determination rules 329

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Fed

eral

fund

rat

e, 1

954-

2006

0510152025

Jul-54Jul-56Jul-58Jul-60Jul-62Jul-64Jul-66Jul-68Jul-70Jul-72Jul-74Jul-76Jul-78Jul-80Jul-82Jul-84Jul-86Jul-88Jul-90Jul-92Jul-94Jul-96Jul-98Jul-00Jul-02Jul-04

FF

rat

e

(b)

(a)

Ban

k of

Eng

land

Rat

e, M

onth

ly A

vera

ge S

erie

s, 1

975-

2006

024681012141618

Jan-75

Jan-77

Jan-79

Jan-81

Jan-83

Jan-85

Jan-87

Jan-89

Jan-91

Jan-93

Jan-95

Jan-97

Jan-99

Jan-01

Jan-03

Jan-05

Mon

thly

_r

Mon

thly

Inte

rest

Rat

e S

erie

s in

Jap

an

0246810121416

1960

1961

1963

1964

1966

1967

1969

1971

1972

1974

1975

1977

1979

1980

1982

1983

1985

1986

1988

1990

1991

1993

1994

1996

1998

1999

2001

2002

2004

2005

Japa

n_r

(c)

Fig.1.Monthly

basicinterest

series

intheUK,USA

andJapan

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the late 1960s; then increased and varied significantlyand unpredictably in the 1970s and 1980s character-izing economic problems and have started changingsystematically in a predictable manner as manywestern economies adopted a rule-based policy ofdetermining these rates after mid-1990. Interest rateshave become a major tool for stabilizing prices andthe markets’ activities more in recent years.

Quarterly fluctuations in the series for the retailprice index, the growth rate of the real GDP and theTreasury bill rates, which represent the wholevarieties of interest rates, from 1970:q2 to 1999:q4are as presented in Fig. 2.

The fluctuations in the rate of interest, inflationand the growth of output were more serious in the1980s than in 1990. The UK economy has beenstabilized and moving more towards its natural rateafter 1995 particularly after the adoption of theinflation targeting rule in 1997 (see Nelson, 2000 formore division between sub-periods). A similar patterncan be obtained from analysis of the annual data ongrowth rates output, rates of inflation and interest forGermany, France, Japan and the US from 1978 to2000 as shown in Fig. 3. The quarterly time seriesshown in Fig. 2 and the annual series in Fig. 3 areused for estimation of the interest rule modelexplained in the previous sections.

IV. Analysis of Results

Many factors other than the output gap and inflationgap influence the rate of interest in an economy.The econometric models incorporate these missingelements in the model given by Equations 1–3including error terms to each equation to representthe influence of these unknown factors. Some of theseomitted factors have positive effects and others havenegative effects. In aggregate the influences ofomitted variables or specification bias elements tendto cancel out each other making their mean to bezero. Further they are assumed to have constantvariance to express no systematic relation among theerrors. Technically speaking these errors aredistributed normally, identically and independently.These assumptions imply that those errors arehomoskedastic and have no autocorrelation andthere is no multi-collinearity among the explanatoryvariables.

yt � y�t ¼ d it�1 � i�t�1� �

þ "1, t ð11Þ

�t ¼ ��t þ c yt�1 � y�t�1

� �þ "2, t ð12Þ

it ¼ i�t þ a yt � y�t� �

þ b �t � ��t

� �þ "3, t ð13Þ

Even if relations may be perfect there is still achance of regressions being spurious as the relationmay be between non-stationary variables. We followDickey–Fuller (1976) Engle and Granger (1987) andJohansen and Juselius (1990) procedures to determinethe existence or absence of unit roots of a variable orcointegration among variables in the model.

Unit root tests suggest that the interest ratevariable is integrated of order one, I(1), and becomesstationary after differencing once. Both output gapand inflation gaps are stationary. The critical andestimated values of coefficients of the unit root forequation of these three variables (from the PC-Giveoutputs) are as shown in Table 1, along withsignificant lag lengths.

Next it is shown how the Engle and Granger (1987)or Johansen (1988) procedure can be used to obtaina non-spurious regression between the interest rateand output and inflation gaps even if the interest rateseries is non-stationary.

The basic relation between the interest rate andinflation gap and output gap for the UK is as givenbelow.

it ¼ 9:446tð32:2ÞSEð Þ 0:29ð Þ

� 0:183ð�1:1Þ

0:13ð Þ

ðyt � y�t Þ þ 0:370ð2:84Þð0:181Þ

ð�t � ��t Þ

Normality test : �2 (2Þ ¼ 11:279½0:0036���

The interest rate rises with an increase in inflationand the coefficient is significant. The coefficient in theoutput gap does not have the expected positive signand it is also statistically insignificant. The aboveresult also suggests that the interest rate is moreresponsive to the inflation rate than to the output gapsince the coefficient on output is not significant.These test results are comparable to those ofBerument and Jelashi (2002) and Silvapulle andHewarathna (2002).

In addition the normality test in line with the Engeland Granger (EG) cointegration test suggests thatresiduals from the above regression of the interestrate on output gap and inflation gap are non-stationary and hence this is a spurious regression.

What about estimating the interest rate rule interms of the reduced autoregressive second-orderdifference equation equivalent to that given inEquation 5? This gives reasonable results as:

itt-ratios

¼ 1:630ð2:71Þþ 0582ð6:42Þ

it�1 þ 0:244ð6:69Þ

it�2

r-square ¼ 0:62 Durbin�Watson ¼ 2:0104

All coefficients of the reduced form equation ofinterest for the UK have expected signs. It is however,

Interest rate determination rules 331

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051015202530

1970

-2 1971

-1 1971

-4 1972

-3 1973

-2 1974

-1 1974

-4 1975

-3 1976

-2 1977

-1 1977

-4 1978

-3 1979

-2 1980

-1 1980

-4 1981

-3 1982

-2 1983

-1 1983

-4 1984

-3 1985

-2 1986

-1 1986

-4 1987

-3 1988

-2 1989

-1 1989

-4 1990

-3 1991

-2 1992

-1 1992

-4 1993

-3 1994

-2 1995

-1 1995

-4 1996

-3 1997

-2 1998

-1 1998

-4 1999

-3

RP

I

Ura

te

TB

ILLS

Sou

rce:

The

se s

erie

s w

ere

obta

ined

from

the

mac

ro ti

me

serie

s da

ta a

rchi

ve in

Ess

ex, a

vaila

ble

at w

ww

.dat

a-ar

chiv

e.ac

.uk

Fig.2.Growth

ratesofoutput,interest

rate

andinflationin

theUK

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−10−5051015

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

Percent per year

Rus

Rjp

Rgr

Rfr

Ruk

Fig.3.Realinterest

ratesin

majorindustrialeconomies

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difficult to retrieve the structural parameters of the

original model from the estimates of the reduced form

equation.The alternative is to use a recursive estimation

method where the output gap is estimated as a

function of the lagged interest rate and then the

inflation gap estimated on the lagged output gap and

finally the interest rate rule equation estimated with

the predicted values of the output gap and inflation

gap. The output gap is influenced by the interest rate,

and the inflation gap is determined by the output gap

and then that is determined by the interest rate. The

recursive simultaneous equation estimation from UK

time series that removes the simultaneity bias looks as

follows:

Interest rate : i ¼ 4:969ð7:74Þðy� y�Þ � 5:182

ð�7:27Þðp� p�Þ

Output gap : y� y� ¼ 0:08ið7:75Þþ 0:504ð5:21Þðp� p�Þ

Inflation : p� p� ¼ �0:071ið�7:27Þ

þ 0:421ð5:21Þðy� y�Þ

System R-Square ¼ 0:8637

The result of the simultaneous equation model has

a better overall fit even that of the results from the

autoregressive model given above. Now the

model explains about 86% of the variation in the

interest rate.The above model can be estimated following the

Johansen and Juselius (1990) procedure for a coin-

tegrated VAR model. The validity of this approach is

based on the rank of the cointegration matrix of the

structural coefficients that is crucial for determining

the number of cointegration vectors in the model.

Consider a VAR model for the above three variables.Yt ¼ A1Yt�1 þ "t

where Yt is the vector of interest rate, output gap and

inflation gap and "t is the vector of normally and

identically distributed random error terms. By

subtracting Yt�1 from both sides

�Yt ¼ ðA1 � IÞYt�1 þ "t

�Yt ¼ �Yt�1 þ "t where � ¼ ðA1 � IÞ

Here � is the matrix of parameters showing the total

long-run relationship between variables. By using the

cointegration procedure this matrix can further be

decomposed into adjustment coefficients (�) and

cointegrating vectors (�) as �¼ ��0. The matrix �denotes the long-run steady state relationship

between variables and � is the dynamic process of

adjustment towards that equilibrium. The estimation

on interest rate, output gap and inflation gap for the

UK for 1972:2 to 1999:4 obtained using the PC-Give

(Doornik and Hendry, 2001) yields the following

results:

� ¼

�0:03654 �0:12011 0:18569

0:02982 �0:24185 �0:08453

�0:09433 �0:07224 �0:53655

264

375

� ¼

0:01810 0:09100 0:01089

0:01667 �0:00346 �0:00208

�0:00793 �0:20723 0:00692

264

375

� ¼

1:0000 0:19498 �6:6460

�13:850 1:0000 3:6355

�4:3680 2:7897 1:0000

264

375

The number of cointegrating vectors in the

Johansen procedure is determined by

�traceðrÞ ¼ �TXni¼rþ1

ln ð1� �̂iÞ and

�maxðr, rþ1Þ ¼ �T ln ð1� �̂rþ1Þ statistics

where �i denotes the eigenvalues of the characteristic

matrix �¼ (A1� I) and r is an indicator for a reduced

rank in (k� r) for k number of explanatory variables.

The calculated values of these statistics are compared

with the theoretical critical values from Johansen and

Juselius (1990) to ascertain the number of cointegrat-

ing ranks as shown in Table 2.The cointegration results (Table 2) are comparable

to those found in other applied works such as Brooks

and Skinner (2000), Camarero et al. (2002), Cheung

and Westerman (2002), Mills and Wood (2002),

Silvapulle and Hewarathna (2002), Yamada (2002),

Valente (2003).The order of the rank of � suggests the number of

cointegrating vectors in �. Above �trace(r) and

�max(r, rþ1) tests suggest that at least there are

two cointegrating vectors in the above model.

Table 1. Stationarity of variables in the model

Interest rate 1st difference of the interest rate Output gap Inflation gap

Coefficient �2.723 �6.463�� �6.160�� �7.428��

Lags 2 2 3 1

Note: ADF tests (T¼ 116, Constant; 5%¼�2.89 1%¼�3.49).

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The long-run relationship between these variables is

shown by a very good fit of the predicted and actual

series of the above three variables.The fit of the predicted and the actual interest rate

is almost perfect as shown in Fig. 4.The model estimated above can be used to analyse

the impacts of shocks to each of the above equations

in terms of incremental and cumulative impulse

responses as shown in Figs 5 and 6.A unique shock to the interest rate by one standard

unit reduces the output gap immediately with a

lagged response in the rates of inflation; a similar

shock to the output equation reduces the interest rate

immediately and has lagged response in the rate of

inflation as shown by graphs in the second row; an

unit shock to inflation reduces the interest rate

immediately and has lagged response in output gap

as shown by the graphs in row 3 of Figs. 5 and 6.

Though the model converges to the steady state over

periods, each of these shocks has its own pattern of

impacts. The cumulative shocks corresponding to

each of three unit shocks on Treasury bills rate,

output and inflation gaps in Fig. 5 are shown by

cumulative response graphs in respective positions in

Fig. 6. Dynamic forecasts along with their confidence

bands are shown in Fig. 7.Studying the time profiles of these shocks it

becomes obvious that it may take from 4 to 24

quarters for an economy to realize the impact of a

shock to the interest rate. The above shocks can

further be divided into real shocks to the output and

the nominal sector shocks in terms of the interest rate.

1970 1980 1990 2000 1970 1980 1990 2000

1970 1980 1990 2000

1970 1980 1990 2000

1970 1980 1990 2000

1970 1980 1990 2000

5

10

15TBILLS Fitted

−5

0

5

10Outgap Fitted

−5

0

5

10Inflagap Fitted

0

50Vector 1

−20

0

20Vector 2

25

50

75

100

125Vector 3

Fig. 4. Prediction from a VAR model: actual, fitted and cointegrating vectors

Table 2. Cointegration test results

Rank: H0 Trace test (Prob.) Max. test (Prob.) Trace test (T-nm) Max. test (T-nm)

r¼ 0 56.86 (0.000)�� 34.38 (0.000)�� 55.43 (0.000)�� 33.52 (0.000)��

r� 1 22.48 (0.003)�� 12.68 (0.087) 21.91 (0.004)�� 12.36 (0.097)r� 2 9.80 (0.002)�� 9.80 (0.002)�� 9.55 (0.002)�� 9.55 (0.002)��

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050

100

050

100

0.0

0.5

1.0

TB

ILLS

(T

BIL

LS e

qn)

−0.2

−0.10.0

Out

gap

(TB

ILLS

eqn

)

050

100

0.0

0.1

0.2

0.3

Infla

gap

(TB

ILLS

eqn

)

050

100

0.0

0.1

0.2

TB

ILLS

(ou

tgap

eqn

)

050

100

0.0

0.5

1.0

Out

gap

(out

gap

eqn)

050

100

0.00

0.05

0.10

0.15

Infla

gap

(out

gap

eqn)

050

100

−0.2

−0.10.0

TB

ILLS

(in

flaga

p eq

n)

050

100

−0.2

−0.10.0

Out

gap

(infla

gap

eqn)

050

100

0.0

0.5

1.0

infla

gap

(infla

gap

eqn)

Fig.5.Im

pulseresponse

analysis

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2.5

5.0

7.5

Cum

TB

ILLS

(T

BIL

LS e

qn)

−3−2−10C

um o

utga

p (T

BIL

LS e

qn)

24

Cum

infla

gap

(TB

ILLS

eqn

)

12

Cum

TB

ILLS

(ou

tgap

eqn

)

1.5

2.0

Cum

out

gap

(out

gap

eqn)

12

Cum

infla

gap

(out

gap

eqn)

050

100

−4−20C

um T

BIL

LS (

infla

gap

eqn)

050

100

−1.0

−0.50.0

Cum

out

gap

(infla

gap

eqn)

050

100

050

100

050

100

050

100

050

100

050

100

050

100

24

Cum

infla

gap

(infla

gap

eqn)

Fig.6.Cumulative

impulseresponse

analysis

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More detailed estimates for various sub-periodsbetween 1973 and 2000 can be found in Nelson(2000) or in Castelnouvo (2003).

The estimation of the interest rate rule model forone economy can be extended to a group ofeconomies taken together. An attempt is made inTable 3 to test it for five major industrial economiesFrance, Germany, Japan, the UK and the USA usingthe annual dataset on growth rates of output,inflation and interest rates obtained from the WorldBank (2002). Three steps are involved in applying thisinterest determination model to five major industrialeconomies. The first step involves estimation of thecurrent output gap as a function of the actual interestrates in the previous period relative to a trend interestrate, and estimation of the current inflation gap asa function of the output gap in the previous period.These predicted series of output and inflation gapsare used to estimate model generated interest rates foreach period in the second step. A comparison is madebetween the series of the actual interest rates to thosepredicted by the model in the third stage. Then thequality of predictions of the model are judged usingtest statistics and studying whether a model-basedprediction can track actual interest rates well anddecompose the sources of changes in the interest rateinto the real or supply-side factors as representedby the output gap and the demand-side factors asrepresented by the inflation gaps.

The explanatory power of this model in analysingthe behaviour of the interest rate in each economy is

quite remarkable as shown by significant t-values forcoefficients and higher values of R-square statistics.The sizes of the coefficients of output gap varysubstantially across these countries reflecting the linkbetween the interest rate and growth rate of theeconomy comparable to those found in other studies(Cheung and Westerman, 2002; Yamada, 2002).These output gap coefficients are significant foreach of the above countries at 1% level of significanceas shown by the t-statistics. These economies reducethe interest rate whenever the actual output growthrate is below the trend growth rate and raise itwhenever the actual growth rate is above the trend

1999 2000 2001 2002

5

10Forecasts TBILLS

1999 2000 2001 2002

−2.5

0.0

2.5

5.0Forecasts Outgap

1999 2000 2001 2002

−5

0

Forecasts Inflagap

Fig. 7. Dynamic forecasts of Interest rate, output gap and inflation gap

Table 3. Test of interest determination rule for five major

economies

Outputgap

Inflationgap Constant R2

France �6.641 0.670 5.900 0.766(�14.778) (1.341) (1.341)

Germany �10.732 4.335 5.339 0.752(�15.187) (4.953) (11.898)

Japan �6.775 �1.794 �1.312 0.641(�6.554) (�7.061) (�3.487)

UK �2.941 1.006 7.416 0.574(�5.885) (2.848) (10.203)

USA �1.794 0.360 5.337 0.696(�7.061) (0.408) (18.955)

Note: Estimates from a 3SLS method; values in parenthesesrepresent t-statistics.

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growth rate in order to avoid inflationary conse-quences. There is dissimilarity however, regarding thelink between the interest rate and inflation gapamong these countries in terms of both the sizeof the coefficients and their significance. Allcountries except Japan have the expected sign of thecoefficient on the inflation gap but that is notsignificant for the US. Despite this the predictivepower of each equation suggests the presence of aninterest rate rule among these economies during thestudy period.

When we study the interdependence in the interestrate determination between the G5 major economiestreating them as one by pooling cross-section andtime series data for the entire 1978–2000 period, itgenerates the following result:

it ¼ 6:25� 0:29ðyt � y�t Þ þ 0:115ð�t � ��t Þ

t-ratios ð0:80Þ ð�3:30Þ ð1:33Þ

R2 ¼ 0:43; F ¼ 5:5; N ¼ 100

From an economic point of view this result is notvery sensible. In theory the interest rate should risewhen output is above its trend but here the estimatedcoefficient has a negative sign showing a reverseresult. The coefficient on inflation gap has theexpected positive sign but it is not significant at 5%level of significance. These results do not support thehypothesis of an interest rate determination rule ataggregate level by the G5 countries for this period,contrary to the literature (Asimakopoulos et al., 2000;Lee, 2002; Ghazali and Ramlee, 2003; Buch, 2004;Butter and Jenson, 2004; Bhattarai, 2003). Each ofthe G5 countries was acting independently indetermining its own interest rate.

Analysis on how the interest rate is determined andhow it affects other economies requires more detailedspecification of demand, production, portfolioallocation and trade structure of the monetaryeconomy in line with Tobin (1969), Altig et al.(1995), Vickers (1998), Corsetti and Pesenti (2001).Such modelling is a subject for further study butbeyond the scope of this paper.

V. Conclusion

An analytical solution for interest rate rules in a threeequation model is found using a second-orderdifference equation technique in terms of modelparameters. Those parameters were estimated usingthe quarterly time series data on Treasury bills rates,the growth rate of output and inflation rates for theUK, the annual time series during the last three

decades for Germany, France, Japan, the UK and theUS. The evidence suggests the existence of an interestrate rule. This empirical model is then applied forimpulse response analysis and forecasting.

Acknowledgements

The author appreciates comments and suggestionsreceived from the editor of Applied FinancialEconomics and participants of the EcomodConference 2006 in Hong Kong. An earlier versionof this paper was published as a ResearchMemorandum from the Centre for Economic Policyof the Hull University Business School.

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Appendix

Table

4.Summary

ofrecentapplied

studiesonthedeterminationandinterdependence

ininterest

ratesandtheirim

pact

oneconomies

Authors

Economic

issues

Method

Summary

Journal

Asimakopouloset

al.(2000)

USandmajorEuropeanequity

markets

Spectralanalysis

Lead-lagrelationin

stock

returnsin

EU

andUS

AE

Bacchetta

andBallabriga(2000)

Theim

pact

ofmonetary

policy

and

banks’balance

sheets:some

internationalevidence

VAR

Strongrelationbetweentheinterest

rate

andoutputin

theUSand13

EU

economies

AFE

BrooksandSkinner

(2000)

Whatwillbetherisk-freerate

and

benchmark

yield

curvefollowing

Europeanmonetary

union?

Linearfactormodel

UK

3-m

onth

yield

curvebest

approxim

atesothersin

EU

AFE

BerumentandJelashi(2002)

Fisher

hypothesisamulti-country

analysis

ADF,OLS,ARCH-LM

Support

forFisher

hypothesisfor13of

26countries

AE

MillsandWood(2002)

Wages

andpricesin

theUK

VECM

Wagegrowth

does

notpredictinflation

AE

Silvapulleand

Hew

arathna(2002)

Robust

estimationandinflation

forecasting

ECM

Support

forFisher

effect

oninflation

forAustralia

AE

Camarero

etal.(2002)

Monetary

transm

issionin

Spain

S-C

VAR

Support

forendogenouspolicy

reaction

ofmonetary

policy

AE

CheungandWesterm

an(2002)

Outputdynamicsin

G7countries:

stochastic

trendsandcyclical

movem

ents

VAR

cointegration

Existence

ofcommonbusinesscycles

amongG7countries

AE

Lee

(2002)

Realinterest

rate

inregional

economic

blocks

VEC,ARIM

ALong-runrelationin

realinterest

ratesof

APEC,EU

andtheUS

AE

Castelnouvo(2003)

Taylorrules,omittedvariables,andthe

interest

rate

smoothingin

theUS

OLSin

firstdifferences

Testofforw

ard

lookingTaylorrule

intheUS

EL

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Yamada(2002)

Realinterest

rate

equalization:some

evidence

from

threemajorworld

financialmarkets

VAR

cointegration

Departure

from

long-runrealinterest

rate

equalizationisnotverylarge

AE

FerrisandGalbraith(2003)

Indirectconvertibilityasamoney

rule

forinflationtargeting

Relativeprice

concept

How

indirectconvertibilitybringsprice

stability(fixingabasket/unitofmoney)

AFE

Valente

(2003)

Monetary

policy

rulesandregim

eshifts

MS-V

AR

Tim

evaryingparameter

andMarkov

SwitchingVAR

model

forpolicy

rule

AFE

GhazaliandRamlee(2003)

Alongmem

ory

test

ofthe

long-runFisher

effect

intheG7countries

ARIM

A,ARFIM

ALong-runrelationbetweeninterest

rate

andinflationin

G7countries

AFE

Wetherilt(2003)

Money

market

operationsand

short-term

interest

rate

volatility

intheUnited

Kingdom

GARCH

andVECM

Reductionin

thevolatility

ofmarket

ratesalongwiththatin

reporates

inUK

AFE

Buch

(2004)

Cross-border

bankingandtransm

ission

mechanismsin

Europe:

evidence

from

Germandata

Creditdata

analysis

Activitiesofcommercialbankscause

transm

issionofshocksacross

countries

AFE

Staikouras(2004)

Theinform

ationcontentofinterest

rate

futuresandtime-varyingrisk

premia

VAR

cointegration

Tests

speculativeefficiency

hypothesisand

supportsprice

discoveryhypothesis

AFE

Butter

andJenson(2004)

Anem

piricalanalysisof

Germanlong-term

interest

rate

ARIM

A,ECM

Fourtheories

ofinterest

rate

explain

Germanshort-term

rate

AFE

Gerlack-K

risten

(2005)

Toolittle

toolate,interest

rate

settingandthecost

ofconsensus

Vote

andsimulation

Majority

vote

betterthanconsensusin

settingpolicy

EL

SugoandTeranishi(2005)

Optimalmonetary

policy

rule

under

thenon-negativityconstrainton

nominalinterest

rates

Constrained

optimization

Policy

rule

optimaleven

innon-negative

constraintoninterest

rate

EL

Notes:AE¼Applied

Economics,AFE¼Applied

FinancialEconomics,EL¼EconomicsLetters.

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