OCT 201S87
U4
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Findings of Forward Discount Bias Interpreted
in Light of Exchange Rate Survey Data*
by
Kenneth
A.
Froot
and
Jeffrey A. Frankel
Working Paper #1906-87
June 29, 1987
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Findings of Forward Discount Bias Interpreted
in Light of Exchange Rate Survey Data*
by
Kenneth A. Froot
and
Jeffrey A. Frankel
Working Paper #1906-87
June 29, 1987
Findings of Forward Discount Bias Interpreted
in
Light of Exchange Rate Survey Data
*
Kenneth A. Froot
Sloan School of Management,
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Jeffrey A. FVankel
Department
of
Economics
University of California, Berkeley
Berkeley, California 94720
June
29, 1987
Abstract
Svirvey data on exchange rate expectations are used to divide the forward discount into
expected depreciation and a
risk
whether the forward discount
is
We
premium.
Our
starting point
is
the
an unbiased predictor of future changes
common
in the
use the surveys to decompose the bias into a portion attributable to the risk
and a portion attributable
to systematic prediction errors.
test of
spot rate.
premium
The survey data suggest that
our findings of both unconditional and conditional bias are overwhelmingly due to sysRegressions of future changes in the spot rate against the
tematic expectational errors.
forward discount do not yield insights into the sign,
as
is
\isually
thought.
find support for
in that
expected depreciation.
in
is
it
We
test directly the
The "random-walk" view
premium. Investors would do better
This
is
with forward market data, but
the
now
if
is
reflect,
one for one. changes
that exjiected dejneriation
is
even significantly more variable than the
xero
risk
they always reduced fractionally the magnitude of
same
it
premium
hypothesis of perfect substitutability, and
changes in the forward discount
thus rejected; expected depreciation
expected depreciation.
size or variability of the risk
result that Bilsoii nu<]
cannot be attributed to a
many
risk
other? have found
prenuum.
an extensively revised version of NBER Working Taper No. 19C3. We wouM like to thank Oreg Conunr, Alherio
JoeMattey and many other partiripants at various seminars for helpful comments; Barbara Bnier,
John Calverley, Louise Cordova, Kathryn Dominguez, Laura Knoy, Stephen Marris, ami Thil "Soung for help in obtaining data,
the National Scienre Foundation (under grant no. SES-82183nO), the Institute for Busines.; and Eronomir Researrh at U. C.
Berkeley, and the Alfred P. Sloan Foundation for researrh support.
*
This
is
CJiovannini, Robert. Hodrirk,
^^^ 2 1,987
Findings of Forward Discount Bias Interpreted
in Light of
Exchange Rate Survey Data
Kenneth A. Front
Sloan School of Management,
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Jeffrey A. Frankel
Department
Economics
of
University of California, Berkeley
Berkeley, California 94720
Introduction
1.
The forward exchange
rate
is
the
first
to
surely the jack-of-all-trades of international financial economics.
a variable representing investor exi)ectations of future spot rates, the
Whenever researchers need
forward rate
is
come
mind.
to
On
the other hand, the forward rate
in efforts to extract the empirically elusive foreign
These two conflicting
discoimt
is
roles are
most evident
exchange
an unbiased predictor of the future change
They tend
to disagree, however,
it,
frequently used
jn-emium.
in the large literature testing
studies that test the unbiascdness hypothesis reject
bias.
risk
is
in the sjiot
exchange
whether the forward
rate.'
Most
of the
and they generally agree on the direction of
about whether the bias
is
evidence of a risk premium or of
a violation of rational expectations. For example, studies by Longwortli (1981) and Bilson (1981a)
assume that investors are
in excess of the
On
risk neutral, so that the systematic
forward discoimt
is
intei-jireted as
component
time-varying risk premium that separates the forward discount
Fama
same systematic component
fro!ii
'Rpfercncps inrlude
Tnon
premium
is
exjiertcd depreciation.
prrmium,
liut also as
evidence
greater than the variance of exjiected depreciation. Bilson
(1979), Levirh (1979), Bilson (1981a), LonRworili (1981), Hsieh (1984), Fams (1984), Huang
For a rerpnf siirvry of flic literature and additional citations see
and Hodrirk and Srivastava (198C).
Boothe and Longworth (1986).
(1984), Park (1984)
to a
(1984) and Hodrick and Srivastava (1980) have recently gone a step
further, interpreting the bias not only as evidence of a non-7,ero risk
that the variance of the risk
exchange rate changes
evidence of a failure of rational expectations.
the other hand, Hsieh (1984) and most others attribute the
Investigations by
of
1
(1985) interprets this view as a
expectations:
new
"empirical paradip:m"
that roine? rinse to assuming static
changes in expected depreciation are small or
and changes
7,ero,
in the
forward
discount instead reflect predominantly changes in the risk jneminm. Often cited in support of this
view
the work of Meese and RogofT (1983),
is
who
find that a randc^n
walk model consistently
forecasts future spot rates better than alternative models, including the forward rate.
But one cannot address without additional information the hasir
expectational errors or the risk
whether
of the forward discount (or
risk
premium
is
premium
it
surveys: one conducted by
in
whether systematic
are alone responsible for the rejieatedly l)iased forecasts
is
some combination
more variable than expected depreciation.
exchange rate expectations
issues of
of the two), let alone
paper we use survey data on
In this
The data come from
an attempt to help resolve these issues
American Express Banking Corjjoration
whether the
of
London
irregularly
three
between
1976 and 1985; another conducted by the Economist'?. Finnnrinl Rrpnrf. also from London, at
Money Market
regular six-week intervals since 1981; and a third conducted by
Redwood
City, California, every
two weeks beginning
in .lanuary
October 1984. Frankel and Froot (1985, 1987) discuss the data and use
of
how
discoimt into
light
its
two components
-
paper we use the stuveys
expected depreciation and the
on the proper interpretation of the large literature that
forward
is
risk
it
to estimate
to divide the
premium
in
models
forward
order to shed
finds bias in the predictions of the
to be skeptical of the accuracy of the survey data, to allow for the possil^ility that they
measure true investor expectations
of ways.
of
rate.
We want
that
In this
(MMS)
1983 and every week beginning
in
investors form their expectations.^
.Services
We
will follow
witli error.
Such measurement error could
the existing literature in talking as
arise in a
number
there exists a single expectation
if
homogeneously held by investors, which we measure by the m('(han survey response. But,
in fact, different
true expectation,
survey respondents report different answers, suggesting that
it
is
expected depreciation
at precisely the
measured with
series
is
same moment
error.
Another possible source
may
that the expected future spot rate
as the
contemporaneous spot
rate
is
if
there
is
a single
of measurcitient error in our
not be recorded by the survey
recor'led.'
Our econometric
tests
'Anothpr pappr that, uses the MMS stincys i« Domingupz (1986).
"To mpasure the rontpmporaiicoiis spot rato. we rxperimontPtl with rliffrrcnt approximations to the prrcisp siincy and
forerast dato': of thf AmPX suney, whirh was r ondiirtpd hy mail ovpr a ppriod of up to a mnntli. Wp used thp avpragp of thp 30
days during thp suney and also thp midpoint of tlip snrvpy ppriod to ronstnirt rpforpnro sots. Doth gavp vppy similar rpsidts,
so that only rpsults from thp formpr samplp wprp roporipd. In thp rasp of tlip Ernnomi.ot and MMS snrvpys, which constitntp
allow for mcasTiremeiit error in the data, provided the error
random. There
is
an analogy with
is
the rational expeetations approach which uses pt pnut exchange rate changes rather than survey
data, and assumes that the error in measuring tnie expected depreciation, usually attributed to
"news,"
is
random. One of our findings below
is
that the expectational errors contained in ex
sample exchange rate changes are not uncorrelated with the forwaid fhscount. This,
be consistent with a failure of investor rationality, but
it
is
also consistent
pnitt
of course, could
with 'peso problems,"
A
learning on the part of investors, or the presence of nonstationarities in the sample.
second
advantage to our approach to measuring expectations: aside from fhe question of bias, the absolute
magnitude
of our
measurement
errors
is
much
less that that of the exjiectational errors in
the ex
post changes.
The paper
is
organized as follows.
We
forward discoimt bias.
4 in turn test
we j)erform the standard regression
test of
then use the surveys to sejiarate the bias into a comjionent attributable
to systematic expectational errors
and
In Section 2.
and a component attributable
whether the component attributable to the
Sections 3
to the risk ]iremium.
i)remium and the component
risk
attributable to systematic expectational errors are statistically significant. Section 5 concludes.
2.
The RegrGSslon
of
The most popular
Forward Discount Bias
test of
forward market unbiasechiess
is
a regression of
tlie
future change in
the spot rate on the forward discount:
A.v+A-
where Afi^i,
is
= « +
and fd^
is
the forward rate minus the log of the spot rate).
=
+
in the null
hypothesis as well.)
the forward rate plus a purely
random
(1)
r?f+<.
the percentage depreciation of the currency
of foreign exchange) over k periods
include a
/?/rif
(tlie
change
in the log of the spot price
the current A--i)erind forward discount (the log of
The
null hy])othesis
is
that
/?
=
1.
(Some authors
In other words, the realized sjiot rate
error term, ni+k
A
is
equal to
second but equivalent sjiecification
is
a
regression of the forward rate prediction error on the forward discount:
/< most of our Hafa
set, this issue
A.^,+it
= ni+
hanlly arises to boEin wifh, as
<liry
ftifd)
+
r7,+^
werr romlnrfoil
l>y
(2)
(rippliniir
nn
,t
knnwii Hay.
= —a and
where aj
variable
is
Most
tlian one.
=
/?i
1
—
The
/?.
rrnll
hypothesis
nnw
is
that
rti
=
fli
=
D:
the left-hand side
purely random.
tests of equation (1)
have rejeeted the null hypothesis, fiudiuR
Often the estimate of
/?
is
/?
to
be significantly
less
Authors disagree, however, on the
close to zero or nep;ative.^
reason for this finding of bias. Longworth (1981) and Bilson (1981a). for example, assume that there
is
no
risk
so that the forward discoinit accurately measures investors" exjiectations; they
premium,
therefore interpret the bias as a rejection of the rational expeetaticms hypothesis. Bilson describes
the finding of
investors
/?
closer to 7,ero than to one as a finding of "excessive speculation,"
would do better
In the special case of
to choose ^f'l^ic
not
=
=
/?
<
magnitude
the excliange rate follows a
of their exi)ected
landom walk, and
Fama
(1984) and Hodrick and Srivastava (198G) rlescribe the finding that
premium accounts
for
more
discount than does expected depreciation. In the special case of
expectations implies that
2.1.
a// of
A.i^^j.
=
of the variation in the forward
/?
=
f),
the assumption of rational
Bilson (1985) points out, the risk
0, so that, as
premium would
the variation in the forward discoimt.
Standard Results Reproduced
We
begin by reproducing the standard
OLS
regre.ssion
periods that correspond precisely to those that we
these results, in part, to
show that the
be attributed to small sample
In these
one
against the dollar.
in the
survey data.^
on sample
We
report
the survey data below caimot
prepared to attribute the usual finding of
MMS
jiool across currencies
survey are the
The other two surveys include
"Tlip fincline fhat forward ratc^ arp poor prciIirt,or« of
In thrir study of the Pxprrtations liypothpsis of thf
ronrliide that rhaiiRPs in the
when we use
(1)
si7,e.
and subsequent regressions, we
(The four currencies
also
is
results for equation
will l»e usinji; fnr the
results obtained
size unless
forward discount bias to small sample
si/.e.
investors would do better
the other hand, Hsieh (1984) and most others assume that investors did
1/2 as a finding that the risk
account for
exchange rate changes.
prediction errors in the sample; they interpret the bias as evidence of a time-
varying risk premium.
ft
0,
On
0.
make systematic
to reduce the absolute
meaning that
fuvler to inaximi7,e
sample
mark, Swiss franc and yen. each
these four exchange rates and the French franc as
fiitiiri>
spot ratre
term stnirturp,
premium paid on longer-term
jiouufl,
i!i
Iiills
fnr
is
not limitpil tn
example,
over short-term
Sliiller.
liills
tlip
foreign cxcianEi- market
('ampl>ell
ami Srboenholt? (1983)
are useless for jireilirting future
changes
iu short-term interest rates.
''DRl provided us with daily forward and spot exrhange rates. romp\ited as the average of the noon-time hid and ask rates.
wrll.)
Ap npual, we must allow
for
in addition to allowing for the
>
(it
1).
We
contemporaneous correlation
moving average
in
the
ermr terms across currencies,
error process indnced hy overlapping ol^servations
report standard errors that assume conditional lioiiioskedasticity, in this case be-
cause they were consistently larger than the estimated standard errors that allow for conditional
heteroskedasticity.
We
also at times jiool across different forecast horizons to maximi7,e the
of the tests, requiring correction for a third kind of correlation in the errors.
of this having
been done before, even
econometric issues
Table
1
is
standard forward
fliscr)unt regression.
are not aware
Each
of these
disctissed at greater length in the appendices.
presents the standard forward discount unbiasedness regressions (equation (1)) for our
sample periods.® Note that
in the
stacked, the standard errors
in
in the
We
power
fell
Economist and
in the
Amex
sets, in
which forecasts horizons were
aggregated regressions by 14
anrl 31 jtercent, resi)ectively,
data
comparison with regressions that used the shorter-term jiredictions alone.
Most
of the coefficients
fall
most
to reject imbiasedness:
into the range reported by previous studii's.
of the coefficients are significantly less
the coefficients are even significantly less than zero, a finding of
two
MMS
data
sets, the coefficients
observation by Gregory and
unstable.
At
The
Theie
than one
many
is
ample evidence
More than
half of
other authors as well, hi the
have unusually large absolute values, lending support to the
McCurdy
(1984) that the regression relation in ecjuation (1)
may be
F-tests also indicate that the imbiasedness hypothesis fails in most of the data sets.
this point,
this interpretation,
we could
it
interpret the results as reflecting systematic jirediction errors.
follows that agents would do better by placing
poraneous spot rate and
less
more weight on the contem-
weight on other factors in forming jueflictions of the future spot rate,
the view discussed by Bilson (1981a).
of a time-varying risk
Under
On
the other hand, we coiiM iiiter]uet the results as evidence
premium. Then the conclusions would be
tliat
changes
in
expected depre-
ciation are not correlated (or are negatively correlated) with changes in the forward rliscount and,
from equation
(3),
that the variance of the risk
premium
is
gicatn- than
tin-
variance of expected
depreciation.
^Regressions were estimated with dummies for earli rnuntry, wliirli we do not report to s,i\e spare. For the refressjons which
pool over different forecast horizons (marked Economist Data and Amex Data), eacli country was allowed its own constant
term
for everv forecast horizon.
Decomposition of the Forward Discount Bias Coefficient
2.2.
The survey data, howfver,
let
us go a step further with the results of Tabh'
allocate part of the deviation from the
and the presence
of rationality
equation (1)
hypothesis of
n\ill
The
premium.
of a risk
/?
=
Wc
1.
ran
now
to each of the alternatives: failure
1
jirohability limit of the coefficient
/?
in
is:
—'•''+*'^-'^_' ;-'-p =
^+>-'-^
:::i
(3)
var(/r/*
where
»7*i
market participants" expectational
is
j.
use the definition of the risk
a little algebra to write
As'j_^_^.
is
the market expectation.
We
premium
rr*
and
and
error,
ft
failure of rational expectations,
as equal to
=
1
M-
A.,^+,
(4)
(the null hypothesis)
minus a term arising from any
minus another term arising from the
ft=l-
ftre-
])remium:
risk
(5)
ftrr
where
^
cov(r?*-^.^,/fif)
'"
'''"
With the help
of the survey data,
var(/4)
-
var(M)
systematic prediction errors in the sample, and
more weakly,
The
if
the risk
results of the
compared
to
/?rp,
premium
is
ftrp
=
if
insjiection,
there
is
no
risk
ftr^
=
if
jnemium
there are no
(or,
somewhat
uncorrected with the forward discount).
decomposition are reported
often by
By
both terms are observable
more than an
orrler of
in
Table
First,
2a.
ft^r '^
very large in size
when
hi all of the regressions, the lions
magnitude,
share of the deviation from the null hyjiothesis consists of syst<-matic (-x])ectational errors.
example, in the
ftrp
in
—
£'cor?,omj,i< data,
0.08. Second, while
equation (5) that the
coefficient
ft
in
ftre 'f
our largest survey sample with
greater than zero in
effect of the
survey risk
the direction abnvc one.
share of the forward discoimt's bias.
all
cases,
premium
Is
r)25 (>bs(Mvations, ft^f
ft^r i?
to
positive values for
push the estimate
6
ftrr.
1.49
and
sometimes negative, implying
In these cases, the risk preniia
The
=
For
do not
of the
standard
ex])lain a jiositive
on the other hand, suggest the
possibility that investors tendrd to ovrrrrart to other infonnatinn, in the sense that respondents
mi^ht have improved their forecasting by
and
less
more weight cm the cont(Miii)oraneons spot
placing;
Third, to the extent
weight on the forward rate.
smveys
the
tliat
rate
are from different
sources and cover different periods of time, they provide independent information, rendering their
agreement on the
check
if
relative
importance and sign of the expectational ennis
the level of aggregation in Table 2a
in
every data
set.
interest,
whether we can formally
we would
reject the
risk
(1) are statistically significant;
premium,
i.e.
two hypotheses
2.3.
and
that the point estimates in
column
are
(a) that
that the jioint estimates in
the presence of the
tfi
(2) are statistically significant.
of Expected Depreciation vs. Variance of the Risk
We
test these
1,
/?
is
Premium
significantly less than 1/2.
It
is
on the basis of such estimates that Fama (1984) and Hndrick and Srivastava (198G) have
claimed that expected depreciation
is
less variable
than the exchange
Fama-Hodrick-Srivastava (FHS) interinetation of the results
var(A.t^^H.)
see
i.e.
know whether they
jiolar hyi>otheses:
(b) that they are attributable
Notice that for most of the sample periods in Table
To
to explain
in turn in the following sections.
The Variance
precisely
like to
two olivious
the results in Table 4 are attriliutable to expectational errors,
column
inemium appears
bias.
While the qualitative results above are of
statistically significant,
To
forceful.
Here the results are the same:
expectational errors are consistently large and positive, and the risk
no positive portion of the
more
hiding important rhversity across currencies. Table
is
reports the decomposition for eacli currency
21^
all tjie
how
to write the
<
risk
itreminm.
We
state the
as:
var(r7)/).
(6)
they arrive at this inequality, we use the definition of the risk pn^iiiium in equation (4)
FHS
projiosition as
var(A.tJ^j.)
<
var(r/)f)
+
var(/fff
)
-
2cov(/-/J'.
A.-';_^_^.).
or
coy(fdlAs'„,)<^-^^
-
(6').
Under the assiimpHon that the predirtion
error, n^^/,,
expectations assumption), the ref^ression coefficient
cov(A,'.'
/?,
is
unconelatrd with fd, (the usual rational
by etiuation
as piven
(3),
becomes
r,/rfh
var(/fif)
Thus a
finding of
<
/?
1/2 implies the variance inequalities in rcpiation
offered by recalling the special case
in rp'l,
and not
We
/?
=
D:
Added
(G).
intuition
is
the variation in fd, then consists entirely of variation
at all variation in ^i^'^k-
can use expectations as measured by the survey data to investigate the
without assuming there
no systematic component to the
is
jirediction errors.
FHS
claim directly,
Table 3 shows the
variance of expected changes in the spot rate, as measured by the surveys, and the variance of the
risk
premia, for each data set and broken
rate changes
(column
1)
down by
The magnitude
currency.
dwarfs that of the forward discount (column
2).^
of ej pntt
exchange
For example, the reported
variance of armualized spot rate changes of 2 percent represents a stanrlard rleviation of about 14
percent.
By comparison,
the variance of expected depreciation
amund
.25 percent, a
standard
the variance of the risk
premium
is
deviation of 5 percent.
The variance
(column
4),
and
expectations
Fama
risk
of expected depreciation
is
(A.t'^j,
is
comparable
in si/.e to
larger in 36 of the 40 samples calculated in Tabl(<
—
0)
do not appear to be supported by
in section 3.
We
We
survey data.
tlie
(1984) hypothesis that the variance of expected depreciation
premium
Thus "random walk"
3.
is
less
test formally the
than the variance of the
can see from Table 3 that both arc several times larger than the
variance of the forward discount. Thus the relative stability of the forward discount masks greater
variability in
its
two components, corroborating Fama's finding
tjiat tlir risk
premium
is
negatively
correlated with the expected change in the spot rate.
3.
A
Direct Test of Perfect Substitutability
hi the previous section
we
offered point estimates of the bias in the forward discount,
"Thi« empirirsl rpgnlarity has oftfn lipon noted; p.g., Muisa (1979).
hnwpvpr, liiaserl downward 1)y any mcasiirPinPrt error tliat might he present in the surveys.
purely random, then the rovarianre of expected depreciation and the risk premium may he written as cov (A»J^j,
"Tlii"! roiTolatinnis,
where A«',
and rff are the "true" vahies
measurement error component of the survey.
.
"The low variance
of the risk
in the forvvard rate errors, /rf*
of expected depreciation
premium reported
in
and of
tlie risk
premium,
respectively,
If
which
snrh error
rp,
)
and
is
-VBr(fr).
fr
is
the
Table 3 also has implications for the aliility of tests of serial correlation
See the NBER working paper
a time\-ar>inR risk premium
- At,^.k, <o detect evidence of
version of this paper (p. 10).
8
suRgested that more of the bias was dnc to a failure of rational ox]i('rtations than to a timo-varyiug
In thi? section
risk jireniinm.
In thr next section
wo
will
wr formally
prrminm.
tost for thr rxistrnrr nf the tinio-varyinf^ risk
formally test rational expectations.
Analogously to the standard regression equation equation, we legress our measure of expected
depreciation against the forward discount:
A.iiJ+1
The
=
+
Q2
null hypothesis that the correlation of the risk
implies
=
/92
1
By
column
results in
inspection, P2
(2) of
=
—
I
error
is
identically zero
is
given by
premium
A.iJ_(_j
=
fd,
.
error in the surveys.
the iinobservable market expected change in the
affect instead the ex post distribution of actual
Another way
of stating the null hypothesis
are perfect substitutes in investor "s portfolios.
forward discount
rates
i,
—
t,
i*f
— i]
.
The
/rf,
is
premium with
is
fmm
either: 0^2
We
That
<
1.
=
the forward discoimt
=
/?2
1
(8) also allows
The hypothesis
0.
zero
wo\dd imply that the
Equation
zero.
is
that the risk
should therefore inteipret the regression
sjiot rate.
using the survey data, the properties of the error term,
which
(8)
Table 2a are not statistically fhfferent
random measurement
as
£(
fr
Prp^ so that a fiiiding of
us to test the hypothesis of no constant risk
premium
+
/?2/f'f
is,
A.<'_^j
=
A.t^_^j.
Note also that
will
+
r^,
where
in a test of
A.fJ^j^ is
equation (8)
be invariant to any "peso problems,"
spot rate changes.
the proposition that domestic anfl foreign assets
Assuming
tliat
covered interest parity holds, the
equal to the differential between domestic and foreign nominal interest
null hypothesis then Ijecomes a
statement of uncovererl interest parity:
A.tJ_(_j.
=
hx other words, investors are so responsive to differences in exjiected rates of return as to
.
eliminate them.'"
We
can also use equation
premium
is
we found
to
(8) to test
formally the
FHS
hyjiotliesis that the variance of the risk
greater than the variance of expecterl flei)reciation.
be violated by point estimates
in Talile 3.
This
is
tjie
estimate of their difference.) The probability limit of the coefficient
^
cov(A^J+,,/r/f)
VBT(fd))
'"For tfsts of iiiirovprpd infprcsf parity similar to
Cumliy and Ohstfpld (1981).
sprtion 3, spp
tli'"
test?:
(G).
wliich
(Although random measurement error
the survey data would tend to overstate each of these variances individually,
''
inequality
it
in
does not affect the
/?2 is:
^ cov(A..?^,..K)
'
var(/rf})
of rotiflirioiial
liia* in
the forw
aril (li«rmiiit
that
we
roii'sidorpil in
where we have used the assumption that the measurement error
disromit fd^ (analogously to the derivation of equation (7)).
if /?2
<
FHS
1/2 does the
inequality (6') hold;
of expected depreciation exceeds that of the risk
Table 4 reports the
OLS
'^
all
(with the sole exception of the
is
,
greater than 1/2, the variance
premium.
some
resjierts tlie
Contrary to the hypothesis of a
of the estimates of
MMS
mirorrelatefl with the forward
follows from etjuation (9) that only
If
sigiiifif'^ntly
regressions of equation (8). In
in favor of perfect substitutability.
with the forward discount,
^2
if
t
/?2
risk
data i>rovide evidence
premium
that
is
correlated
are statistically inrlistinguishabie from one
three-month sample). In the Ernnnwixf and
Amex
data sets
which aggregate across time horizons, the estimates are 0.99 and DOC, resjiectively." Expectations
seem
move very
to
strongly with the forward rate.
With the exception
MMS
of the
data, the
coefficients are estimated with surprising precision.
In terms of our decomposition of the forward disco\nit bias coefficient.
values of
from
/?rp
7,ero.
entirely
in
Thus the
by the
the survey risk
premium
column
risk
2 of
Table 2a are statistically far from one
rejection of unbiasedness found in
premium
premium has
at
any reasonable
aiul are not significantly different
jirevions section cannot
level of confidence.
we cannot
substantial magnitude,
Indeed,
reject the
be explained
in spite of
the fact that
hypothesis that the risk
explains no positive portion of the bias.
There
is
strong evidence of a constant term in the risk
statistically greater
than zero. Each of the F-tests reported
a level of significance that
of the risk
premium
is
less
than
0.1 percent.
in
premium however: ^2
Table G rejects the
and 2b should not be taken
to
Thus the
large
f>f
in the
the risk
r(^si>ons(^s iiirlude
forward rate
and
jiarity relation at
qualitatively small values of
imply that the survey
about the future spot rate beyond that contained
is
Charts 1-4 make apparent the high average
(as well as its lack of correlation with the usual nu-asure
the forward discoimt prediction errors).'^
Tallies 2a
th(^
Table 4 shows the
fl^p
level
premium,
rejiorted in
no information
'
''For ihp Economist six-month and (wrlvo-mnntli and (he Amrx wplvcmontli ilata <rt«, tlif rstimatos nfj(ffrom pqiiaticn
do noi exactly correspond to 1 - fi,p in TaMrs 2a and 2li. This is brcanso Tatilr 4 iiirliulps a frw siiriry olispnations for
which actual future spot rates have not yet heen realized, whereas these ohser\ations w ere left out of the decomposition in
Tables 2a and 2b for piirposes of comparability-. If we had used the smaller samples in Tabic 4, the regression coefficierts would
have been .92 and 1.03, for the Economist and Amex data sets, respectively.
"The degree to which the surveys qtialitati\ ely corroborate one another is strikiuR. For example, the risk premium in the
Economist data (Chart 1) is negative dtiring the entire sample, except for a short period from late 1984 until midI985. The
three-month sample (Chart 2) reports that the risk premium did not become jio'^ilivr until the last quarter of 1984, while
one. month data (Chart 3) shows the risk premium then remained positive until niidl980. That the surveys agree on the
nature and timing of major swings in the risk premium is some evidence that the particularities of each croup of respondents
(8)
MMS
MMS
do not influence the restilts.
"In Table 2 of the NBER working paper version of
this study,
10
we reported mean values
of the risk
i>remium
as measiu^ed
Table 4 also reports a
hypothesis that
t-test of the
/?2
=
In six ont
1/2
strongly reject the hypothesis that the variance of the tnie risk iiremiiiiii
to that of tnie expected depreciation;
that
=
/?2
1
we have rather var( A.i^_^j.) >
greater than or equal
var(r;),). Indeed, the finding
premium
is
negative (as
variance of the true risk
Under the
+
=
cov{As';^^..rp';)
(10).
f)
reject the hypothesis that the covariance of true ex])erted depreciation
Thus we cannot
(I is
Fama
premium
foimd), nor can
is
we
random measurement
extreme hypothesis that the
reject the
is
no time-varying
error,
premium and
risk
we can use the /?^s
If)
the survey data.
in
Table 4 are relatively high, suggesting that measurement error
For example, under this interpretation of the
small.
thr legression error
fnuii thr regressions to obtain
an estimate of the relative importance of the measurement error comjioiient
statistics in
/?^s,
same sample period
in
Table
1
(which uses
n
relatively
is
measurement ermr accounts
percent of the variability in expected dejireciation from the Ernnnm.iKt data.
of comparison, the 7?^ for the
and the
is 7,ero.
null hypothesis that there
equation (8)
The R^
nine rases the data
implies that:
var(r/.*)
true risk
is
f)f
about
for
For a standard
post exchange rate
changes as a noisy measure of expectations) implies that 84 percent of the variability in the measure
noise.
is
In Table 5
we
correct for the potential serial correlation jiroblem in the
data sets by employing a Three-Stage-Least-Squares estimator that allows
correlation
(SUR)
results reported in Table 5
all
it
show that
is
this correction
greater than that of the risk
alternative that the variance of the risk
by
tlip
5ur\cy data.
They were
different
from zero
premium
at
contemporaneous
is
consistent here
has the advantage of being asymptotically
is
efficient.
The
does not changi- the nature of the results;
but one of the coefficients remain close to one, and therr
expected depreciation
MMS
predictions by the forward rate and the surveys
because there are no overlapping observations
and
for
3SLS
as well as first order auto-regressive disturbances.*^'
are observed contemporaneously -
Ernnnmist and
is
premium
<
Icai
<>viden((' that the
(whiN'
th(>r<'
is
variance of
no evidence
for the
greater).
(he 99 permit level for almn^l
all <.iir\
f.\
•ionrres, nirrenries
and sample
periods.
'^In both cases, thp /?' statistics inrhidc the explanatory power of
'^Unfort.nnatply, the highly irrepilar sparing of
th*-
Amrx
tli'-
roust ant
tr-riiis
fnr rarli riirr<-nry
data sets did not permit an antorrgresii\
11
r
ami forTast horizon.
rorrertion in this rase.
Tests of Rational Expectations
4.
the previous srrtion we formally tested the hypothesis that there exists no time-varyinp; risk
111
premimn that could explain
test the
would do better
all
we formally
Test of Excessive Speculation
Perhaps the most powerful
to
hi this section
hypothesis that there exist systematic exjiectational errors that ran explain those findings.
A
4.1.
the findings of bias in the forward disrnniit
if
expectations
test of rational
they placed more or
is
one which asks whether investors
weight on the contemjioraneons spot rate as opposed
less
This
other variables in their information set.'®
test
is
performed by a regression of the
expectatioiial prediction error on expected depreciation:
-
AflJ^i
where the
had
we
in
null hypothesis
is
a
=
=
A.v+jt
and d
=
mind, which we already termed a
0.'^
+
a
This
+
(^As'i_^.^
is
(11)
v';'_^_^.
the equation that Bilson (1981a) and others
test of "excessive"
spemlation, with the difference that
are measuring expected depreciation by the survey data instead of
liy
the ambiguous forward
discount.
Our
tests are reported in
investors could on average
less
=
The
6.
findings consistently indicate that d
d
=
The
is
upheld.
0, reject at
0, so
that
hi other words, the excessive speculation
F-tests of the hypothesis that there are no systematic expectational errors,
the one percent level for
results in Table 8
survey expectations.
>
do better by giving more weight to the contemi^oraneous spot rate and
weight to other information they deem pertinent,
hypothesis
a
Table
Up
would appear
of the survey data sets.
all
to constitute a resounding rejection of rationality in the
until this point, our test statistics
have been robust to the presence of
'"FVankcl and Froot (198C) fesf whcthor the stirvpy rxprrtation' plart- (no littli- w oiglit nii ilir rniitomiinraiicou"! "spot ratp
miirli weigh* on sppfifir piercs nf information «iirh a<! thr lagRrd "spot rate, (he lonR-nui rqiiililiriiini cxrlianR'- rate, and
thp lagRfd cxpertpd spot rafp.
"To spp how ihf altcmafivr in oqiiation (11) is too murh or too liltlo Wright on all ^arialilfs in tho information «pt odior
than tlip contpmporanpons .spot rafp, assiinip pxpprtations arp formpd as a liiipar roinliination of thp rurrpnt spot ratp, »i, and
and too
anv linpar romhination of variahlps in thp information
spt, Ir:
'UiIf
the actual prorpss
(II)
+
(1
-
"I
+^
=
iilr
(1
-
ti)'!
(jri
-
Ti)(It
+
-
I'f+yt
can hp rpwrittpn as
^''+^ Rational pxppctations
insuffiripii)
"I'r
is:
(r,
Thpn pqnation
=
wpight on
is
ni
"'
+*
=
1
+
thp casp in whirh thp ropffiripnt !<\ - »j is
and too mnch wpight on otlipr information.
12
-
/.pro.
«i)
A
+
>',
+ k-
posi(i\p \ahip implies a
^
>
^j:
invpstors put
random measurement
hand
error in the survey data lierausi' expertatinns liavr ajijirarefl only on the
But now expectations appear
side of the equation.
under the
null hypothesis,
To demonstrate
measurement
this effect,
of the tnie
and
E((i\A.'>'j_^_i^)
=
The
0.
= ^'Hk +
^
=
_ var(Q) -
which the measurement error accoxmts
- in other words,
surveys
no information
at
OLS
for
all
all
+
cov(f?f^t-
+
in probability to;
As',_^_^.)
var(A.t;_^,.)
estimates toward on<\ hideerl. in the limiting case in
of the variability of ex])ectrd dejireciation in the survey
about the "tnie" market exj^ectatiou
of 15 estimates of d are greater than one; in five cases the difference
However, we
4.2.
shall
measurement
now
error
is
sum
(13).
ryf+t
is
contained in the
the parameter estimate would be statistically indistinguishable from one. In Table
result suggests that
is
(12)
''
equation (11) converges
error therefore biases our
by the survey
error:
A.^J+j^
var(f,)
Measurement
as recorded
spot rate change can then be expressed as the
actxial
market expectation plus a prediction
facts, the coefficient d in
rif^ht-hand side; as a result,
plus an error term:
A.i'_,.;^,
A.v+t
Using these
tli<'
error biases toward one our estimate of d in equation (11).
As'+i
tf is iid
on
suppose again that expected dejtreciation
equal to the market's tnie expectation,
where
also
left-
not the source of our
is
statistically significant.
r(-i(^ction
G,
13
This
of rational expectations.
see that stronger evidence can be ol^tained.
Another Test of Excessive Speculation
Another
test of rational
expectations that
free of the ])r'>bl(-ni of
is
measurement
replace A.'J^j on the right-hand side of eqtiation (11) with the forwaid discount
A.5J^j
where the residual,
ci
—
rji^i^, is
-
the
A.i,_^i
=
rci
+
measurement
flifd,
+
(,
-
error
is
to
/r/,':
(15)
r1,^^..
error in the surveys less the iniexpected change
in the spot rate.
There are several reasons
for
making the
results in section 3 that expected depreciation
siibstitution in equation (15).
is
13
highly correlaterl with /'//
We know
from our
Because fd^
is
free
good candidate
an exogenous "insfniiiiental variable." Indeed,
of measiircmpnt error,
it
we
can look the forward discount
as econonietricians
do so
is
a
A
as prospective speculators.
a speculator could have
made
"bet against the market"
is
for
finding of
> D
/?i
more
practical
expressed as
if
also
in either efjuation (9) or (13) suggests that
But the strategy
excess profits by betting against the market.
far
we can
in the newsjiajier,
uji precisely
if
to
against the (ol)servable) forward
''])ci
discount" than as "do the opposite of whatever you would have otherwise done."
Equation (15) has additional relevance
unbiasedness regression in section
the coefficient,
3:
unbiasedness due to systematic prediction errors,
large positive values of
Table 7 reports
Prr iu Tables 3a
ft^e
OLS
found
in
column
ft^r-
(1) of
Thus equation
Tables 3a anrl
hypothesis of rational expectations,
O'l
of excessive speculation. (Because the
=
0, /9i
=
We now
measurement
are necessarily lower than those of Table 6.)
They
0.
31).
(24) can
reject
error has
Thus the
the forward rate
result that
us whether the
see that the point estimates of
/?,
been
tell
are statistically significant.
The data continue
precision.
f)f
]uecisely eriual to the deviation from
is
/?i,
regressions of equation (15).
and 3b are measured with
our decomposition
in the context of
=
to reject statistically the
0, in
favor of the alternative
jiurgetl, the levels of significance
f^f^
is
significantly greater than
zero seems robust across different forecast horizons anrl different survey samjiles.
In
decomposition of the typical forward rate tmbiasedness
now
hypothesis that
all
of the bias
is
is
survey responrlents. Recall that this finding need not
new exchange
in
Table 3a, we can
attributable to the survey risk premium.
cannot reject the hypothesis that none of the bias
learning about a
test
rate process, or
the error term, then one could not expect
the errors appear to be systematic ex
them
if
due to
mean
there
is
14
reject the
differently,
ie]-)eated (>xi)ectational errors
that investors are irrational.
a "j)eso ])roblem
to foresee errors in
pn.tt.
Put
terms of the
th<'
"
If
we
made by
they are
with the distribution of
sample
jieriod,
even thotigh
5.
Conclusions
Our
general ronchision
is
that, contrary to
what
is
assiunod in ronvmtional practice, the
.'^ys-
tematic portion of forward discount prerhction errors do not capture a time-varying risk premium.
This result was quahtatively clear from the point estimates
can now make several statements that are more precise
We
(1)
reject the hypothesis tliat
premium. This
is
the
same thing
all
in section 2 or
from the charts. But we
statistically.
of the bias in the fnrwaifl fiiscouut
as rejecting the hyjiothesis that
none
is
flue to the risk
of tlu^ bias
is
due to the
presence of systematic expectational errors.
We
(2)
cannot reject the hypothesis that
all of
the bias
is
attributable to these systematic
expectational errors, and none to a time-varying risk premium.
The implication
(3)
changes
in
of (1)
and
(2)
is
that changes in the forward discoiuit reflect, one-for-one,
expected depreciation, as perfect substitutability among assets denominated
in different
currencies would imply.
(4)
We
reject the claim that the variance of the risk
jMemium
is
greater than the variance of
expected depreciation. The reverse appears to be the case: the variance of expected depreciation
is
large in
comparison with lioth the variance of the
risk
premium and
the variance of the forward
discount.
(5)
Because the survey
risk
premium appears
cannot reject the hypothesis that the market
We do
risk
to be uncorrelaterl witli the forward discoimt,
premium we
find a sul)stantial average level of the risk jiremium.
the forward discount as conventionally thought.
15
are trying to
But
it
floes not
measure
is
we
constant.
vary positively with
6.
References
"The Speculative Efficiency Hypofhesis,"'
435-51 (1981a).
Bilsnn, John,
"Profitability
,
and Stability
in Inteniatinnal
Journal nf Business.
Cnrrmry Markets,"
.Inly 1981,
NBER
54,
Working
Paper, No. 664, April 1981 (1981b).
"Macroeconomic
view,
May
Stability
and Flexible Exchanc^r Rates",
Amrrimn
Er.onnmir. Re-
1985, 75, 62-67.
Boothe, Panl and David Longworth, "Foreign Exchange Market Efficient Tests: Implications
of Recent Empirical Findings," Journal of Intcrnntinnnl Mnnry and Finance, 1986.
Cumby, Robert, and Maurice
Differentials:
A
Obstfehl, "Exchange Rate Expectations and Nominal hiterest
Test of the Fisher Hypothesis," Journal of Finance, 1981, 36, 697-703.
Domingne/,, K. "Are Foreign Exchange Forecasts Rational:
Economics
New Evidence from Survey
Data,"
Letters, 1986.
Dombusch, Rudiger, "Exchange Rate Risk and
the Macroeconoi7iirs of Exchange Rate Determination," in R. Hawkins, R. Levich, and (^. Wihiborg. eds.. The Internationalization of
Financial Markets and National Economic Policy, Greenwich. Conn., JAI Press, 1983.
Fama, Eugene, "Forward and Spot Exchange Rates," Journal of Monetary Economics, 1984,
14.
319-338.
Data to Test Some Standard PropoRegarding Exchange Rate Expectations," NBER Working Pai)er, No. 1672. Revised as IBER Working Paper. No. 86-11, University of ralifornia. Berkeley, May 1986.
Frankel, .leffrey A. and Keiuieth A. Froot, "Using Survey
sitions
"Using Survey Data to Test Some Standard Propositions Regarding Exchange Rate Expectations," American Economic Review. March 1987, 77, 1,
,
133-153.
,
talists
and Chartists,"
"The Dollar as a Speculative Bubble: A Tale
Working Paper, No, 1854. March 1986.
of
Fiuidamen-
NBER
Gregory, Allan W., and Thomas H. McCurdy, "Testing the Unbiasedness Hypothesis in the
Forward Foreign Exchange Market: A Specification Analysis." Journal of International
Money and Finance, 1984, 3, 357-368.
Hansen, Lars Peter, "Large Sample Properties of Ceneraliz.ed Method of Moments Estimators,"
Econometrica, 1982, 50, 1029-1054.
Hansen, Lars and Robert Hodrick, "Forward Rates
16
as Oiitimal Predictors of
Futiue Spot Rates:
An
Econoinrtric Analysis," Journal nf Political Ecnnomy. OcU^hcv 1980, 88, 829-53.
No Risk PiTininni in Forward Exchange
Markets," Journal of International Er.onomirn, Angiist 1984. 17, 173-84.
Hsich, David, "Tests of Rational Expectations and
Hodrick, Ro])ert, and Sanjay Srivastava, "An Investigation of Risk and Return in Forward
Foreign Exchange," Journal of International Money and Finanrr. April 1984, 3, 5-30.
,
"The Covariation
of Risk
rreniiuiiis
and Exjjccted
Fiiture
Spot
Rates," Journal of International Mone.y and Finanee. 198G.
Huang, Roger, "Some Alternative Tests of Forward Exchange Rates as Predictors of Future
Spot Rates," Journal of International Money and Finanee Aiigust 1984, 3, 157-67.
Krasker, William, "The 'Peso Problem" in Testing
kets,"
tlie
Efficiency of
Journal of Monetary Eennomicf, April 1980,
Levich, Richard,
"On
G,
Forward Exchange Mar-
2G9-7G.
the Efficiency of Markets of Foreign Exchange," in R. Dornbusch and J.
Eeonnmic Policy. Johns Hojikins Ihiiversity Press, Baltimore:
Frenkel, eds.. International
1979, 246-2G6.
Longworth, David, "Testing the Efficiency of the f/anadian-I'.S. Exchange Market Under the
Assumption of No Risk Premium," Journal of Finance. March 1981, 3G, 43-49.
McCulloch, J.H., "Operational Aspects of the Siegal Paradox,"
nomics, February 1975, 89, 170-172.
Quarterly Journal of Eco-
Mussa, Michael, "Empirical Reg\ilarities in the Behavior of Exchange Rates and Theories of the
Foreign Exchange Market," Carnegie-Rochester Serie.f on Puhlic Policy, Autumn 1979,
11, 9-57.
Newey, Whitney and Kenneth West, "A Simple, Positive D(~finite. Heteroskedasticity and A\itocorrelation Consistent Covariance Matrix," Woorlrow Wilson School Discussion Paper,
No. 92,
May
1985.
Tryon, Ralph, "Testing for Rational Expectations in Foreign Exchange Markets,"
Reserve Board hiteniational Finance Discussion Pajier. No 139. 1979.
17
Federal
CHART 1
FORWARD RATE ERRORS & THE RISK PREMIUM
MONTH ECONOMIST SURVEY
3
30
DATA SMOOTHED
o
0.
a
B
O
u
-10
9
Ok
-20
-
-30
-
-40
23-Jua-81
a
01-Jua-a2
08-May-83
16-Apr-a4
For><«rd rate error
+
19-Mar-85
jy^i.
Premium
CHART 3
FORWARD RATE ERRORS & THE RISK PREMIUM
MONTH MMS SURVEY DATA
SMOOTHED
1
.
3
U
o
a
a
V
o
u
e
0.
24-0ct-a+
CI
27-Feb-85
Forward Rate Error
10-Jul-aS
••
04-Dec-85
Risk
Premium
CHART 2
FORWARD RATE ERRORS
3
THE RISK PREMIUM
Sc
MONTH MMS SURVEY DATA SMOOTHED
u
o
0.
a
o
u
o
ft
05-Jan-e3
a
aO-Jan-a*
18-Jiil-83
Forward Rate Error
*
CHART
FORWARD RATE ERRORS
6
15-Aug-a4
Riak
Premium
A.
THE RISK PREMIUM
Sc
MONTH AMEX DATA
s
3
d
«
b
C
a
d
V
u
k
«
0.
-10
-
-20
-
-30
-
-40 -I
30-Jan-76
a
.
1
31-Jan-77
1
.
Ol-Dec-77
FonrBLrd Rate Error
>
,
Ol-Dec-78
-,
—P
,
30-Jim-82
Risk
j-
29-Juii-64
Premium
TABLE
1
TESTS OF FORWARD DISCOUNT UNBIASEDNESS
I
OLS Regressions of
^{.^.j^
o"
td^
I
F lest
•::,T
i
":ntn
TABLE 2a
COMPONENTS OF THE FAILURE OF THE UNBIASEDNESS HYPOTHESIS
In Regressions of
As
on
fd
TABLE 2b
COMPONENTS OF THE FAILURE OF THE UNBIASEDNESS HYPOTHESIS
In Regressions of
Faikre
or
Ritional
on
As
Existence o*
Rist Freaiui
Expectations
Approxirate
lata Set
EC2N 3 flCNTH
Dates
N
Regression
Coefficient
(i;
(2)
^
^
B^j
fd
B,^
i-di-c)
TABLE 3
COMPARISON OF VARIANCES OF EXPECTED DEPRECIATION
AND THE RISK PRE.MIUM
(x
10
per annum)
TABLE 4
TESTS OF PERFECT SUBSTITUTABILITY
OLS Regressions of
^s
,
on
fd^
F test
Dates
Data Set
B
6/31-12/35
Ec3r.0£i5t iati
t:
0.98B0
B=.5
t:
BM
R
OH
OF
a=0,
BM
Prob
>
F
-0.08
0.8?
554
1,44
:3.il
0.000
til
1.19
0.70
184
!.5i
16,55
0.000
3.14 III
0.19
0.B9
18«
1.37
52.06
0.000
m
-0.48
0.91
134
1.44
65.32
0.000
-0,09
0.21
171
1.02
6.79
0.000
-2.75 tit
0.73
182
1.50
14, :0
0,000
-0.16
0.64
91
0.74
5.38
0.000
1.04
0.71
45
1.45
6.32
0.000
-0.45
0.61
45
0.51
8.10
0.000
3.33 Jtt
(0.1465)
i/ai-i:/35
Econ 3 ,tonth
-
1.3037
3.H
(0.2557)
6/91-12/65
Econ 4 «onth
1.0326
(0.1694)
6/31-12/35
EccR 12 Honth
0.9236
2.06
(0.1499)
nHS
10/34-2/36
flcrth
1
0.8416
0.20
(1.7275)
MS
3
1/83-10/84
Month
-0.1316
-1.59
(0.4293)
1/76-7/85
AHEI Data
0.9605
1.85
t
(0.2495)
m\
1/76-7/85
6 fionth
1.2165
3.44 ttt
(0.2085)
1/76-7/85
AHEI 12 rionth
0.8770
1.37
(0.2755)
Notes: Method of ilorents standard errors are
101 level,
It
m
parentheses,
t
Represents signiHcince at the
and ttt represent significance at the 51 and II levels, respectively.
TABLE
5
TESTS OF PERFECT SUBSTITUTABILITY
3SLS Regressions of
e
on
^^.k.
averaqe
Dates
Data Set
Ecsnosist
3
:ionth
6/31-12/85
B=.5
t:
0.8723
t:
pll)
B=l
fd
k
Proo
DF
/
r
a=0, B=l
2.81 III
-0.?6
0.13
184
0.000
4.33 III
-1.53
0.32
124
0.000
4.26 ttt
-2.04
U
0.27
184
0.000
-2.06 tt
0.21
171
NA
1.59
0.33
179
0.000
(0.1327)
Eccncsist
3
ncnth
4/31-12/35
0.87i3
(0.0730)
Econosist 12 Month
6/81-12/85
0.3373
(0.0793)
nnS
I
10/84-2/86
Honth
(1.0445)
1/B3-10/84
nns 3 nonth
0.4672
-0.10
(0.3354)
(1)
Average
p
is the sean across countries of
the first order auto-regressive coeHicients.
Notes: Asyeptotic standard errors are in parentheses.
101 level,
tt
I
Represents significance at the
and ttt represent significance at the 5J and IZ levels, respectively.
TABLE 6
TESTS OF EXCESSIVE SPECULATION
Regressions of
As^_^j^
- s^^^
on
l^^^^
F test
Djti Set
Econoiist Dati
Dates
B
i/81-12/B5
1.0162
t:
B=0
t:
2.49 tt
BM
R
DF
DK
a^O,
Prob
)
3^0
0.0<
0.49
S09
4.79
0.000
(0.4104)
Econ 3 Honth
i/6!-12/S5
l.iMl
3. 46
tU
1.32
0.26
1E4
2.91
0.010
3.75
Ul
2.27 tl
0.41
174
3.54
0.002
-2.48 It
0.67
14?
6.32
0.000
6.07
O.OOO
3."
002
(0.4664)
Econ 6 Konth
6/31-12/85
2.5325
(0.6746)
Econ 12 rionth
6/S1-12/85
-0.3005
-0.57
(0.5241)
.ins
1
'.ieek,
1
«onth
10/34-2/26
-
1.2561
3.54 III
0.72
0.24
414
3.90
»H
0.50
0.1*
242
7.09
lU
-1.93
0.19
:;9
12. !2
0.000
2.76 IIJ
0.65
0.23
171
Ml
o."10
(0.:544)
"KS
1
Keek
10/84-2/96
1,1476
1.34
(0.2939)
MS
!
Seek.
S'JR
10/S4-2/36
0.7E53
t
(0.1109)
'"1«S
1
north
10/34-2/86
1.3063
(0.4741)
nnS 2 Week, 3 Honth
1/B3-I0/34
1.0474
F
TABLE 7
TESTS OF RATIONAL EXPECTATIONS
OLS Regressions of
As
- As
on
F
Dates
Data Set
B
l.i?03
4/81-12/B5
Eccnoaist Data
t:
B=0
DF
R
1.41
fd
test
a=0,
B=0
Prob
>
F
0.48
509
4.75
0.000
(1.0530)
6/31-12/35
Econ 3 nontn
•
2.5127
1.95
1
0.14
184
1.31
0.25i
1.37
t
0.28
174
l.^s
0.194
0.42
0.67
149
b.Ol
O.O(m)
2.42 It
0.20
171
2.54
0.030
2.60 It
0.66
182
11.93
O.COO
2.72 tit
0.33
56
2.i?
0.005
2.70 III
0.26
45
3.30
0.009
2.40 11
0.25
40
1.48
0.210
(1.2913)
6/81-12/35
Econ i Month
2.??i6
(1.5974)
6/81-12/85
Econ 12 flonth
0.5174
(1.2290)
?.K
I
10/84-2/36
Honth
15.3945
(6.3520)
NHS
3
1/83-10/84
Honth
6.0725
(2.3392)
1/76-7/85
AHEI Data
3.2452
(1.1675)
flUEI
6
1/76-7/85
Honth
3.6346
(1.3437)
AllEX
1/76-7/85
12 Honth
3.1031
(1.2954)
Notes: flethod of Hoeents standard errors are
K'Z level,
II
and
Ml
m
parentheses,
represent significance at the
57.
and
t
Represents significance at the
II
levels, respectively.
Econometric Issues
APPENDIX
GENERAL
1:
Estimation of most of our equations
tries,
and
some
in
is
stack different coun-
The complicated
cases different forecast horizons, into a single equation.
correlation pattern of the residuals, however, renders the
samples.
We
performed using OLS.
OLS
standard errors incorrect in finite
Several types of correlation are present.
First,
there
induced
correlation
serial
is
by
corresponding forecast horizon (up to eight times).
ping observations imply
that,
a
sampling
This
is
interval
the usual case
under the null hypothesis, the error term
is
shorter
m
a
than
the
which overlap-
moving average
process of an order equal to the frequency of sampling interval divided by the frequency of the
horizon, minus one.
Hansen and Hodrick (1980) propose using
a
method of moments (MoM)
estimator for the standard errors in precisely the application studied here.
Second,
order to take advantage of the fact that the surveys covered four or five
in
currencies simultaneously,
we pooled
the regressions across countries. This type of pooling
induces contemporaneous correlation in the residuals.^ Normally, Seemingly Unrelated Regressions should be used to exploit this correlation efficiently.
the serial correlation induced
The
by overlapping observations makes
model may be
basic
is
number of periods
the
=-»^..P +
SUR
SUR
later, here,
however,
inconsistent.
(14)
v,*,
in the forecast horizon
account for the two types of correlation
ance mainx of
use
written as:
)'o
where k
We
in the residuals
and
with a
/
indexes the currency.
MoM
estimate of the covari-
p:
0' =(XNT'XNTr'XsT'iiXNT(^'NT'XNT)"'
'*
Each currency
in
our pooled regressions was given
the differences across currencies
reasonable in view
ot"
ward
Table
di:>cuuiit (see
We
I
of
NBER
m
its
the
Working Paper version
own consiant term. Tins modeling siralcgy seemed mosi
magruiudes of both ex posi spot rau: chances and the lorol this study).
(1-^')
-2-
where X^x
is
the matrix of regressors of size
of the unrestricted covariance matrix,
^''J">= TTFIT
SS
=
otherwise
where
r
is
V,^v,_t^
Q
for
N
(countries) times
T
(time).
The
j)lh element
(/
is:
mr-r<A:<wr+r
;
m=C
N-l
(15)
.
MA
the order of the
process, v,.^^
is
the
OLS
residual,
and k = \i-j\.
We
cases, this imrestricted estimate of £2 uses well over 100 degrees of freedom.'^
estimated a restricted covariance matrix,
aUt+lT, i-k+pT)
=
1
^-^
y"
S
=
These
restrictions
have the
lation functions of the
ance parameters
-
otherwise
down
if
\=p
therefore
and -r < k < r
.
(16)
.
effect of averaging the
OLS
some
with typical element:
(OU+IT, i-k+pT)
s-\s-\
f
Q
In
own-currency and cross-currency autocorre-
residuals, respectively, bringing the
number of independent
covari-
to 2r.
Tests of forward discount unbiasedness also provide an opportunity to aggregate across
different forecast horizons (though
we
are
unaware of anyone who has done
this,
even with the
standard forward discount data), adding a third pattern of correlation in the residuals.
stacking seems appropriate in this case because
forward discount generally, rather than
at
we wish
to study the predictive
any particular time horizon.
Such
power of
Moreover, a
MoM
the
esti-
mator which incorporates several forecast horizons has appeal beyond the particular application
'*
The number of independent parameters
in the
covanattce matru does not affect the asymptotic covariance. a$
long as these parameters are estimated consistently (see Hansen (1982)).
sample properties of
the
MoM
estimator svorscn as the
Nevertheless, one suspects thai the small-
number of nuisance parameters
to be
estimated iiKrcascs.
-3-
studied here because
it
compuialionaliy simpler than competing tecliniques and
is
at the
time can be more efficient than single k-step-ahead forecasting equations estimated with
To
MoM.
demonstrate the precise nature of the correlation induced by such aggregation, con-
sider the stochastic process, Y,,
finite
same
second moments.
We
which
is
stationary
and ergodic
in first differences
denote the k period change in y from period t-k to
t
and has
as
>,*,
and
1-1
the h period
change as
the innovations, vf
where
These
=
>>/'
and v^
]^>'*_,t,
where h = nk for any positive integer n.'^
We
then define
as:
vf
= y,*-E(>-,M 0,^)
vf
=
(17)
>•,*- £(>•,"! <>,_,)
includes present and lagged values of the vector of right-hand-side variables, x*.
^,
facts allow us to write the covariance matrix of the innovations as:
v;
I=£
where the
(/,y)th
element of each submatrix of I
function, evaluated at ^
=
A*^-£(vNf^)
/
=
=E(vNf^) =
=
"
A'^' A"
is
equal to the corresponding autocovariancc
?.*
if
\q\
othenvise
>i;;
if
l<7l
oibcrwise
<k
,
</,
,
The following example can easily be generalized lo allow It and k to be any positive integers. It is also possicombine in a similar fashion more than two different forecast horizons. Indeed, we combine three horizons in
the Economist dau estimates m the regressions below. Because these extensions yield no additional msights and come
at the cost of more complicated algebra, however, we retain the simple example above.
ble to
(18)
-J:
=
Af,^
[^,'<
(19)
-4-
Af^
=£(vy^)
= X^
= E( vNf^) =
XJ*
=
In this context consider the
if
0<<7 <Jt
if
-/I
<
otherwise
<
<?
.
aggregated model:
=x,p +
y,
where
=
y,'
the usual
b',***'
y/'+v.l, x,'
MoM estimate
s
is
[.r*'
x!"]
and
v,'
=
v,
[v,*^'
(21)
The OLS estimate of P then has
vf^l.
of the sample covariance matrix:
Q
where Z
(20)
—
(XzNT X;nt)
a consistent estimate of I, and
is
X2nt^X2«jt(X2NT Xjnt)
formed by using the
OLS
residuals to estimate the
autocovariance and crosscovariance functions in equations (19) and (20).
One might
think that by stacking forecast horizons, as
asymptotic efficiency always results than
words, that 0' - 0^
is
if
we
in equation (21), greater
only the shorter-term forecasts are used, in other
positive semidefinite.
only additional estimates
we do
After
all,
the
sample size has doubled, and the
require are nuisance parameters of the covariance matrix.
inmition would be correct for asymptotically efficient estimation strategies, such as
likehhood.
reflect
the
But because
OLS
weights each observation equally, the
average precision of the data.
It
follows
that
if
the
MoM covariance
This
maximum
estimates
longer-term forecasts are
sufficiently imprecise relative to the shorter-term forecasts, the precision of the estimate of p
drops:
we could
actually lose efficiency by adding
this potential loss in
cast horizons.
more
asymptotic efficiency, and show
Efficiency
is
most
find a
the
Economist and
marked increase
Amex
how
it
is
we demonsu^ate
related to the disparity in fore-
likely to increase if the longer-term forecast horizon
relatively small multiple of the shorter-term horizon.
we
In the appendix
data.
in precision
samples), but
or
a
Indeed, in the forthcoming regressions
from stacking across forecast horizons when
little
is
no increase
in precision
when
r
= 2
(in
= 4 or 6
(in
r
-5-
MMS
the
samples).
above
Finally, the
estimates of the covariance matrix need not be positive definite
Newey and West
in small samples.
that discounts
MoM
the jth order autocovariance by
positive definite in finite sample.
question of
we
sions
how
tried
small
m
=
2:
to guarantee positive definiteness.
Newey and West
value of
m
In the
themselves suggest) and
m
upcoming regres-
= 2r; we report
because they were consistently larger than those using
we show how
the asymptotic efficiency of the
by aggregating over forecast horizons. Consider
affected
method-of-moments
>•*+*
y, £(€f+t
regressor,
I
= Y,^ -
Y,
and the error term
j:,V,*-i ,..-.y*,>',*-i ....)
j:*,
but
may
=
0.
yU
For the two aggregated
'
orthogonal to the present and past values of x and
Our example below considers
....,)*,>*_, ,...)
are jointly covariance stationary, then the
'
(Al)
the simple case of a single
easily be extended to a vector of righthand-side variables.
innovations \>,^ = £(>'*l-r,V*.,
iid
is
MMS
data sets
and
=
r|,^
E(j:*I.v*a,-i
Wold decomposition
= £5,u,^., + £Y,r|,^_, +
in
Table
resulted in a nonpositivc semi-deftniie covariance matrix.
6,
a value oi
m
=
r
,.).
If
-v
"
Tlie regressioru reported in the text
We
estimated sundard errors.
errors.
assume homosccdasticity.
D*
tJie
Concern often
tried a hctcrosccdaslicity-consistcnt estimator arxJ
We
choose
tlie
and y
implies that:
was used
Tins correction reduced
y*,>'*-i
Define the
(A2)
after finding that
standard errors
m
in these
= 1r
two
re-
gressions by an average of only 3 percent.
the prediction errors.
esti-
the model:
y,V=-r,*p + ef^
where
stan-
EFFICIENCY AND POOLING OVER FORECAST HORIZONS
In this appendix
is
covanance matrix
the
Nevertheless, for any given sample size, there remains the
must be
latter
l-0/('" + l)). making
}^
APPENDIX
mator
m
(which
r
dard errors using the
the former. ^^
(1985) offer a corrected estimate of the covariance matrix
arises about iKleroscedasticity in
found that the correction resulted
in
lower
conservative route of reporting the results with the larger estimated standard
-6
where D^
y.
We
That
is,
is
the deterministic
component of y and
,
are primarily concerned with the case in
which x^
is
the best imbiased forecast of >*+*.
under the null hypothesis of forward discount unbiasedness, fJ*
Thus we assume
tor of the future spot rate change, 4Lr,+t.
information for forecasting
We
h = nk.
includes past and present values of x and
<>,
so that £0'*+*
v,*^,
define analogously the h
!<!),)
=
an efficient predic-
is
that a* already contains all relevant
£"(>'*+* 'v,*).
period change in Y,^
as v/U
=
^ ^
y,*^_^t_,
,
where
Using equations (Al) and (A2) we then have:
y/U = X£s,u,v,-,t-, + XXv.Tl.w,-;*-, + o;
(A2')
n-1
£^(y*^l«t>,)=
These
facts
imply
that the k
tionary with finite second
£ X
it
moments.
-1 order
Z Z
+
and h period prediction
we assume
If
ary with finite variance, then E{q^q^^j)
be expressed as a
5.'»-^-;t-.
=
for ;
moving average
Similarly, from equation (A2')
we have
>
errors, ef^
that q^
A-,
Y.^.^-;*-. +^y'
and
ef+v,.
= ef^.v* and
and E(q^q^^^) =
respectively, are sta-
<?/"
= ill^x^ are
for ;
station-
> h. Thus ^* can
process:
I
<?,*=
£a,v,^_,
thai
q^
may
(A3)
be written as a /;-! order moving average
process:
n-l h-\
^,''=
ZZ
^.V,-**-;*-,
(A3')
(A4)
+*->*-
n-2
where
c,,_^_
= X£/(„^)t_.
The covariance generating
function for ^*
is
denoted by
X'^(r),
Pl=y
where
t-i
i-l
*-1
(A5)
Using equation (A4), the covariance generating function
X''(z)
where
= \^{z)+
-^^%^
\''(z)+
for
"^"
c^l"
can be written
'^
;^
X'"'\z) is a complicated generating function of the a,'s
specify here.
Finally,
the covariance generating function,
+
as:
(A5')
2X''^'u-)
and
<r,
's
X'"'{z)= a;
which we need not
^
X'^;^;*'
^'^ ^^
rewritten as:
X"^
where
X'"^\z)
Now
is
(r
)
=
i^
X-"
(.- )
+
-^-^
X^ +
another generating function of the a,'s and
consider the asymptotic
MoM covariance
V^\z
(
c,'s.
matrix of
VF (p -
P)
from equation (A I):
e' = {Kr\''(z)
where X^ = ,'^ T'^'^x^xf.
If
we add
(A7)
longer-term forecast data, our model
in the
A6)
is
that
of
1=1
equation (21) above, with asymptotic covariance matrix:
0= =
By
substitution,
we have
that
(X,';
+
©' > 0-
Kr'
if
(X'iz
)
and only
r
I
+ x^co +
if:
2X"^(.-))
(A8)
> n^
'A
K
1+
+ n
X"(--)
Equation (A9) says
same
K
3+-
1/6
X'
(r
)+\'
(.-
H2(?i'"'
that the variance
of the longer-ierm data,
,
if
term forecasts to data sets of only shorter-term forecasts.
we
)+X'"\z ))-X^
>i''(--)
as or greater than the relative forecasting interval, n
increases,
'(--
?i5,
we
must increase
at a rate
the
are to gain by adding longer-
Thus as
the
forecastmg interval
require correspondingly greater variability of the regreSsors in order to
compen-
sate for the greater variability of the forecast errors.
One might
think that the result in equation (A9)
is
a
consequence of weighting the more
imprecise longer-term predictions equally with the predictions of shorter-term.
downweighted the longer-term
is
we would always
data,
not the case. In the remaining space,
and show
that the efficiency
of
we
gain in efficiency.
It
Perhaps
if
we
turns out that this
construct a consistent, optimally weighted estimator
this estimator
may
still
worsen asymptotically by adding
in the
longer-term forecasts.
In
most circumstances,
GLS
have different levels of precision.
overlapping observations.
is
represents the optimal weighting strategy
GLS
when
the data
however, inconsistent when used on a model with
is,
Thus we consider
instead a weighted least squares estimator which
optimal within a class of consistent estimators.
Consider a transformation of the model
in
equation (il), which stacks the shorier- and longer-term data:
IV y,
where
W
is
sistent for
a diagonal matrix.
The
= VVx,P + \V\,
MoM
any arbitrary diagonal matrix
W
estimate of p
.
To
(AlO)
in
equation (AlO),
see this, note that the
MoM
pn-
,
will be
con-
estimate of equa-
tion (AlO), p^v, inay be written as:
-1
Vr(pH,-p) =
''2NT ^'^'''INT
(All)
9-
V
7T
Z'
-
X^.-H-,,-
2T
^
X-r,v,»v,r
=1
The
term in equation (All) converges
final
in equation
£(v, Ix,)
(A 10)
=
is
in probability to zero,
provided that the error term
conditionally independent of the contemporaneous value of the regressor,
(this is just the
estimating equation (AlO)).
Gauss-Markov assumption required
Suppose now
that
we choose
W
where 0^
is
the
MoM
That
is:
0^
©' -
(A12)
asymptotic covariance matrix of pn
©' = (x,^TWh,^j)-\,^j'\V'i^W\,^j(x,f,j'Whi,^j)-
By
normalizing the weight on every shorter-term data point to one,
show
that the
in
optimally in order to maximize
the gain in efficiency from adding longer-term forecasts to our shoner-lerm data.
w
OLS
for the consistency of
optimal weight placed on each longer-term observation
(A13)
it
is
straightforward to
is:
\'^
X^'-(.-)-X^-'^(r)
^h =
(A 14)
X*>.^(--)-XiX'^(.-)
Note
that
m-^*
will
always be positive
if
the data sets are uncorrelated,
i.e.
if
X'^(r) = 0.
In
other words, appropriately weighted independent information can always improve efficiency,
no matter how imprecise
the
new
may
information
be.
But, the nature of the correlation
between contemporaneous longer-term and shorter-term predictions implies
weight given to longer-term data
in
may be
equation (A 14) becomes negative.
tive variance
shown
zero.
This occurs
of the longer-term forecasts.
that a sufficient condition for
In particular,
n',,'
to
(w
if
n
is
yi\
that the
will be zero if the
is:
1)
^
—
-I-
>
numerator
too large in comparison with the rela-
Using equations (A5"), (A6) :md (A 14),
be zero
optimal
it
can be
(A15)
-
Thus, while the standard errors reponed
10-
in the text indicate that for
obtain improvements in efficiency, this result
is
with considerably longer forecast horizons, even
not likely to apply
when
the data are
for the greater variance of the longer-term forecast errors.
this potential loss in efficiency is a direct
mation strategy.
small values of n one
It
is
MoM
estimation of data
downweighted
worth stressing
maximum
to
account
in closing that
consequence of our limited information
Full information techniques, such as
may
MoM
esti-
likelihood estimation, will
consistently achieve nonzero gains in asymptotic efficiency with the addition of longer-term
dau.
L
534 065
Date Due
FEB.
2 8 If3
Lib-26-67
3
TDfiD
DOS 13M 553