Modelling Real Exchange Rate Behaviour with the Taylor Rule: An Empirical Analysis
Bulent Guloglu
Pamukkale University, Department of Economics, Denizli, Turkey
and
Fuat Erdal
Adnan Menderes University, Department of Economics, Nazilli, Aydin, Turkey
Foreign exchange rate, interest rate and inflation rate are the main intermediate tools in many
policy modeling and their interdependence are always debated. The aim of this study is to analyze the
link between interest rate and real exchange rate by using a modified Taylor rule. The modified rule
expresses nominal interest rate in terms of inflation gap, output gap and real exchange deviation, and
relates current real exchange rate to expected real exchange rate on the one hand and to inflation and
output on the other in conjunction with interest rate parity.
The modified Taylor rule is tested for DM-Turkish Lira real exchange rate as Germany is the
main trading partner of Turkey with its 23 % share in Turkish foreign trade. Using monthly data from
1987:01-2004:12, the vector auto-regressive econometric technique is used to explore the link between
Taylor rule and exchange rate, and confidence intervals of auto-regressive coefficients are improved
by applying Klian (1998) non-parametric bootstrap after bootstrap procedure. The findings are
expected to illuminate the interdependence of the intermediate instruments in especially monetary
policy modelling.
Introduction
The main objective of the theoretical models of exchange rate determination is to
produce a clear understanding of the economic mechanisms governing the actual behaviour of
exchange rates in the real world. In addition, they should give satisfactory explanations for the
relationship between exchange rates and other important economic variables. Many models of
the real exchange rate are reduced to test the Purchasing Power Parity. Breuer (1994) and also
Bleaney and Mizen (1993) provide excellent surveys of the empirical studies. Equilibrium
value of the real exchange rate is also investigated by using behavioral models such as
Edwards (1988, 1989), Elbadawi (1994), Cottani et al (1990) and Ghura and Grenness (1993).
A new strand of literature attempts to model the real exchange rate with the Taylor rule in
which the real exchange rate is determined by expected inflation differentials and output gap
differentials. In this paper, an attempt is made to analyze the behavior of the Deutsche Mark –
Turkish Lira exchange rate with the Taylor Rule. Next section provides a brief review of
literature on the Taylor Rule. This is followed by the introduction of the model and the
variables. After the discussion of the empirical results, the paper is finalized by some policy
remarks.
Taylor Rule and the Real Exchange Rate
Taylor rule is originally an interest rate reaction function which specifies the monetary
policy rule of a central bank. Taylor (1993) expresses the rule as
it = i* + 1.5(t- *) + 0.5yt.
where it is the federal funds rate in quarter t, i* is the long-run rate, measured as average funds
rate (4%),  is four-quarter inflation * is targeted inflation rate, yt is the output gap.
It is later modified to accommodate some other policy instruments including exchange rate
because interest rate has been accepted as the main instrument of monetary policy of many
central banks to reach inflation stability, output stability and even exchange rate stability.
Engel and West (2004) and Clarida et al (1998) are among those who employ this approach to
explain the behaviour of the dolar-DM exchange rate.
Foreign exchange rate, interest rate and inflation rate are the main intermediate tools in
many policy modelling and their interdependence are always debated. The role of interest rate
differentials in the determination of the real exchange rate is not new. Uncovered interest
parity is often used for that purpose. However, modelling the interest rate with the Taylor rule
introduces a multivariate structure that produces a richer set of dynamics and interest rate
forecasts that may be more accurate that those obtained from univariate time-series
specifications (Mark, 2005). Fundamental determinants of the exchange rate are relative
inflation gaps and relative output gaps when central banks conduct monetary policy by setting
interest rates according to Taylor rule. Mark (2005) presented evidence that the real dolar-DM
exchange rate is linked to Taylor rule fundamentals.
McKinnon and Ohno (in de Andrade and Divino, 2005) argue that Bank of Japan has
given more emphasis on exchange rate targeting instead of inflation targeting and that
exchange rate is a forcing variable and the domestic prices level an adjusting variable. De
Andrade and Divino (2005) provide further evidence towards that argument in their empirical
analysis.
As an alternative policy rule, Us (2004) proposes Monetary Conditions Index (a
combination of the changes in the short-term real interest rate and in the real effective
exchange rate in a single variable). His analyses on the Central Bank of Turkey indicate that
the economy stabilizes much more quickly and shows significantly less volatility under the
MCI rule than the Taylor rule.
Gerlach and Schnabel (2000) mentions about two advantages of Taylor rule: Firstly, it
provides a degree of macroeconomic stabilisation close to that offered by an optimal rule.
Secondly, using a rule known to the public may help reduce uncertainty about the future
course of monetary policy and help avoid unnecessary macroeconomic instability.
Model
A two-country model is used in order to to explain the link between interest rate and
real exchange rate according to a modified Taylor rule. Following Engel and West (2004), the
monetary rules in the home and foreign countries can be described as:
iht =et +1 Eth t+1+1yh t+uht
(1)
ift = 2 Etf t+1+2yf t+uft
(2)
where et = st-( ph-pf ) and E is mathematical expectations conditional on a period t information
set.
In these equations, iht and ift stand for respectively interest rate for home and foreign
country, h and f are inflation rates in home and foreign country, ph and pf are domestic and
foreign prices(in logarithmic terms), yh and yf represent output gap. In the first equation, et is
the natural logarithm of the real exchange rate and s is the nominal exchange rate. Finally u ht
and uft are monetary shocks in two countries.
The first equation is a standard Taylor rule, while the second is a “modified” Taylor
rule. Unlike Engel and West, we assume that the two countries have different monetary policy
parameters. According to standard Taylor rule, we expect that the parameters of output gaps
(1 and 2 ) and the parameter of real exchange rate () will be positive. We also expect the
parameters of inflation rates (1 and 2) to be greater than unity. It is assumed that the
monetary authority raises interest rates when the real exchange is above its long-run level.
Note that when equation 1 is written without constant and trend, it gives the long-run level as
zero (Engel and West, 2004)1. The first equation may be seen as the specific form of a Taylor
rule which includes the nominal exchange rate (s) and its target (s*). The target is equal to the
difference between domestic and foreign prices (ph-pf ).2
If we subtract the second equation from the first one, we can get
idt =et +1 Eth t+1 -2 Etf t+1 + 1yh t -2yf t +udmt
(3.a)
Where idt = iht - ift and udmt=uht -uft
According to the uncovered interest rate parity we can write
it = Etst+1-st (4)
1
2
In our empirical work we add a constant to both equations.
A constant is also added to this difference in empirical work.
Denoting d, the difference between home and foreign country inflation rates and
subtracting the expected value of next period’s inflation differential ( d ) from both sides of
(3) we can obtain
idt - Etd t+1=Etet+1- et
thus
et = Etet+1 -idt + Etd t+1 (5.a)
where d =h -f
In equation (5), we used the definition of real exchange rate. Finally substituting (3)
into (5) produces:
et = Etet+1 - et -1 Eth t+1 +2 Etf t+1 - 1yh t +2yf t -udmt + Etd t+1
rearranging this equation we can obtain the real exchange rate as
et=[Etet+1 -1 Eth t+1 +2Etf t+1 + Etd t+1 - 1yh t +2yft ] (6.a)
where =1/(1+) , 0<<1.
Data Sources and Definitions
In this study, Turkey is the home country and Germany is the foreign country. Since
Germany is main trading partner of Turkey with its 22,68 % part in Turkish foreign trade, we
examine Deutschmark-Turkish Liras real exchange rate behaviour. Moreover, Kesriyeli and
Yalçın(1998) for Turkey, and Clarida et al. (1998) for Germany showed that Taylor rule may
be valid for Turkish and Germany monetary policies. This study uses monthly data covering
the period 1987:01-2004:12 for Turkey and the period 1980:01-2004:12 for Germany.
All data for Turkey were obtained from the web site of the Central Bank of Turkey.
Data for Germany were obtained from the International Financial Statistics and from
Datastream. Interest rate is measured as annual money market rate, inflation as the first
differences of the logarithm of consumer prices index. To calculate the real exchange rate, the
difference of the logarithm of consumer prices index of two countries was taken and then this
difference was subtracted from the nominal exchange rate defined as TL per DM. The output
gap was calculated as the residual from quadratically detrended industrial production index,
which is a proxy for output.
Empirical Analyses
Some arrangements are made before the estimation of the above model. Firstly,
following Clarida et al.(1998, Engel and West
(2004), we substituted expected annual
inflation Et(ph t+12– ph t ) and Et(pf t+12– pft ) for monthly inflation in the monetary policy rules
equations (1 and 2). With this change, the interest rate differentials equation (3) becomes
idt =et +1 Et(ph t+12– pht ) -2 Et(pf t+12– pft ) + 1yh t -2yf t +udmt
(3.b)
Secondly, since the interest rates for both countries were measured at annual rates and
monthly data used throughout, we can rewrite uncovered interest parity equation in real terms
as
( idt/12) - Etd t+1=Etet+1- et
from which we can deduce the real exchange rate as follows
et =* [12 Etet+1 + 12Etd t+1 -1 Et (ph t+12– pht ) +2 Et (pf t+12– pft ) - 1yh t +2yf t -udmt ] (6.b)
where *=1/(12+)
Finally if this difference equation ( 6.b) is solved recursively forward 3, the solution
can be written

et =1/(12+)  b j Et [12 d t+1 -1 (ph t+12– pht ) +2 (pf t+12– pft ) - 1yh t +2yf t -udmt ] (7)
j 0
Estimation Technique
We initially aim to obtain the fitted real exchange rate using equation (7) and compare
it with the actual exchange rate. For this purpose, firstly equations (1) and (2) are estimated
with a constant and a dummy variable using OLS technique. Dummy variable for Turkey
allows for financial crises in April 1994, November 2000 and February 2001. From this
estimation, we get the coefficient of nominal exchange rate (), the inflation rates coefficients
(1and 2 ) and the output gaps coefficients (1 and 2 ). The coefficients for Turkey are
=0.11, 1=3.49 and 1 =0.32. Those for Germany are 2=1.36 and 2 =0.26. The magnitudes
and signs of the coeefficients are as expected. The coefficients for Germany are very close to
those estimated by Clarida et al.(1998). After having estimated the interest rates equations, we
3
For details of this solution see Hamilton (1994,p.39)
checked the stationarity of the variables using Dickey-Fuller. The results are illustrated in
Table 1.4
Table 1: Augmented-Dickey Fuller Unit Root Tests Results
No trend or constant
Constant
Constant and Trend
Test statistic
ih t
qt
ht
ift
ft
Lag length Test statistic
length
-2.47***
-0.51
-1.07
-2.57**
-1.19
1
1
11
4
11
-5.32***
-2.85**
-8.77***
-2.01
-12.73***
Lag Test statistic
length
1
1
0
4
0
-5.29***
-3.47**
-8.62***
-3.91***
12.92***
Lag
1
1
1
14
0
Lag lengths were chosen using Schwarz criterion.
** and *** indicate the rejection of unit roots at 0.01 and 0.05 significance level respectively.
Probabilities are obtained from MacKinnon (1996).
After having estimated monetary policy rules parameters we can “fit” equation (7)5.
To do so, we forecast monthly inflation and output gaps with standard vector auto-regressive
(VAR) technique. VAR estimations have been made using interest rates, monthly inflation
and the output gap as endogenous variables and constant and dummies as exogenous. Unlike
Engel and West (2004) who initially estimated a VAR using interest rates, inflation and
output gap differentials, we estimate a separate VAR for Turkey and Germany. We also
estimated a separate VAR with monthly and annual inflation rates. To sum up we have been
estimated following VAR models.
VAR1= (iht, ht , yh)
VAR2 =( ift ,ft, yf)
VAR3 =(iht, ph t+12– pht , yh )
VAR4 =(ift, pf t+12– pft , yf )
Using VAR1 VAR2 forecasts we calculate monthly inflation differentials ( dt)
between two countries. Then forecasts from VAR3 and VAR4 were used to compute fitted et
( ê t ). Thus interest rates are used to predict the future inflation and output gaps.
4
5
These results are also confirmed by KPSS tests.
We omit monetary shock udmt because of lack of an independent time series
Let x denote a VAR model of order p
xt = c + A1xt-1 + A2 x t-2 +....Ap x t-p + ut
This VAR(p) model can be written in VAR(1) form(i.e in companion form) as
Xt = C + AXt-1 + Ut
Where Xt=(xt, xt-1, xt-2... xt-p+1), C=(c, 0,....0), U=(ut,0,0...0) and A is the companion matrix
(Lutkepohl, 1991).
It can be shown that ê t is linear in Xt. As long as inflation and output gap differentials
are mean reverting, ê t will be so as well.
As argued above, our purpose is to compare the properties of ê t with actual et. To do
so, we first calculate the correlation between these series and construct 95 % percentile
confidence interval using Kilian’s (1998) boostrap-after boostrap method. We use Kilian
procedure because bias and skewness in the small-sample distribution of OLS coefficients can
cause traditional confidence intervals to be extremely inaccurate (Kilian, 1998). Obtaining
bias-corrected coefficients involves following steps:
1) Estimating the VAR(2) with the actual and the fitted real exchange rate (et, ê t ).
et = a1+b1et-1 +c1et-2 + d1 ê t -1 + f1 ê t - 2 + u1
ê t = a2+b2et-1 +c2et-2 + d2 ê t -1 + f2 ê t - 2 + u2
2) Resampling residuals estimated in step 1 (with replacement).
3) Computing new values of et and ê t using resampled residuals and estimated VAR
coefficients.
4) ReestimatingVAR(2) model using new values of et and ê t .
5) By repeating this procedure 1000 times, one can obtain standard nonparametric bootstrap
coefficients.
After having obtained standard nonparametric bootstrap coefficients, one can estimate
and correct bias as follows.
6) Calculate the bias by  = B -B, where B is the bootstrap coefficients vector and B is the
vector containing the averages of these coefficients.
7)Compute the modulus (m) of the largest root of companion matrix associated with B.
8)If m1 let B* (bias-corrected coefficient estimate) be equal to B ( B* =B). Otherwise
B*=B-
9)Calculate the modulus (m*) of the largest root of companion matrix of B*.If m*1, let 1 =
and 1=1 and denote i+1= ii and define i+1=i-0.01.
10) Set B*=B*i after iterating on B*i =B- i , i=1,2... until the modulus of the largest roots of
companion matrix associated with B*i less than unity.
11)Using B*, repeat 2000 times the steps 2-10 and calculate pearson correlations between et
and ê t .
12) Construct 95 % percentile confidence interval for correlation coefficients.
Finally we repeat above procedure to get the correlation between the first difference of actual
and fitted real exchange rate (et and  ê t ) on the one hand and st and  ŝ t on the other.
Empirical Results
Table 2 illustrates autocorrelations of model-based (fitted) and actual nominal and real
exchange rate. As usual, the actual real exchange is highly autocorrelated. The first order
autocorrelation of the actual exchange rate is very high (0.95). The growth rates of the
nominal and the real exchange rates seem to have the first order autocorrelation. As for model
based series, the real exchange rate is also serially autocorrelated. However unlike the actual
real exchange rate, the growth rate of fitted real exchange rate is serially uncorrelated. The
growth rate of fitted nominal exchange rate is less autocorrelated than that of the actual
nominal exchange rate.
Table 2: Autocorrelations of actual and fitted nominal and real exchange rates
Lag
et
et
st
ê t
ê t
ŝ t
1
0.927
0.057
2
0.841
-0.069
3
0.761
-0.061
4
0.689
-0.095
5
0.622
0.005
6
0.56
-0.005
ŝ t = ê t +dt and st=et+dt
0.215
0.101
0.047
-0.019
0.021
-0.029
0.956
0.887
0.819
0.756
0.695
0.643
0.35
-0.004
-0.126
-0.077
-0.085
0
Table 3: Cross correlations of fitted nominal and real exchnage rates
Series
ydt
dt
ê t
ê t
ŝ t
ê t
1
0.16
0.24
0.14
0.15
ê t
0.16
1
0.88
-0.32
0.16
ŝ t
0.24
0.88
1
0.12
0.15
0.14
-0.32
0.12
1
-0.032
dt
d
yt
0.15
0.16
0.15
-0.03
1
d
d
 t and y t are actual inflation and output gap differentials respectively.
0.412
0.084
-0.001
-0.015
-0.0014
0.071
Table 4: Cross correlations of actual series rates
Series
et
et
st
et
1
0.13
0.32
0.13
1
0.83
et
0.32
0.83
1
st
d
0.38
-0.048
0.50
t
d
yt
-0.011
0.051
0.028
dt
0.38
-0.048
0.50
1
-0.028
ydt
-0.011
0.051
0.028
-0.028
1
Table 5: Bootstrapped correlations between fitted and actual nominal and
real exchange rates and their 95 % percentile confidence intervals
Series
ê t
ê t
ŝ t
et
0.81
(0.76) (0.90)
0.16
et
(0.0036) (0.21)
0.051
st
(0.0031) (0.128)
The numbers in parenthesis are lower and upper limits of 95 % percentile confidence intervals
The cross-correlations between the fitted real exchange rate (et) and the growth rates
of real and nominal exchange rates (et and st ) are close to those between actual values.
(Tables 3-4). However the correlation between the fitted real exchange rate and inflation
differential (0.14) is lower than the correlation between the actual real exchange rate and
inflation differentials. Also the correlation between the fitted real exchange rate and output
differentials is very different than between the actual real exchange rate. The same can be
argued for the nominal exchange rate. So our model fits less successfully the correlation
between the real exchange rate on the one hand and the output gap and inflation differentials
on the other.
The bias corrected bootstrap correlations and their 95 % percentile confidence
intervals are shown in Table 5. As can be seen, the correlation between the fitted and the
actual real exchange rate is very high (0.81). Also the lower and the upper limits of 95 %
percentile confidence interval are very close to each other (0.76-0.90).
Finally Figure 1 plots the fitted and the actual real exchange rate series. It can be seen that the
real exchange rate is successfully fitted by the model. In figure 1 two series move very close
to each other.
Figure 1:Fitted (etf) and Actual (et) Real Exchange Rates
1330
1320
1310
1300
1290
1280
1270
1260
1250
1240
88
90
92
94
96
et
98
00
02
04
etf
Concluding Remarks
In this study we analyzed the empirical implications of Taylor rules for the DMTurkish Lira exchange rate. For this purpose, we used a modified Taylor rule. This modified
rule expresses the nominal interest rate in terms of inflation gap, output gap and the real
exchange deviation and relates the current real exchange rate to expected real exchange rate
on the one hand and to inflation and output on the other in conjunction with interest rate
parity.
The model-based or fitted nominal and real exchange rates have similar patterns with
the actual nominal and real exchange rates. The correlation between the fitted real exchange
rate and the actual real exchange rate is quite high and our model predicts well the real
exchange rate behavior. Also the correlation between the model-based nominal exchange rate
and the actual nominal exchange rate is modest. However our model is less successful in
reproducing the correlations between the real exchange rate on the one hand and the output
gap and inflation differentials on the other.
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