consistently predicted depreciation after two years of consistent,
albeit volatile, spot appreciation.

A second explanation for the biased forecasts, which has been
widely discussed in the literature, is the peso problem: the existence
of a small probability of a large depreciation which did not occur over
the sample period used (Krasker 1980). The forecasters may actually
have rationally incorporated the true probability of depreciation in
their predictions, but because the three-month-ahead forecast sample
ended before the spot rate began to depreciate, the test statistics are
able to reject unbiasedness.

Dornbusch (1985) states that if we think of the dollar
appreciation as a single rational speculative bubble (where the smal]
probability event is the bursting of the bubble24) this probability
should have been fully incorporated in the interest differential paic¢
on dollar assets. While the interest differential did rise sharply
after 1980 (when the dollar began its appreciation), both long and
short-term interest differentials declined after mid-1984 yet the
actual and forecasted dollar continued to appreciate through February
1985. Frankel (1985) shows that the short-term interest differential
was insufficient to sustain a rational dollar bubble between 1/85 and
3/85. He finds that the cumulative probability of noncollapse over this
period would have been only three percent.

A final explanation for the survey expectations may well be that

24The probability of depreciation could be based on the fact that
fundamentals (see Appendix A for examples) were stacked against the
dollar by mid-1984 or in response to record U.S. budget deficits over
the sample three years.

18


Frankel, Jeffrey, Six Possible Meanings of "Overvaluation": The 1981-85
Dollar, Essays in International Finance, No. 159, 1985, Princeton
University, New Jersey.

Frankel, Jeffrey and Ken Froot, "Using Survey Data to Test Some
Standard Propositions Regarding Exchange Rate Expectations," NBER
Working Paper #1672, 1985.

Frooz, Ken "Exchange Rate Survey Data: The Role of Expectational
Errors, the Risk Premium, and Measurement Error," 1985, unpublished.

Frenkel, Jacob, "A Monetary Approach to the Exchange Rate: Doctrinal
Aspects and Empirical Evidence," Scandinavian Journal of Economics, May
1976, 78, pp.200-24.

Frenkel, Jacob, "Further Evidence on Expectations and the Demand for
Money during the German Hyper-inflation," Journal of Monetary
Economics, 1979, 5, pp.81-96.

Friedman, Milton, "The Case for Flexible Exchange Rates," in Essays in
Positive Economics, Chicago: University of Chicago Press, 1953.

Granger, C.W.J., "Co-Integrated Variables and Error-Correcting Models,"
UCSD Discussion Paper 83-13a.

Granger, C.W.J. and Robert F. Engle, "Dynamic Model Specification with
Equilibrium Constraints: Co-Integration and Error-Correction," mimeo,
UCSD, August 1984.

°
Granger, C.W.J. and A.A. Weiss, "Time Series Analysis of Error-
Correction Models" in Studies in Econometrics, Time Series, and
Multivariate Statistics, ed. by S. Karlin, T. Amemiya, and L.A.
Goodman, Academic Press, 1983, pp.255-278.

Hakkio, Craig, "Expectations and the Forward Exchange Rate,"
International Economic Review, 1981, No.22, pp.663-678.

Haltiwagner, John and Michael Waldman, "Rational Expectations and the
Limits of Rationality: An Analysis of Heterogeneity," American Economic
Review, June 1985, 75, pp 326-340

Hansen, Lars P., "Large Sample Properties of Generalized Method of
Moments Estimators," Econometrica, 1982, 50, pp.1029-1054.

Hansen, Lars P. and Robert Hodrick, "Forward Rates as Optimal
Predictors of Future Spot Rates: An Econometric Analysis," Journal of
Political Economy, October 1980, 88, pp.829-853.

Hansen, Lars P. and Robert Hodrick, "Risk Averse Speculation in the
Forward Foreign Exchange Market: An Econometric Analysis of Linear
Models," in Exchange Rates and International Macroeconomics, ed. by
Jacob Frenkel, University of Chicago Press, (for NBER) 1983.

39


Figure 6
One-Week $/DM
Expected and Actual Depreciation

0.008

-0.01

~0.018

184 toas-1 1988-1 198s—I 1pas—v 1988-1

0 MMS WEEK—PREM + $/0M—CEPRECUTION

One-Month $/DM
Expected and Actual Depreciation

coos 4

co 4

oot Lay
o4

poor 4 jv j

-voa 4

-vos

AAAERSAERARR ARR RSSR RRR SERED ES RRO ROR RESRELESSERLESLS
1oaay 1pas—| 1985-1 1985-10 19as—v 1oae-!
© ums Mo-PREM + $/OM-DEPRECATION


expected premia, can generally reject the null hypothesis at the .01
level. These test results suggest that foreign exchange rationality

tests should be conducted using integrated (or co-integrated)

transformations rather than the levels of these series.

Table 1(a) Unit Root tests on level series.

DEP.VAR. HORIZON

ett 1-WK
&t+2 2-WK
ety 1-MO

-087 0.107 0.083

(0)
(0)
oO.
0.125 0.108 1.250%**
0
0.128 0.122 0.113
0

DEP.VAR. HORIZON SMPL $/STLG $/DM $/SWF $/YEN

Cegj-ep  L-WK 84-86 1.920% §— 1.943%" 1,933" 1, 712%
esg-eg  2-WK 83-84 -1.833%* 1.567%" 1,699%% 1, 645%
epsyn-eg  1-MO-— 84-86 (0.562% = 0. 511* = 0.498* 0. 505*
ee4127"e¢ 3-MO 83-84 0.563* = 0.4800. 489 0.477

DEP.VAR. HORIZON SMPL $/STLG $/DM $/SWF $/YEN

fe 4n-et 1-MO 84-86 «= .2.173** = -2,181** = 2.122% — 2.024%
Ee’i2-ee « 3-MO. 83-84 0.509% = 0.490 0.463 0.480

Epepg1-ee 1-WK 84-86 2.141% = 2.201% 2.022%" — 1, B45**
Epeps-e¢ 2-WK = 83-84 1.455** = 1,498%* 1,193%* = 1,188%*
Eree44-e¢. 1-MO 84-86 = 0.849%" §=— «1.667% 1.515** — 1, 363**
Eree+127e¢ 3-MO 83-84 = 0.743%* =—-1. 656%* 1.859% — 0,991 **

* denotes rejection at the .05 level and ** denotes rejection at the
.01 level of the simple random walk hypothesis. Confidence limits are
presented in Sargan and Bhargava (1983) pp.157.

12


predicted larger depreciation than did the one-month forward rate.

IV. Summary and Conclusions

This paper tested the rational expectations hypothesis in foreign
exchange markets using a newly available set of survey data collected
by Money Market Services. Tests examined whether these expectations
series are unbiased and whether they incorporate information available
from past forecast errors and the forward rate.

The estimation procedure employed on regressions with overlapping
data allowed both for conditional heteroscedasticity and serially
correlated forecast errors. Seemingly unrelated regressions were used
in tests of the short-horizon non-overlapping data. The test results
found that the MMS forecasts fail consistently at predicting future
changes in the spot rate. Indeed, the MMS forecasts do no better than
the current spot rate in forecasting the future spot rate. In change
form, all forecasts are biased, and the three-month-ahead MMS forecasts
were found to violate strong-form RE.

A number of possibilies were proposed to explain the behavior of
the survey forecasts. It is possible that, although the tests presented
reject unbiasedness, the early sample period was not long enough to
bear out the forecasters' "rational" expectation of depreciation.
However, rationality arguments, whether they are in the context of a
bayesian or bubble process, do not seem convincing given the direction
and magnitude of forecast errors over the two sample periods.
Certainly, however, a direct test of the bayesian and bubble hypotheses

is warranted in future research.

22


Hayashi, Fumio and Christopher Sims, "Nearly Efficient Estimation of
Time Series Models with Predetermined But Not Exogenous Instruments,"
Econometrica, May 1983, 51, pp.783-798.

Hodrick, Robert J. and Sanjay Srivastava, "An Investigation of Risk and
Return in Forward Foreign Exchange," Journal of International Money and
Finance, 1984, pp.5-29.

Hooper, Peter and John Morton, "Fluctuations in the Dollar: A Model. of
Nominal and Real Exchange Rate Determination," Journal of International

Money and Finance, 1982, pp.39-56.

Hsieh, David A., "Tests of Rational Expectations and No Risk Premium in
Forward Exchange Markets," Journal of International Economics, 1984,
17, pp.173-184.

Johnson, Harry, "The Case for Flexible Exchange Rates," Federal Reserve
Bank of St. Louis Review, 1969, 51, pp.12-24.

Miskin, Frederic S., "Are Market Forecasts Rational?," American
Economic Review, June 1981, 71(3), pp.295-306.

Meese, Richard A. and Kenneth Rogoff, "Empirical Exchange Rate Models
of the Seventies: Do They Fit Out-of-Sample?," Journal of International
Economics, February 1983, 14, pp.3-24.

Meese, Richard A. and Kenneth Rogoff, "Was It Real? The Exchange Rate-
Interest Differential Relation, 1973-1984," Federal Reserve Board
International Finance Discussion Paper, No.268, August 1985.

Meese, Richard A. and Kenneth J. Singleton, "On Unit Roots and the
Empirical Modeling of Exchange Rates," Journal of Finance, September
1982, pp.1029-1035.

Mussa, Michael, "Empirical Regularities in the Behavior of Exchange
Rates and Theories of the Foreign Exchange Market," in Policies for
Employment, Prices and Exchange Rates, ed. by Brunner and Meltzer,

Carnegie Rochester Conference Series on Public Policy, Vol.11, 1979.

Muth, John F., "Rational Expectations and the Theory of Price
Movements," Econometrica, July 1961, 29, pp.315-335.

Nordhaus, William D., and Steven Durlauf, "Empirical Tests of the
Rationality of Economic Forecasters: A Fixed Horizon Approach," Cowles
Foundation Discussion Paper No. 717r, November 1984, pp.1-40.

Sargan, J.D. and Alok Bhargava, "Testing Residuals from Least Squares
Regression for Being Generated by the Gaussian Random Walk,"
Econometrica, 1983, pp.153-174.

40


relative to expected equilibrium prices (or rates of return). In other
words, in an efficient market investors should not be able to earn
excess profit using publically available information, where excess
profit is defined relative to an expected equilibrium rate of return.
In the context of foreign exchange markets we therefore first require a
model of equilibrium exchange rates. Conditional on the equilibrium
model we can then derive expectations which are model-consistent or
rational. The efficiency test is consequently of a joint hypothesis,
that the equilibrium model specified is correct and that expectations
are rational, conditional on the equilibrium model.

The main problem confronting this approach to testing foreign
exchange market efficiency lies in the first part of the joint
hypothesis: specifying the equilbrium model. While a number of models
have been developed to describe the determination of flexible exchange
rates, none of the models have gained general acceptance by the
profession4. Empirical tests of structural models in the literature
have had little success in isolating the important explanatory
variables». Indeed Meese and Rogoff (1983, 1985) find that neither
structural models nor univariate time series techniques improve on a
random walk model of exchange rates in spite of the fact that they base

thei: forecasts on realized values of future explanatory variables.

4see Appendix A for an example of the fundamental variables
typically included in equilibrium exchange rate models and the central
role the expectations formation process plays in such models.

5 See, for example, Frenkel (1976), Branson, Halttunen, and Mason
(1979), Hooper and Morton (1982), Edison (1985).


Figure 4
Two-Week $/Yen
Expected and Actual Depreciation

0.018
ore
0.018
0.012

oot
0.008
o.coe
0004
9.002
°

0.002

0.004

0.008

~o.c0a
-oot

-o.012

apts TTT EBeEARees
1983-1 1083-" 1983-1 = 1983-1 eas) toa4—n tose 1984

MMS Z—Wk Premiums + 9/YEN Depreciation

Three-Month $/Yen
Expected and Actual Depreciation

-o03 4

ay \v

-o08
TTT TTT
1983-1 1083-" 1oBs- $983 1984-1 1984-1 1984—  19a4—1y

MMS 3-MO Premiume + S/YEN Depreciation

33


the forecasts are biased, that forecasters over-weighted the small
probability of depreciation in their three-month-ahead forecasts and
under-weighted the probability of depreciation in their one-month
forecasts.

B, Tests of Weak- and Strong-Form RE

The second type of RE test examines whether expectations
incorporated available information. As a preliminary check for weak-
form RE, the Durbin-Watson statistic on the previously presented non-
overlapping short-horizon unbiasedness tests do not suggest the
presence of serial correlation in the regression forecast errors for
all but the one-week-ahead $/Yen equation. A direct check for serial
correlation involves regressing the current period forecast error on
the previous period error (equation (4) presented earlier). Table 4
presents the forecast error regression results, a significant slope
coefficient would indicate that forecasters did not incorporate past

errors in their current forecast.

19


differential

(5) rea > Fta = fea > et

where feyi is the i-period-ahead forward rate. Foreign and domestic
bonds are assumed to be perfect substitutes and international arbitrage
ensures that speculative efficiency holds, i.e., that the i-period
ahead forward rate equal the i-period ahead expectation of the spot
rate

(A6) fyi = Ep (egy)

where E; represents the mathematical expectation at t. If we combine
(A5) and (A6), substitute this in (A4), and let z=(m - m*) + a(yTM - y),
we arrive at an expression for the exchange rate in terms of relative
monies, income and the future exchange rate.

(A7) eg = ze + B(EL(eg43) - e¢)

We can now rearrange terms and solve for the reduced form of the

exchange rate by process of forward iteration.
(A8) ey = 1/ (148) 525 (8/(148) Ee (243)

In the monetarist model, the current spot rate depends on current
expectations of all important driving variables (the z's) from now into
the indefinite future. The model, while not successful empirically2>,

exemplifies the view that exchange rates are fundamentally dependent on

beliefs concerning the future course of monetary policy.

25Meese and Rogoff (1983, 1985) find that out-of-sample exchange
rate forecasts using the monetary model do no better than a random walk
model.

25


$/STLG 3-MO 83-84 -.039 -7.89 5 +37: 128.8**
(.006)** (1.03)**

1-MO = 84-86— «012 -1.26 1 +03 5.9
(.017) (1.56)
$/DM 3-MO = 83-84 «.003 -4,96 5 -08 36.9%
(.053) (4.33)
1-MO 84-86 — 019 -1.14 1 .03 6.4%
(.008)** (1.18)
$/SWE 3-MO 83-84 —«.065 -7.89 5 +26 69.4%
(049) (3.19)**
1-MO = 84-86_—S—«w 015 -1.15 1 05 3.5
(009) (1.11)
$/YEN 3-MO  —- 83-84— «003 -3.26 5 06 6.9%
(.035) (3.59)
1-MO = 84-86—s «015 -.357 1 +01 3.1

(008) (.783)
* denotes rejection at the .05 level and ** at the .01 level for the

hypotheses that cg=0 and c,=0. The value of the chi-square statistic
pertains to the test of the joint hypothesis that cg=c,=0.

These regression results indicate that the forward premium
contains additional information for the three-month-ahead forecasts.
While the $/DM one-month-ahead forecast also rejected the efficiency
hypothesis, it is the constant term, not the forward premium which lead
to the rejection. The three-month-ahead forward rates over the early
sample period generally (wrongly) predicted spot depreciation as did
the survey predictions but by a smaller percentage. The one-month-ahead
forward rates predicted spot depreciation throughout the later sample
period. The MMS appreciation forecasts were therefore more correct for
the first five months of the sample. The forward rate then "beat" the
MMS forecasts through August 1985 as the survey continued to predict

appreciation incorrectly. After September 1985 the survey correctly

21


Figure 5
One-Week $/Sterling
Expected and Actual Depreciation

0.01

002

-o03

TTT Tt
1oas-v tne) 1oes—H 19e8—in 1988-1 1938-1
MMS WEEK—PREMIVH + = $/37.0-CEPREoUTON
One-Month $/Sterling
Expected and Actual Depreciation
0.07 7

°
9
3
+1

mal,

OOS St THOTT TTT TT TTT TTT
eae soes-1 1988-11 1928-11 1985-1 sone—!

MMS MO—PREM + $/S7L9-DEPRECIATION

34


These tables show unbiasedness rejected at the .01 level for all
equations but the one-month-ahead $/Yen regression, which almost
rejects at the .05 level. The a; coefficients are particularly
striking. In the short-horizon equations a; is generally positive and
close to zero for all but the one-week-ahead $/Yen equation. In the
longer-horizon equations a, is generally negative and close to -.5,
implying that forecasters both over-predicted the size of spot
depreciation and also got the direction of the exchange rate movements
wrong. Further, across both short- and long-horizon equations a; is
insignificantly different from zero for all but the one-week-ahead
$/Yen equation, suggesting that the forecasts do no better than the
contemporaneous spot in predicting future spot rate changes.

Figures 1 through 8 in Appendix D present graphs of MMS predicted
and actual spot depreciation for the currencies over the four forecast:
horizons. A number of empirical regularities are notable:

(1) In all cases the variance of ex-post depreciation (appreciation)
was markedly greater than predicted changes.
(2) The short-horizon forecasts generally predicted very small changes

22,

relative to the longer-horizon forecasts Both the one-week and two-

week-ahead forecast error variances were found to be smaller than the
one-month and three-month-ahead forecast error variances@3,

22 Discussions with MMS indicate that the survey changed from
two-week and three-month-ahead forecasts to one-week and one-month-
ahead forecasts specifically because the forecasters felt more
comfortable reporting shorter-horizon forecasts.

23 In contrast, Engle and Kraft (1983) show in another context

that forecast error variances estimated from ARCH models for several
periods ahead can be less than one period inflation forecast error variances.

16


available regressors are included, the tests examine the semi-strong,
form of RE. Semi-strong rationality tests take the form:
(5) Cetsi - Beretta) = 60 + S12 + Ub,i
where z_ is a variable which embodies available information relevant. to
the forecasts. The hypothesis of informational efficiency requires that
the coefficient on z, and the constant be insignificantly different
than zero.

Results of these three rationality tests will be presented in the
following section. The remainder of this section will describe the
data and econometric technique used in the regression estimation.

B. Description of the Data

Money Market Services surveys bilateral (dollar) exchange rate
forecasts of the British pound, the West German mark, the Swiss franc
and the Japanese yen. From January 1983 to October 10, 1984 forecasters
were asked every other week (bi-weekly) to predict spot rates two weeks
and three months from the day of the survey. Since October 24,1984
forecasters have been asked weekly for their one-week-ahead forecasts
and bi-weekly for their one-month-ahead forecasts. Throughout its
existence, the survey has typically been taken after 12 pm EST: the
forecasters thus have information on the noon New York spot rate and
can calculate (but are not told) their forecast error from their
two-week or one-week previous short-horizon forecast. If the forecast
date falls on a holiday or weekend, the date is switched to the next:
business day. In contrast to the one and two week-ahead cases,
forecasters will not be able to calculate their previous one- or three-

month-ahead forecast error before the next survey.


o.or2

0.01
0.008
0.008
0.004

0.002

~0.002
-e.004
0.008
0.008
-0.01
0.012
~oo16
0.018
-o.ote

-o02

MMS 20k

Figure 2
Two-Week $/DM
Expected and Actual Depreciation

1083-1 1983-0 1oas—m = 1983 1984.1 1984—-N 1984-1 1084-1
Premiums, + 3/0M Depreciation

Three-Week $/DM
Expected and Actual Depreciation

= £4 paren

o ty

von df

-oaa

-o03 4
-oos 4
ee eo
1983-1 1983-u t9as— = 1983—-V 1084—1 1084—" 1984 1084
MMS 3-MO Premiums + 8/0M Depreciation

31


respect to lagged forecasts are constants. Cumby and Obstfeld (1983)
provide a formal test for conditional homoscedasticity. Under the
assumption that the survey forecasts are unbiased predictors of future
spot rates, the expectation of the square of the forecast error from
the OLS unbiasedness regression:

(c4) erty “et = ag + az (Epecyy - ee) + Vega

with respect to all instruments available at time t, Ze» should be
constant.

(65) ES y |e) = 0

This can be tested by estimating an equation of the form:

(06) tag = do + dy (Beeps ~ et) + do(Etetss - et)? + utes

and testing the hypothesis that dj=d7=0. Table 6 provides results of
the conditional homoscedasticity tests of one- and three-month-ahead
survey forecast errors. The instrumental variables used were the time t
expected depreciation (Epetaa - e,) and the same variable squared for

the four countries.

$/STLG 3-month 83-84 1.58
l-month 84-86 5.71*

$/DM 3-month 83-84 1.31
l-month 84-86 1.10
$/SWF 3-month 83-84 1.56
l-month 84-86 1.82
$/YEN 3-month 83-84 0.77

l-month 84-86 6.40*

The chi-square statistic pertains to the joint hypothesis that d=
* denotes rejection of the joint hypothesis at the .05 level.

28


specific risk/return relationship and rationality.

Empirical tests of formal models of exchange rate risk have had
little success in characterizing the nature of the risk premium.©
Consequently, recent work has generally tested for risk neutrality and
rationality, consistently rejecting this joint hypothesis’, but it
remains unclear whether this rejection shows expectations to be biased
and inefficient, or whether it reflects the existence of a time-
varying (and elusive) risk premium.

C, Survey Expectations

The benefits to be derived from studying direct observations of
exchange market expectations should be clear. Survey data allow single-
hypothesis, model-free tests of rationality in foreign currency
markets. Studies which reject the joint hypothesis of spot and forward
market efficiency cannot distinguish whether it was the failure of the
equilibrium model or rational expectations which led to the rejection.
In order to test the rationality hypothesis directly, therefore,
expectations data is essential.

Although survey data have become increasingly accepted in
empirical work, some economists remain wary of potential bias. The

methodology and respondent sample for the Money Market Services survey

€ Hansen and Hodrick (1983) and Hodrick and Srivastava (1984)
provice weak evidence in support of a time varying risk premium using a
single beta latent variable model. Domowitz and Hakkio (1985) find
little evidence for the existence of a risk premium based on the
conditional variance of market forecast errors which are assumed to
follow an ARCH process.

7 See, for example, Tryon (1979), Hansen and Hodrick (1980),
Hakkio (1981), Cumby and Obstfeld (1981, 1983) and Hsieh (1984).

5


B. Forward Market Efficiency

One path that researchers have taken to side-step the "lack of an
equilibrium rate" problem is to examine the forward exchange rate,
testing whether it is an unbiased predictor of future spot rates.
Forward market studies can exploit Fama's "fair-game" approach by
testing whether there exist unusual profit opportunities in the forward
exchange market.

An investor can sign a forward contract at time t to purchase
foreign currency at time t+i at a price, fei =e, + (rE,4 oa)
where the variables are the forward rate (F), spot exchange rate (E)
and the interest rate (R); capital letters denote the level and lower
case letters the logarithm; and foreign variables are denoted with an
asterisk *. At time t+i the investor can turn around and sell the
foreign currency at the spot price ey4;. Presumably the "rational"
investor will only enter into a forward contract if he expects that:
C1) Eg(eeay) 2 ee + (rea > FYbya) = feya
where E; is the expectation conditional on information available at
time t. Further, if the forward market is efficient, then expected
forward market profits should be zero; Ep (egsi) - ft, = 0. However,
forward market efficiency does not preclude the existence of a risk
premium, defined as the excess expected return demanded by investors
for assuming the risk of future changes in exchange rates. So the
relevant forward market equilibrium condition may include an additional
term :

(2) Ey (egyy) - risky; - fe,4 = 0-

Forward market tests therefore involve the joint hypothesis of a


(3) The three-month-ahead forecasts from 1/83 through 10/84
consistently predict depreciation of between one and two percent for
the four currencies while, throughout the sample period, the actual
spot rate appreciated by as much as six percent in some three-month
periods.

(4) The one-month-ahead forecasts consistently predicted appreciation
from 10/84 until 8/85, while actual spot depreciation began in 3/85
for the four currencies. Since 10/85 the one-month-ahead forecasts have
been much closer aligned to actual spot movements.

Why did the three-month-ahead forecasts consistently predict
depreciation in 83-84, while the actual spot rate continued to
appreciate? Why did one-month-ahead forecasts not begin predicting
depreciation until late summer 1985? A few explanations are possible.
First, the MMS forecasts may be consistent with a "rational" bayesian
learning process model. After four years of appreciation, one-month-
ahead forecasts of appreciation in early 1985 may have reflected a
skepticism that the exchange rate would reverse itself so quickly. The
new "depreciation regime" may not have been fully legitimized until the
September 1985 G-5 meeting where a policy of dol lar weakening was
officially sanctioned. Further, if the forecasters perceived shifts in
the exchange process over the estimation sample, the underlying
covariance matrix estimator assumption - that changes in the exchange
rate are egodic (every sequence is equally representative of the whole
process) may not be justified, in which case the standard errors
presented in Table 3 may be suspect. This explanation, however, is less

convincing for the earlier sample three-month-ahead forecasts which

7


where 244; is the i-period ahead spot exchange rate, E, is the
expectation at time t and uy 4 is the forecast error. (Eye¢4; has
typically been proxied with the time t, i-period-ahead forward rate.)
Unbiasedness requires the joint hypothesis that ag=0 and a;=1.

The second type of test checks that relevant information is included
in the forecast. Clearly if any information available at time t is
systematically excluded from the forecast and would improve the
forecast, then the forecast is not optimal, and therefore also not
ratioral. In order to test whether forecasts do incorporate relevant
information we regress forecast errors on specific data! that were
available to the forecasters. One piece of information which a rational
forecaster should consider is his most recent forecast error. This can
be tested by running the regression:

(4) Cepag - Eeetta) = Bo + Bilee ~ Ee-1ee) + Ut,i

If the forecast errors exhibit a significant non-zero mean and serial
correlation! (significant B,) then this implies that the information
contained in past forecast errors was not fully utilized in forming
future predictions.

When the information set is restricted to a constant and the current
or lagged set of forecasts, the tests examine what is commonly called

the weak version of rational expectations (RE). When other, publicaly

10In the strictest sense, the forecast errors should be regressed
on "sll available information" to test for rationality. In the interest
of parameter parsimony, however, only those variables which models
woulc. predict to influence exchange rates are generally included.

llthis, more specifically, tests only that the forecast error,
P’ y

(eta; - Eee +i) is white noise, not that it is an innovation with

respect to (ez, Ep-ete+++)+


Finally, it seems that there is a great deal to be learned from
subjecting survey data on expectations to rigorous tests. This seems
particularly so for high-quality data covering the expectations of
expert participants in asset markets. Further work with the MMS and
similar data, therefore, should generate additional useful perspectives

on exchange rate behavior.

23


International Finance Discussion Papers
Number 281

May 1986

ARE FOREIGN EXCHANGE FORECASTS RATIONAL?
NEW EVIDENCE FROM SURVEY DATA

by

Kathryn M. Dominguez

NOTE: International Finance Discussion Papers are preliminary
materials circulated to stimulate discussion and critical comment.
References in publications to International Finance Discussion Papers
(other than an acknowledgement by a writer that he has had access to
unpublished material) should be cleared with the author or authors.


level-form of actual and expected variables!5, If, however, the
stochastic processes generating the series are non-stationary, the
usual asymptotic theory invoked to construct hypothesis tests is not
justified. While a number of foreign exchange-market studies have
constructed rationality tests using first differences of variables
rather than their levels in order to avoid problems of nonstationarity,
with the exception of Meese and Singleton (1982) and Meese and Rogoff
(1985), none have explicitly tested for unit roots!®,

Granger (1983)17 defines a series to be integrated of order d
(expressed as I(d)), if the series must be differenced d times in order
to achieve stationarity. Similarly, two nonstationary series are co-
integrated of order (d,b) if there exists a linear combination of them
which jointly achieves stationarity!8, We can test whether series, in
level or difference (integrated) form, are stationary by testing for
unit roots.

The Sargan and Bhargava (1983) unit root test, using the Durbin-
Watson statistic, tests the null hypothesis that a series is a Gaussian

random walk against the alternative that roots are strictly less than

15see, for example, Bilson (1978), Frankel (1979) and Frenkel
(1979).

l6yussa (1979) was,one of the first to describe the process
generating the logarithm of spot exchange rates as a random walk.

\7see also, Granger and Engle (1984) and Granger and Weiss (1984).

18r¢ e, and f, are both I(d), there may exist a co-integrating
vector a such that: aey + (l-a)fp~I(d-b), b>0.

10


Temporal alignment of exchange rate data is extremely important as
the rates can vary as much within a day as across days. The actual
market spot and forward rates used in the regression analyses are taken
from Data Resources Inc. and the New York Federal Reserve Bank
International Balance of Payments Data Base. All series are in
logarithms!2 and are the average of the New York certified noon bid and
ask rates. Each MMS median!3 prediction was carefully aligned with the
corresponding spot rate, and, in the one- and three-month-ahead cases,
the one- and three-month-ahead forward rate. The relevant institutional
features of forward contract delivery timing have been taken into
account as discussed in detail by both Meese and Singleton (1980) and
Hsieh (1984)14,

C. Data Stationarity - A Pre-Test

A number of rationality tests in the literature have involved the

12m order to avoid Siegel's (1972) Paradox (which arises because
the expectation of an inverse does not, in general, equal one over the
expectation of the original variable), spot, forward and survey
expectations are in logarithms, thereby ensuring that results are
independent of whether exchange rates are expressed in units of home or
foreign currency.

13~he median is the appropriate value to use for RE tests with the
MMS: data because the thirty individual forecasts are not, in general,
symmetrically distributed. If a distribution is symmetric the mean will
equal the median but, if not, the median measure gives distributional
outliers less influence than does the mean.

14 Forward contracts are set for the same date; however, many
months ahead, (i.e., Jan. 1 to Feb. 1 for an one-month contract) but
delivery takes place two business days later (one day for Canada).
However, unless the trader is holding a covered position, she must
cover for the above example on Feb. 1 as her spot transaction also
takes two business days. This paper assumes that forward contracts are
uncovered so that spot rates are aligned two days before actual
delivery of the forward contract.



IFDP
NUMBER

281

280

279

278

eTT

276

275

274

273

272

271

International Finance Discussion Papers

TITLES

1986

Are Foreign Exchange Forecasts Rational?
New Evidence from Survey Data

Taxation of Capital Gains on Foreign
Exchange Transactions and the Non-neutrality
of Changes in Anticipated Inflation

The Prospect of a Depreciating Dollar
and Possible Tension Inside the EMS

The Stock Market and Exchange Rate Dynamics
Can Debtor Countries Service Their Debts?
Income and Price Elasticities for Exports
of Developing Countries

Post-simulation Analysis of Monte Carlo
Experiments: Interpreting Pesaran's (1974)

Study of Non-nested Hypothesis Test Statistics

A Method for Solving Systems of First Order
Linear Homogeneous Differential Equations
When the Elements of the Forcing Vector
are Modelled as Step Functions

International Comparisons of Fiscal Policy:
The OECD and the IMF Measures of Fiscal
Impulse

An Analysis of the Welfare Implications of
Alternative Exchange Rate Regimes: An
Intertemporal Model with an Application

1985
(partial listing)

Expected Fiscal Policy and the Recession
of 1982

Elections and Macroeconomic Policy Cycles

Please address requests for copies to International Finance Discussion
Papers, Division of International Finance, Stop 24, Board of Governovs of

the Federal Reserve System, Washington, D.C.

20551.

42

AUTHOR(s)

Kathryn M. Dominguez

Garry J. Schinasi

Jacques Melitz

Michael K. Gavin
Jaime Marquez

Caryl McNeilly

Neil R. Ericsson

Robert A. Johnson

Garry Schirasi

Andrew Feltenstein
David Lebow
Anne Sibert

William H. Branson
Arminio Fraga
Robert A. Johnson

Kenneth Rogoff
Anne Sibers


Appendix D. Graphs of Expected and Actual Depreciation

Figure 1
Two-Week $/Sterling
Expected and Actual Depreciation

-0.008
-0.01
0.015
0.08 PTT TTT TTT TT TT TTT TT
1083-1 1083-0 1983-I  1983-V 1984-1 toes tose toeK—V
MMS 26k Premiums. + $/Stig Depreciation
Three-Month $/Sterling
Expected and Actual Depreciation
cos
02
° AN
ws
001
~0.02
-0.03
~008
~008
{aS 19a3—u teas tas 1984-1 1084—N 19Ms—I 108
MNS 3-MO Premium + 3/374 Depreciation

30


In two of the eight cases, the null hypothesis of conditional
homoscedasticity is rejected. These results suggest that Hansen and
Hodirick's (1980) asymptotic covariance matrix may not always be
justified in conducting hypothesis tests on the coefficients of
unbiasedness tests with the survey data. Further, Hsieh's (1984)
results with non-overlapping data suggest that tests assuming
conditional homoscedasticity when the assumption is not justified often
underestimate the standard errors of the OLS coefficients, biasing
toward rejection.

Hansen (1982) and Hansen and Hodrick (1983) provide a
heteroscedasticity consistent covariance matrix estimator which does
not require knowledge of the nature of the heteroscedasticity. This
alternative estimate of 2, can be obtained by calculating a consistent
estimate of the spectral density matrix at frequency zero of the vector
stochastic process Leg '¥] (where ve are, again, the OLS residuals of
the unbiasedness regression).

The one- and three-month-ahead equations in the paper were
estimated with Hansen's (1982) case(v) heteroscedasticity consistent
covariance matrix; assuming a first-order moving average process for
the bi-weekly one-month-ahead disturbances and a fifth-order moving

average process for the bi-weekly three-month-ahead disturbances.

29


should, however, inspire unusual confidence. The thirty respondents are
professional exchange rate forecasters; most work in foreign currency
trading divisions of major commercial banks. The standard arguments

8 - in particular, that they do not adequately reflect

against surveys
the decisive players in the market - are clearly less problematic with
this sample than in most surveys. There is equal reason to have
confidence in the survey's design and execution. Unlike many surveys
which are administered through the mail, the MMS survey is conducted by
telephone each Wednesday afternoon eastern standard time (EST). Such
accurate timing minimizes the problem of different information sets

9

across forecasters.

Il.Description of the Tests, the Data, and Estimation.

A. Tests of the Rational Expectations Hypothesis

The rational expectations hypothesis states that rational
participants in the market have expectations which are optimal
forecasts using all available information. The usual rationality tests
found in the literature have been of two basic sorts: the first tes:s
whether the forecasts are unbiased estimates of the actual series, and
the second, whether forecasts incorporate available information.
Commonly, unbiasedness is tested by running the OLS regression:

(3) egg = 00 + ay Eeepey + Ue, i

8see Appendix B for further discussion of the usefulness of survey
data.

%Frankel and Froot (1985) and Froot (1985) use a semi-annual mail
survey published by Amex Bank Review and the Economist Financial Report
to examine exchange rate expectations. Although their results are
consistent with this papers findings, because the respondants were not
polled simultaneously and the number of survey dates was small, the

power of their tests is probably considerably lower.

6


TABLE 4 (e443 - Ere,

CURRENCY HORIZON SMPL

$/STLG 2-WK 83-84 0023.43
1-WK 84-86 001 1.85
$/DM 2-WK 83-84 001 6.31%
1-WK 84-86 001 1.07
$/SWF 2-WK 83-84 +001 10. 09%%*
1-WK 84-86 001 0.29
$/YEN 2-WK 83-84 -014 4.85
1-WK 84-86 -001 3.64

* denotes rejection at the .05 level and ** at the .01 level for the
hypotheses that bg=0 and b,=0. The chi-square statistic pertains to the
joint hypothesis that bo=b)=0

While three of the two-week-horizon equations show significant
constant terms, all equations show insignificant serial correlation.
This indicates that although the short-horizon forecasts were generally
biased, they did efficiently incorporate information from past errors.
For the one-month and three-month-ahead cases, since the previous
forecast errors are not available before the next forecast, an
analogous test is not feasible.

The one- and three-month-ahead forward rates provide strong-form
tests of rationality. Since the MMS survey is taken after 12pm EST; <he
noon N.Y. forward rate is available to forecasters. Table 5 presents
the estimated regression results of the MMS forecast error on the
appropriately aligned one- and three-month-ahead forward premium
(fe,i > ep).

20


Figure 8
One-Week $/Yen
Expected and Actual Depreciation

-0.005

0.01

0016

1984 foas-1 toes 1988-1 1988-1 1o8e—1

MMS WEEK—PREW + $/VEN-vePRECUTION

One-Month $/Yen
Expected and Actual Depreciation

8B TTT TTT Td
1984—v 1pes-1 1988-1 1985-1 1988-1 pee!

5 MMS MO-PREN + | $/YEN-DEPRESIATION

37


REFERENCES

Akerlof, George A. and Janet Yellen, "Can Small Deviations from
Rationality Make Significant Differences to Economic Equilibria?,"
American Economic Review, September 1985, 75, pp.708-735

Bilson, John, "Rational Expectations and the Exchange Rate," in The
Economics of Exchange Rates ed. by Frenkel and Johnson, Addison-Welsey,
1978.

Branson, William H. and H. Halttunen and P. Masson, "Exchange Rates in
the Short Run: The Dollar-Deutschemark Rate," European Economic Review,
December 1977, 10, pp.303-324.

Carlson, John A. "A Study of Price Forecasts," Annals of Economic and
Social Measurement, No.1, 1977, pp.27-56.

Cumby, Robert and Maurice Obstfeld, "A Note on Exchange Rate
Expectations and Nominal Interest Differentials: A Test of the Fisher
Hypothesis," Journal of Finance, 1981, 36, No.3, pp.697-704.

Cumby, Robert and Maurice Obstfeld, "International Interest Rate and
Price Level Linkages under Flexible Exchange Rates: A Review of Recent
Evidence," in Exchange Rate Theory and Practice, ed. by Bilson anc.
Marston, University of Chicago Press, (for NBER) 1984, pp.121-149.

Dominguez, Kathryn, "Are Foreign Exchange Forecasts Rational? New
Evidence from Survey Data," Economics Letters, 1986, forthcoming.

Domowitz, Ian and Craig S. Hakkio, "Conditional Variance and the Risk
Premium in the Foreign Exchange Market," Journal of International
Economics, 1985, pp.47-66.

Dornbusch, Rudiger, "Flexible Exchange Rates and Interdependence," IMF
Staff Papers, March 1983, pp.3-30.

Edison, Hali J., "The Rise and Fall of Sterling: Testing Alternat:ive
Models of Exchange Rate Determination," Applied Economics, 1985, ~7,
pp. 1003-1021.

Engle, Robert and D.F. Kraft, "Multiperiod Forecast Error Variances of
Inflation Estimated from ARCH Models," in Applied Time Series Analysis
of Economic Data, ed. by A. Zellner, 1981, Bureau of the Census.

Fama, Eugene F., "Efficient Capital Markets: A Review of Theory and
Empirical Work," Journal of Finance, May 1970, 25, pp.383-417

Frankel, Jeffrey, "On the Mark: A Theory of Floating Exchange Rates
Based on Real Interest Differentials," American Economic Review, 1979,
69, pp.810-822.

38


APPENDIX B. On the Usefulness of Survey Data

Although many readers will not require justification for using
survey data, such data have often been dismissed by economists claiming
that surveys are not representative of the agents who drive the market.
The usual example given, is a market with a few rational agents whose
actions ensure rationality at the margin despite the irrationality of
the majority (Miskin 1981).

The rationality on the margin argument may bear relevance.
However, no one has been able to provide convincing theoretical
evidence which shows how such a marginal condition might apply. Indeed,
newer work2® concludes that only under a very restricted set of
conditions is rationality of some but not all agents sufficient to
ensure rationality in the market result overall.

There is little debate that use of massive surveys along the

lines of the Livingston series2/

is likely to bias the tests of
rationality improperly towards rejection (in the sense of examining the
rationality of macroeconomically irrelevant expectations). However, one
way around the problem is to use surveys of professional forecasters.
Since the success and indeed livelihood of a professional forecaster is
presumably a function of accuracy, the use of professional forecasters
as representative agents will ensure that those examined are those most

26 See, for example, Akerlof and Yellen (1985), Haltiwagner and
Waldman (1985).

27Numerous papers have examined this bi-annual data set on
inflation forecasts compiled by Joseph Livingston of the Philadelphia
Inquirer. See, for example, Carlson (1977) for tests of unbiasedness
and efficiency. Struth (1984) finds that a simple Kalman filter using
only past price information can out-perform the Livingston Forecasts.

26


The first section of the paper describes the joint hypothesis
problem implicit in tests of market rationality and the consequential
usefulness of direct observations of market expectations to circumvent
the problem. Section II describes the rationality tests, the data and
the method of estimation used in this study. Section III presents the
regression results of the rationality tests using concensus forecasts.
Finally, section IV provides a summary and conclusions.

I. The Joint Hypothesis Problem and a Circumvention

A. Exchange Rate Efficiency

The decision to change from the Bretton Woods fixed exchange rate
system to market-determined exchange rates in 1971 dramatically changed
the research focus of empirical work on exchange rates. The economic
argument for changing to a floating exchange rate regime centered
largely on the issue of market efficiency?. In an efficient market,
exchange rates should fully reflect all available information, and
rational economic decisions based on these relative prices should
insure an efficient allocation of resources. After the change to
flexible exchange rates, empirical research therefore focused on
testing the efficiency hypothesis, i.e., testing whether floating
exchange rates did indeed serve as efficient aggregators of
information.

One approach to testing exchange rates' informational efficiency,
advocated by Fama (1970), argues that efficiency requires that

actual prices (or rates of return) follow a "fair game" process

3geminal papers on the topic include: Friedman (1953) and Johnson
(1969).


IF DP
NUMBER

270

269

268

267

266

265

264

263

_ 261

260

259

253

257

International Finance Discussion Papers

TITLES

Assertion Without Empirical Basis: An
Econometric Appraisal of Monetary Trends

in the United Kingdom by Milton Friedman
‘and Anna J. Schwartz

Canadian Financial Markets: The Government's
Proposal for Reform

Was It Real? The Exchange Rate Interest
Differential Relation, 1973-1984

The U.K. Sector of the Federal Reserve's
Multicountry Model: The Effects of
Monetary and Fiscal Policies

Optimal Currency Basket in a World of
Generalized Floating: An Application to
the Nordic Countries

Money Demand in Open Economies: A Currency
Substitution Model for Venezuela

Comparing Costs of Note Issuance Facilities
and Eurocredits

Some Implications of the President's Tax
Proposals for U.S. Banks with Claims on
Developing Countries

Monetary Stabilization Policy in an Open
Economy

Anticipatory Capital Flows and the Behaviour
of the Dollar

Simulating Exchange Rate Shocks in the MPS
and MCM Models: An Evaluation

Trade Policy for the Multiple Product
Declining Industry

Long Memory Models of Interest Rates, the
Term Structure, and Variance Bounds Tests

Currency Substitution and the New Divisia
Monetary Aggregates: The U.S. Case

The International Transmission of Oil Price
Effects and OPEC's Pricing Policy

43

AUTHOR(s )

David F. Hendry
Neil R. Ericsson

Garry J. Schinasi
Richard Meese
Kenneth Rogoff

Hali J. Edison

Hali J. Edison
Erling Vardal

Jaime Marquez

Rodney H. Mills

Allen B. Frankel

Marcus H. Miller

Arnold Kling

Arnold Kling

Catherine Mann

Gary S. Shea

Jaime Marquez

Jaime Marquez


D. The Estimation Method

In the following section, all models using one- and two-week-ahead
forecasts were estimated over the four currencies using seemingly
unrelated regressions (SUR). Currency arbitrage implies that error
terms across currencies are likely to be comtemporaneously correlated.
Zellner's (1962) joint estimation technique resulted in an efficiency
gain because individual equation disturbances were contemporaneously
correlated. If the error terms had not been correlated, SUR would be
identical to four stacked OLS regressions.

While the one- and two-week-ahead forecasts are nonoverlapping,
the one- and three-month-ahead forecasts are surveyed bi-weekly and
therefore do overlap. If the sample size were not already small,
constructing a nonoverlapping data set would be preferable. Hansen and
Hodrick (1980,1983) and Hayashi and Sims (1983) discuss the econometric
problem which arises when the sampling interval is finer than the
interval over which forecasts are made. In our case, because the
actual spot rate corresponding to the preceeding period one- and three-
monti-ahead forecast is not available for another two or six periods
respectively, the disturbances are no longer guaranteed to be serially
uncorrelated. The OLS estimate of a, in a regression of the form
ect, = OQ + OpELeegy + Uta should remain consistent as long as Ut i
and Epey44 are not correlated, but the standard errors will be biased
when any lagged values of Up, i are not in the information set
(ie. Elug ilut-z,i] is non-zero for all i>l and k<i, where i=forecast
horizon and k=sampling interval).

In order to correct for bias, all the one- and three-month-ahead

13


$/STLG 2-WK = 83-84. -.005 034 1.99 .003 96.2%
(.003)* (.119)**
1-WK 84-86 001 -.171 1.85 .05 44.3%
(.003)  (.181)**
$/DM 2-WK 83-84 -.004 +122 1.79.03 155.3%*
(.002)* (.095)**
1-WK 84-86 002 049 1.85 .01 48.8%
(.003) (.137)**
$/SWF 2-WK 83-84. -.004 101 1.80 .01 84.0%%
(.002)* (.091)**
1-WK 84-86 002 +064 1.88 .01 62.3%*
: (.003) (.118)**
$/YEN 2-WK 83-84 -.001 166 2.05 .02 69.3%*
(.002) (.100)**
1-WK 84-86 +003 502 1.59.07 12.8%*
(.002)  (.146)**
TABLE 3 (e¢4; - ep) = ag + ay(Eee¢sy - e¢) + ue,y 54,12 (WK)
CURRENCY HORIZON SMPL ag ay MA R2_—Chi2(2)
$/STLG 3-MO 83-84 -.029 -.450 5 01 39.23%*
(.015)* (.395)**
1-MO 84-86 = -.001 -.505 1 +05 21.44%
(.006) (329) **
$/DM 3-MO 83-84 -.043 412 5 01 23.26%*
(.016)** (.529)
1-MO 84-86 :014 = -.248 1 +01 15.07%*
(.007) (392) **
$/SWE 3-MO 83-84 -.033 +054 5 -001 135.38**
(.009)** (099)
1-MO 84-86 012 -.374 1 02 11.52%*
(.008)  (.425)**
$/YEN 3-MO 83-84 .003 -.457 5 +02 9.38%%*
(.016) (.626)**
1-MO 84-86 015 +341 1 +01 4.98
(.008) (.359)

The numbers in parentheses are the estimated standard errors of the

coefficients, * denotes rejection at the .05 level and ** at the .01
level. for the hypotheses that ag=0 and aj=1. MA is the moving average
assumption for the disturbances. The chi-square statistic pertains to

15


Are Foreign Exchange Forecasts Rational ?
New Evidence from Survey Data
by

Kathryn M. DominguezTM

INTRCDUCTION

For two decades, theories of expectations formation --
particularly the rational expectations hypothesis -- have been at the
forefront of economic research. More recently empirical work has begun
to focus on testing the operational validity of rational
expectations! Survey evidence on the formation of expectations has,
as a result, become an important data source. This paper examines the
rationality of a newly available set of survey data on foreign exchange

rate forecasts provided by Money Market Services (ms )2.

*This paper represents the views of the author and should not be
interpreted as reflecting the views of the Board of Governors of the
Federal Reserve System or other members of its staff. This paper is
based on a chapter of my Ph.D. thesis and was written while I was an
intern in the International Finance Division of the Federal Reserve
Board. I am greatly indebted to my advisors Ray Fair and Matthew
Shap: ro for their support, advice and detailed comments. Special thanks
are also due to Neil Ericsson, Hali Edison and Michael Gavin for
comments and help in drafting this version of the paper.

lRationality in the Muth (1961) sense is defined as an expectation
which is "own-model-consistent". This definition requires that there be
a specific model which agents both know and use to formulate their
expectations. Typically, however, tests of rationality are not based on
a specific model but require that the prediction error be uncorrelated
with "all available information".

2~hanks to Mark Porter and David Broder from MMS for providing me
with the survey data.


o

0.02

~o01

-e02

Figure 7
One-Week $/Swiss-Franc
Expected and Actual Depreciation

TTT TTT ed

s9a4—1v 1oas-1 pes 1985-0 1985-1 toee-1
MMS WEEK—PREW + 9/Swr-DEPREcIATION
One-Month $/Swiss-Franc
Expected and Actual Depreciation
0.06
00s
203
O02
201
r)
-o01
-0.02
-003 4
8.8
1984 ypes-1 toas—i 1oas—in 1pas—1v ‘98-1
MMS WO-PREW + 3/Swr—CEPRECIATION


Appendix A. The Monetary Approach to Exchange Rate Determination

The standard monetary approach to exchange rate determination is
illustrative both of the fundamental variables typically included in
equilibrium models and of the central role of the hypothesized
expectations formation process.

In the monetary approach the exchange rate is viewed as the
relative price of one country's money in terms of another. The money
demand equations conventionally used for both the domestic and foreign
countries depend on income and interest rates and, for simplicity, many
models assume that the elasticities with respect to income and interest
rates are identical for both countries:

(Al) m

p tay - Br

(A2) n* p* + ay* - Br*®

where, the variables are money (M), prices (P), income (Y), the
interest rate (R), and the exchange rate (E); capital letters denote
the level and lower case letters the logarithm; and foreign variables
are denoted with an asterisk *. Goods prices in the two countries are
assumed to be perfectly flexible and substitutable so that purchasing
power parity holds instantaneously.

(A3) e =p-p
We can solve equations (Al) and (A2) for prices and then substitute in
equation (A3) to yield the exchange rate as a function of relative
money supplies, income and interest rates.

- m*) + aly* - y) + B(r - x)

(A4) e =
We can further refine equation (A4) by including time subscripts and

substituting the covered interest parity condition for the interest

24


ABSTRACT

Tests of rational expectations in foreign exchange markets have
been inconclusive because of disagreement over the underlying asset
pricing model. This paper uses a newly available set of data on foreign
exchange forecasts to examine directly expectations formation in four
foreign currency markets. Generally, results do not support the simple

rational expectations hypothesis.


Figure 3
Two-Week $/Swiss-Franc
Expected and Actual Depreciation

0.012
0.01
2.000
0.006
0.004
0.002

0.002
0,004
~o.c0e
-0.008

-0.01
0.012
-oo14 4

coor 4

-ooe

1983-1 1983-1 pas — 1983_W 1pa4—1 1984-1 1984 1984-1

MMS 2—"k Premiuma + 3/SWF Depreciation

Three-Month $/Swiss-Franc
Expected and Actual Depreciation

oo ln oil an, PeaeeePes d

-eot 4

-002 4

-003 +

~ooe 4

1983-1 1983-1 toas— 1983 1984-1 1084" 984m = 1984—Wv

0 MMS 3-MO Premiums + 3/SWF Depreciation

32


Siegel, Jeremy J. "Risk, Interest, and Forward Exchange,"
Quarterly Journal of Economics, 86, 1972, pp- 303-9.

Struth, Friedrich K., "Modelling Expectations Formation with Parameter-
Adaptive Filters: An Empirical Application to the Livingston
Forecasts", Oxford Bulletin of Economics and Statistics, August 1984,
46(3), pp.211-239.

Tryon, Ralph, "Testing for Rational Expectations in Foreign Exchange
Markets," Federal Reserve Board International Finance Discussion Paper,
No. 139, 1979

Zellner, Arnold, "An Efficient Method of Estimating Seemingly Unrelated
Regressions and Tests of Aggregation Bias," Journal of the American
Statistical Association, 57, pp.348-368.

4l


likely to have the most rational expectations. The performance of
professional forecasters therefore may be taken to define an upper
bound on the expectational accuracy of agents in the economy as a whole

(Nordhaus and Durlauf 1984).

Apper.dix C. Estimation with Overlapping Data

With the MMS data, at time t when MMS respondents are bi-weekly
forecasting the month ahead forecasts, E,e¢4,, they will not know what
their previous periods forecast error was because e447 is still a
forecast period (two weeks) away. Therefore, it is likely that
Elue,4le-2,4] is non-zero. Hansen and Hodrick's (1980) method
(generalized latter in Hansen 1982) is first to obtain consistent
estimators of E(epe¢4,') and E(ug,iMtte,i' ) for k=-itl,...,0,..,i-1.

They show that if e, is stationary and ergodic and k>0,

T
(cl) 1/T £ e,er_,' > E(epey4,') (almost surely) and
rope otet-k t&ttk
T 4 “ 1 '
(c2) A/T, Be kU t-k, > E(ug, pue¢k,i') (almost surely)

where aes is the OLS residual for observation t with a sample size T.
Using these relationships they can then obtain a consistent estimate of
the asymptotic covariance matrix :
(C3) T(Ep! Ep) Ep SpE p(Ep' Ep)!

A irl 6T A
where 1/T(Ep'QpEp)= = E Gpep'e,-,0¢-, and Ep is the matrix of T
e U/T(Ep'OpEp)= FR Geet ee-Ke-k T

stacked observations on ey.
However, this asymptotic covariance matrix is not justified in our

case, unless the conditional covariances of forecast errors with

27


one, a stationary process!9, The test involves regressing the relevant
series on a constant and comparing the D.W. for the residuals to the
bounds provided in Sargan and Bhargava (1983).

Table 1(a) presents unit roots tests on the level of the spot
(et4;), forward (f,,4), and expectations (Epet4;) series. With the
exception of the one-week and three-month-ahead $/SWF spot expectation
series, the tests cannot reject the hypothesis that unit roots are
present in the first-order autoregressive representations of the
logarithms of the spot, forward and expectations variables for all the
level series. Table 1(b) presents tests using rates-of-change of the
one- and two-week and one- and three-month-ahead spot rates. In
contrast to the level-form results in Table l(a), the change-form
variables do appear stationary29, Finally, Table 1(c) presents tests of
whether the forward rate and the expectations series are co-integrated
with the contemporaneous spot exchange rate. Intuitively we are testing
whether the actual and expectations series tend to move together over
time. Unit root tests on the differences between each of the two

expectations series and the spot exchange rate, termed the forward and

19 , number of unit root tests of the random walk hypothesis are
available in the literature. The Sargan and Bhargava (1983) test
results are presented here because their alternative hypothesis,
(stationary process) is highly restrictive. Other unit root tests have
as alternative hypotheses that roots do not equal one (potentially
stationary) or that the the data generation process is not white noise.

20 The tests for stationarity were performed on the rate-of-change
of the variable between the forecast and the current period because
this form is easily interpreted in regressions. However, because lags
can always be re-parameterized (ie. (ey¢42 - e¢) = (et42 - ete) +
(ey-41 - e¢)), omission of lagged variables does not vitiate the tests.

11


equations in the following section are estimated with the Hansen (1982)
heteroscedasticity consistent asymptotic covariance matrix, assuming a
moving average process of order one for the bi-weekly one-month
disturbances, and of order five for the bi-weekly three-month
disturbances2!,

III. The Test Results

A. Tests of Unbiasedness

Tables 2 and 3 present regressions of actual spot depreciation
(e444 - ep) on forecasted depreciation (Erets¢a - e,). Rationality
requires that the coefficient on forecasted depreciation be one, the
constant be zero, and the disturbances be innovations with respect to
the complete set of information available at time t. In addition to the
finding that the variables in change-form appear stationary, Tryon
(1979) shows that using the change in the spot rate, rather than the
level, constitutes a more stringent test of rational expectations. The
change-form test of unbiasedness is to estimate the regression
coefficents in:

(6a) (ety; - eg) = ap + ay (Eeeegy - eg) + upg

Adding the current spot, e;, to both sides and rearranging terms we
obtain:

(6b) egyg = ag + ayEpeeyy +(1-az dee + upg

The change-form regression, by explicitly including the current spot
rate in the regression, allows us to distinguish whether it is the

current spot rate or the forecast that actually has predictive power.

21se60 Appendix C for a more detailed discussion of the method
of estimation.

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