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S , = (Xf)™1 (Sf-1)™1 (St-2)Q™ ( ˜ 1 - 3 ) ” (Sr-4 144

The above representation assumes an exogenous money supply but we can al
model under the assumption of interest rate smoothing

where !P(> 0) measures the intensity of interest rate smoothing. The simula
the exchange rate under interest rate smoothing for a given chaotic solutio
in Figure 13.1 and has the general random pattern that we associate with
exchange rate data. De Grauwe et a1 then take this simulated data and test fo
walk (strictly, a unit root).
St = aSt-1 Et

and find that they cannot reject (x = 1 (for a wide variety of parameters of
which result in chaotic solution). Hence a pure deterministic ˜fundamentals
mimic a stochastic process with a unit root.
Figure 13.1 Chaotic Exchange Rate Model with Money. Source: De Grauwe et a1 (1
duced by permisssion of Blackwell Publishers

If covered interest parity holds, the forward premium ( F I S ) , is equal to
differential (1 + rr)/(1 + r:). De Grauwe et a1 use the simulated values of
differential as a measure of ( F I S ) , and regress the latter on the simulated
( S ,+ 1 /st 1:
(S,+l/s,>= HFIS),

They find b < 0 which is also the case with actual real world data. Hence i
model the forward premium is a biased forecast of the change in the spot rate
we have risk neutrality at the level of the market (i.e. UIP holds). The above
resolved when we recognise the heterogeneity of expectations of the NT and SM
domestic interest rates rt leads to an immediate appreciation of the current spot
domestic currency (i.e. S, falls as in the Dornbusch model) and via covered int
a rise in ( F I S ) , . However, if the market is dominated by NT they will extr
current appreciation which tends to lead to a further appreciation in the domest
next period (i.e. E,Sr+1 - S, falls). The domination by extrapolative NT, howe
that on average a rise in ( F I S ) , is accompanied by a fall in E,S,+1 - S,. If there
SM in the market then UIP indicates that the spot rate would depreciate (i.e. E
increases) in future periods after a rise in rf as in the Dornbusch model.
De Grauwe et a1 also simulate the model when stochastic shocks are allow
ence the money supply (which is now assumed to be exogenous). The mone
assumed to follow a random walk and the error term therefore represents ˜ne
long time period the simulated data for S, broadly moves with that for the mo
as our fundamentals model would indicate and a simple (OLS cointegrating)
confirms this:
S, = 0.02 0.09 M ,
(0.5) (32.8)
In the simulated data the variability in S, is much greater than that for Mr. Th
data is then split into several subperiods of 50 observations each and the
S , = (x + BM, is run. De Grauwe et a1 find that the parameters a and B are high
data using 50 data points forecasts better than either the random walk or th
model. Thus, data from a chaotic system can be modelled and may yield
forecasts over short horizons. These results provide a prima facie case for th
success of error correction models over static models or purely AR models
world exhibits chaotic behaviour. The static structural regression S, = a B
the long-run fundamentals and the linear AR(3) model mimics the non-linea
Of course, the (linear) error correction model is not the correct representation
non-linear chaotic model but it may provide a reasonably useful approximat
finite set of data.
Chaotic models applied to economic phenomena are in their infancy so
expect definitive results at present. However, they do suggest that non-li
economic relationships might be important and the latter may yield chaotic
In addition one can always add stochastic shocks to any non-linear system
tend to increase the noise in the system. Hence although much of economic
been founded on linear models or linear approximations to non-linear mode
tractability and closed form solutions, it may be that such approximation
important non-linearities in behavioural responses (see Pesaran and Potter (1
A key question not yet tackled is whether actual data on exchange rates exh
behaviour. The obvious difficulty in discriminating between a chaotic determini
and a purely linear stochastic process has already been noted. There are s
available to detect chaotic behaviour but they require large amounts of data (i.e
data points) to yield reasonably unambiguous and clear results. De Grauwe et a
several tests on daily exchange rate data but find only weak evidence of chaoti
for the yen/dollar, pound sterling/dollar daily exchange rate data and no evi
for chaotic behaviour in the DM/dollar exchange rate over the 1972-1990 p
may be due to insufficient data or because the presence of any stochastic ˜n
the detection of chaos. They then test for the presence of non-linearities in th
rate data. Using two complementary test statistics (Brock et al, 1987 and Hi
they find that for six major bilateral rates, they could not reject the existence o
structures for any of the bilateral rates using daily and weekly returns (i.e. the
change in the exchange rate) and for monthly data they only reject non-line
case, namely the pound sterling/yen rate. Of course, non-linearity is necessary
behaviour but it is not sufficient: the presence or otherwise of chaotic behavio
on the precise parameterisation of the non-linear relationship. Also the above
tell us the precise form of the non-linearity in the dynamics of the exchange
that some form or other of non-linearity appears to be present. We are still l
somewhat Herculean task of specifying and estimating a non-linear structur
the exchange rate.

It is important that the reader is aware of the attempts that have been made
movements in spot exchange rates in terms of economic fundamentals, not le
performance. At present it would appear to be the case that formal tests of
models lead one to reject them. Over short horizons, say up to about one yea
fundamentals generally do not help predict changes in the spot rate. Over long
of four years fundamentals do provide some predictive power for some curren
1995). The latter is consistent with the view that purchasing power parity and t
for) money-income nexus hold in the long run (i.e. the relevant variables are co
However, on balance it must be recognised that there exists a great deal of
regarding the underlying determinants of the spot exchange rate in industrialise
with moderate inflation. The concepts and ideas which underlie these models
UIP relative money supplies) do still play a role in guiding policy maker
because they have little else to go on other than their ˜hunch™ about the
interest rate-exchange rate nexus to apply in particular circumstances. The
firm policy implications which arise because of the statistical inadequacy o
models has resulted in policy makers trying to mitigate the severity of wide sw
real exchange rate by coordinated Central Bank intervention, a move toward
zones and even the proposal to adopt a common currency.
The lack of success of ˜pure fundamentals™ models in explaining movements
exchange rate has recently led researchers to consider non-linear models that
in chaotic behaviour. The underlying theory in these models usually involves b
traders and some form of noise-trader behaviour. They are not models that
agents being rational and maximising some well-defined objective function. N
their ad-hoc assumptions are usually plausible. The main conclusions to emerg
literature as applied to exchange rates are:

Deterministic non-linear models are capable of generating apparently
random and irregular time series which broadly resemble those in real w
The addition of ˜news™ or random ˜shocks™ provides additional random im
The empirical evidence on exchange rates is ambivalent on the presence
(deterministic) chaos. However, it does quite strongly indicate the presen
linearities in the data generation process.
The challenge now is to produce coherent theoretical models that are
generating chaotic or behavioural non-linear models that are subject to rand
Non-linear structural (fundamental) models need not necessarily rule out
expectations. However, models involving the interaction of NT and SM,
not fully (Muth) rational, also seem worthy of further analysis.
Non-linear theoretical models will need to be tested against the data an
provide further econometric challenges.

The problem with chaotic models and indeed non-linear stochastic models is
small changes in parameters values can lead to radically different behaviou
casts for the exchange rate. Given that econometric parameter estimates from
models are frequently subject to some uncertainty, the range of possible fore
such models becomes potentially quite large. If such models are not firmly
Macroeconomics texts such as Cuthbertson and Taylor (1987) and Burda a
(1993) provide an overview of theories of the determination of spot exchan
an intermediate level. Other intermediate texts which deal exclusively wit
rates are Copeland (1994) who also has a useful chapter on chaos, and
(1988) who provides copious references to the empirical literature. MacDonald
(1992) and Taylor (1995) in their survey articles concentrate on what might
described as macroeconomic models of the exchange rate. Froot and Thaler (1
the anomalies literature in the FOREX market while De Grauwe et a1 (199
accessible account of chaos theory applied to the FOREX market.

Tests of the EMH using the VA

Tests of the EMH in the FOREX market and for stocks and bonds outline
chapters have in part been concerned with the informational efficiency assumpt
by rational expectations. Empirical work centres on whether information at time
can be used to predict returns and hence enable investors to earn abnormal
variance bounds approach tests whether asset prices equal their fundamental
anomalies literature frequently seeks to test for the existence of profitable tr
in the market. All of these tests are, of course, conditional on a specific econo
(or ad-hoc hypothesis) concerning the determination of equilibrium expected
This part of the book examines some relatively sophisticated tests of the EM
to stocks, bonds and the FOREX market which have recently appeared in th
The generic term for these tests is the ˜VAR methodology™. Some readers will
aware that tests of the informational efficiency requirement of the EMH resul
tions on the parameters of the model under investigation (see Chapter 5). Th
this instance usually consists of a hypothesis about equilibrium returns plus
forecasting equation (or equations) which is used to mimic the predictions fo
servable rational expectations of agents. Early tests of these cross-equation
were undertaken by estimating the model both with and without the restrictio
and then seeing how far the ˜fit™ of the model deteriorated when the rest
imposed. A likelihood ratio test was often invoked to provide the actual metric
statistic. The VAR methodology tests these same RE restrictions but, as we sha
only estimate the unrestricted model and this is usually computationally much
In testing the EMH on asset prices in previous chapters we have used varia
inequalities. In the VAR methodology these variance inequalities are replaced
equalities. For example, consider the expectations hypothesis EH of the term
Using the VAR methodology we can obtain a best predictor of future chang
term interest rates EtAr,+,: call this prediction S . Under the EH RE the
Si should mimic movements in the actual spread between long and short rates
the variance of S, should equal the variance of the best forecast Si. Henc
methodology provides two ˜metrics™ for evaluating the PEH RE, first a se
equation) restrictions on the estimated parameters of the VAR prediction equ
asset price levels.
So far we have concentrated on the econometrics of the tests using the VA
ology. However, it is important to note that the (cross-equation) parameter
referred to above do have an economic interpretation. Only if these restriction
the case that investors make zero abnormal profits and that the (RE) forecas
independent of the information set used by agents. The latter conditions are o
the heart of the EMH and informational efficiency.
Some readers may find the material in Chapter 14 more difficult than tha
chapters. However, it is my contention that the material is not analytica
although the algebra sometimes seems a little voluminous and on first read
be difficult to see the wood for the trees. The latter is in part due to a desi
the reader from simple specific cases to the more complex general case which
the burgeoning literature in this area. Your perseverance will have a high pay
understanding the literature in this area. To aid the exposition a simplified o
the main analytic issues is presented first.

To throw some light on where we are headed in relation to tests already d
earlier chapters consider, by way of example, our much loved RVF for stoc
and the expectations hypothesis for the long rate Rr (using spot yields):

Pr = v, y = 1/(1+ k )
i= 1


The ˜fundamental value™ V, denotes the DPV of expected future dividends and
rate y is assumed constant (0 .c y < 1). TV is the expected terminal value of
rolled-over one-period investment in short bonds. Under the EMH the actual
equals V, and the long rate equals T V I , otherwise profitable (risky) trades a
and abnormal profits can be made.
The regression-based tests and the variance bounds tests of the EMH
applying the RE assumption:

to (1) and (2), respectively, where qr+l and vr+1 represent ˜news™ or ˜surprises™
equations imply that stock prices only change because of the arrival of new
between t and t 1 hence stock returns (i.e. basically the change in stock
unforecastable using information available at time t or earlier (Q,). Similarly
(2) and (4) imply that the excess return from investing in the long bond ra
into the RHS of (1) so we have
+ 0,
P; = P ,
where P: is the perfect foresight price calculated using ex-post actual valu
Since by RE, P, is uncorrelated with w,, yet var(w,) must be greater than zer
the usual variance bounds inequality. A similar argument applies to the ter
equation (2). Hence the variance bounds tests do not seek to provide explicit
schemes for the unobservable expectations E,D,+i and Etr,+i but merely assum
are unbiased and that forecast errors are independent of Q, . (This is the ˜errors i
approach which will be familiar to econometricians.)
In contrast, the methodology in Chapter 14 seeks to provide explicit
equations for D,+; and rt+i based on regressions using a limited informat
These forecasting equations we may term weakly rational since they do not
use the full information set R, as used by agents. However, given explicit
dividends and assuming we know y then we can calculate the RHS of (1) a
an explicit forecast for V, which we denote Vi. For ease of exposition it is u
point to assume the econometrician has discovered the ˜true™ forecasting mod
used by agents, namely an AR(1) process:

By the chain rule of forecasting:
E,D,+j = aiD,
and the best forecast of the DPV of future divideilds using the true informatio

Knowing y and having an estimate of a from the regression (6) a time series
be constructed. (As we shall see Pi is referred to as the ˜theoretical price™: it
estimate of the DPV of ˜fundamentals™ given by our theoretical valuation mo
RVF and (6) are ˜true™ then we expect (i) P , and Pi to move together over
be highly correlated, (ii) the variance ratio, defined as:
VR = var(P,)/ var(P;)
to equal unity and (iii) in the regression
+ PlP: + 4
P, = P O
we expect PO = 0, = 1. By positing the ˜true™ expectations scheme for D
mating this relationship, we have been able to move from a variance bounds in
a relationship between P , and P: based on a variance equality™. The relationsh
P , and P: allows the three ˜metrics™ in (i) to (iii) to be used to assess the val
EMH based on the RVF plus RE.
dends and (6) will be an approximation. In this case we do not expect P, t
forecast P: exactly. The reason is that the equation chosen by the econom
forecast Dr+i may be based on a limited information set ( A l c S2,) and th
econometrician™s forecasting equation will not equal the true (rational) expect
cast, as formulated by investors. Clearly the closer (6) is to the ˜true™ equatio
we expect the conditions (i)-(iii) to hold˜™). Also, even if the econometricia
form of the true model to forecast Dt+j his estimates would be subject to sam
and hence the conditions (i)-(iii) would hold, for any given sample, only with
statistical confidence limits.
It was stated above that conditions (i)-(iii) will hold even with a limited
set. In fact, this is only true if the forecasting equation for D,+j depends on P
term structure model if forecasts of r,+j depend on R,. The key element in ob
results (i)-(iii) with a limited information set is that the LHS variable in the
efficient markets relationship (e.g. P, or R,) is used in the forecasting equat
RHS variables,(i.e. D,+j or r f + j ) . This then constitutes a VAR system wh
analysed in much of the rest of this section of the book.

From AR to VAR
Let us return to our story of trying to predict r,+i and assume that r,+l de
and R,:
+ +
B2Rr &,+l
Tr+l = Blrr

Having estimated (11)™ we now wish to use it to forecast E,r,+2 so that we ca
this value in the RHS of (2) in order to calculate TV,. see that

The values of (r,,R,) are known at time t; however we do not have a value
and hence we cannot as yet obtain an explicit forecast for Errf+2. What w
forecasting equation for R,+l and so assume (somewhat arbitrarily) this is o
general form as that for r,+l in (ll), that is:

Now, having estimated (13) we can obtain an expression for E,R,+1 in terms
known at time t, namely:
E&+1 = W r *2R,
Equations (11) and (14) taken together are known as a vector autoregression
order 1) and they allow one to calculate all future values of E,rr+i to input
econometrician uses only a limited information set At(C $2,). The reason is
depends on R, (as well as r,). Hence all future values of r,+j depend on R, (
is TV;= f l ( / 3 , Q)R, f 2 ( / 3 , Q)r, where f l and f 2 depend on the estimated
of the VAR. The expectations hypothesis then implies R, = TV; which can
if f l ( P , Q) = 1 (and f 2 @ , Q) = 0). But if f l = 1 (and f 2 = 0) then TV;
naturally must move one-for-one with actual R,. Hence even with a limited info
we expect the correlation between R, and TV;to equal unity and V R = var(R,
to also equal unity. This basic insight is elaborated below.
Turning briefly to the RVF for stock prices, note that if we allow both
discount factor yt = 1/(1 k , ) to vary over time then a VAR in D,and k, is
forecast all future values of E,D,+, and E,k,+; so that we can calculate our fo
the RHS of (1). There is an additional problem, namely the RHS of (1) is n
k, and D,. As we shall see in Chapter 16, linearisation of the RVF provides a
this technical problem.
There is another way that the VAR methodology can be used to test the EM
involves so-called cross-equation restrictions between the parameters of each
equations (11) and (14), respectively. However, a brief explanation of these
possible here and they are discussed below.
If the reader thought at the end of the last chapter that he had probably a
given a near exhaustive (or exhausting!) set of tests of the various versions o
then it must be apparent from the above that he is sadly mistaken. These ˜new™
on the VAR methodology have some advantages over the variance bounds tes
based on the predictability of returns. But the VAR methodology also has som
drawbacks. It does not explicitly deal with time varying risk premia (this is
of Chapter 17). It relies on explicit (VAR) equations to generate expectations
linear with constant parameters and which are assumed to provide a good app
to the forecasts actually used by investors. In contrast, all the variance b
require is the somewhat weaker assumption that whatever forecasting schem
forecast errors are independent of R,. Hence to implement the variance bound
econometricianhesearcher does not have to know the explicit forecasting mode
used by investors.
We discuss the VAR methodology first with respect to the bond market in
since concepts and algebra are simpler than for FOREX and stock markets
discussed in the following chapters. At the outset the reader should note tha
the above examples of the VAR methodology have been conducted in terms o
of the stock price and dividends and the levels of the long rate and short ra
EH, in actual empirical work, transformations of these variables are used in o
and ensure stationarity of the variables. In the case of bonds we have alread
the EH equation (2) can be rewritten in terms of the long-short spread S, =
changes in the short rates Ar,+,. For stocks the RVF is non-linear in dividen
time varying discount factor and therefore the transformation is a little mor
involving the (logarithm of the) dividend price ratio and the (logarithm of the
dividends. These issues are dealt with at the appropriate point in each chapter
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The Term Structure and
Y Market
C n
This chapter outlines a series of procedures used in testing the EMH in the b
using the VAR methodology. For the bond market the VAR algebra is some
than for the stock market but some readers may still find the material difficu
be made of the simplest cases to illustrate points of interest: extending the ana
general case is usually straightforward but it involves even more tedious an
algebra. In the rest of the chapter we discuss:
How cross-equation parameter restrictions arise from the EMH and RE.

The relationship between the likelihood ratio test and the Wald test of

using the VAR methodology.
How the parameter restrictions ensure that (i) investors do not systemati

abnormal profits and (ii) that investors (RE) forecast errors are independe
mation at time t used by agents in making the forecasts.
Illustrative empirical results at the short and long end of the maturity sp

government bonds, using the VAR methodology are presented.
Beginning with the simplest model of the EH of the term structure, the forecast
for the short rate is assumed to be univariate. The analysis is repeated fo
autoregressive system (VAR) in terms of the spread S, = R, - rf and the chan
rates Ar, . It is then shown how the cross-equation parameter restrictions can be
in matrix notation, so that any VAR equation can be used to provide forec
theoretical spread Si and hence allow the comparison between Si and the ac
S f using variance ratios and correlation coefficients. Finally, a brief survey o
work in this area is examined.

The pure expectations hypothesis (PEH) implies that the two-period interes
given by:
+ rf+@
R, = 0 - f

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