ñòð. 33 
where we have used F  T;* = n  n* (the â€˜international Fisher effectâ€™ which is
the hyperinflation FPMM). The crucial elements in the Frankel model are the e
equation (13.14) and the distinction between the shortrun and longrun dete
the exchange rate. Substituting for S from (13.16) in (13.17) we obtain Frankelâ€™s
formâ€™) exchange rate equation:
m*  @(? ?*)  (l/O)(r  r * )+ [ ( l / O ) + h ) ( n 71*)

s = rn  

= m  m *  @(Y  ?*I + a ( r  r * ) + p(n  n*)
˜
where a = (l/@ and = ( l / @+ A. We can now characterise our three
models in terms of the parameters a and p.
It is evident from Table 13.1 that in the Frankel model we obtain a Dorn
result (as/ar < 0) if interest rates increase while inflation expectations remai
This situation is likely to correspond to an unanticipated change in the mo
which has an immediate impact on interest rates (to â€˜clearâ€™ the money ma
not immediately perceived as permanent and hence does not influence II. O
hand, an equal increase in the nominal interest rate, r, and inflationary exp
+
cause a depreciation in the exchange rate ( p a > 0)  a FPMMtype result
adding an ancillary assumption to the Dornbuschtype model, namely equati
an anticipated increase in the money supply is likely to lead to an expecte
tion and (the Frankel model then predicts) an actual depreciation. Implicitly
model highlights the possible differential response of the exchange rate to anti
unanticipated changes in the money supply and interest rates.
Table 13.1 The Frankel Real Interest Rate Model
Model Parameters
< 0, B I 1 > IQ1
S
Frankel > 0;
Q
FPMM =0
> 0,?!/
Q
FPMMhyperinflation = 0, >0
Q
< 0, p
DornbuschSPMM =0
Q
The Portfolio Balance Model
The current and capital account monetary models which have been the sub
of the preceding sections make at least two important simplifying assumption
portfolio balance model (PBM) is determined, at least in the short run, by
demand in the markets for all financial assets (i.e. money, domestic and foreign
the PBM a surplus (deficit) on the current account represents a rise (fall) in n
holdings of foreign assets. The latter affects the level of wealth and hence
demand for assets which then affects the exchange rate. Thus, the PBM is an
dynamic model of exchange rate adjustment which includes behavioural inte
asset markets, the current account, the price level and the rate of asset accumu
reduced form equations used in testing the PBM therefore include stocks of a
than money. For example, domestic and foreign bonds and stocks of overseas
by domestic and foreign residents (usually measured by the cumulative curre
position) influence the exchange rate.
A General Framework
The above models can be presented in a common framework suitable for empir
by invoking the UIP condition as a key link to the â€˜fundamentalsâ€™ in each mo
this, note that the UIP condition in logarithms is:
 s, = rr  r:
E,s,+l
Now assume that some model of the economy based on fundamentals z, impl
interest differential depends on these fundamentals:
r,  r: = yfzr
From the above:
= m t + 1  YfZf
Sf
and by repeated forward recursion and using the law of iterated expectations
fl1
i=O
+
Hence movements in the current spot rate between t and t 1 are determin
sions to expectations or â€˜newsâ€™ about future fundamentals Z r + j . In this RE
exchange rate is volatile because of the frequent arrival of news. The funda
in equation (13.22), vary slightly depending on the economic model adopte
flexprice monetary model FPMM we have
 p: = (rn  m*), a ( y  y*If + /?(r r*lt
st = p t
substituting for r  r* from the UIP condition and rearranging we have
+ P)lzr + â€œ(1 + B)I&Sr+l
=[ W
3,
where z, = (rn  rn*)t  a ( y  y * ) f . By repeated forward substitution
Pt  PI1 = axt
and excess demand is high when the real exchange rate depreciates (i.e. st in
+ P:  p r )
Xr = q 1 ( s t
Using UIP and the money demand equations this gives rise to a similar form
except that there is now inertia in the exchange rate:
00
+
= e1St <1
6 2 ˜ t1zt+ j
i (@1,@2)
st
j=O
where zr depends on current and lagged values of the money supply and outp
The portfolio balance model (PBM) may also be represented in the form
noting that here we can amend the UIP condition to incorporate a risk prem
depends on relative asset holdings in domestic B, and foreign bonds BT.
+ f(Br/B:)
rt  r: = ErSr+1  sr
The resulting equation (13.29) for the exchange rate now has relative bond
the vector of fundamentals, zt.
Forecasts of the future values of the fundamentals Zt+j depend on informa
t and hence equation (13.29) can be reduced to a purely backward looking
terms of the fundamentals (if we ignore any implicit crossequation RE restri
this is often how such models are empirically tested in the literature as w
below. However, one can also exploit the full potential of the forward terms
which imply implicit crossequation restrictions, if one is willing to posit an
of forecasting equations for the fundamental variables. This is done in Cha
the FPMM to illustrate the VAR methodology as applied to the spot rate in t
market.
13.5 TESTING THE MODELS
As one can see from the above analysis, tests of SPMM involve regressions of t
on relative money stocks, interest rates, etc. and tests of the PBM also include
stocks. If we ignore hyperinflation periods, then these models have not proved
in predicting movements in bilateral spot rates, particularly in post 1945 dat
the models do work reasonably well over short subperiods but not over the wh
Meese (1990) provides a useful â€˜summary tableâ€™ of the performance of such
estimates a general equation which, in the main, subsumes all of the above th
+ a l ( L ) ( m m*), a 2 ( L ) ( y y*)t + a3(L)(r r*)t + a4(L)(n 
+
st = a0
where F = stock of foreign assets held by domestic residents and F* = stock
assets held by foreign residents. Meese (1990) repeats the earlier tests of
mean square forecast errors from (13.30) with those from a benchmark prov
â€˜nochangeâ€™ prediction of the random walk model of the exchange rate. It is
Table 13.2 that the forecasts using the economic fundamentals in (13.30) are
worse than those of the random walk hypothesis.
Meese (1990) dismisses the reasons for the failure of these models based on
tals as mismeasurement of variables, inappropriate estimation techniques or e
variables (since so many alternatives have been tried). He suggests that th
such models may be due to weakness in their underlying relationships such
condition, and the instability found in money demand functions and the mountin
from survey data on expectations that agentsâ€™ forecasts do not obey the axioms
expectations. He notes that nonlinear models (e.g. chaotic models) that invo
changes (e.g. Peso problem) and models that involve noise traders may pr
insights into the determination of movements in the spot rate but research in
is only just beginning.
A novel approach to testing monetary models of the exchange rate is p
Flood and Rose (1993). They compare the volatility in the exchange rate and i
fundamentals for periods of â€˜fixed ratesâ€™ (e.g. Bretton Woods, where permitte
a b l e 13.2 Root Mean Square Error (RMSE) OutofSample Forecast Statistics 
1980 through June 1984 (44 Months)
Exchange Horizon Random Mod
Rate (Months) Walk 1
3.1 3.1
1
6 7.9 8.4
12 8.7 11.1
1 3.5 3.3
6 7.0
7.8
12 9.0 7.5
RMSE OutofSample Forecast Statistics  November 1976 through June 1981 (56 M
Exchange Horizon Random Forward M
Rate (Months) Walk Rate 1
log(DM/$) 1 3.22 3.20 3.65
6 8.71 9.03 12.03
12 12.98 12.60 18.87
log(yen/$) 1 3.68 3.72 4.11
6 11.58 11.93 13.94
12 18.31 18.95 20.41
(a) Model 1: Equation (13.30) with a5 = 0. Model 2: Equation (13.30) with all parameters fre
Both models are sequentially estimated by either generalised least squares or instrumental
a correction for serial correlation. RMSE is just the average of squared forecast errors fo
three prediction horizons.
Source: Meese (1990).
floated. For nine industrialised (OECD) countries, Flood and Rose find th
the conditional volatility of bilateral exchange rates against the dollar alters d
across these exchange rate regimes, none of the economic fundamentals e
marked change in volatility. Hence one can legitimately conclude that th
fundamentals in the monetary models (e.g. money supply, interest rates, inf
output) do not explain the volatility in exchange rates. (It is worth noting, howe
latter conclusion may not hold in the case of extreme hyperinflations where th
in relative inflation rates (i.e. fundamentals) across regimes might alter subst
A more formal exposition of the Flood and Rose methodology in testing
may be obtained from equation (13.4) with @ = @*, h = A* and substituting
from the UIP condition
+
Sf = TF, h(s;+l  Sf)
where TF, = (m  m*)t  +(y  y*), is â€˜traditional fundamentalsâ€™. Equa
defines â€˜virtual fundamentalsâ€™ (VF) as VF, = st  h ( r  r*)r. Now, under
equation (13.4), we expect the variability in virtual fundamentals VF, to
variability in traditional fundamentals TFt. To obtain a time series for TF, and
requires is a representative value for the structural parameters of the demand
function h and 4. As reported above, Flood and Rose find that while the
VF increases dramatically in the floating rate period, the volatility of funda
changes very little. (This result is invariant to reasonable values of E, and 4 an
when they consider the SPMM.) Flood and Rose speculate that since few mac
variables undergo dramatic changes in volatility, which coincide with changes
rate regimes, then it is unlikely that any exchange rate model based only o
fundamentals will prove adequate. For nine OECD countries, they also c
average monthly variance of the exchange rate over successive twoyear hor
against the variance of various macroeconomic variables. They find that
correlation between a2(S)and either the variability in the money supply or in
or FOREX reserves or stock prices and only a rather weak negative correlati
variance of output. Hence in moving from a floating exchange rate regime to
the reduced volatility in the exchange rate is not reflected in an increase in
other macroeconomic variables. (A similar result is found by Artis and Taylo
European countries which moved from a floating rate into the Exchange Rate
in the 1980s.) The balance of the argument on fixed versus floating based o
evidence would seem to favour some kind of target bands rather than a p
regime. The evidence also suggests that there is some change in the trading b
agents when there is a move from flexible to â€˜fixedâ€™ rates (e.g. do noise trad
to other â€˜unrestrictedâ€™ speculative markets? does the credibility of the fixed
play an important role in influencing expectations and hence trading activity?
Rational Bubbles
We noted in Chapter 7 that there are not only severe econometric difficultie
for rational bubbles but such tests are contingent on having the correct equilib
bubble term results in an equation for the spot rate of the form:
00
+ PIâ€™ + BO[(P + ˜)/B)Iâ€˜
sr = (1 ˜ / ( 1 P)Eizr+i
+
i=O
where 2, = set of monetary variables and Bo = value of bubble at t = 0. The
bubbles is Ho: BO = 0. However, if the 2, variables are a poor representation
fundamentals then the estimate of Bo may be different from zero, as it is the
candidate left to help explain the dependent variable. Testing Bo = 0 in (13
problematic because of the â€˜exploding regressorâ€™ problem. However, the test
(1987a) avoids the latter problem and provides a test for any form of bubb
stochastic or deterministic.
Meese (1986) uses the FPMM as his maintained fundamentals model fo
exchange rate (19731982) and rejects the nobubbles hypotheses. West (19
second type of West (1988a) test and augments the FPMM of Meese to inc
demand errors which may pick up other potential â€˜fundamentalsâ€™. He finds
presence of bubbles for the $/DM rate (19741984).
The Peso problem poses an additional difficulty when testing for bubbles in
exchange market. It is widely believed that the monetary authorities frequentl
in the FOREX market and that the authorities often try and mitigate large swi
and) nominal exchange rates (e.g. Plaza and Louvre agreements in the 198
participants are likely to form expectations of such events and these expe
unlikely to be measured correctly by the econometrician. The latter, couple
poor performance empirically of exchange rate models based on fundament
that there is little one can say with any degree of certainty about the presence o
of rational bubbles in spot exchange rates.
Random Walk Reappears
The failure of structural models of the spot rate led researchers to fall back on
parsimonious statistical representations. To a reasonable approximation, dai
exchange rates follow a martingale
 s,1 = 0
E,lS,
where the forecast error q, = S,  E,1S, has a nonconstant variance (Baillie
slev, 1989 and Baillie and McMahon, 1989). Hence the model is:
+ rlr
= Sr1
Sr
and the time varying variance of q, denoted of seems to be well approxim
autoregressive structure of the form
+ alorl+ a2qr1
2 2
=a0
0
:
which is known as a GARCH(1,l) process (see Chapter 20). In a recent s
(1995) has reexamined the usefulness of â€˜fundamentalsâ€™ in explaining chan
 sr = J k k @k(st  zr)
Si+k
Mark finds that the R2 in the above regression and the value of /?k incre
horizon k increases from 1 to 16 quarters. (He uses quarterly data on the
against the Canadian dollar, Deutschmark, yen and Swiss franc, 1973 1991.
ofsample forecasts at long horizons ( k = 16) outperform the random walk f
and Swiss franc. The above ana!ysis is not a test of a â€˜fully specifiedâ€™ mone
but demonstrates that â€˜monetary fundamentalsâ€™ may provide a useful pred
exchange rate, over long horizons (although not necessarily over short hor
above exchange rate equation is a simple form of error correction model, w
correction term (s  fiis model is reexamined in the context of coin
Chapter 15. Ciearly, if one is looking for a purely statistical representatio
also consider nonlinear models (e.g. Engel and Hamilton (199(ijj, neural ne
Trippi and Turban (1993)) and chaos models of the exchange rate. Below
discuss the latter.
13.6 CHAOS AND FUNDAMENTALS
The RE hypothesis when applied tomodds of speculative prices and return
an Euler equation which can contain an exogenous bubble term (De Grauwe e
The model is therefore not fully selfcontained because we need to select
either with or without the bubble and this choice is not determined by the
requires and adhoc assumption using information which is exogenous to the
there is, in a sense, an adhoc element in RE models; they are not â€˜complete
the expectations process.
RE models of the exchange rate attribute the volatility in the spot rate to
of new information or news. However, using high frequency data the study b
(1989) finds that most exchange rate movements appear to occur in the absenc
able news. Ad4 to this evidence from cointegration studies that for many curr
is no longrun (cointegrating) relationship between the spot rate and economic
tals (e.g. Boothe and Glassman (1987) and Baillie and Selover (1987)) then m
on fundamentals begin to look rather weak. There is also evidence that at sho
(e.g. one month) the spot rate is positively autocorrelated whereas at long
there is significant negative serial correlation (Cutler et al, 1989) which is i
NT with extrapolative predictions at short horizons and with fundamentals mo
inant at !onger horizons. Finally, there is evidence using survey data that expe
not rational and they may exhibit bandwagon effects at short horizons with m
sion at !onger horizons (Frankel and Froot, 1988 and Takagi, 1991). Also, F
(1989) demonstrate that the chain rule of RE doesnâ€™t apply, in that the forwa
of expectations over short horizons do no: equal those for the equivalent lon
An eclectic view of the evidence on exchange rate behaviour therefore
numberrof anomalies relative to the central paradigm of a model based o
fundamentals in which ait agents use RE. Models of chaos suggest that appar
nonlinearities rather than in applying RE to such models (which analyticall
very difficult because the mathematical expectations operator cannot evaluate
such as E ( x , / y , ) which appear in the nonlinear systems).
The spirit in which we approach models of chaos is one where we hope the
to provide alternative insights into the somewhat anomalous behaviour of th
rate (and other asset prices) rather than provide a complete theory of such m
This section examines the contribution that chaos theory can make in acc
the empirical results discussed earlier, which seem to indicate a failure of fu
in explaining movements in the spot rate and for an apparent failure of RE
efficiency. The work of De Grauwe et a1 (1993) is used extensively to sh
specific illustrative chaotic model can provide a starting point, at least, into t
anomalous behaviour of the FOREX market. This is done using the stickyp
tary model (SPMM) to determine the equilibrium exchange rate as viewed b
money (SM) and is combined with the model of heterogeneous expectation
traders (NT) and SM presented in Section 8.3 which provides further nonli
the model. The model gives rise (under certain parameter values) to chaotic
of the exchange rate. It is then possible to show that the simulated exchange
from this chaos model:
approximates a random walk
0
yields a regression in which the forward premium is a biased predictor o
0
change in the exchange rate
yields a regression of the exchange rate on fundamentals (i.e. money suppl
0
the economic fundamentals provide a poor predictor of future movem
exchange rate.
All of the above empirical results are observed in the real world data and
chaotic model is at least capable of mimicking the behaviour of real world
FOREX market. The final part of this section briefly outlines the results of som
chaos and for the presence of nonlinearity in relationships in the FOREX ma
StickyPrice Monetary Model
We have already discussed this model so we can be brief. In the long run th
rate is governed by PPP
s; = p;/pf"
where P: is the domestic steady state price level. Changes in the real exchang
to changes in real (excess) demand (i.e. net trade) and hence in the rate of in
The parameter k measures the speed of adjustment of prices to excess dem
full employment is assumed, equilibrium is realised through price changes. Th
In this model (unlike that in Section 8.3) the exchange rate has a feedback e
domestic price level via equation (13.37). The steady state is defined as Pf =
Y, = 1 and E,S,+l = S,. Hence from the UIP condition ri = 0 and from (13.38
The solution for the model at time t is found by substitution for P, from (13.37
+
and then solving for (1 rt) and substitution in the UIP condition (13.39) to
where
x ,  M t y ta p  â€˜ l / â€˜ l + k â€
 t1
and 81, 8 are functions of the other structural parameters (and 0 < 82 < 1)
2
spot rate is determined by the fundamentals that comprise X,. The term E,(
marketâ€™s expectation and provides an important nonlinearity in the model. T
expectation is determined by a weighted average of the behaviour of NT an
Section 8.3).
E ( S t + l / S r  l ) = f ( S r  1 9 ˜ r  2 ,. . .)mr(S:,/Srl)(lmf)a
where the weight given to noise traders is
Equations (13.42), (13.43) and (13.40) yield the following nonlinear equat
exchange rate
ñòð. 33 