This study investigates the relationship between the euro-dollar exchange rate and its underlying fundamentals by adopting non-linear time series modelling. We found that this relationship is episodically unstable. We also found that an equilibrium-distorting shock is likely to have a greater effect on the exchange rate during periods when the deviation between exchange rate and fundamentals is large; as a consequence, when the exchange rate is close to its equilibrium value it tends to be less sensitive to any shocks in the fundamentals.Non-linearity, Markov-switching Model, Fundamentals

Carlo Altavilla

English

Research Papers in Economics

 
Department of Economic Studies 
University of Naples “Parthenope” 
   
 
 
Discussion Paper 
 
 
No.6 
 
 
 
Title: The (Un -) Stable Relationship between 
          The Exchange rate and its Fundamentals 
 
 
 
Author: Carlo Altavilla   
              
 
 
 
Affiliation:  University of Naples Parthenope 
                     
 
 
 
 
  2006 
 
 
 
 
 
  
   1
The (UN-) Stable Relationship between the 
Exchange Rate and its Fundamentals 
 
Carlo Altavilla
*  
 
 
Abstract 
This study investigates the relationship between the euro-dollar exchange rate and its 
underlying fundamentals by adopting non-linear time series modelling. We found that this 
relationship is episodically unstable. We also found that an equilibrium-distorting shock is 
likely to have a greater effect on the exchange rate during periods when the deviation 
between exchange rate and fundamentals is large; as a consequence, when the exchange rate 
is close to its equilibrium value it tends to be less sensitive to any shocks in the fundamentals.  
 
Keywords: Non-linearity, Markov-switching Model, Fundamentals   
JEL Classification: C32, F31  
 
 
1. Introduction 
There exists a large body of empirical literature focusing on whether floating exchange 
rates are ultimately determined by movement in underlying fundamentals.  
Most of the literature, including Baillie and Selover (1987), McNown and Wallace (1989), 
and Baillie and Pecchenino (1991), applies linear econometric frameworks. The results 
suggest a slow adjustment of the exchange rate to news in fundamentals.  
More recently, a number of studies (Taylor and Peel (2000) and Taylor et al. (2001)) 
explicitly incorporate a non-linearity between exchange rate and fundamentals by employing 
Multiple Regime Smooth Transition Autoregressive models (MR-STAR)1. Other studies (see 
e.g. Taylor (2004) and Taylor (2005)) employ Markov-switching models to analyse the 
dynamic relationship between exchange rate and fundamentals. As suggested by Kilian and 
Taylor (2003), non-linearities in the relationship between exchange rate and fundamentals 
might also reflect the poor out-of-sample forecasting performance of standard linear models 
when compared with the naïve random walk forecasts.  
In the present study we account for non-linearity in the exchange rate by adopting a 
multivariate extension of Hamilton's (1989) Markov-switching model.  
 
                                                 
* University of Naples "Parthenope", Via Medina, 40 - 80133 Naples (Italy). E-mail: 
Altavilla@uniparthenope.it; Phone: (+)39 0815474733, fax (+)39 0815474750.  
1 See for example, Micheal et al. (1997); Sarno (2000); Sarantis (1999) and Baum et al. (2001).   2
2. Linearity vs. Non-linearity hypothesis 
In order to test for non-linearity in the exchange rate dynamics we proceed as follows. 
First, we estimate a linear VECM with the maximum likelihood technique. Then, we check 
the non-linearity of the residuals by employing a battery of standard tests. 
The sample period goes from 1979:1 to 2004:4. The data are seasonally adjusted quarterly 
observations and were drawn from DataStream. The pre-EMU exchange rate is 
approximated by a synthetic Euro/US dollar rate.  
2.1 A Linear Mean-reverting Model 
The benchmark linear model is a finite-order VECM: 
[1]   () ()
α
α
−
−−
−
ΔΔ ⎡ ⎤ ⎡⎤ ⎡ ⎤ ⎡⎤
=Γ + − +⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ΔΔ ⎣⎦ ⎣ ⎦ ⎣⎦ ⎣ ⎦
11 1
11
12 1
e
tt t
tt f
tt t
ee u
Le f
ff u
 
Where  [] ΔΔ
'    tt ef  is a vector of non-stationary variables containing the euro/dollar 
exchange rate ( t e ), and a measure of fundamentals in the Euro Area relative to the USA 
( t f ): 
[2]   ( ) ( ) ( ) ( ) π π =−−−− − +−
** * * ee
tt t t tt t t t fm m y y i i  
where  t m  represents the log-level of money supply;  t y  denotes the log-level of GDP;  t i  
is the nominal interest rate; and 
e
t π  denotes the expected inflation rate which is proxied by 
long term interest rate; α11and α21 indicate the speed of adjustment of each variable back 
to its long-run value; starred variables indicate the equivalent variables for the foreign 
economy. Table 1 reports the ML estimates.  
Table 1: VECM Estimates 
Coeff St. Error Coeff St. Error
c -0.01 [0.00] -0.0021 [0.01]
-0.26 [0.09] 0.11 [0.15]
-0.13 [0.06] -0.40 [0.10]
0.06 [0.02] 0.03 [0.07]
Fundamentals Exchange rate 
1 t f − Δ
1 t e − Δ
α  
 
For the fundamentals equation the adjustment coefficient ( 11 α ) is significantly different 
from zero, meaning that the fundamentals adjust to restore the long-run equilibrium. By 
contrast, in the exchange rate equation the error correction term is not significant. The 
exchange rate is then said to be long-run weakly exogenous with respect to the long-run 
equilibrium.   3
The estimated error-correction coefficient in the fundamental equation is quite low (0.06). 
After almost three years, 50 percent of the disequilibrium gap created by the shock has been 
closed by the adjustment in fundamentals. The results obtained in the analysis corroborate 
the so-called exchange rate disconnect puzzle (see Obstfeld and Rogoff (2001)).  
 
2.2 Testing Linearity 
We check the non-linearity of the residuals by using the BDS, the Reset and Tsay test to 
the residuals of each equation in the VECM system. As shown in Table 2 and table 3, all 
tests reject the null hypothesis of a linear generating mechanism for the residuals of the 
selected variables. This evidence suggests the presence of non-linearity. 
Table 2: Tsay and Reset Test statistics 
Distribution Statistic P-value Statistic P-value
Tsay F( 1 , 98 ) 25.15 2E-06 36.34 3E-08
Reset F(3,96) 2.763 0.04619 27.093 9E-11
Fundamentals Exchange rate 
 
Table 3: BDS Test statistics 
Residuals from Fundamentals Equation
Dimension BDS Statistic Std. Error z-Statistic Asymptotic Bootstrap
2 0.02 0.01 2.24 2.5E-02 6.3E-02
3 0.09 0.02 5.18 0.0E+00 8.0E-04
4 0.14 0.02 6.28 0.0E+00 0.0E+00
5 0.20 0.03 7.26 0.0E+00 0.0E+00
6 0.24 0.03 7.88 0.0E+00 0.0E+00
Residuals from Exchange rate Equation
Dimension BDS Statistic Std. Error z-Statistic Asymptotic Bootstrap
2 8.88E-04 4.17E-04 2.13 3.32E-02 4.59E-02
3 2.26E-04 1.04E-04 2.17 3.04E-02 5.70E-02
4 -1.05E-04 1.96E-05 -5.35 0.00E+00 5.66E-02
5 -1.04E-05 3.22E-06 -3.24 1.20E-03 9.27E-02
6 -9.80E-07 4.90E-07 -2.00 4.55E-02 7.56E-02  
 
3. Modelling Non-linear Mean Reversion 
We account for non-linearity by estimating a Markov-switching model. The asymmetry of 
the effects is captured by allowing for state-dependent parameters where the latent state 
variable follows a Markov-switching process. The idea behind this class of models is that the 
parameters underlying the data generating process of the observed time series vector depend 
upon the unobservable regime variable  t s , which represents the probability of being in a 
different state of the world. Specifically,  t s  is governed by a discrete state Markov stochastic 
process, which is defined by the following transition probabilities:   4
() + == = 1 Pr ij t t ps j s i        
⎡ ⎤
⎢ ⎥
⎢ ⎥ =
⎢ ⎥
⎢ ⎥
⎣ ⎦
 
 
      
…
11 12 1
21 22 2
12
m
m
mm m m
pp p
pp p
P
pp p
 
where  ij p  is the probability that state i is followed by state j and P is the corresponding 
transition matrix. We estimate a Markov-switching vector equilibrium correction model: 
 [6]      αβε
−
−−
=
Δ= + Γ Δ + + ∑
1
'
1
()
k
tt i t i t k t
i
yc s y y  
where the residuals are conditionally Gaussian,εtt s ~ ( ) ( ) Σ 0, t NID s . Following the two-
stage procedure suggested by Krolzig (1997) the results obtained in the linear VECM 
concerning the cointegration analysis are used in this stage of the analysis. The ML estimates 
are reported in table 4.  
Table 4: ML estimates of the MSIH(3)-VECM(2) 
Coeff St. Error Coeff St. Error
c 1 -0.03 [0.010] -0.05 [0.006]
c 2 0.01 [0.003] -0.003 [0.008]
c 3 0.02 [0.009] 0.03 [0.008]
0.14 [0.093] -0.16 [0.096]
-0.07 [0.099] -0.08 [0.098]
-0.07 [0.055] 0.08 [0.092]
-0.03 [0.056] -0.06 [0.079]
0.07 [0.025] -0.10 [0.028]
SE (Reg.1) 0.04 0.02
SE (Reg.2) 0.01 0.04
SE (Reg.3) 0.05 0.03
Fundamentals Exchange Rate
1 t f − Δ
2 t f − Δ
1 t e − Δ
2 t e − Δ
α
 
 
Standard bottom-up procedure hints three as the number of regimes. The results 
associate regime 1 with a decreasing exchange rate. By contrast, Regime 3 is associated with 
an increasing exchange rate, i.e. a depreciating euro. The close-to-zero intercept term of 
Regime 2 reflects near-to-fundamentals behavior of the exchange rate. The speed of 
adjustment coefficients are larger than those estimated by using the linear VECM: the 7% of 
the adjustment takes place each period. As a result, fundamentals restore 50% of the pre-
shock long-run equilibrium level after almost 10 quarters.  
The transition probabilities (table 5) suggest that regimes are quite persistent. Table 6 
shows that whereas overvaluation has an expected duration of almost 3.5 quarters, the near-
to-fundamentals state and the undervaluation state have a persistency of 14.48 and 6.56 
quarters, respectively. 
   5
Table 5: Transition probabilities                         Table 6: Regime Properties 
Regime 1 Regime 2 Regime 3
Regime 1 0.72 0.14 0.15
Regime 2 0.07 0.93 0.00
Regime 3 0.15 0.00 0.85  
 
nObs Prob. Duration
Regime 1 24 0.25 3.52
Regime 2 36 0.51 14.48
Regime 3 28 0.24 6.56  
We can now characterize the timing of the estimated regimes (Figure 1).  
 
Figure 1: Smoothed Regime Probabilities 
Regime 1
1980 1984 1988 1992 1996 2000 2004
0.00
0.25
0.50
0.75
1.00
Regime 2
1980 1984 1988 1992 1996 2000 2004
0.00
0.25
0.50
0.75
1.00
Regime 3
1980 1984 1988 1992 1996 2000 2004
0.00
0.25
0.50
0.75
1.00
 
Regime 1 coincides with the euro-appreciation of the second half of the 1980s, as well as 
the recovery of the exchange rate following the currency crises of the 1992.  The estimated 
smoothed probabilities are close to one, indicating a high probability of being in state one, 
also for 2004. Regime 2 characterizes near-to-fundamentals exchange rate behaviour, while 
Regime 3 represents euro-depreciation episodes. The years 1981-85, 1991-93, and 2000-2001 
are identified as periods of depreciation of the euro against the US dollar. 
Following Krolzig and Toro (1998) we compute the effects of a regime-wide shock on 
the exchange rate and fundamentals (Figure 2). A 40-quarter horizon is considered. We first 
compute the response of all the variables in the system to a shock that induces a movement 
from the ergodic2 distribution to a specific regime.  
The first column of Figure 2 shows the response of the variables when they move from 
steady-state probabilities to the three estimated regimes. The second column of Figure 2 
reports the response of the variables in the fundamentals-exchange rate space. We observe 
the different dynamics that govern the exchange rate and fundamentals during overvaluation 
and undervaluation periods.  
 
 
 
 
 
                                                 
2 Note that as the Markov chain is stationary, the conditional distribution of the different states 
converges to the ergodic distribution.   6
Figure 2: The Response to Regime Shifts 
LEXR_1 FUN_1
Shift to Regime 1
0 5 10 15 20 25 30 35 40
-0.100
-0.075
-0.050
-0.025
-0.000
LEXR_2 FUN_2
Shift to Regime 2
0 5 10 15 20 25 30 35 40
-0.10
-0.05
0.00
0.05
LEXR_3 FUN_3
Shift to Regime 3
0 5 10 15 20 25 30 35 40
0.00
0.04
0.08
0.12
0.16
Shift to Regime 1
LEXR
F
U
N
-0.100 -0.075 -0.050 -0.025 -0.000
-0.100
-0.075
-0.050
-0.025
-0.000
Shift to Regime 2
LEXR
F
U
N
-0.036 -0.018 0.000
-0.1
0.0
0.1
0.2
Shift to Regime 3
LEXR
F
U
N
0.00 0.04 0.08 0.12 0.16
0.00
0.05
0.10
0.15
0.20
 
The two upper graphs (shift to regime 1) represent the reaction of exchange rate and 
fundamentals to the information that  =1 t s . Then, the time profile of the reactions illustrates 
the pattern of exchange rate appreciation and the subsequent fundamentals movements 
when the system moves to Regime 1. More interestingly, the upper-right graph shows the 
dynamics of the exchange deviation rate from fundamental equilibrium. There are clear 
asymmetries in the reaction of the selected variables when the currency becomes overvalued 
(shift to regime 1) with respect to when the exchange rate moves to the undervalued regime 
(shift to regime 3). In particular, the occurrence of regime 3 produces a larger deviation from 
the fundamental equilibrium than when there is a shift to regime 1. In other words, a shock 
that leads to currency undervaluation has a bigger impact on the fundamental equilibrium 
than a shock that induces currency overvaluation. This evidence corroborates the existence 
of a non-linear relationship between the exchange rate and fundamentals. The response of 
the variables when there is a shift to regime 2, i.e. the near-to-fundamentals regime, illustrates 
the small reaction of exchange rate when the currency is close to fundamental equilibrium.  
Figure 3 depicts the dynamics of the exchange rate and fundamentals when there is a 
change in the existing regime (from the near-to-fundamentals regime to the over- and under-
valuation regimes).   
The time profile of the estimated responses suggests that when the exchange rate moves 
from regime 2 to regime 1, i.e. it becomes overvalued, the speed and the magnitude of the 
depreciation that follows is lower with respect to the exchange rate appreciation that follows 
the shift to regime 3, i.e. when the exchange rate becomes undervalued.    7
It follows that a shock that produces a shift from the steady state probabilities to the 
estimated regimes will have a greater effect when it leads to depreciation than when the 
shock produces an appreciation. Finally, when the exchange rate is close to its equilibrium 
value it tends to be less sensitive to regime-shifting shocks. 
Figure 3: The Response to Regime Transitions 
LEXR_2_1 FUN_2_1
Transition Regime 2 to 1
0 5 10 15 20 25 30 35 40
-0.100
-0.075
-0.050
-0.025
-0.000
LEXR_2_3 FUN_2_3
Transition Regime 2 to 3
0 5 10 15 20 25 30 35 40
0.00
0.05
0.10
0.15
0.20
Transition Regime 2 to 1
LEXR
F
U
N
-0.100 -0.050 -0.000
-0.100
-0.075
-0.050
-0.025
-0.000
Transition Regime 2 to 3
LEXR
F
U
N
0.00 0.05 0.10 0.15 0.20
0.00
0.05
0.10
0.15
0.20
 
4. Conclusions 
This paper investigated whether the dynamic interaction between the exchange rate and 
its fundamentals is time-varying. The results obtained by using linear VECM framework 
corroborate the so-called exchange rate disconnect puzzle.  
However, when introducing non-linearities in the relationship between the exchange rate 
and its fundamentals the results change substantially. We found that the exchange rate 
dynamic can be modelled by a nonlinear error-correction models where deviations from the 
long-run equilibrium are mean-reverting but occasionally follow a nonstationary process.  
 
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The (Un-) Stable Relationship between The Exchange rate and its Fundamentals

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The (Un-) Stable Relationship between The Exchange rate and its Fundamentals

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