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

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acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

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/

-2-

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./

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 II 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

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. II. 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/

1 = * e C1) FL YQ * MFe t+ YoStag * Up

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

forward contracts so that (1) becomes

1)! F. = * e (1) 1" Yo* YF. + YS, + Uy,

where F, is the one-month forward rate, FY is the corresponding interest-

e

parity forward rate and S41

is the value of the spot exchange rate expected to prevail at the end of next month. Spot and forward exchange

rates are expressed as the mark price of U.S. dollars. The MI 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_Rt where 3, is the current spot exchange rate and

7

l¢+i

(2) R. = Tea

a fet

with it and it 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.7/

To obtain parameter estimates of the coefficients in equation (1)', McCallum and Callier both employed an instrumental variable technique (see McCallum [1976]). This 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 to 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¢+}.

First, assume that the structural distrubance u; in (1)' is a white noise process with zero mean, finite variance, 0 and Efu,uy_;] = 0 for j #0.

The assumption of rational expectations implies

_ ¢e (3) Sten 7 Stet * Mea e where Se4] = E(S,,,11,] and Neay? the "true" forecast error, is serially

uncorrelated. Also, E(n.., 11,1 = 0, by the properties of the linear least

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

* 4) FL = Yo * YyFe * YoSea * %t

where V, =U, - Yon,,,- While the structural disturbance uz and the

forecast error n are each serially uncorrelated alone, the new

t+l composite disturbance v; is serially correlated. This is because uz the structural disturbance term (or innovation) occurring at time t, will be correlated with ",, the forecast error between periods t-1 and t.2/ The composite disturbance vy will be correlated with the variable S¢4) so that estimates of Y> will be inconsistent unless proper estimation procedures are employed.

The problems introduced by the term Stel are further complicated

when the structural disturbance u; follows a first order autoregressive

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

2 . and ee (0, o.), with E(e,€,_,) = 0 for all s # 0. Applying the usual

Cochrane-Orcutt transformation yields 6 F = . * . * . (6) t* Yotl-P) * PF) + YyFe - OYyFe iy + YoSea. - PY2S, + ¥

where vy; = e&. * PYM, - YoNe4,° The composite error is now serially

10/ correlated because it contains a moving average component (PY oN, - Yo rte. ).

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

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

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

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

t? Reo trot? Feep

} and project the explanatory variables in’ (6)

Specifically, they use the information set YF {R R

Feige eee Spye Spupe

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

. ef e e e (7) °S Sear t Star 7 See

t+] ~ tel )

S417 (E{S, 114) - ES,43!%))-

h S = e where Seal E(s.,,1¥,J> | See t+

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

Neer’ This is the forecast error due to using ¥. rather than I,. Next given (8) Fl = Fre (FE- FY)

where Fr = E(Fr|Y¥,]

and Feel” Fea = 0 ;

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:

a a a a

a * * (9) FL = yg(l-0) * Fey * ¥yFe > OvyFey * Y2%ee1 PYS, * 24

where Ze =e. + FY. f*) - * of e 2

e

+ py, (S, - §.)

t

-7-

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

correlazed with Friel and all variables dated at time t(Fe, S.) because of

* *

. * a the terin (Feat Fp = Fed

- E(Fr_iI¥,_,]- Moreover, §, |, will be

es. | correlated with the (S, - S,) 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

E(zl¥, iJ = 0 which implies that to get consistent estimates in (9) involves

* *

t? Feep Sea] and Son an adequately large subset t-l- {Reiy> Regs eee y S.-2) S._3> ooo 9» Fii2? Fio3? e¢ 1 12/

fitted values Frew Fie Feed > Sea? Ss. .--/ In sum, the order of

the projection of F F

t-1’ of ¥

.. } to obtain the the moving average process generating Zz (an MA(9) in the present case) will 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 (R,)- 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 2 and ¥eej? are orthogonal for j 2 1,4t is not the case that Z, and Yee5 will be uncorrelated for j <0. Thus, the explanatory variables are not necessarily uncorrelated with z, at all leeds and lags. it is for this reason that standard GLS techniques cannot be used to obtain consistent estimates of the coefficients and asymptotic

eo

standard errors.

-8-

An additional problem not addressed by McCallum and Callier is that the composite disturbance term Zz, 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[zel Yt, Yd oor |] # 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

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 disturbance will follow a third-order moving average process, u. = e. + ae.) + a5€,_5 + ase. 3 where Ee, = 0, Ee = o* and Ee;e, = 0 for all ij. In principle such a moving average process can be handled by the est:imation procedures to be outlined below. 22! 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 Sta ndard Errors for the (MT) Equation

In this section, a different procedure for estimating the parameters

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

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 cistribution theory for parameter estimates under conditions where disturbances are serially correlated.and instruments need not be strictly econometrically

17/ exogenous. 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-

18 turbances,”— III.(a) A Consistent but Inefficient Estimation Procedure

a ene nemenh ur anerricient 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 Stel? Ss. onto y . 2 as r* aa ; Ps = Yel to obtain the fitted values Feel? Fy Fra Se4] and S,’. Then

form the estimating equation

v § 1 t+17 OX 2Se +

(10) F, = y)(1-9) + PPL + FE - py, Ft_j} + ¥98 where d, 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' = (ep, Yor Yq: Yo) one would want to impose the nonlinear restrictions on the parameters Yy(l- 0), p, PY,» Yo and PY: This can be done by rewriting

¢

equation. (10) in matrix notation as

(l0)' F = Q fly) + a Txl tx6 6x1 txl

- 10 -

where T is the number of observations,

Cr " ~ oh ne

t-l t t-l tel t

and f(y)' = i (l= —),° Oy Yr, -PYa5. - ~ 0 ’ » Ty? Yy? Yo PY> ° : 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 (11) Cy) = (F - Q£(y)) (CE - G(x)

which is equivalent to obtaining the nonlinear instrumental variables

estimator for Y as in proposition 1 of Cumby et al. [1980] 22/ The normal

equations can be written as

ef (y)' . _. (12) = (-Q'TE + Q'QF(y)) = 0

wn<> ov ~ _ fo.) bad _ oy ~ ~

Y 4x6

Yo is) Yy Yo af (y) . where. 5 = | l-6 “Yo 0 0 6x4 ; 0 1 0 0 0 0 1 O

Thus any acceptable gradient method can be used to minimize o(y) where equation (12) provides one with the relevant gradient vector 22 Minimizing (y) allows us to obtain the yector y' = (9, Yo: Yh Yo) of consistent parameter estimates. Hansen's [1979] theorem yeu suggests that

Y is asymptotically normally distributed as

(13) FF (y-y) 4 N(o, 24s 277) , af(y)]}" BF CY) where Ios 5 E[q, q.J = 4x4 Y Y 4x6 'Y 6x6 6x4 'Y

~ 1 - ee ze 1 6 é 1 q E Fooly Fee Fey > Seal? 5. and the partial

1x6 ’ @o der:.ative matrix is defined as above. In addition, S, = £ Ro (1), . tet -)

, f(y)" R(t) = E(z, za) and z, = d, a ’ 4x4 4xl - -

qe

-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 evaluated at the "true" minimum value of Y; since the results in (13) are asymptotic.

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

af(y)'| | needed. To obtain’ S_, form zt = e, ——| q, where @,, a consis- : 4x4 4x1 ~ Yoyo, tent estimate of the t-th residual, is calculated as a =: . 3. 2 _— . 5 * aa * _ 4 as a4) Ge = Fe > Yo(h- 0) - OF, 1 = YyFe + PYFEly - YoSear * PY25e-

McCallum [1979B] notes that e. 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., s.) can be obtained by forming the spectral density matrix of ZF and evaluating the elements in this matrix at the zero frequency .22/ ‘This is done by obtaining estimates of both the spectral and cross spectral density functions through the use of the Daniel estimator. To obtain a consistent estimator of £(.g., £) Hansen's [1979] theorem 3

implies that

af(y)' af (y) T+ “y ly y ly . 1 FM'} . . bf) Thus in finite samples £ = TTI. Q' Q > . where all 4x4 Y Y Y ly

-13-

matrices are defined as before. For hypothesis testing we use

g- Nyy, k22 8) ft) = NG, TE?

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

4 1

35, = say 81 where SI°is the matrix of smoothed periodogram and cross

periodogram ordinates evaluated at the zero frequency and multiplied

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

(15) var(#) = pi si £74) 23 4x4 4x4 4x4 In the present:case the construction of S, 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

e : 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.

- 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 presen<ed

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 2urocurrency 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, covariance 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,,,) is aligned with the forward rate (ff), 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

- 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 (Fr) and spot rates (S,,,) accordingly would introduce a form of heteroskedasticity 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 distribution is derived for the estimated coefficients (7) 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 representation of s(t), f(t), i(t) and ix(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.

- 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 used.-= Also, 30-day eurocurrency interest rates were used instead of treasury bill rates, since these interest rates are

2 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 R? is quite high and the t-ratio associated with the coefficient on F. is very high. The coefficient on S441 (15) 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 Y2 would be inconsistent and asymptotically biased toward zero if an OLS procedure was applied. This is because vy and-Si41 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 that

-17-

Table I

ESTIMATES OF THE (MT) EQUATION FOR THE DM/DOLLAR RATE MARCH 1973 - JUNE 1979 USING MCCALLUM'S INCONSISTENT PROCEDURE® .

eer en NEE A LL

Explanatory Variables

Case I

Case II

Case III

Case IV

Case V

renee I A

constant std.error t-stat

std.error t-stat

S + t 1 od.error

t-stat

F t-l...g error

t-stat

S.E.R.

D.W.

a b

Cc R-

-.01879

"(.001611)

(-11.667)*

1.0472 (- .015626) (67.0143)*

.0061

(.014906) (.41202)

.999 .00158

. 3672

-.0222 (.004075) (-5.44722)*

.77926 (.146564) (5.31689) *

27848

(.144819) (1.92297)

994 003683

1.1023

R? value is shown for F. not Fi oF

value is shown for Fe not Fi oF

-.02256 (.007463) (-3.0237)*

.52112 (.400936) (1.29975)

53443

(.396917) (1.34645)

982 006733

1.2491

t-1

t-1

- .0276

(.00538) (-5.131)*

1.075 (.0341) (31.500)* -.0018 (.03074) (-.06096) 8095 (.06967) 11.6177 .999> 0009

1.9151

-.0449 (.01218) (-3.689)*

1.0490 (.04502) (23.305)* . 0636 (.0419) (1.5172) .9450 (.03882) 24.3475 .999° .0003

1.678

nn

* Indicates that the coefficient estimate is significant at the .0S significance level.

- 18 -

appear as regressors in the first stage regressions used to generate

the fitted values F* and §.., are S.y, Sy_a> Feige Feige Ree Rena and a constant term. In Case III, a) and Rea were dropped from the set of variables, following McCallum [1977]. In both cases, the constant term is significantly different from zero, whereas, 1) is significantly different from zero in Case II but not in Case III. Also, the constant terms are small while the coefficients on Fr and Seal? Y,* Y2» approximately sum to 1.0.22/ The D-W statistics again reflect the fact that first order serial correlation is present in the error term. Moreover, we know from the analysis in sections II and III the use of instrumerts lagged only one period will result in inconsistent estimates of the coefficients 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 Res Rego Feozs Frege Feog3> Steere Sto» St-3 and a constant term. In

case V, R and S,_3 are dropped from this list. In both cases,

t-1° Fees ¥1* ¥ approximately sum to unity while the coefficients Yo and ¥, are significantly different from zero. Also Y> 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 js 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

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

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

a

Explanatory Variables Case I Case II Case III Case IV . constant "2.026004 -.01623 - 01732 - 91695 std. error .00928 .01451 .01420 .0124 t-stat -2.80216 -1.11825 -1.21929 -1.36921 F. 56753 .13592 1545 .1289 std. error .05822 . 143209 .13902 .1277 t-stat 9.74766* 6949115. ~=—-11.112 1.0099 Sel .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 R2 .973 9537. ~—s«w 952 98560 S.E.R. .0115 .01516 .0149 .0148 D.W. 1.2857 2.2789 2.2546 1.9913

In

®@ 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 2: R* is shown for F.. not F.- OF. 3:

* . Indicates that the coefficient estimate is significant at the .0OS significance level.

- 20 -

to those obtained in Table I. The coefficient estimates for Yy and Y are (.5673, .50715) and both coefficients are significantly 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 estimators are shown in-cases II, III and IV. The first stage regressions used to form the fitted values for Fr and S,,, 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 (y,) 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 Sa (YQ) are large (.9017, .8863) and statistically significant whereas in Table I these coefficients 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 presented 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 Yo is large and statistically significant while the coefficient 1) is small and statistically insignificant. Also, in contrast to the results in Table I, the autoregressive parameter (p) on Fre. 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

~ 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 IT) were absent. Even if it was assumed that the structural error term a, (eq. (2)) was serially uncorrelated, serial correlation can arise in the error term due to the presence of the term stad (see section II). Thus the coefficient estimates and standard errors obtained in Tables I and II 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 samp].e veriods Table IJI and IV were created as follows: Each series me =e 1 E

was cletrended and the fitted values F F F S

S l were formed t-l’ t’ t-l?’ Se

t+)’ in first stage regressions where the regressors were F,_y, Fie3? S.o2? S423? Rey Rio? R 3° Equation (10) was then estimated subject to the nonlinear restrictions developed in section IIa. The Davidon Fletcher Powell Algorithm was used to obtain the estimates of ¥ in Table III and 1.22! The "corrected" standard errors and t-statistics reported were obtained using the procedure suggested in section III. (a).

The estimated spectral density matrix of Ze evaluated at the zero frequency, §*(0) was obtained as follows. First, the consistently estimated

,

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

.matrix for z, was formed where estimates of the spectral and cross spectral

- 22 - .

densities at the zero frequency were obtained by using a lag window (Daniell) of width 7 and 9 ordinates, depending upon sample size.2>/ 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 $7 (0). Finally, the Var(y) is computed 4s 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 %y and 6 are statistically significant in Table III. This is in contrast to Table I where Yo? Y and 6 were all statistically signi-

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

standard errors presented in Table III for (Yq ¥y> Yo 6) are all greater than the standard errors computed for these coefficients using Fair's method (see cases IV, V, Table I). 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 procecure 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 prope1:

procedures are pursued. The coefficient estimates for Yq; yy and Y> are

- 23-

Table 1114

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

nnn eae

‘Explanatory "Corrected" "Corrected" Variables Coefficient Standard Error t-stat constant -.0121570 . 13821789 - .0879553

FY .9963831 3010503 3.309689" Stel - .023768027 .0571002 - 4162506 | Foy 902132 1168834 7.71822"

cnn

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

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

- 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

EEE

Explanatory "Corrected" "Corrected" Variables Coefficient Standard Error t-stat aC constant - 0464825 -692322 -.06714 FY . 636763 1.627486 . 39126 sé -415037 1.78551 .23245 t+] ’ Foy - 55076 2.11135 . 26085

rN EL

- 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 insignificant in contrast to the results in Table II which suggested that Y> 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 developed by McCallum [1979A] and Kohlhagen [1979]. Specifically, these papers have suggested two different sets of conditions under which

. = sa so that perfect collinearity among the explanatory variables occurs iin the equation being estimated. The first set of conditions sufficient to obtain this result suggested by McCallum, are (a) covered interes: arbitrage, so that Fe = Fy and (b) strong market efficiency so that Fe =s - The second set of conditions, suggested by Kohlhagen, 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=/ and thai: 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.

- 26 -

V. 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.

APPENDIX 1

It is important to realize that the estimator of Ss, obtained by Cumby et al. [1980] (e.g., %) 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

(Al.1) § = co'wow'w) ~2w'Q) “2Qwow'w) “Wry they have

- d gtwow'w)72wrg. 7? Qtw (WW) 7” Wtw (Al.2) 7-97 OS) oT

YT

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

W'w

(A.1.5) $ N(O, plim = (Whww'H)),

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 /T (6- 8) as

- A2 -

wo Pe. CtwCWTK)FWrQ)=? Qt CHIN)” (Al.4) YT (6-6) + N(O, plim ‘ a a

Too

g (win) * weg Qrwu ny” 1wig” —— *)

W'ww'W

where 2 = plim —>

T+

Now consider the estimator of the present paper

canes) B= (QrQe) Tatty = 8 + (QQ) GW

thus we have

*10* -1 xt (Al.6) VT (8-6) = vT (Qe) t*tw = (Q ae 7

Now

(A1.7) ees N(O, plim = (Q**ww'Q*))

Tao

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

*tnk* -1 *1O* -1 (Al. 8) VT (é- 5) 2 NCO, plim (grrgr) 98 a) )

Tae

; * Yn kOe where now 2 = plim oe

T+2

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 G@ will be asymptotically equivalent. However, in

- A3 -

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

two alternative methods of computing 2.

- 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) = —_(t) . The techniques

l+i

_ (t)

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 parameter space expands. In performing these tests the equation estimated

appears as

N R(t- jo, (3) + 2% S(t- j)o,(3)

(A2.1) R(t) =

J

te

+

: Pe 830) + €; (t)

wuz

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

- A5 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(r, n-k) = “QRes7(n- k)

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

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

- values 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

¢

k* Allowing for a deterministic seasonal and deseasonalizing, the

series in question before conducting the exogeneity tests does not alter the results reported here.

- Ab - Table A2

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

R(t) S(t) . F(t) Lags Coefficients Coefficients . Coefficients

1973:3 - 1979:6

1 0.939 ( 7.8349) .3095 ( 1.205) £3343 (-1.311) 2 -0.164 (- .96814) -.3932 (-1.532) .4039 ( 1.586) 3 0.107 (° .6702)

4 0.008 ( .0516)

5 -0.030 (- .1946)

6 -0.258 (-1.655) F(4,55) = 2,6492*

7 0.302 ( 1.903)

8 -0.0817 (- .7009)

LS SE

1973:7 - 1980:7

0.8750 ( 7.098) £1395 (1.0787) -.3484 (-1.833) -0.2989 (-1.894) -.0311 (-.2640) .0532 ( .3108) 9.1255 ( .7621)

2477 (-1.560)

0.0677 ( .4302)

0.0641 (- .4139) F(4,65) = 3.2175"

0.1330 ( .8667)

-0.1825 (-1.506)

on nun & WA N FF t Oo

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.

?

- A7 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.

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 rates

used as regressors are of different maturities than the left hand side forward rate so that a certain form of heteroskedasticity would be introduced 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).

- F2 -

7/ It should be observed at this point that the estimation procedures attributable to Hansen [1979, 1980] which are applied below do not require that Ry 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 conditions. 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

. - t ; economet:rrician only usés a subset ¥ of the information in I. in forming

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

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

the forecast error due to using y, 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 urilize 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 is the forecast error between period t and ttl and u. is a structural Nee] P t

white noise disturbance, as of time t. Now we can form Elv ve_4] so

Elv Vez] = El(u,- Yoneey) Ug_y 7 ¥2",)] = Eluyuy_s] - 2 Y2Eturn] - YoETn Ye] + 2 En. 417] #0 _ Note that: although E{ne 44-1) = 0 and Efuu. i] = E{n. yn] = 0 by

assumption, the term E[u.n,] #0. Also Ely Va] # O0for similar reasons. This point has been made by Hayashi [1980] and Cumby et al. [1980]. Also Elv,"] # O and Elviv. ] = 0 for j > 1 or j < -1 so that Y, follows a first

eo

order moving average process (MA(1)).

- F3 - 10/ To see this note that Ej[v v = - [V.%pa] = Ele, + PY 2M, - YoMe ay) (Egy * OYOM | - Yor)

= Ele ,e,_y) + ovgECe.ny_1) - yoE(epn,) + ovgE (ne, 4)

+

22 2., 2 PY. E(meng_y) - PY, Elm) - yoE (ney &e 4) + py.ZE( ) + y,7E(n...n,) # 0

PY> Eee Mead Yo FAMe a Me

2, _ 22 _because E(e,n,) # O and E(n, ) = 67.

Moreover, Elv vied] # 0 for similar reasons.

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 I> I

t-l t-1?

and Yee 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 [19&0].

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 bilateral 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 developad

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.hand vis a vis the number of parameters 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 GMM estimators, see Hansen [1980] or Hansen and Sargent [1981]. For an example of the application

- FG -

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

(1) e*(y) = CY = Q&(y))" wOW'W) 7H! CY - QECY)) ~ Txl TxN Nxl 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) whilch is defined as

(2) Q@ = www) twa TXN TxK KxT TxK

Finally, W is a (TxK) matrix of instruments and Y is a (Nxl) vector of coefficient:s to be estimated. The first order conditions for the criterion function in (1) can be written as

avy) af (y)' 1 5 = -2 = Q'WwcW'W) "Wry Y y l¥ af(y)"| -1 . + 2——|_ Q'W(WIW) “WIQE(y) = 0 Y Y

~

or after some manipulation

Ber(y) — 9£(y) (3) sa. Y Y Y

~ ~

gtw¢w'w) +w Ly - QF (9) ]

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

ae*(y) af(y) 5 9 Y Y

— 1 - VE7))

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.

- 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, a.) = a.'£(y) where a. 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 oo

g,(s) = 2 s?R (3). Now evaluating the autocovariance generating function jJ=-©

2

: * ° 4 lw : : on the unit circle (s=e) where w now signifies frequencies results in

i iw} : WW) s Ee PR) which is the definition of the spectral density

g_ (Ce matrix of the z vector. Now note that evaluating the spectral density

matrix at the zero frequency (w=0) results in Ss. = £ R_(j). j=-°

Second, the reader may be wondering why the S. is not estimated

using time domain procedures, as

1 a a 5 E pe? = R(-1) + R,(0) + RQ),

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 autocowariances. 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 S_ results in an estimate of S| which will be positive definite by construction. Frequency domain procedures do have the disadvantage.that they do not exploit a finite number of autocovariances in forming consistent estimates of the weighting matrix 3_. 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 frequency 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 S, matrix computed in this paper. See Appendix 1 for a proof 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, equat‘'on (15) is the correct expression for the variance of y when the equa:ion being estimated is nonlinear in the parameters. 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

: af (y) that because the partial derivative matrix —>—_ is nonstochastic, an 6x4 Y alternative and computationally less burdensome expression’ for the variance of Y is . 1 ..-1 9FCY)' af(y)} oy (15) Var(y) = s-[f © —— r T r) “ y |¥ 6x6 *y Y 4x6 where SI is now a 6x6 spectral density matrix of z. = e.4, instead of the 4x4 matrix obtained when we use 2,5 e, >|. q, as defined in Y Y axe OX?

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

_ f(y) difference due to the nonstochastic nature of =

a

Y 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

- 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) * Yo = 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, Y, the behavior of arbitraguers and Y2 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 developing 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.

BIBLIOGRAPHY

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

weocee » 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, "Iwo-Step Two-Stage Least Squares Estimation in Models with Rational Expectations," forthcoming in the Journal of Econometrics.

ceeeee » Obstfeld, M., 1980B, “Exchange Rate Expectations and Nominal Interest Differentials: A Test of the Fisher Hypothesis," forthcoming 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,

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.

- B2 -

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

Glaessner, T. C., 1982, "Theoretical and Empirical Essays on the Determination 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,

Canadian Journal of Economie: 7, pp. 496-501.

Haas, R. D., Alexander, W. E., 1979, "A 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.

co--c- , 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. l.

corre , 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.

eo

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

- 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 Determination: 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.

weeee- » 1.977, "The Role of Speculation in the Canadian Forward Exchange Market: Some Estimates Assuming Rational Expectations," Review of Economics and Statistics, 59, pp. 145-151.

ceeene » 1.979B, "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, PPpe 1-48,

Stockman, A. C., 1978, "Risk Informatign 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.

he

-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.