Download the real exchange rate and real interest differentials

THE REAL EXCHANGE RATE AND REAL
INTEREST DIFFERENTIALS: THE ROLE OF
NONLINEARITIES
Nelson C. Marka;
a
;y
and Young-Kyu Mohb
University of Notre Dame and NBER, USA
b
Tulane University, USA
Abstract
Recent empirical work has shown the importance of nonlinear adjustment in
the dynamics of real exchange rates and real interest di¤erentials. This work
suggests that the tenuous empirical linkage between the real exchange rate and
the real interest di¤erential might be strengthened by explicitly accounting for
these nonlinearities. We pursue this strategy by pricing the real exchange rate
by real interest parity. The resulting …rst-order stochastic di¤erence equation
gives the real exchange rate as the expected present value of future real interest
di¤erentials which we compute numerically for three candidate nonlinear processes.
Regressions of the log real US dollar prices of the Canadian dollar, deutschemark,
yen, and pound on the fundamental values implied by these nonlinear models are
used to evaluate the linkage. The evidence for linkage is stronger when these
present values are computed over shorter horizons than for longer horizons.
JEL CODE: F30, F31
KEY WORDS: Real exchange rate; real interest di¤erential; nonlinear adjustment
Correspondence to: Nelson C. Mark, Department of Economics and Econometrics, 434 Flanner
Hall, University of Notre Dame, Notre Dame, IN 46556, USA. Tel.: +1 574 6310518; fax: +1 574
6314783.
y
E-mail address: [email protected]
1
Introduction
This paper investigates whether accounting for nonlinear adjustment in the real interest
di¤erential helps to strengthen the evidence linking the real interest di¤erential to the
real exchange rate. Because today’s real exchange rate depends on the public’s expectations of future real interest di¤erentials, a potential pitfall in earlier studies lies in
misspeci…cation of real interest rate dynamics. In our analysis, we price the real exchange rate by real interest parity. This …rst-order stochastic di¤erence equation then
gives the real exchange rate as the expected present value of future real interest di¤erentials with a unit discount factor. We evaluate this expected present value for three
candidate nonlinear processes governing the real interest di¤erential. They are the band
threshold autoregression (BTAR), and two smooth-transition variants–a cosine smooth
transition autoregression (CSTAR) and the exponential smooth transition autoregression (ESTAR) model. Numerically computed expectations of future interest di¤erentials
are used to generate implied fundamental values of the real exchange rate. The link between the real interest di¤erential and the real exchange rate is assessed by regressing
the log real exchange rate on the fundamental values implied by these nonlinear models.
Economic theory predicts that the real interest di¤erential is a key determinant of
the real exchange rate [Dornbusch (1976), Obstfeld and Rogo¤ (1995)] but empirical research has struggled to …nd a statistically signi…cant link. While a variety of econometric
methods and modeling strategies have been employed in the empirical literature these
analyses have typically centered on linear regressions of real exchange rates on the contemporaneous real interest di¤erential. Frankel (1979) compares the predictions of a real
interest parity model to monetary and Keynesian formulations, Campbell and Clarida
(1987) model the ex ante real interest di¤erential as an AR(1), Meese and Rogo¤ (1988),
Edison and Pauls (1993), and Edison and Melick (1999) employ cointegration analysis
and Baxter (1994) employs frequency domain decompositions to isolate co-movements
at business cycle frequencies.
Our paper is motivated by recent research that has detected signi…cant nonlinear
dynamic adjustment in both real exchange rates and in real interest rate di¤erentials.
Most of this work has studied either the real exchange rate or the real interest di¤erential separately but not jointly. This body of work does suggest, however, that failure to
account for these nonlinearities may underlie the di¢ culties in establishing the linkage
between them. In international …nance, empirical work investigating nonlinear adjustment in the real exchange rate includes Obstfeld and Taylor (1997), Parsley and Wei
(1996), Michael et. al. (1997), Sarno and Taylor (2001), and Bec et. al. (2002) who
motivate nonlinear adjustment through transportation costs that de…ne a neutral band
that de…ne the boundary where pro…table commodity arbitrage opportunities exist. Kilian and Taylor (2003), on the other hand, obtain nonlinear adjustment from interactions
1
of market participants with heterogeneous beliefs about exchange rate valuation.
Several studies on nonlinear adjustment in domestic interest rates have examined
the Markov-switching model.1 Less work has been done on interest di¤erentials or
their interaction with exchange rates. Nonlinear adjustments in both the real exchange
rate and the real interest di¤erential will be linked if central banks choose the interest
rate defense of its currency over direct intervention with foreign currency reserves [e.g.,
Drazen and Hubrich (2003) and Flood and Jeanne (2000)].2 Balke and Wohar (1998)
and Peel and Taylor (2002) study threshold e¤ects in deviations from covered interest
parity and Mancuso et. al. (2003) has examined the dynamics of real interest rate
di¤erentials and …nd important threshold nonlinearities in their dynamics. Nakagawa
(2002) investigated nonlinear regressions of the real exchange rate on the ex post real
interest di¤erential and found stronger statistical evidence of a link between the real
interest di¤erential and the real exchange rate than in earlier studies that worked in
a linear environment. Here, we go one step further and ask to what extent the real
exchange rate can be priced by the expected future real interest di¤erentials, which are
assumed known to follow nonlinear dynamics.
To summarize our results, we …nd evidence of the real interest di¤erential–real exchange rate link but they are not uniformly strong. For a particular model of real
interest rate dynamics, the statistically signi…cant linkage varies across countries and
across horizons.
The remainder of the paper is as follows. The next section outlines our strategy.
Section 3 describes the data used in this study. Estimation of the threshold and smooth
transition models of the real interest di¤erential are reported in section 4. Section 5
reports the results on estimating the linkage between the fundamental value implied by
the nonlinear real interest rate models and the real exchange rate. Section 6 concludes.
2
Real Exchange Rate Pricing by Real Interest Parity
Let st be the log spot exchange rate expressed as the home currency price of a unit
of foreign currency and it (it ) be the 1-period home (foreign) Eurocurrency rate. The
uncovered interest parity condition which equates expected nominal yield in terms of
1
Hamilton (1988), Sola and Dri¢ ll (1994), Ang and Bekaert (2002), and Bekaert, Hodrick and
Marshall (2001) examine Markov-switching models for interest rates but not for real interest di¤erentials.
2
Mark and Moh (2003) exploit this idea to explain the forward premium anomaly as resulting from
unanticipated central bank foreign exchange interventions that work through interest rates rather than
reserves. See also Baillie and Osterberg (2000) who …nd that the magnitude of deviations from uncovered
interest parity increase during intervention periods.
2
the home currency from holding a 1-period domestic Eurocurrency deposit is
Et (st+1
st ) =
(it
it ):
(1)
Let qt = st + pt pt be the log real exchange rate where pt and pt are the log home and
foreign price levels respectively and let t be the real foreign-home interest di¤erential
where t rt rt = (it it ) Et (pt+1 pt ) (pt+1 pt ) . Then real interest parity
is obtained by subtracting the expected in‡ation di¤erential from both sides of (1),
qt =
+ Et
t
+ Et qt+1 ;
(2)
which is a stochastic di¤erence equation in qt . We include a constant to allow for a
possible trend in the real exchange rate perhaps induced by Balassa-Samuelson e¤ects
which would not be captured in real interest di¤erentials.
Forward iteration of (2) gives the real exchange rate as the undiscounted present
value of expected future interest di¤erentials and the expected long-run real exchange
rate,
qt
where
t = q^t;k + (Et qt+k
!
k 1
X
q^t;k
Et
:
t+j
(t + k)) ;
(3)
(4)
j=0
A solution based on interest di¤erentials requires a terminal condition that expected
long-run real exchange rate is a constant. To achieve identi…cation, we appeal to a
variant of what Edison and Melick (1999) refer to as the ‘standard case,’ and assume
that Et (qt+j
(t + j)) = E(qt
t) = C for all j
k: For = 0; this is consistent
with long-run implications of standard exchange rate models such as Dornbusch (1976)
and Frankel (1979).3
In (3); q^t;k is the (real interest di¤erential) fundamentals value of the real exchange
rate. Our empirical work focuses on examining the empirical linkage between the implied
fundamentals determinants and the real exchange rate. In the next section, we discuss
our empirical measures of this fundamentals value. Then regressing the real exchange
rate on this empirical measure, q^t;k ; we ask if there is a statistically signi…cant link
between the two variables and if so, whether the choice of k matters. That is, we are
interested in assessing how far into the future market participants look in pricing the real
3
Another strategy [e.g., Meese and Rogo¤ (1988)] draws on Dornbusch (1976) to motivate adjustment
dynamics Et qt+1 q t+1 = (qt q t ) ; where q is the shadow ‡exible price equilibrium real exchange
rate and 0 < < 1. In the absence of real shocks, Et q t+k = q t . Substituting these two expressions
1
into (2) gives qt = (
1) Et t + q t :
3
exchange rate. If events far into the future weigh heavily, we will require large values of
k and a focus on current real interest di¤erentials will tell only a small part of the story
about real exchange rate dynamics. On the other hand, if we …nd small values of k are
more appropriate then it suggests that the di¢ culty in assessing the impact of current
news on future prices hinders the evaluation of the exchange rate’s fundamental value.
3
The Data
Our data set consists of monthly observations from 1978:9 to 2002:4 for Japan (JPN)
and from 1975:1 to 2002:4 for Canada (CAN), Germany (GER), United Kingdom (UK),
and the United States (USA).4 1-month Eurocurrency rate data are from Datastream.
The US is taken as the numeraire (home) country with exchange rates quoted as dollar
prices of the other currencies. Exchange rates and consumer price indices are from the
International Financial Statistics.
Figure 1 plots the log real exchange rate along with the 1-month real Eurocurrency
di¤erentials in our sample. Some co-movement between the series is visually evident
at what appears to be the low-frequency components. There are also some sizable departures between the contemporaneous real interest di¤erential and the real Canadian
dollar at the beginning and end of the sample. Similar departures can be seen for Japan
during the early 1990s and for Germany towards the end of the sample.
4
Nonlinear Models of the Interest Di¤erential
We consider three alternative nonlinear mean-reverting models for the real interest di¤erential t . In all three models, the persistence in t is inversely related to the magnitude of
recent values of the real interest di¤erential. The economics behind such an adjustment
pattern are straight forward to motivate. International mobility of real capital creates a
long-run constraint on international real interest rate divergence whereas international
monetary policy coordination provides a short-to-medium run constraint. If policy interventions tend to occur when the interest di¤erential is large but not when it is small,
the interest di¤erential will tend to behave unsystematically as in a random walk process
when the di¤erential is small. Policy interventions drive t towards its mean value when
the di¤erential becomes large. These considerations suggest that the interest di¤erential
will be mean reverting where the speed of adjustment is directly related to the size of
the di¤erential.
4
Following the adoption of the euro, we use imputed values of the deutschemark.
4
4.1
Models
Band-Threshold Autoregression (BTAR) In the BTAR model, the real interest
di¤erential follows a driftless random walk when it lies between the …xed bands [c2 ; c1 ]:
It follows a band-reverting autoregression when it lies outside the bands. The BTAR
model is speci…ed as,
t
=
8
< c1 (1
:
t 1
c2 (1
1)
+
1 t 1
+ vt
+ vt
2)
+
2 t 1
+ vt
if
if
if
t 1
c2 <
t 1
> c1
t 1
< c1 :
(5)
< c2
In the BTAR model, the transition from the autoregression to the random walk occurs
abruptly at the bands.
We estimate the BTAR model by sequential conditional least squares. This involves
doing a grid search over the bands and delay parameter (in our case it is 1). For given
values of [c2 ; c1 ] and delay parameter, we estimate (5) by least squares and the …nal
estimates are those that minimize the overall sum of squared errors.5 As a diagnostic
test, we employ Hansen’s (1999) test of the null hypothesis that t follows a linear AR(1)
against the alternative hypothesis of the nonlinear BTAR model. The test statistic
is H = (sL sN ) =sN where sL is the sum of squared errors from the linear AR(1)
model and sN is the sum of squared errors from the nonlinear BTAR model. The test
statistic H does not have an asymptotic chi-square distribution because it depends on
the particular speci…cation under the null hypothesis (in our case, the AR(1) model)
and on the particular alternative speci…cation. In addition, the threshold parameters
(c1 ; c2 ) and delay parameter are not identi…ed under the null hypothesis.6 As a result,
we follow Hansen’s suggestion for obtaining the distribution of H via the parametric
bootstrap. The p-value of the test is obtained by a relative frequency count of the
values of bootstrapped H statistics that exceed the value of H obtained from the data.
The BTAR estimation and test results are presented in Table 1. Here, it can be seen
that estimates of the autoregressive parameters ( 1 ; 2 ) are highly statistically signi…cant.
The negative estimates of 1 for Canada, Japan, and the UK indicate very rapid reversion
5
We do this over the range lower 5th c1
lower 100th observation, upper 100th c2
upper
5th observation.
6
Our model is the BTAR model with threshold parameter = (c1 ; c2 ) and delay parameter d = 1.
Consider a general BTAR model of a form
Yt =
0
1 Xt 1 I1t (
; d) + ::: +
0
m Xt 1 Imt (
; d) + et ;
(6)
where Xt 1 = (1 Yt 1 Yt 2 ::: Yt p ) and = ( 1 ; :::; m ) with 1 < 2 < ::: < m . Ijt ( ; d) =
_
I( j 1 < Yt d
is threshold parameter, and d is delay
j ) where I( ) is the indicator function,
parameter with 1 d p. Thus in the general BTAR framework another nuisance parameter is the
delay parameter. In our case, the delay parameter is 1.
5
towards the upper band (large positive values of t = rt rt ).7 Hansen’s test rejects
the null hypothesis of linearity at very small signi…cance levels for each of the four real
interest di¤erential pairs.
Smooth Transition Models We begin our development of smooth transition models
by testing the null hypothesis that the interest di¤erential follows a linear AR(p) model
against alternative of smooth transition AR(p) model. The test, suggested by Escribano
and Jordá (1999), proceeds by estimating the auxiliary regression,
yt = b0 xt + b1 xt zt + b2 xt zt2 + b3 xt zt3 + b4 xt zt4 + et ;
(7)
where yt =
t = t
t 1 , xt = (
t 1 ; :::;
t p ) and zt =
t d is the transition
(sL sA )=4p
8
variable. The test statistic is F = sA =(T 5p) where sL is the sum of squared errors
from the linear AR(12) model and sA is the sum of squared errors from the auxiliary
regression. Under the null hypothesis of linearity H0 : b1 = b2 = b3 = b4 , the test
statistic has an approximate F distribution with 4p and T 5p degrees of freedom.9
We obtain the following results: F48;268 = 196:7; (Canada), F48;268 = 348:0 (Germany),
F48;225 = 165:6 (Japan), and F48;268 = 267:4 (UK). The Escribano-Jordá test rejects the
linear AR(12) model at very small levels of signi…cance.
The …rst of two smooth transition models that we consider uses the cosine function to parameterize the transition coe¢ cient. We call it the cosine smooth threshold
autoregressive model (CSTAR)
0 v
1
!
u
p
X
u 1 X̀ 2
@ t
A t 1+
+ vt :
(8)
t = cos
t j
j
` j=1 t j
j=1
The cosine term is a time-varying transition coe¢ cient that depends on an ` period
moving window measure of t volatility which modulates the speed at which t makes
its transition back to 0. This term is close to 1 when t has been near 0 during the
recent past so that the process behaves locally as a unit-root process. When t exhibits
large deviations from its mean, the cosine term shrinks and the process behaves locally
like a stationary autoregression. After extensive experimentation, we set ` = p = 12: We
estimate the CSTAR model by nonlinear least squares and report the results in Table
7
The asymmetry in the bands does not con‡ict with our requirement that the unconditional mean
of the interest di¤erential be zero since the proportion of time that t spends in each of the regions is
di¤erent.
8
The auxiliary regression can be obtained by applying Taylor approximation to the transition function of smooth transition models. The auxiliary regression (7) can be used to test linearity null against
smooth transition autoregression alternatives. See van Dijk et. al. (2002).
9
In our sample we have T = 328 for Canada, Germany, and UK and T = 285 for Japan.
6
2. Here, our main interest rests with the speed of adjustment parameter ; so we do not
report estimates of the lagged coe¢ cient values ^j : As can be seen from the table, the
estimates of are correctly signed and are highly signi…cant.
Our second smooth-transition model takes the transition coe¢ cient as an exponential
function of the volatility estimate,
0 v
1
!
u
p
X
u 1 X̀ 2
@ t
A t 1+
+ vt ;
(9)
t = exp
t j
j
` j=1 t j
j=1
iid
where vt
N (0; 2v ). This model has been employed by Kilian and Taylor (2003) to
model the real exchange rate. Here, the time-varying transition coe¢ cient is speci…ed
as exponential term that depends on the ` period moving window of t volatility. As
with the CSTAR model, we set ` = p = 12: Our nonlinear least squares estimates
of the ESTAR speed of adjustment parameter are reported in Table 3. Again, the
estimates are all highly statistically signi…cant. The mean squared errors from ESTAR
and CSTAR are roughly equivalent.
4.2
Comparing Fundamentals Measures
We now turn to estimation of q^t;k which proceeds as follows. At time t 1 and conditional on t 1 ; simulate the k future values according to the particular nonlinear process
(BTAR, CSTAR, or ESTAR). Call the simulated sequence t ; t+1 ; : : : ; t+k and use
P
them to form the sum kj=0 t+j . Repeat this 50; 000 times. The mean value of these
sums gives our estimate q^t;k : Next, advance the time subscript and repeat the procedure
conditional on t to get q^t+1;k : Continue in this fashion through t = T; to obtain the
T
sequence q^t;k t=1 :
Having obtained estimates of the expected partial sum of future real interest di¤erentials, we look for evidence that the real exchange rate is priced by the sum of the
expected future real interest di¤erentials as implied by real interest parity.
We examine three alternative values of k = 12; 120; and 1000 which correspond to
forecast horizons of 1; 10; and 83 years respectively. Taking k = 1000 emphasizes the
long-horizon features of the data as in Alexius (2001), and Chinn and Meredith (2004)
who …nd evidence that long-horizon linkages between the nominal exchange rate and
nominal interest di¤erentials are stronger than they are at short horizons.
To explore some basic properties of the alternative q^t;k ; Table 4 reports the standard
deviation and …rst two autocorrelations. In the CSTAR and ESTAR models, the volatility of q^t;k increases quite dramatically with k: This contrasts sharply to the BTAR model
where the rapid reversion to the bands limits range and volatility of the q^t;k : The volatil7
ity of q^t;k implied by BTAR is fairly stable across our choices of k and is substantially
smaller than volatility implied by either of the smooth-transition models. The BTAR
q^t;k are also generally a bit more persistent than those from the smooth transition models. For example, the …rst-order autocorrelation coe¢ cient for the German BTAR q^t;12
is 0.38 whereas the largest …rst-order autocorrelation from a smooth-transition model is
0.09, for Germany’s ESTAR speci…cation of q^t;1000 :
Table 5 shows the correlation matrix of the alternative q^t;k for each country where
it is seen that the BTAR q^t;k are highly correlated across horizons whereas the CSTAR
and ESTAR q^t;k are largely uncorrelated across k: For a given horizon, the BTAR q^t;k
are moderately correlated with the CSTAR and ESTAR q^t;k for Germany (average correlation 0.21). For the other countries, the BTAR measures of q^t;k are uncorrelated with
either CSTAR or ESTAR measures. For a given horizon, the average correlation between
the CSTAR and ESTAR q^t;12 is a rather high value of 0:81. This average correlation
drops to 0:34 for k = 1000: Although the CSTAR and ESTAR models appear to have
quite similar in sample …t, the information content for long-run appears to be somewhat
di¤erent.
5
Expected Future Real Interest Di¤erentials and
the Real Exchange Rate
We are now ready to investigate the extent to which expected future real interest di¤erentials are priced into the real exchange rate. We begin with the regression
qt =
0
+
1t
+ q^t;k :
We perform system estimation generalized method of moments using the current value
and three lags of q^t;k across each of the four currencies as instrumental variables. We
also report constrained GMM estimates for those cases where homogeneity restrictions
on the slope coe¢ cient across currencies are not rejected at the 10 percent level. Robust
asymptotic standard errors computed with VAR(1) pre-whitening and with Newey-West
(1994) automatic lag-length selection are used to form the asymptotic t-ratios.
Table 6 reports estimation results both with and without trend. The regressions yield
slope coe¢ cients that are correctly signed in every case except one–that of the yen for
the BTAR q^t;1000 : The estimated coe¢ cients under BTAR without trend are signi…cant
(at the 10 percent level) at the medium and long horizons (k = 120; 1000). At the short
horizon (k = 12) without trend, the slope is signi…cant only for Germany while the
constrained slope estimate is highly signi…cant. When a trend is added, the estimated
slope is signi…cant both for Canada and for Germany.
8
Under the CSTAR fundamentals, we obtain signi…cant slope estimates only at the
short horizon k = 12: These results are robust to whether or not a time trend is included
in the regressions. We note that the magnitude of the slope coe¢ cient point estimates
declines as the horizon increases. This occurs on account of the higher volatility of q^t;k
as k increases under the smooth transition models.
Turning now to the regressions on ESTAR fundamentals, we …nd at the short horizon
the slope coe¢ cients are signi…cant for the deutschemark and the pound and these results
are robust to whether or not a time trend is included. The constrained estimate with
trend is also marginally signi…cant. At the medium horizon k = 120, the estimates are
signi…cant at the 10 percent level for the Canadian dollar, deutschemark and the yen
without trend, but is only signi…cant for the yen when a trend is included. At the long
horizon, the constrained estimates are signi…cant both with and without trend.
We included a time trend to account for potential Balassa-Samuelson e¤ects that
would not be re‡ected in the real interest di¤erentials. However, if such an e¤ect is
present and the productivity di¤erentials for the countries in our sample evolve with a
stochastic trend, then the regression should be run using …rst-di¤erences in the (log)
real exchange rate rather than including a time trend.
To explore this possibility, we report the GMM estimation results from regressions of
qt on q^t;k in Table 7. Here, under BTAR, all point estimates have the predicted sign.
Individual coe¢ cient estimates are signi…cant for the deutschemark and pound for all
three horizons as are the constrained estimates. Under the smooth transition models,
statistical evidence of linkage is mixed. Here, we obtain some point estimates that have
the wrong sign but they are not signi…cant. Coe¢ cient estimates for the pound are
signi…cant both for k = 12 and k = 1000; and are signi…cant for the yen at k = 120:
6
Conclusions
Nonlinear adjustment in log real exchange rates and real interest di¤erentials are a
signi…cant and prominent feature of the data. Statistical evidence that the real exchange
rate is linked to the real interest di¤erential is tenuous. In this paper, we look for evidence
of a link when a standard pricing relationship in which the real interest di¤erential is
assumed to follow a nonlinear process. The evidence, while not uniformly strong, does
suggest the existence of such a link when the pricing relationship is evaluated over the
relatively short horizon of one year.
9
Acknowledements
Mark thanks the National Science Foundation for support under grant no. 0339139.
Moh gratefully acknowledges the COR fellowship from Tulane University.
10
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[30] Peel D, Taylor MP. 2002. Covered interest rate arbitrage in the interwar period
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13
14
Figure 1: Log real exchange rate and contemporaneous real interest di¤erential.
15
Figure 2: Standardized values of qt and q^t;12 from the BTAR model.
16
Figure 3: Standardized values of qt and q^t;12 implied by the CSTAR model.
17
Figure 4: Standardized values of qt and q^t;12 implied by the ESTAR model.
Table 1: BTAR interest rate di¤erential
CAN
GER
JPN
UK
-0.145
0.060
-0.299
-0.232
1
(-1.979) (1.124) (-4.511) (-2.805)
-0.209
0.068
-0.018
0.153
2
(-3.705) (1.131) (-0.262) (3.013)
c1
2.653
1.887
1.439
4.093
c2
-0.992
-2.153
-2.682
-1.338
MSE
18.301
16.817
38.129
49.926
H
241.437 164.187 212.733 178.390
p-value
0.000
0.000
0.000
0.000
Note: Asymptotic t-ratios are in parenthesis and MSE is the mean squared error of the
model. For Japan, sample period is 1978:9- 2002:4. H is the test statistic for the linearity
test and p-value is the marginal signi…cance level from 1,000 bootstrap replications of
Hansen’s (1999) linearity test.
Table 2: CSTAR interest rate di¤erential
CSTAR
MSE
CAN
GER
JPN
UK
0.183
0.159
0.127
0.091
(8.793) (12.149) (8.267) (11.221)
15.423
12.809
24.614
28.506
Notes: Asymptotic t-ratios are in parenthesis. MSE is the mean squared error of the
…tted model.
Table 3: ESTAR interest rate di¤erential
ESTAR
MSE
CAN
GER
JPN
UK
0.121
0.081
0.090
0.081
(2.909) (3.115) (2.753) (3.917)
15.495 13.268 24.707 28.496
Notes: Asymptotic t-ratios are in parenthesis. MSE is the mean squared error of the
…tted model.
18
19
GBR
2/76-4/02
2
2
1
s.d.
2
1
s.d.
2
1
s.d.
Note: s.d. is the standard deviation.
JPN
GER
9/79-4/02
2/76-4/02
1
Sample
Country Statistic
2/76-4/02
CAN
s.d
j
CSTAR
Med
27.108
0.011
-0.060
26.618
0.030
0.028
35.839
0.028
-0.033
39.234
-0.005
-0037
ESTAR
Long Short
Med
23.283 7.887 25.935
0.009 0.041 -0.016
0.002 0.045 -0.029
21.969 7.470 23.963
0.073 0.043 -0.010
0.017 0.057 -0.029
30.728 10.640 33.474
0.090 -0.005 -0.017
-0.033 -0.051 -0.053
36.082 11.311 35.220
-0.024 0.009 -0.008
0.021 -0.057 -0.042
is the j th order autocorrelation (j = 1; 2).
BTAR
Short Med
Long Short
1.635 1.676 1.642 9.052
0.157 0.148 0.164 0.012
0.176 0.180 0.191 0.067
3.217 3.172 3.190 7.899
0.379 0.374 0.371 0.008
0.231 0.238 0.215 0.050
2.027 1.853 3.284 10.466
0.066 0.060 -0.076 -0.024
-0.096 -0.125 0.041 -0.039
3.414 3.423 3.424 11.795
0.183 0.165 0.177 -0.026
0.054 0.040 0.051 -0.019
Table 4: Basic features of alternative q^t;k
Long
70.167
0.085
0.028
65.111
0.088
0.033
87.740
0.0131
0.021
95.371
0.087
0.033
20
1.000
0.974
0.982
0.020
-0.023
0.056
0.045
0.014
0.026
1.000
0.988
0.992
0.208
0.009
0.031
0.223
0.064
0.027
BTAR12
BTAR120
BTAR1000
CSTAR12
CSTAR120
CSTAR1000
ESTAR12
ESTAR120
ESTAR1000
BTAR12
BTAR120
BTAR1000
CSTAR12
CSTAR120
CSTAR1000
ESTAR12
ESTAR120
ESTAR1000
BTAR12
1.000
0.983
0.213
0.022
0.035
0.220
0.069
0.023
1.000
0.962
0.006
0.005
0.061
0.049
0.026
0.018
BTAR120
1.000
0.203
0.019
0.034
0.221
0.070
0.019
1.000
0.017
-0.000
0.060
0.044
0.029
0.013
BTAR1000
1.000
-0.059
0.064
0.780
-0.045
0.081
1.000
0.096
-0.074
0.749
0.050
-0.023
US-Germany
1.000
-0.032
-0.033
0.696
-0.017
1.000
0.024
0.034
0.755
-0.077
CSTAR12 CSTAR120
US-Canada
Table 5: Correlation matrices q^t;k
1.000
0.047
0.015
0.373
1.000
-0.014
-0.026
0.345
CSTAR1000
1.000
-0.035
0.008
1.000
-0.043
-0.023
ESTAR12
1.000
-0.052
1.000
-0.061
ESTAR120
21
1.000
0.836
0.502
0.040
-0.028
-0.061
0.096
-0.031
-0.053
1.000
0.979
0.986
0.035
0.074
0.055
0.022
0.019
0.015
BTAR12
BTAR120
BTAR1000
CSTAR12
CSTAR120
CSTAR1000
ESTAR12
ESTAR120
ESTAR1000
BTAR12
BTAR120
BTAR1000
CSTAR12
CSTAR120
CSTAR1000
ESTAR12
ESTAR120
ESTAR1000
BTAR12
1.000
0.969
0.033
0.090
0.057
0.025
0.030
0.009
1.000
0.538
0.050
0.005
-0.032
0.097
-0.025
-0.029
BTAR120
1.000
0.027
0.074
0.054
0.019
0.029
0.005
1.000
0.025
0.016
0.024
0.056
-0.031
0.020
BTAR1000
1.000
0.027
0.081
0.832
-0.010
0.046
1.000
0.026
0.039
0.887
-0.007
-0.042
US-UK
1.000
-0.009
0.012
0.718
-0.018
1.000
0.044
0.039
0.720
0.077
CSTAR12 CSTAR120
US-Japan
(Table 5 continued)
1.000
0.065
-0.028
0.308
1.000
0.076
-0.034
0.333
CSTAR1000
1.000
-0.017
0.008
1.000
0.025
-0.057
ESTAR12
1.000
-0.050
1.000
-0.029
ESTAR120
Table 6: Real exchange rate and interest rate fundamentals: qt =
Model
BTAR12
BTAR120
BTAR1000
CSTAR12
CSTAR120
CSTAR1000
ESTAR12
ESTAR120
ESTAR1000
CAN
slope
trend (asy-t)
no
21.002
(0:538)
yes
18.023
(2:684)
no
20.384
(4.400)
yes
16.624
(1.054)
no
16.585
(5.452)
yes
22.250
(1.580)
no
1.148
(3.654)
yes
0.654
(2.159)
no
-0.026
(-0.119)
yes
-0.027
(-0.192)
no
0.081
(0.687)
yes
0.041
(0.349)
no
0.395
(1.020)
yes
0.312
(0.893)
no
0.256
(1.760)
yes
0.092
(0.761)
no
0.042
(0.870)
yes
0.098
(2.311)
0
+
1t
+ q^t;k
GER
JPN
UK
Constrained
slope
slope
slope
Wald
slope
(asy-t)
(asy-t)
(asy-t) (p-value)
(asy-t)
18.898
17.797
5.966
0.231
8.473
(2:693) (0:145) (0:224)
(0:972)
(3:383)
13.878
17.028
2.666
8.579
–
(3:293) (0:955) (0:950)
(0:035)
–
19.499 17.678
5.387
35.986
–
(5.530) (2.201) (2.203) (0.000)
–
18.475
29.244
0.923
0.372
6.062
(1.140) (0.336) (0.031)
(0.946)
(0.621)
13.811 23.567
4.400
25.740
–
(6.146) (1.904) (3.633) (0.000)
–
5.904
-121.736
3.403
0.348
3.198
(0.138) (-0.213) (0.080)
(0.951)
(0.118)
2.953
3.146
1.061
15.592
–
(4.951) (1.229) (3.866) (0.001)
–
2.699
7.855
0.793
16.907
–
(4.067) (2.794) (2.621) (0.001)
–
0.072
1.705
0.078
0.872
0.042
(0.296) (0.830) (0.557)
(0.832)
(0.394)
0.009
0.493
0.057
0.391
0.060
(0.055) (0.429) (0.570)
(0.942)
(0.950)
0.392
1.814
-0.036
9.193
–
(1.940) (2.364) (-0.442) (0.027)
–
0.218
1.141
0.021
3.385
0.095
(0.904) (1.695) (0.273)
(0.336)
(1.392)
3.614
2.191
1.078
23.687
–
(4.298) (1.155) (3.757) (0.000)
–
2.426
0.889
0.726
5.245
0.869
(2.126) (0.610) (2.077)
0.155)
(1.949)
0.457
2.275
0.086
8.920
–
(2.248) (2.278) (0.836)
(0.030)
–
0.294
1.394
0.045
4.065
0.133
(0.913) (1.890) (0.372)
(0.255)
(1.380)
0.040
0.896
0.030
3.754
0.053
(0.592) (1.952) (0.867)
(0.289)
(1.852)
0.085
0.496
0.004
6.528
0.052
(0.788) (1.232) (0.111)
(0.089)
(1.849)
Note: Wald is the test statistic to test the
22 homogeneity restriction across currencies.
Bold face indicates that estimates are signi…cant at the 10 percent level.
Table 7: Real exchange rate depreciation and interest rate fundamentals:
Model
BTAR12
BTAR120
BTAR1000
CSTAR12
CSTAR120
CSTAR1000
ESTAR12
ESTAR120
ESTAR1000
qt =
0+
CAN
slope
(asy-t)
0.213
(0.474)
0.239
(0.535)
0.239
(0.497)
GER
JPN
UK
Constrained
slope
slope
slope
Wald
slope
(asy-t)
(asy-t)
(asy-t) (p-value)
(asy-t)
1.572
2.225
0.979
6.068
1.154
(4.785) (1.171) (2.506) (0.108)
(4.439)
1.623
3.181
1.058
6.819
1.214
(4.789) (1.572) (2.662) (0.078)
(4.598)
1.559
4.159
1.022
7.421
1.210
(4.316) (2.056) (2.418) (0.060)
(4.545)
-0.449
(-1.594)
0.356
(1.673)
-0.058
(-0.306)
-0.306
(-0.525)
-0.301
(-0.922)
-0.369
(-1.200)
-0.048
1.623
(-0.071) (3.467)
0.462
0.396
(2.048) (1.362)
0.402
0.518
(1.463) (2.125)
18.102
(0.000)
3.665
(0.300)
8.413
(0.038)
–
–
0.312
(2.508)
–
–
-0.174
(-0.513)
-0.140
(-0.685)
0.050
(0.690)
0.313
0.549
1.670
(0.493) (0.786) (3.118)
0.372
0.780
0.036
(0.942) (2.059) (0.109)
0.430
-0.127
0.386
(1.977) (-0.897) (2.762)
9.404
(0.024)
4.974
(0.174)
6.936
(0.074)
–
–
0.248
(1.407)
0.099
(1.590)
q^t;k
Note: Wald is the test statistic to test the homogeneity restriction across currencies.
Bold face indicates that estimates are signi…cant at the 10 percent level.
23