~ 19 -
Table II
ESTIMATES OF THE (MT) EQUATION FOR THE DM/DOLLAR RATE

JULY 1973 - JULY 1980 a
USING MCCALLUM'S INCONSISTENT PROCEDURE” .

Explanatory :
Variables Case I Case II Case III Case IV
constant - 026004 -.01623 -.01732 -.91695
std. error -00928 .01451 01420 0124
t-stat -2.80216 -1.11825 -1,21929 -1.36921
Fr -56753 ~13592 «1545 +1289
std. error -05822 + 143209 «13902 .1277
t-stat 9.74766* -949115 © 1.112 1.0099
See -507151 «901773 8863 9095
std, error 050461 .12173 ~11824 - 10890
t-stat 10.0503* 7.4076* 7.4957* 8.3512*
Fey -.1608
std. error ~1089
t-stat -1.476
2 556°
R’ +973 -9537 9552 9556
S.E.R. -0115 -01516 0149 -0148
DW. 1.2857 2.2789 2.2546 1.9913

3 the data series used here were 30 day eurocurrency interest
rates and forward and spot exchange rates quoted on the interbank
market (see section III.A for a further discussion).

b

R2 is shown for Fe not F. - oF

t-1l°

Indicates that the coefficient estimate is significant at the
.05 significance level.


- F7 -

log or level of the spot exchange rate was rejected. Thus the procedure
followed seems satisfactory.

28/ Interbank Forward and Spot exchange rate data was obtained from the
Federal Reserve Bard data base. These exchange rate series were bid
quotes at noon of the given day. As John Wilson and Ralph Smith have
pointed out, these interbank rates are not subject to the sort of "limits"
imposed on the movement of forward and spot rates in the IMM market,.

29 The Eurocurrency rates were obtained from DRI and are London Interbank
offered rates (LIBOR) for Eurodollars and Euromarks.

30/ The fact that Y)* YQ * lwhereas Y) is close to zero is described by
McCallum [1977] as the Stoll Kesselman Haas (s-k-HO condition. These are
restrictions on the coefficients suggested by the MT of forward foreign
Exchange. The term Yo represents the behavior of hedgers, Yy the behavior
of arbitraguers and v2 the extent of the effect of speculation on the
determination of the forward exchange rate.

31/ The procedures adopted to estimate the forward exchange rate equation
(10) would be more efficient if the weighting matrix (reflecting the MA(1)
structure of the error term) was used in a second stage criterion function
which can be minimized to obtain parameter estimates and standard errors. These
procedures have been outlined in Glaessner [1982], Cumby et al. [1981] and

in Hansen [1979, 1980]. Use of these somewhat more efficient procedures
did not tend to change the results reported below.

32/ This subroutine was obtained from Kent Wall at the University of
Virginia. The author would especially like to thank Richard Meese for help
in devéloping the software to do many of the calculations performed here.

33/ A window of 7 ordinates was used for the small sample case of 1973-1979
whereas a window of 9 ordinates was used for the sample period 1973-1980.

34/ See Herring and Marston [1976] who have argued that covered interest
arbitrage holds for eurocurrency deposits issued by a given bank.


- F3 -

10/ To see this note that

E[v.v = -
¥e-1) = Eley + ergne - YpMe4y) Copy * PYaMey > YQMe)

= Elepes 3) + Pv.E(E My) - YoE(E,n,) + PYZE(M Le, 3)

22 2 2
+ ey, E(mi ney) - PY, E(n.”) - YoE (ME Ee)

2 2
+ pry E(me,y Mey) * V2 E(n. 437.) # 0

2, _ <2
because E(e,n.) # 0 and E(n, ) = 6h.

Moreover, E{v.v_,,] # 0 for similar
reasons. beet

1l/ This distinction was first introduced by Hansen and Sargent [1980,
1981] and is adopted here both as an expositional device and to explain
the presence of an error term in the equation.

12/ This is true by virtue of the fact that ¥ is a subset of Ths I

t-1 tel?

and oe .« See Glaessner [1982] for a further explanation.

13/ This point is made in more detail by Hansen [1979, 1980] and Hansen and
Sargent [1981], Cumby et al [1982], McCallum [1979B], and Scott [190].

14/ This point is discussed in Hansen [1982], Obstfeld [1982], and Cumby
and Obstfeld [1983].

15/ Cumby and Obstfeld [1983] develop a test for conditional heteroskedas+:
ticity which is applied to forward rate forecast errors for many bilate:ral
currencies. They find that in most cases the forward rate forecast error
is conditionally heteroskedastic,.

16/ The procedure developed by Hansen is designed to yield consistent but
not necessarily asymptctically efficient estimates of coefficients when the
error term follows a general moving average process. See, for example,
Hansen and Hodrick [1980] or Meese and Singleton [1980] where three month
forward contract data is used, given a much smaller sampling interval.

17/ The class of gneralized method of moments (GMM) estimators developed

by Hansen [1979, 1982] and Hansen and Sargent [1980, 1981] explicitly

utilize the orthogonality conditions implicit in various theoretical models
to form estimates of a particular parameter vector. The types of nonlinear
instrumental variables procedures pursued in the present paper make use of
the orthogonelity conditions between the instruments and the residuals to
form consistent estimates.of the parameters in the model. Indentificat‘on

of particular parameters can be cast in terms of the number of orthogonality
conditions present in particular case at.shand vis a vis the number of para-
meters the investigator wishes to estimate. It turns out that the number of
orthogonality conditions depends upon (1) the number of instruments, and (2)
upon the lag lengths chosen for each instrument. All of these considerations
are important in the procedures described below. For a much more explicit
ttreatment of these issues, and for a derivation of optimal GYM estimators, see
Hansen [1980] or Hansen and Sargent [1981]. For an example of the application


-B4-

Geweke, J., 1978, "Testing the Exogeneity Specification in the Complete Dynamic
Simultaneous Equation Model", Journal of Econometrics, pp. 163-185.

Geweke, J., Dent, W., 1978, "On Specification in Simultaneous Equation Models",
SSRI, University of Wisconsin, Madison.

McCallum, B.T., 1979A, "Testing for the Role of Speculation in the Forward
Exchange Market: A Reply" Review of Economics and Statistics, pp. 611-612.



-13-

matrices are defined as before. For hypothesis testing we use

sae sels gel
§ it) = wy, TE? 8,

The expression for the var(y) can be simplified further when we write

$= shy 1 where SIis the matrix of smoothed periodogram and cross

periodogram ordinates evaluated at the zero frequency and multiplied

by the scalar 27T. Thus, the Var(¥) becomes

(15) var(Y) = Ai si g74) 23/
4x4 4x4 4x4
In the present:case the construction of & will involve the estimation
of ten spectral-and cross spectral density functions.

In sum, the procedure developed above suggests how consistent
estimates of the coefficients and asymptotic standard errors can be
obtained. However, interpretation of the empirical results employing
these procedures are subject to several qualifications. First, consistent
estimation requires that the investigator know the disturbance correlation
length. Sims [1980] has argued that this a priori identifying information
is often unavailable. Thus hypothesis testing frequently involves tests
of joint hypotheses including assumptions about disturbance correlation
lengths as is the case in the empirical application below, where an AR(1)
process on ut has been assuméd. Second, a requirement of the procedures
discussed above is that the regressors follow a jointly erodic covariance

’
stationary process. Meese and Singleton [1980] have recently conducted
tests for unit roots in univariate autoregressive representations of the

log of the spot and forward exchange rate for several countries.


- 23 -
Table iit4

ESTIMATES OF THE NONLINEAR (MT) EQUATION (5.10)
FOR THE DM/DOLLAR FORWARD RATE ‘ -
MARCH 1973-JUNE 1979
USING MCCALLUM'S CONSISTENT PROCEDURE

Explanatory "Corrected" "Corrected"
Variables Coefficient Standard Error t-stat
constant -.0121570 - 13821789 - 0879555

Fy 9963831 . 3010503 3.309689"
Stay - 023768027 .0571002 -.4162506
, Fie 902132 1168834 7.71822"

AThe software used to perform these calculations makes use of a
version of the Davidon Fletcher Powell minimization routine provided
by Kent Wall.

*
Indicates that the coefficient estimate is significant at the
.05 significance level.


BIBLIOGRAPHY

Callier, P., 1980, "Speculation and the Forward Foreign Exchange Rate:
A Note," Journal of Finance.

serene » 1981, "Speculation, Interest Arbitrage, and the Forward Foreign
Exchange Rate of the Canadian Dollar: Updated Evidence," Journal of
Macroeconomics, Spring, Vol. 3, No. 2, pp. 293-299.

Carvalho, J. L., Grether, D. M., Nerlove, M., 1979, Analysis of Economic
Time Series A Synthesis, Academic Press.

Cumby, R. E., Huizinga, J., Obstfeld, M., 1980A, "Two-Step Two-Stage
Least Squares Estimation in Models with Rational Expectations,"
forthcoming in the Journal of Econometrics.

wereee » Obstfeld, M., 1980B, "Exchange Rate Expectations and Nominal
Interest Differentials: A Test of the Fisher Hypothesis," forth-
coming in the Journal of Finance.

Cumby, R., Obstfeld, M., 1983, "International Interest-Rate and Price
Level Linkages Under Flexible Exchange Rates: A Review of Recent
Evidence." In Exchange Rates: Theory and Practice, edited by
John F. 0. Bilson and Richard C. Marston. Chicago: University of
Chicago Press for the National Bureau of Economic Research.

Dickey, D. A., Fuller, W. A., 1979, "Distribution of the Estimators for
Autoregressive Time Series with a Unit Root, "Journal of the American
Statistical Association, 74, pp. 427-431.

Dooley, M., Isard, P., 1980, "Capital Controls, Political Risk, and Devi-
ations from Interest Rate Parity," Journal of Political Economy,
Vol. 88, 21, pp. 370-384.

Driskill, R. A., McCafferty, S., (1981) "Spot and Forward Rates in a
Stochastic Model of the Foreign Exchange Market." Paper presented
at the Winter 1981 Econometric Society Meetings, Unpublished.

Fair, R. C., 1970, "The Estimation of Simultaneous Equation Models with
Lagged Endogenous Variables and First Order Serially Correlated
Errors," Econometrica, 38, pp. 507-516.

Fletcher, R., Powell, M.J.D., 1963, "A Rapidly Convergent Descent Method
for Minimination," Computer Journal, Vol. 6, pp. 163-168.

Flood, R. P., Garber, P. M., 1979, "A Bitfall in Estimation of Models
With Rational Expectations," Journal of Monetary Economics, forth-
coming.



with it and ip denoting the interest rate (in terms of monthly rates of
return) in Germany and the United States, respectively. Finally, like
McCallum [1977] it is assumed that Ry is exogenous and that Fy and S¢
are simultaneously determined endogenous variables L/

To obtain parameter estimates of the coefficients in equation (1)',
McCallum and Callier both employed an instrumental variable technique (see
McCallum [1976]). his procedure must be used in order to overcome the
classical errors in variable problem which is present in (1)' due to the:
appearance of the explanatory variable, S_4). However, it also leads tc
inconsistent parameter estimates of (1)' and to inconsistent asymptotic
standard errors. This becomes evident by focusing attention upon the
complications in estimation introduced by the term S-4).

First, assume that the structural distrubance u, in (1)' is a white
noise process with zero mean, finite variance, oe and Efu,uy_ 5] = 0 for j #0.
The assumption of rational expectations implies

sé

(3) Ste. = cH * Mey

\t,] and n, the "true" forecast error, is serially

t+1’
uncorrelated. Also, E(n..,/1,] = 0, by the properties of the linear least

e
where Seed = E[Se 41

squares operator. Agents are assumed to have full current information, where

the information set is written as 1, .8/

Substituting (3) into (1)' results in

*
“ Feet Yo* YaFe * YoSte1 * Ye


FOOTNOTES

*/ Economist, Division of International Finance, Board of Governors of
the Federal Reserve System. The views expressed in this paper are solely
those of the author and should not be interpreted as those of the Board

or other members of its staff. The paper is from the author's doctoral
dissertation ("Theoretical and Empirical Essays on the Determination of
Spot and Forward Exchange Rates," University of Virginia - Charlottesville,
1982). The author is greatly indebted to both of his thesis advisors
Robert P. Flood and Richard Meese for their support and help in the
writing of this paper. The author would also like to thank Richard Cervin,
Robert Cumby, Marjorie Flavin, Richard Haas, Peter Hooper, Bennett McCallum,
Maurice Obstfeld, and Louis Scott for helpful comments on earlier drafts
of this paper.

1/ Not all of the work in this area has assumed that expectations are
formed rationally. See, for example, the work of Stoll [1968] and Kesselman
[1971] Tsaing [1959] and Sohmen ]1969]. In addition, Callier [1980] not
only estimates the MT equation incorrectly, he also writes the forwarc.

rate as a function of expected future forward rates. The forward rat
used as regressors are of different maturities than the left hand side
forward rate so that a certain form of heteroskedasticity would be int:ro-
duced in the error term of the equation estimated. This is a further
complication which Callier [1980] does not allow for in his work.

2/ McCallum has recognized some of these problems in a recent note to

‘the Review of Economics and Statistics (see McCallum [1979A] and also in
[1979B]. These problems have been discussed in some detail in Hansen [1979,
1980], Cumby et al [1982], Hayashi [1980], Flood and Garber [1979], and
Stockman [1978].

3/ The estimator developed in this paper is a member of a wider class of
general method of moments estimators described by Hansen [1980]. For more
elaborate applications of this estimator see Glaessner [1982] chapter IV.

4/ Readers of McCallum [1977], Kesselman [1971] and Stoll [1968] may
wonder why the modern floating period is used as the sample period ra‘her
than the canadian floating period 1953-1960. The reason for this is the
nonavailability of one month forward contract data for this period. ‘The
reasons why one month forward contracts are used are explored in Section
III.

5/ See McCallum [1977], Driskill and Mcafferty [1981], Callier [1980,
1981] and Kesselman [1977].

6/ Equations like (1) below are derived by making assumptions about the
interaction of a trichotomy of agents» speculators, traders and arbitraguers.
See Tsiang [1959], Grubel [1966], McCallum [1977] or Driskill and McCafferty
[1981] for a more detailed derivation of equation (1).


- 20-

to those obtained in Table I. The coefficient estimates for Y) and Y2
are (.5673, .50715) and both coefficients are significantiy different
from zero, in contrast to the results for Case I in Table I. However,
as in Table I the Durbin Watson statistic does suggest that serial
correlation is present in the error term. Instrumental variables esti-
mators are shown in-cases II, III and IV. The first stage regressions
used to form the fitted values for Ff and S,,1 were the same as those
described for cases II, III and IV in Table I. In contrast to the results
in Table I, the constant terms are insignificantly different from zero
in cases II, III and IV. The coefficient attached to the interest rate
parity forward rate (1) is small (.1392 and .1545) and insignificant
for cases II and III vs. values close to 1.0 which are significant in
Table I. In addition, the coefficients on S441 (%9) are large (.9017,
.8863) and statistically significant whereas in Table I these coeffi-
cients are small and insignificant. Finally the Durbin Watson statistics
for cases II and III in Table 2 suggest that the structural error term
may not be serially correlated. This is in contrast to the results pre-
sented in Table I which suggested that serial correlation was present.

Case IV, Table II presents the results of applying Fair's [1970]
procedure. In contrast to the Table I results, the coefficient 5 is
large and statistically significant while the coefficient %y is small
and statistically insignificant. Also, in contrast to the results in
Table I, the autoregressive parameter (p) on F._y is small (-.1608) and
insignificant.

In sum, the results of Table I and II suggest that the sample

period and the nature of the data series used can make a large difference


- 26 -

Vv. Conclusion

The findings of this study can be summarized very briefly. Use
of the statistically appropriate procedures outlined in section III
yielded significantly different results than those obtained using the
incorrect procedures of McCallum [1977] and Callier [1980, 1981] for the
Dollar/Mark rate. Specifically they yielded much higher asymptotic
standard errors particularly for the sample period from July 1973-July 1980.
In light of various criticisms of the modern theory of forward foreign
exchange developed by McCallum [1979A] and Kohlhagen [1979], the empirical
results presented in this study tend to support the view that the modern
theory of forward foreign exchange is not a robust theory of DM/Dollar
forward exchange rate determination. Finally, this study has also suggested
that the frequently adopted assumption that interest rates follow an ex-

ogenous process may not hold for the DM/Dollar case.


- 22 -

densities at the zero frequency were obtained by using a lag window

(Daniell) of width 7 and 9 ordinates, depending upon sample’ size.22/

It
should be observed that in general the cross-periodogram ordinates are
not real, because the cross covariogram is nonsymmetric. However, at
the zero frequency the cross periodogram ordinates are conjugate
symmetric. Thus when we average over the ordinates the complex parts
cancel out and we obtain real values for the off diagonal elements in
$5 (0). Finally, the Var(¥) is computed as in equation (15). Obtaining
consistent asymptotic standard errors and coefficient estimates does make
a substantial difference particularly in the sizes of standard errors for
both sample periods.

Table III presents the results of applying Mccallum's consistent
procedure to the (MT) equation for the March 1973-June 1979 period. Note
that only the coefficients aay and 6 are statistically significant in Table III.

This is in contrast to Table I where Yo Y and p were all statistically signi-

ficant. Perhaps more important is the finding that the "correct" asymptotic

standard errors presented in Table III for Fg» 4) 1p) 6) are all greater
than the standard errors computed for these coefficients using Fair's
method (see cases IV, V, Table 1). Such a finding suggests that when
asymptotic standard errors are computed correctly multicollinearity may
become a serious problem in contrast to the arguments of McCallum [1977],
(footnote 18, p. 149). Finally the magnitudes of coefficient estimates
did not change markedly when correct procedures were pursued.

Table IV presents the application of McCallum's consistent proceclure
for the July 1973-July 1980 sample period. A comparison of the results
presented in Table II and Table IV reveals that both the magnitudes of
coeffficients and asymptotic standard errors change markedly when proper

procedures are pursued. The coefficient estimates for Yo? yy and Y2 are


- F2 -

Z/ It should be observed at this point that the estimation procedures
attributable to Hansen [1979, 1980] which are applied below do not require
that R, be exogenous. However, this assumption would seem important to
the "modern theory" if we are to call this a theory of forward exchange
rate determination rather than the combination of several arbitrage con-
ditions. Granger causality tests (see Appendix 2) suggest that the
assumption of an exogenous R. may be untenable for both treasury bill
rates ani eurocurrency rates.

8/ Agents are assumed to have full current information I> however, the

econometrician only usés a subset ‘ of the information in 1, in forming
e

tt1*

ysis, when a distinction is made between the true forecast error n and

predictions of S This point is developed more fully in later anal-

the forecast error due to using WY, rather than I,. A problem in defining
the information set suggested by the modern theory arises, because the
modern theory does not really suggest a complete model of the simultaneous
determination of spot and forward exchange rates. Thus, which variables
agents utilize to form predictions of the future value of the spot exchange
rate are not model specific. However, this is a problem which this paper
does not address since we are only concerned with estimating equations like
(1) properly, given the specification of the model. In contrast the paper
by Haas and Alexander [1979] does present an explicit model of spot exchange

rate determination.

9/ To see why the composite disturbance is serially correlated, note that

Nee] is tthe forecast error between period t and t+l and uy is a structural
+

white noise disturbance, as of time t. Now we can form Elvive_y] so

Elvave ig) = El(uy = yongyy) (Uy) - yang] = Eluu, 1]

+

2
~ YE(UyM] - yZElMe. Yep] + YQ°Eln ym] # 0

Note that: although El Mee] = 0 and E(u,y. i] = Elm" J = 0 by
assumption, the term Efu.n,] #0. Also Ely y ia] # Ofor similar reasons.
This point has been made by Hayashi [1980] and Cumby et al. [1980]. Also

Ely, I # O and Elvi, 7} = 0 for j > 1 or j < -1 so that % follows a first

order moving average process (MA(1)).


-17-
Table I

ESTIMATES OF THE (MT) EQUATION FOR THE DM/DOLLAR RATE

MARCH 1973 - JUNE 1979
USING MCCALLUM'S INCONSISTENT PROCEDURE*

Explanatory :
Variables Case I Case II Case III Case IV Case V
constant -.01879 -.0222 -.02256 -.0276 -.0449
std.error (.001611) (004075) (.007463) (00538) (.01218)
t-stat (-11.667)* (-5.44722)* (-3.0237)* (-5.131)* (-3.689)*
Fi 1.0472 . 77926 .52112 1.075 1.0490
std.error (-.015626) (.146564) (.400936) (.0341) (.04502)
testat (67.0143)* (5.31689)* (1.29975) (31.500)* (23.505)*
Spot 0061 27848 53443 -.0018 0636
std.error (.014906) (.144819) (.396917) (.03074) (.0419)
testat (41202) (1.92297) (1.34645) (-.06096) (1.5172)
Fey - - - .8095 .9450
sid.error - - - (.06967) (.03882)
t-stat - - - 11.6177 24.3475
R2 .999 .994 .982 .999> .999¢
S.E.R. ,00158 003683 006733 -0009 .0003
D.W. -3672 1.1023 1.2491 1.9151 1.678
a
b

2

© R* value is shown for Fo not Fie oF

R? value is shown for F, not F. - oF

t-1

t-1

*
Indicates that the coefficient estimate is significant at the

.05 significance level.


-9-

can be thought of as a member of a wider class of general method of moment
(GYM) estimators whose large sample properties have been developed by

Hansen [1982]. Specifically, Hansen [1979, 1982] has derived a consistent
estimator of the asymptotic variance covariance matrix and an asymptotic
distribution theory for parameter estimates under conditions where disturbances
are serially correlated-and instruments need not be strictly econometrically
exogenous 2! The nonlinear instrumental variables procedure described below
makes use of the asymptotic distribution theory developed by Hansen [1982]

in order to obtain the correct asymptotic variance covariance matrix for

the estimated coefficients. This instrumental variables procedure also
accounts for the possible conditional heteroskedasticity of equation dis-
turbances =

III.(a) A Consistent but Inefficient Estimation Procedure

In the present context, the application of the estimation procedure
suggested by McCallum [1979B] and Hansen [1982] to obtain consistent parameter

estimates can be described as follows: Project Fee Fr, Frey See? Ss, onto
B, #1,

i F 61
to obtain the fitted values Feep te Peay Sead and Ss, . Then

eed

form the estimating equation

(10) FL = yg(1-) + oF t YF

eo ee Soe B
to PMFey * YoSpar 7 P¥2Se + dy

where 4; is, by construction, a composite error term that is uncorrelated with
the regressors. OLS estimates of (10) will be consistent. To obtain estimates
of Y' = (p, Yor Yy> Y2) one would want to impose the nonlinear restrictions on
the paremeters are) ~ 0), p, PY» Yo and PY: This can be done by rewriting

’
equation. (10) in matrix notation as

(10)' F = Q f(y) + 4
Txl = =tx6 6x1 txl


- Ab -
Table A2

COEFFICIENT ESTIMATES FOR THE GRANGER TEST a/
THAT F(t) AND S(t) DO NOT GRANGER CAUSE R(t)—

R(t)

Lags Coefficients

S(t) .
Coefficients

F(t)
Coefficients

1973:3 -

0.939 ( 7.8349)
-0.164 (- .96814)
0.107 ( .6702)
0.008 ( .0516)
-0.030 (- .1946)
-0.258 (-1.655)
0.302 ( 1.903)
-0.0817 (- .7009)

Ory nun wWwne

1979:6

+3095
- .3932

( 1.205)
(-1.532)

+3343
+4039

(-1.311)
( 1.586)

F(4,55) = 2,6492"

1973:7-

0.8750 ( 7.098)
-0.2989 (-1.894)
0.1255 ( .7621)
-0.2477 (-1.560)
0.0677 ( .4302)
0.0641 (- .4139)
0.1330 ( .8667)
-0.1825 (-1.506)

ony nn rk wn

1980:7

+1395
-.0311

(1.0787)
(-.2640)

-.3484 (-1.833)
.0532 ( .3108)

F(4,65) = 3.2175"

2/statistics in parentheses are ratios of coefficient estimates to

their asymptotic standard errors.

Also, the author would like to thank

C. Crosby for efficient research assistance in the creation of this

table.

*
Indicates significance at’ the .05 level but not at the .01] level.

,


- AB -

obtaining 2 when sample size is small, differences may arise under these

two alternative methods of computing 2.


Both McCallum and Callier use a subset () of the information in

I, to obtain fitted values of the explanatory variables in (6) Lu

Specifically, they use the information set %F {R_, R. » F.

t? tel? Rea t-l’

F } and project the explanatory variables in‘ (6)

s S.

ter? cet e Spgs Splge cee

onto the relevant set of instruments in Yee Thus, they have

et é e e
7) “Sry * Spar * Seer 7 Stor?

e

hi $.. = s
mere Stel E(Se4,|¥q]> Stel Ste

1 = CESg4y [Tg] - ElSpeg!¥4])-

The forecast error in (7) is not equivalent to the "true" forecast error

n, This is the forecast error due to using Ye rather than Th Next given

t+1*

*

(8) Fi Fr + (FE- FD

ne
where Fe = ELFrI¥,]

and Feel - Fea =0 7

The equations both McCallum and Callier estimate can be derived by
substituting (7) and (8) into (1)' and performing the Cochrane Orcutt

transformation to obtain:

- ne ne .
(9) FL = yp(l-0) + OF.) * YyFe 7 OYyFeay * YaSeer ~ Pr2S, * 2,

where ze =e. + FY- ft) - * pt € S
7 Vy CFe Fe) = vy Fey Fey) * yp (Seay > Seay)

’

+ py (s°-§)
aut

t


where V, = u While the structural disturbance uy and the

Me 7 YoMeer

forecast error N, are each serially uncorrelated alone, the new

t+l
composite disturbance vy; is serially correlated. This is because uy the
structural disturbance term (or innovation) occurring at time t, will be
correlated with 1, the forecast error between periods t-1 and .2/
The composite disturbance v, will be correlated with the variable S+)
so that estimates of Yo will be inconsistent unless proper estimation
procedures are employed.

The problems introduced by the term Stel are further complicated

when the structural disturbance ut follows a first order autoregressive

process (Ar(1)), as both McCallum and Callier assume, where

G) 4, = Pu 4 ee» fe] < 1.0

2 :
and fe. (0, on) with E(e,€._.) = 0 for all s # 0. Applying the usual
Cochrane-Orcutt transformation yields

6 . ; +
(6) Fe Yo(t- 0) * PFLy * Fe = Ov Feiy * YoSeay > P1258: * vy

where v; = €, + PY2M - YoN,4]- The composite error is now serially

: ; ; 10/
correlated because it contains a moving average component (PY Ny - YN).

Moreover, unless instruments are chosen very carefully when forming the
fitted values of the explanatory variables, Ye will also be correlated with

the explanatory variables in (6). This can be shown as follows:


- 18 -

appear as regressors in the first stage regressions used to generate

the fitted values Fr and 8), are S. ys Sy_o> Feiys Frege er Rea

and a constant term. In Case III, Frio and Rea were dropped from the
set of variables, following McCallum [1977]. In both cases, the constant
term is significantly different from zero, whereas, y is significantly
different from zero in Case II but not in Case III. Also, the constant
terms are small while the coefficients on FE and S

30/

mately sum to 1.0. The D-W statistics again reflect the fact that

te? Yy + Yo» approxi-

first order serial correlation is present in the error term. Moreover,
we know from the analysis in sections II and III the use of instruments
lagged only one period will result in inconsistent estimates of the co-
efficients and inconsistent asymptotic standard errors in the second
stage regressions reported in cases II and III. Finally, cases IV and
V replicate McCallum's [1977] use of Fair's [1970] procedure in order to
account for the serial correlation in the error term. In case IV we
follow McCallum by using a set of first stage regressors consisting of
Re» Reqs Feoq> Feeos Feeg> St-a> Stia> St.3 and a constant term. In

case V, Riy, F and S,__3 are dropped from this list. In both cases,

t-3
ttt approximately sum to unity while the coefficients Yo and % are
significantly different from zero. Also Ty is very small (.0018 and .0636)
and insignificantly different from zero in both case IV and V. Finally
as in McCallum's [1977] results for the Canadian/Dollar rate the D-W
statistic improves substantially and the standard error of the regression
falls markedly when Fair's technique is employed.

Table II presents the results of performing McCallum's incorrect

procedure for the sample period 1973-1980 using a different set of data

(see section IV(a) above). The results are substantially different relative


-8-

An additional problem not addressed by McCallum and Callier is
that the composite disturbance term z, may be conditionally heteroskedastic..
To the extent that the composite disturbance in their work is not coaditionally
homoskedastic with respect to the set of instruments used (i.e. E[22| Yt,
Yop )# 02) estimates of the variance covariance matrix will not be
consistent 24/ Recent work by Cumby and Obstfeld [1983] suggests that this
may not be a problem of minor importance 22/

A final problem in the work of McCallum and Callier occurs due to
the use of monthly data with three month forward contracts. In this case
the sampling interval (monthly) and the forecasting horizon (3 months) are
not the same. Hansen and Hodrick [1980], Garber [1978] and Stockman [1978]
have suggested that the overlapping of successive forecast periods will lead
to serial correlation in the error term. Specifically, the structural dis-
turbance will follow a third-order moving average process, uy. = e, + Be
+ a2€,_> + %5€,_5 where Ee, = 0, Ee? = o7 and Be,e, = 0 for all i#j.
In principle such a moving average process can be handled by the est:mation
Procedures to be outlined below, 26! However, to see how sensitive McCallum's
and Callier's results are to the estimation procedure alone it is useful to
work with data series which do not result in these complications. In the
Present study, forward contracts having a maturity of one month are employed;
so that the forecast horizon and the sampling interval are both one month.
Hence, the problems associated with the overlapping of successive forecast

periods are eliminated.

III. Procedure for Obtaining Consistent Estimates and Asymptotic Standard
Errors for the (MT) Equation .

In this section, a different procedure for estimating the parameters

in the (MI) equation is described. The estimator obtained using this procedure


- 24-
Table IV

ESTIMATES OF THE NONLINEAR (MT) EQUATION (5. 10)
FOR THE DM/DOLLAR FORWARD RATE
JULY 1973-JULY 1980
USING MCCALLUM'S CONSISTENT PROCEDURE

Explanatory "Corrected" "Corrected"
Variables Coefficient Standard Error t-stat
constant - 0464825 -692322 -.06714

Fr «636763 1.627486 -39126
e

Stel -415037 1.78551 ~23245

F - 55076 2.11135 + 26085

t-1



International Finance Discussion Papers
Number 206

April 1982

THE MODERN THEORY OF FORWARD FOREIGN EXCHANGE:

SOME NEW CONSISTENT ESTIMATES UNDER RATIONAL EXPECTATIONS

by

Thomas C. Glaessner

NOTE: ‘“nternational Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an

acknowledgment by a writer that he has had access to unpublished material)
should be cleared with the author or authors.


2 Yo eo acl Y2 _
af (y) . -
where 3 =| 1-6 -¥9 0 0
tly
6x4 0 1 0 0
0 0 1 0

0 “YO

Thus any acceptable gradient method can be used to minimize amy) where

20
equation (12) provides one with the relevant gradient vector 20/

Minimizing oy) allows us to obtain the yector Y' = (6, Yo Ye 4) of

21
consistent parameter estimates. Hansen's [1979] theorem 32)/ suggests that

1 is asymptotically normally distributed as

(13) FF -y 4 No, 2's 274)
af (y) |' _. 3€(y)
where Ios E(q. 41] ——
3 Ce a)
4x4 Y Y
4x6 'Y 6x6 oxa |Y

~ = Ee ce 1 6 @1
9," i. Fey Fe Feel , Sean? 5, and the partial

’ o
deriative matrix is defined as above. In addition, Ss, = f R(t),

'
R(t) = E(z, Zeeql> and ze
4x4 ; 4x1



-10-

where T is the number of observations,

and f(y)'= [rot PJs" Oy Vy» -PYys Yo “ora
1x6

Given the orthogonality of the regressors and the error d,; consistent

estimates of the coefficients in (10)' can be obtained by minimizing the

sum of squared residuals. This results in
GY 8) = & - €Q))'E - GM)

which is equivalent to obtaining the nonlinear instrumental variables
estimator for Y as in proposition 1 of Cumby et al. [1980] 1o/ The normal

equations can be written as

at(y)'
)
Y
4x6

(-VE + VQ) = 0
6x1 6x1 4x1

(12)

me


- BB -

Haza, D. P., Fuller, W. A., 1979, "Estimation for Autoregressive
Processes with Unit Roots," Annals of Statistics, pp. 1106-
1120.

Herring R., Marston, 1976, "The Forward Market and Interest Rates in
The Eurocurrency Rates in the Eurocurrency and National Money
Markets." in Eurocurrencies and the International Monetary System,
edited by Stem, Makin and Logue, AEI, 1976.

Kesselman, J., 1971, "The Role of Speculation in Forward Rate Deter-
mination: The Canadian Flexible Dollar 1953-1960," Canadian
Journal of Economics, 4, pp. 279-297.

Kohlhagen, S. W., 1979, "Testing for the Role of Speculation in the
Forward Exchange Market: Some Problems if There are Fisherian
Expectations," 4, Review of Economics and Statistics., pp. 608-10.

Kubarych, R.M., 1978, Foreign Exchange Markets in the United States,
Federal Reserve Bank of New York.

McCallum, B. T., 1976, "Rational Expectations and the Natural Rate
Hypothesis: Some Consistent Estimates," Econometrica, 44, pp. 43-52.

wrens » 1977, "The Role of Speculation in the Canadian Forward Exchange
Market: Some Estimates Assuming Rational Expectations," Review of
Economics and Statistics, 59, pp. 145-151.

--, 1979B, "Topics Concerning the Formulation, Estimation and Use
of Macroeconometric Models with Rational Expectations," in American
Statistical Association, 1979, Proceedings of the Business and
Economics Statistics Section.

Meese, R. A., Singleton, K. J., 1980, "Rational Expectations, Risk
Premi.a and the Market for Spot and Forward Exchange," unpublished
manuscript, Board of Governors of the Federal Reserve System,
Washington, D.C.

Parkinson, P., 1981, "The Usefulness of Treasury Bill Futures for
Forecasting and Hedging," Ph.D. Thesis, University of Wisconsin.

Scott, L., 1980, "The Demand for Money with Rational Expectations: An
Inefficient Estimation Strategy," unpublished manuscript, University
of Virginia.

Sims, C. A., 1980, "Macroeconomics and Reality," Econometrica, 48,
pp. 1.-48.

Stockman, A. C., 1978, "Risk Information and Forward Exchange Rates,"
in The Economics of Exchange Rates, Frenkel and Johnson, Addison-
Wesley.

Stoll, H. R., 1968, "An Empirical Study of the Forward Exchange Market
Under Fixed and Flexible Exchange Rate Systems," Canadian Journal
of Economics, Feb., pp. 55-78.


- AS -
values of the spot rate and the forward rate are likely to have non-zero
coefficients. A constant term and a linear trend were also included in
the regression equation (A2.1), but are not reported in Table A2 below.
-In addition, all variables are in levels and all data series havenot been
deseasonalized.”” Table A2 below reports coefficient estimates and F
statistics which are used to test the null hypothesis that F(t) and S(t)
do not granger cause R(t). Equation (A2.1) is estimated for two sample
_ periods, 1973:3-1979:6 and 1973:7 - 1980:7. Over the shorter sample
period Forward and Spot rate data was obtained from the International
Money Market and Treasury bill rates for the U.S. and Germany were used
in forming R(t). Over the longer sample period eurodollar and euromark
rates were used to form R(t) and the spot and forward exchange rate data
are interbank rates, not IMM quoted rates.

The F Statistics reported in Table A2 test the null hypothesis
that all past values of the forward exchange rate and true spot exchange
rate have zero coefficients in equation (A2.1). Specifically, we form

(RRSS - URSS)/r

F(z, n-k) = “QRss7in-k)

where RRSS is the restricted (or constrained) residual sum of squares

obtained in estimation equation (A2.1) with the coefficient on lagged

-  yalues of S(t) and F(t) constrained to be zero. URSS is the uncon-

strained residual sum of squares, r equals the number of restrictions,

n is the number of observations, and k is the number of regressors

>

ae

Allowing for a deterministic seasonal and deseasonalizing, the
series in question before conducting the exogeneity tests does not
alter the results reported here.


- AG -

APPENDIX 2

This appendix presents the results of performing tests of the
exogeneity specification of the modern theory of forward foreign ex-
. change. This is done by testing whether F(t), the forward rate, ard

1+i,,,*

S(t), the spot rate, granger cause R(t) = _ . The techniques

L+icgy

used in this appendix are explained more fully in Glaessner [1982],
Geweke [1978] or in Dent and Geweke [1978]. When performing granger
causality tests the researcher must choose a parameterization that
offers a compromise between the criteria of unbiasedness which suggests
a generous parameterization vs. power which will diminish as the param-

eter space expands. In performing these tests the equation estimated

appears as
M N
(A2.1) R(t) = ZR(t-j)o,(3) + 2 S(t-j)e,(3)
j=l j=l .
N
+ DF(t-j)es(3) + e,(t)

j=l

where we choose a generous parameterization for the lag length of the
hypothesized exogenous variable (M= 8) to insure that €, (t) is serially

uncorrelated. To preserve power, a choice of N=2 is made since if the

‘null hypothesis is false it seems reasonable that the first few lagged


-7-

The problem with this equation is that the composite disturbance Zz will be

correlated with F,_4 and all variables dated at time tre, Se) because of

*

. * ae .
the term (Fey - Foy = Fey

- ECFE_y|¥,_3]- Moreover, Seat will be

correlated with the (st - .) component of the composite error z,. Estimates
of the coefficients in (9) will be inconsistent unless instruments are

chosen very carefully. -Specifically, by the law of iterated expectations
Elz. |¥,_1] = 0 which implies that to get consistent estimates in (9) involves

the projection of F F*, F* Ss. and Ss, on an adequately large subset

*
tea? Per Feeae Seat
of Yer = fRea> Rea» sees Si-2? Sii3> weeny Fiiz? Fiz? .

= se te 1 = 1 12
fitted values Fev Fi Frail) Stel? g.2/ In sum, the order of

the moving average process generating a (an MA(9) in the present case) will

.. } to obtain the

determine the appropriate set of instruments to use (e.g. in terms of the
number of lags etc.) in estimation.

In McCallum's and Callier's work the instruments employed in forming
the fitted values of the explanatory variables in (6) were the endogenous
variables (F,, s.) lagged one period and the current value of the exogenous
variable (Ry) + Thus, the parameter estimates and asymptotic standard errors
obtained in this work are inconsistent. Several recent papers have suggested
that lagging the instruments properly does not insure that consistent asymp-

13/

totic standard errors can be obtained. Although Ze and Yeaje are orthogonal
for j > 1,it is not the case that ze and ej will be uncorrelated for j <0.
Thus, the explanatory variables are not necessarily uncorrelated with 2, at
all leads and lags. it is for this reason that standard GLS techniques cannot
be used to obtain consistent estimates of the coefficients and asymptotic

.
standard errors.


-~21-

in the magnitudes of coefficient estimates and the extent to which errors
are serially correlated. It should be observed that evidence of little
serial correlation in the error term, after performing the instrumental
variables procedures in cases II and III of Table II, does not suggest
that the problems inherent in these estimation procedures (see section II)
were absent. Even if it was assumed that the structural error term u,
(eq. (2)) was serially uncorrelated, serial correlation can arise in the
error term due to the presence of the term see1 (see section II). Thus
the coefficient estimates and standard errors obtained in Tables I and IT
are inconsistent.

Iv.(c) Application of a Proper Estimation Procedure

Tables III and IV below show the results of applying a consistent,
but inefficient, estimation procedure (see section III) to the modern
theory equation (10) for the DM/Dollar forward rate for two different

31/
sample periods. | Table IJI and IV were created as follows: Each series

was detrended and the fitted values Feo Fr. Fry Seay

in first stage regressions where the regressors were Frio Fo 3? Ss. 2? St_3>

5) were formed

Rey R. R Equation (10) was then estimated subject to the nonlinear

t-2? “t-3°
restiictions developed in section IIa. The Davidon Fletcher Powell Algorithm
< 32
was used to obtain the estimates of Y in Table III and w.22/ The "corrected"

standard errors and t-statistics reported were obtained using the procedure
suggested in section III. (a).
The estimated spectral density matrix of 2 evaluated at the zero
frequency, 85(0) was obtained as follows. First, the consistently estimated
,

residuals were obtained using equation (14). Second, the spectral density

*
matrix for 2, was formed where estimates of the spectral and cross spectral


- F5 -

21/ Hansen's theorem 3 is relevant in the present context for two reasons.
First, linear least squares predictors have been assumed to be equivalent
to conditional expectations. Second, the nonlinearities in the present
problem are only in the parameters or in terms of Hansen's [1979] general
notation f(y, 4) = a, '£)) where q,' is the first row of the Q matrix

and f(y) is a vector capturing the nonlinear restrictions in the parameters.
22/ Several points need to made here. First, to see why this claim is

correct write the autocovariance generating function of the z's as
o

8,(s) = t sR, (3). Now evaluating the autocovariance generating function
=-0
on the att circle (se?) where w now signifies frequencies results in
eo 2.
2,(e") = Ra eR) which is the definition of the spectral density
matrix of the z vector. Now note that evaluating the spectral density
matrix at the zero frequency (w=0) results in S, = t R,(j)-
=-0

Second, the reader may be wondering why the 5, is not estimated

using time domain procedures, as

1 n a «
85 2 RO ACD +8 +R),

since the error term d_ follows an MA(1) process. Using a time domain
procedure has the disadvantage that there is no assurance that S_ will be
positive definite when it is computed from a finite number of autocovariances.
See (Hansen [1979] p. 12-14), or Hansen and Singleton [1981] who point this
out. In contrast using the frequency domain procedures in order to consis=
tently estimate Ss, results in an estimate of S_ which will be positive
definite by construction, Frequency domain procedures do have the disadvan-
tage.that they do not exploit a finite number of autocovariances in forming
consistent estimates of the weighting matrix an In fact, Hansen and
Singleton [1981] have pointed out that when the'number of observations is
small and the number of orthogonality conditions is large use of frecuency
domain procedures, which involve a greater loss of degrees of freedom relative
to time domain procedures, may result in a deterioration in the precision. of
estimates of S,.

Third, it is important to realize that the estimator of S, obtained
by Cumby et al. and suggested by Hansen and Sargent [1981] in a different
context (e.g., §) is asymptotically equivalent to the §, matrix computed
in this paper. See Appendix 1 for a prdof of this proposition in the case
of the linear model. The results generalize for the nonlinear case.


- F6 -

23/ Three points deserve to be made about the expression for the variance
derived here, First, equation (15) is the correct expression for the
variance of ¥\when the equa:ion being estimated is nonlinear in the para-
meters. This is in contras~ to equation (16) p. 68 in McCallum [1979B]
which expresses the relevant variance covariance matrix the investigator
would want to compute if the ecuation being estimated was linear in the
parameters, which it is not (see eq. (15) in McCallum [1979B]). Second,

it can be shown (with the use of equation (2) in footnote 19 that equation
(22) in the text is equivalent to the expression for the Var(¥) obtained

by Cumby et al. [1980] in their proposition I. Third, it should be observed

ak Cy)
that because the partial derivative matrix a . is nonstochastic, an
6x4 IY
alternative and computationally less burdensome expression’ for the variance
of ¥ is
ye FQ)! attr) |
(15) -Var(¥) = zqlt 5 . 5 a
Y ly 6x6 yy fy¥
4x6
where SI is now a 6x6 spectral density matrix of z, = ea, instead of
_ 3) - 6x1

the 4x4 matrix obtained when we use 2, = €, —>——|_ 4, as defined in

YY 6x2

4x6

the text. In the present study computational ease suggested the com-
putation of (15)'. However, which method is used should not make a

af(y)

E
Y

difference due to the nonstochastic nature of

Y

24/ For example, a monthly forward-contract written in February will
be for 28 days while a contract written in September will be for 31
days, I would like to thank Ralph Smith for pointing this out to me.

25/ When Tuesday-Thursday pairings were not available due to holidays,
Wednesday-Friday or Monday-Wednesday pairs were used. See Stockman [1979]
or Meese and Singleton [1980] for a more detailed treatment of these issues.

26/ The fact that heteroskedasticity can occur when contract maturities
change at different points in time has been shown in work on T-Bill futures
markets (see Parkinson [1981]).

27/ The reader of Meese and Singleton [1980] and Haza and Fuller [1979]
will notice that the test for unit roots in applicable to series where a
time trend has already been removed, so that the procedure followed here
would seem incorrect. However, because transforming the model would
fundamentally change the equation estimated, other transformations to
stationanize the series were not undertaken. Also the choice of the DM/
Dollar rate was made exactly because the hypothesis of a unit root for the


APPENDIX 1

It is important to realize that the estimator of S, obtained by
Cumby et al. [1980] (e.g., Q) is asymptotically equivalent to the S_
computed in the present paper. To see this, note that in Proposition

1 Cumby et al. shows that given

(ana) B= cqrwowtwy72weg)“Pqrwowewy “Wey
they have

3 d g'wow'w) wg Low ww) ww
(A1.2) vT (8-6) + [ 7 J

To

where W, Y,and Q are defined as in footnote 19, and Y = Qé + w where w
is the structural disturbance and 6 is a vector of coefficients. As
Cumby et al. [1980] show in their Appendix, p. 30, Hansen's [1979] re-

sults suggest that the last term in (Al.2) converges in distribution to
tw d
(Aas) LES noo, plim * (W'ww'W)) .
. YT Tro
Since all the other terms in (Al.2) converge to some finite constant
due to the assumption of joint covariance stationarity and ergodicity

of the variables in Q and W (see Cumby et al. [1980]), we obtain the

asymptotic distribution of vT (é- 8) as


-32-

procedure discussed in section III in order to discern the effect of using
proper estimation techniques for both the magnitudes of estimated coefficients

and standard errors. Finally, section V summarizes the results.

Il. Problems with Past Estimation Procedures in Tests of the Modern Theory
oi Forward Foreign Exchange

McCallum [1977] and Callier [1980, 1981] estimate the following
equation for the forward exchange rate6/

*

ql) FLt Yo * YFe

e
t * 2843 *

Initially the analysis which follows can be simplified by working with one-month

forward contracts so that (1) becomes

GQ)" FL = Yq + YF + 72841 su,

where F, is the one-month forward rate, Ft is the corresponding interest-
parity forward rate and Seal is the value of the spot exchange rate ex-
pected to prevail at the end of next month. Spot and forward exchange

rates are expressed as the mark price of U.S. dollars. The MT defines the
interest rate parity forward rate FE as that forward exchange rate which
eliminates any yield incentives for covered capital flows given spot exchange
rates and domestic and foreign interest rates. More formally, Fy = S_Re

where 3, is the current spot exchange rate and

1+i*
® RF ieay

a let




-14-

Their findings suggest that care must be taken in making the assumption
that the regressors follow a jointly covariance stationary process when
lagged values of the forward and spot exchange rate are used as ins<ruments,
as in the present study. Third, the assumption that R(t) is exogenous

(see McCallum [1977], Driskill and McCafferty [1981], and Callier [1980,
1981]) would seem very questionable in light of the findings presented

in Appendix 2, where it is shown that F(t) and S(t) Granger cause R(t).
Thus, the necessary conditions for R(t) to be strictly econometrically
exogenous do not hold. In addition, this rejection of Granger noncausality
of S(t) and F(t) to R(t) occurs when either treasury bill rates or euro-
currency rates are employed as the relevant interest rate series.

Iv. Empirical Results

Iv.(a) Data Alignment, Covariance Stationarity and the Choice of
Data Series

Before proceeding with a discussion of the empirical results, it
is useful to discuss briefly the three problems of data alignment, co-
variance stationarity of the regressors, and the choice of data series
within the context of the present study. First, Meese and Singleton
[1980] have suggested that how the spot rate in period ttn (S,4,) is
aligned with the forward rate «Dp, of maturity n is an important factor
in determining the coefficient estimates an investigator obtains. Problems
arise because in actual practice the length of a given forward contract
(let us say of one month maturity) will vary according to the month. in

24/

which the forward contract is written.— In addition the agreed payment

’ :
date or "value date" for a given spot transaction is usually two dzys after
the day on which the transaction originated (see Kubarych [1978]). In the

present study and in other work (see Meese and Singleton [1980] or Hansen


- 16 -

Third, as will become clear below (see section IV(c)) the results
obtained are very sensitive to both the data series used and to the sample
period over which the MT equation is estimated. In the results which
follow, two sample periods are examined for the DM/Dollar rate. Over the
sample period from March 1973 to June 1979, Forward and Spot rate data
were obtained as bid prices reported in the International Money Market
Year books. The interest rate series were 90-day treasury bill rates
obtained from the IFS Data Tapes. The other sample period examined is
from July 1973 - July 1980. In this case, Interbank Forward and Spot
exchange rate data was usea 22 Also, 30-day eurocurrency interest rates
were used instead of treasury bill rates, since these interest rates are
not subject to political risk (see Dooley and Isard [1980] ) .22/

Iv.(b) Results of Applying McCallum's Incorrect Procedure for
the DM/Dollar Rate for Two Different Sample Periods

Table I presents the results of performing McCallum's and Callier's
incorrect procedure for the DM/Dollar rate over the 1973-1979 period. The
Case I figures shown in the first column are OLS estimates of equation (4).
As is readily apparent, the value of x is quite high and the t-ratio
associated with the coefficient on r is very high. The coefficient on
Stel (FQ) is .2784, however, it is insignificantly different from zero.
Moreover, the Durbin-Watson statistic is very small suggesting that the
error is serially correlated. This is of course not surprising in light
of the discussion in Section II which suggested that estimates of ‘2 would
be inconsistent and asymptotically biased toward zero if an OLS procedure
was applied. This is because vy and S47 in equation (4) are correlated
much like an errors in variables model.

Accordingly, we follow McCallum [1976,1977] and use instrumental

variables estimators in cases II-V. In Case II, the variables thet


The purpose of this paper is to reveal explicitly how the problems
mentioned above will result in inconsistent parameter estimates of the
coefficients in the MT equation anc in inconsistent estimates of asymptotic
standard errors if typical estimation procedures are used. An alternative
estimation procedure suggested by McCallum [1979B] and Hansen [1979, 1980] is
presented, and implemented for the (MT) equation of the forward exchange
rate.3/ The coefficient estimates of the (MT) equation and the standard
errors obtained using a correct estimation procedure are then compared with
those obtained using McCallum's [1977] original procedures for the DM/dollar
rate over the current floating exchange rate period.4/

The proper estimation of the DM/Dollar forward exchange rate equation
suggested by the MT results in marked changes in estimated coefficients and
generally greater asymptotic standard errors. The common assumption that
interest rates are generated by an exogenous stochastic process>/ is
rejected using both eurodollar and treasury bill interest rates for the
DM/Dollar case. These two conclusions suggest that recent attempts to explain
the simultaneous determination of the spot and forward exchange rate by
adopting both the assumption of rational expectations and the theoretical
framework suggested by the MT@s in Driskill and McCafferty [1982]) are
potentially misleading.

This paper is organized as follows: Section IT reveals how the
estimation procedures recently used to test the MT leads to inconsistent
parameter estimates. Section III develops a method for obtaining consistent
parameter estimates and asymptotic standard errors for the MT equation.

Section IV implements McCallum's [1977] incorrect procedure and the proper


~ FA -

of GMM procedures and for a more explicit derivation of criterion functions
for obtaining parameter estimates given a set of population orthogonality
conditions implied by economic theory, see Glaessner [1982] or Hansen and
Singleton [1981].

18/ See Hansen [1982] or Cumby and Obstfeld [1983] who point out that the
estimation procedures pursued below does not require the investigator to
even specify an explicit form for the conditional heteroskedasticity.

19/ The criterion function for the nonlinear instrumental variables
estimators used by Cumby et al, [1980] can be written as

(1) ety) = CY = ey)" wew'w) “WH CY - ECD)
~ Tx] TxN Nx] TxK

where Y is a (Tx1) vector of endogenous variables, Q is a TxN matrix of
regressors (not to be confused with Q in the text (the fitted value
matrix) which is defined as

(2) Q = www) twig
TXN TxK KxT TxK

Finally, W is a (TxK) matrix of instruments and Y is a (Nx1) vector of
coefficient:s to be estimated, The first order conditions for the crite-
rion functiion in (1) can be written as

agy) 3£(y)'

a PG
Y Y
at(y)'| “1 «

j_ omarHD THRE =

y

-1

Q' WWW) "TWtY

Y

+2

3

Y

or after some manipulation
aer(y) af Cy)

(3)
ay y

_ awww) tw fy = QF]
7

Using equat:ion (2) in conjunction with (3) results in

aer(y) f(y)

3 3
Y Y

_ (W- VE)

Y

which is equivalent to equation (12) in the text.

20/ The Davidon Fletcher Powell (1963) algorithm was used in order to
minimize the various criterion functions,developed in the text.



- 25 -

(.0465, .6368, .4150) in Table IV vs. (-.0169, .1289, .9095) for case

Iv of Table II. Also the associated standard errors are (.0124, .1277,
-10890) for Case IV of Table II vs. (.6923, 1.627, 1.785) in Table IV
when the correct standard errors are obtained. Thus, due to the large
rise in standard errors, all the estimated coefficients are statistically
insigniiicant in contrast to the results in Table II which suggested that
9 was statistically significant.

In sum, when the consistent procedure of McCallum [1979B] is
applied, multicollinearity becomes a severe problem regardless of the
sample period. This result is not completely surprising in light of
recent criticisms of the modern theory of forward foreign exchange deve-
loped by McCallum [1979A] and Kohlhagen [1979]. Specifically, these

papers have suggested two different sets of conditions under which

*

e
to s. wv so that perfect collinearity among the explanatory variables

occurs ‘in the equation being estimated. The first set of conditions
sufficient to obtain this result suggested by McCallum, are (a) covered
interes: arbitrage, so that FP = Fr and (b) strong market efficiency

so that Fr = s° « The second set of conditions, suggested by Kohlhagen,

t+1
are a) absolute purchasing power parity and b) equality of real interest
rates across countries. Seperate tests of McCallum's conditions (a) and

(b) suggest that the former tends to hold, particularly for eurocurrencies*4/
and that: the latter although a reasonable approximation in some contexts
does not, because of the possible existence of a small time varying risk
premium (see Hansen and Hodrick [1981]). These results would tend to
suggest that the finding of high collinearity in the explanatory variables

of equat:ion (1) should not be too surprising.


The Modern Theory of
Forward Foreign Exchange:
Some New Consistent Estimates Under Rational Expectations

by

Thomas C. Glaessner*

I. Introduction

In recent papers, B. T. McCallum [1977] and P. Callier [1980, 1981]
have estimated an equation implied by the modern theory of forward foreign
exchange (MT), assuming that expectations about future spot exchange rates are
formed rationally 1/ Their estimates of the (MT) equation are subject to
several problems that may arise when estimating rational expectations models.
First, the presence of the expectation of the future value of a variable
conditional on information available at time t, within the equation being
estimated, has been found to lead to serial correlation in the error term.
Second, the use of standard GLS techniques to correct for serial correlation
in the structural disturbance will lead to inconsistent parameter estimates
and to inconsistent asymptotic standard errors. Third, the use of monthly
data on all series in conjunction with forward contracts of three months
maturity results in successive forecast periods overlapping and attendant
complications in estimation. Finally the possibility that the disturbance
term in the MT equation is not conditionally homoskedastic can cause the

standard covariance matrix estimators to be inconsistent .2/


- AT -
including the constant term. Since both F statistics are significantly
different from zero at the .05 significance level, we reject the null
hypothesis that S(t) and F(t) do not Granger cause R(t). In sum, R(t)
does not meet the necessary conditions for strict econometric exogeneity

whether it is defined using treasury bill rates or eurocurrency rates.


- B2 -

Garber, P. M., 1978, "Efficiency in Foreign Exchange Markets: Inter-
preting a Common Technique," unpublished manuscript, University
of Virginia.

Glaessner, T. C., 1982, "Theoretical and Empirical Essays on the Deter-
mination of Spot and Forward Exchange Rates," University of Virginia,
unpublished Ph.D. Thesis.

Grubel, H., Forward Exchange, Speculation and the International Flow of
Capital, Stanford University Press, 1966.

Haas, R. D., 1974, "More Evidence on the Role of Speculation in the
Canadian Forward Exchange Market," Canadian Journal of Economics,
7, pp. 496-501.

Haas, R. D., Alexander, W. E., 1979, "\ Model of Exchange Rates and
Capital Flows, Journal of Money Credit and Banking, Vol. 11,
No. 4, pp. 467-482.

Hansen, L. P., 1979, "The Asymptotic Distribution of Least Squares
Estimators with Endogenous Regressors and Dependent Residuals,"
unpublished manuscripts, Carnegie Mellon University.

--, 1980, "Large Sample Properties of Generalized Method of
Moments Estimators," unpublished manuscript, Carnegie Mellon
University.

Hansen, L. P. and Hodrick, R. J., 1980, "Forward Exchange Rates as
Optimal Predictors of Future Spot Rates: An Econometric Analysis,"
Journal of Political Economy, Vol. 88, No. 5, October.

Hansen, L. P. and Sargent, T. J., 1980, "Formulating and Estimating
Dynamic Linear Rational Expectations Models," Journal of Economic

Dynamics and Control, Vol. 2, p. 1.

corres , 1981, “Instrumental Variables Procedures for Estimating Linear
Rational Expectations Models," forthcoming in the Journal of Monetary
Economics.

Hansen, L. P., Singleton, K., 1981, "Generalized Instrumental Variables
Estimation of Nonlinear Rational Expectations Models," forthcoming
in Econometrica.

Hayashi, F., 1980, "Estimation of Simultaneous Equation Models Under
Rational Expectations: A Survey and Some New Results," unpublished
manuscript, Harvard University.

,

Hayashi, F., Sims, C., 1980, "Efficient Estimation of Time Series Models
with Predetermined, But Not Exogenous, Instruments," unpublished
manuscript, University of Minnesota.


- A2 -

wow! Ho)7) on ctw)?
(A1.4) YT (8- 8) £ x(0; pin aie *) “WQ) a = -

.cvtwyTM! weg (grncnwy2wg)"*)
T T T

W'ww'W

where Q = plim T

Tro

Now consider the estimator of the present paper

canes) B= QQ erty = + (QQ te

thus we have

igs} ont
(al.6) 0 ¥T 8-8) = FF (Qriqn ‘gt = a ad
Now

(A1.7) oes > NCO, plim ¢ & (Q*tww'Q*))

Tro

by the theorems in Hansen (1982]. Also we have

etary) getgey7t
(Al. 8) WT (8 - 8) $ NCO, plim (Q e) 9 LQ g*) )

Too

‘ww**
where now 2 = plim grrwergr

T+°

But by construction of Q* the expression obtained for the asymptotic
variance of the estimator (Al.8) is equivalent to (Al.4), thus the two

- methods of computing §& will be asymptotically equivalent. However, in


-12-

where qd, is the t-th disturbance term and the other terms in z, are
defined as before except that the partial derivative matrix is eval-
uated at the "true" minimum value of y, since the results in (13)
are asymptotic.

To perform hypothesis tests, consistent estimates. of Ss, and = are

a «  , 2fQ)"Y . .
needed. To obtain Sy form 2 ee It where é,, a consis-
4x4 4x1 ~ nf

tent estimate of the t-th residual, is calculated as

oe salt ae fete abet og x
4) ee Fm gh P= OF FE + PYF > T28per * PTS

McCallum [1979B] notes that é

must be formed by using the actual values

of the explanatory variables instead of the fitted values in equation

(10). Hansen [1979, 1982] has suggested that consistent estimates of

Ss, (e.g., §) can be obtained by forming the spectral density matrix of
22/

zt and evaluating the elements in this matrix at the zero frequency.

This is done by obtaining estimates of both the spectral and cross spec-
tral density functions through the use of the Daniel estimator. To

obtain a consistent estimator of L€@.g., i) Hansen's [1979] theorem 3

implies that

1 fC)" af (y)
plin ra 5 =f
T= ’y |y y ly
m 1 2FM)' af (y)
Thus in finite samples & = tI. Q' Q > . where 211
4x4 y |v YIy


-15-

and Hodrick [1980]) the value of n is usually set equal to some constant
value regardless of the month one is in. In this study, the value of
n chosen was 30 days so that a forward contract written on the Tuesday of
the beginning of the month was aligned with the Thursday spot rate 30
days hence if available.2=! Aligning the forward and spot rates in this
fashion introduces a form of measurement error into the series for the
spot exchange rate in period t+n. However, letting n vary according to
the month in which the contract is written and aligning forward rates
(FE) and spot rates (Stin) accordingly would introduce a form of heter-
oskedasticity into the error term.22/ In sum, it should be recognized
that different methods of data alignment result in different problems.
Second, Hansen's [1979] theorem 3, in which an asymptotic distri-
bution is derived for the estimated coefficients 62) requires that all
variables have mean zero and be jointly covariance stationary. Meese
and Singleton [1980] have attempted to see whether this assumption holds
in several countries by using a test for unit roots in univariate autoe
regressive processes developed by Haza and Fuller [1979] and Dickey and
Fuller [1979]. Meese and Singleton reject the hypothesis that the log
of the mark dollar spot exchange rate has a unit root, however, this
hypothesis cannot be rejected for a forward rate. More importantly,
tests for the existence of unit roots in the joint autoregressive repre-
sentation of s(t), £(t), i(t) and i*(t) have not yet been developed so
that it is not clear that the assumption of covariance stationarity has
been violated. Thus, when proper estimation procedures are used (see
section IV(c) below), all data series were detrended and the resulting
series had zero-mean and were assumed to be jointly covariance

27/

stationary.—