- 19 -
where
atte) = [(1-,), 0, -(1 - 9,L) fanB'¥ 7
080 Py) 0, -(1 ~ 91) [4281 ¥9 (Eq)

2x6 ' ,
/* 428) * 4382) ¥** (Eq) ]

‘o, (L- poh), -(1- gh) [4,8 8) +458 82) ¥** (55)

+ 4)8)9"(5)]

rs 550) = |0, 0, -(1- pyb) [48195 (Eq) +. (498) +458 8); remy)
“ox6

0, 0, -(1- pob) [(428] + 4582) ¥5" (Sq) + 4 845 (Eq)

for j=l, ... yl and 2 = (s., X.). Observe that N=2 so that

Poe -t
given P=3 and h=4 there are only sixteen orthogonality restrictions.
In sum, this procedure will not result in estimates of parameters

which will be as efficient as those obtained if one could estimate all

twenty-four structural parameters jointly as in (21) .30/

Iv. Estimation of the Flex-Price Model for Canada and the U.S.

1973:3 - 1980:8

4.5(a) Description of Estimation Procedure

Tables I and II below present the results of estimating the flex-
price model represented by the orthogonality restrictions in equation (26)
given a choice of lag length for the "forcing" variables (see Appendix 2) of r

= 1, and a maximum lag length for the instruments of P = 3. Thus the set of


- FS -

assumption which is spelled out in more detail in Glaessner (1982A) Chapter IV,
concerns the definition of the information set 2. In the context of estimating the
model it was assumed that2, = {X,, X apo 5 where,both economic agents
and the econometrician useQy to form optimal predictors of x +k? Assuning that
agents observe a different information set than the econometrié fan, (see Shiller
[1972]) introduces an aaditionai forecast error into the set-up described below.
However, it can be shown that allowing for this forecast error will not alter the
orthogonality restrictions derived below.

22/ The set of instruments could also have included lagged values of the variables
Pr and S_ in addition to the lagged values of the "forcing" variables. Thus, an
implication of this observation would be to use a different set of instruments in
estimation, Future work may address this issue.

23 EIVEIO, 1] = EM, - E19 319.) = 0.
4x1

24/ Note that E(vi,1%, 5) = EIQ. - 25% 4.31% 5] =0 for i=1,2

if it is assumed that the elements in QF {Xo Xpuge ee ) are

uncorrelated with Qt and Qed for i=1,2. This will be the case

since the fundamental noises ae i=1,2 are not assumed to be
serially correlated. Note, however, that in general the MA(1) error

is not projected on a4 because we allow for contemporaneous corre-

lation between the elements of a. and Q. i=1,2.

-1 it-1’

25/ In Glaessner [1982A] a more detailed explanation for the notation adopted here is
given. Also the observant reader will have noticed that one orthogonality condition
is missing in equation (21) to the extent that we want to use the fact that

X. : .

EVE, JI = 0 for v > 1 instead of v > 2. This

4x1
consideration could be allowed for in estimating the model. There is,
however, no sense in which using v > 2 will lead to inconsistent pa-
rameter estimates. —

r-1 :
_26/ For example, to see how the coefficients i) in ¥¢L) = 5 vty) were

. . j=0
derived, note that from equation (13)' and (14)' we can derive expressions of the form

r-1 : “1 r-l r kj :
YL) = © wtp 2 a ()Ir+ ECE on PAL]
j=0 J jel k=j+l


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 Emprical Essays on
the Determination of Spot and Forward Exchange Rates," University of
Virginia - Charlottesville, 1982). The author is particularly
indebted to his thesis advisors Robert Flood and Richard Meese for
encouragement and support. Kenneth Rogoff also made many helpful
comments on an earlier draft of this paper presented at the 1981
Winter Econometric Society Meetings. Discussions with Robert Cumby,
Dale Henderson, Robert Hodrick, Peter Hooper, Maurice Obstfeld,
Kenneth Singleton and Steve Symansky led to improvements in the
paper.

1/Although, these authors adopt a methodology similar to the one used
‘in the present study in order to derive the cross-equation
restrictions implied by the rational expectations assumption, their
work is aimed at estimating different models of exchange rate
determination than the flex-price model developed in the present
study. (See Glaessner [1982A] Chapter III). Moreover the methodology
used in estimating their models, the joint assumptions of exogeneity
of the "forcing" variables and the nonallowance of complicated
interactions between the "forcing" variables in the vector
autoregression are not necessary in the present study.

2/Authors such as Frenkel [1976, 1979], Bilson [1978] and Frankel
[1979, 1981] have also made simplifying assumptions with respect to
(a) and (b) in their empirical work. Frankel (1979] does admit the
possibility of a quasi-reduced form error term in the spot exchange
rate equation following a first order autoregressive process. However
Hakkio [1981] finds evidence which suggests that the stochastic
process generating the error term is somewhat more complicated than an
ARC 1).

3/This was the case for the models of Bilson [1978], Hartley [1982],
Frankel [1979] and others. (See Glaessner (1982A] Chapter II).

4/ See Obstfeld [1983] and Cumby and Obstfeld [1983] for a discussion
of how and why conditional heteroskedasticity of error terms can
present problems.


- b3-

Hansen, L., 1982, "Large Sample Properties of Generalized Method of Moments
Estimators," Econometrica, (forthcoming) .

Hansen 1,., 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., Hodrick, R. J., 1981, "Speculation and Forward Foreign Exchange
Rates: Some Tests of Cross-Currency Restrictions," unpublished manuscript,
Carnegie Mellon University.

Hansen, L. P, and Sargent T. J., 1980, “Formulating and Estimating Dynamic Linear
Rational Expectations Models," Journal of Economic Dynamics and Control, Vol.
2,1.

Hansen, L. P., and Sargent, T. J., 1981, "Instrumental Variables Procedures for
Estimating Linear Rational Expectations Models," unpublished manuscript,
Journal of Monetary Economics (forthcoming).

Hansen, L. P., Singleton, K., 1982, "Generalized Instrumental Variables Estimation
of Nonlinear Rational Expectations Models," Econometrica, (forthcoming).

Hartley, P. R., 1982, "Rational Expectations and the Foreign Exchange Market," NBER
Working Paper No. 863.

Henderson, Dale W., 1979, "The Dynamic Effects of Exchange Market Intervention
Policy: Two Extreme Views and A Synthesis," International Finance Discussion
Paper, No. 142.

Henderson, Dale W., 1980, "The Role of Intervention Policy in Open Economy Financial
Policy: A Macroeconomic Perspective," Prepared for the Pennsylvania State
Conference on the Political Economy of Domestic and International Monetary
Relations.

Hodrick, R. J., 1979, "On the Monetary Analysis of Exchange Rates: A Comment,
Carnegie Rochester Conference Series on Public Policies for Employment, Prices,
and Exchange Rates, Vol. 11, Supplement to Journal of Monetary Economics.

Hodrick, R. J. 1978, "The Monetary Approach to the Determination of the Exchange
Rate: Theory and Empirical Evidence," in Frenkel and Johnson (eds.), The
Economics of Exchange Rates: Selected Studies (Addison-Wesley, Reading,
Mas3.).

Hooper, P. and Morton, J., 1980, "Fluctuations in the Dollar: A Model of Nominal and
Real. Exchange Rate Determination," International Economic Review,
(forthcoming).



-13-
Specifically, write

(16), QQ- gh).
it” ToD) Ge

5 i= i =
where i=1,2,0< los | <1, 0< ig; | <1, and £(Q,,) = 0 for allt,

FIQ;.Q2_5] = 0 for all j#0, i=1,2. Also, Elq.ai] = o> where
x.
qn = (Q,> Q,) so that the reduced form error terms in the spot ex-

change rate and price equations are allowed to be contemporaneously

correlated.

To derive the orthogonality restrictions present in the flex-price

model, let us define the information set %_) = Xp Xo ve.) Which
is composed of predetermined values of the forcing variables.22/ The
definition of the innovations in the vector autoregression cv) stated in
section II suggests that E[velo, i] = 0. 23/ This observation implies

the following set of orthogonality conditions for the four equation vector

autoregression represented by equation (15)

(17) Xt =
EL YER! J = 0

for all v > 1,
4x1 4x4

The orthogonality restrictions for the price and exchange rate
equations can be obtained by multiplying equations (13) and (14) by Q.-p)1)

and (1 - 9L) respectively and rearranging to obtain

(18) ye] 12 k
Vig = (2-0, 1) (S, + 458) eg? ELK. .y 1%)

~ (458) + 4585) mg) ELK, ay 12.)


- 21 -

The parameter estimates presentred in Table II were obtained by using
the initial estimates Aur to form a second criterion function which
was weighted by an optimal weighting matrix based on the orthogonality
restrictions in equation (26). Heuristically, these parameter estimates can
be thought of as more efficient than those in Table I because the weighting
matrix reflects the investigators assumptions concerning the structure of the
error vector, 3L/

The parameter estimates in Table II are not appreciably different in
sign or magnitude than those in Table I. However, a (the semi interest
elasticity of money demand) now has the wrong sign, ao (the income elasticity
of money demand) is now significant at the .2 level and o (the domestic
consumption share of the domestic good) is significant at the .05 and .01
significance levels. It is possible to conduct a joint test of (a) the
structure of the flex-price model (including assumptions made about the serial
correlation properties of reduced form error terms) and (b) the nonlinear
rational expectations restrictions. This test statistic is distributed a (8)
since there are sixteen orthogonality restrictions (R = 16) and eight
structural parameters (Q = 8) given the representation of the orthogonality
restrictions in equation (26) with a maximal lag for the instruments of
p=3.32/ the x2(8) statistic of 21.473 rejects the theoretical
restrictions implied by the flex-price exchange rate determination model.

The findings above may in part be explained by the recent work of
Meese and Singleton [1981] which. suggests that the log of the Canadian dollar
rate may not be a covariance stationary process (i.e. the univariate process

on St may have a unit root). Several other considerations of a more


- F6 -

Note thet when j= 0 we have At) so % = atin), however, for
jel,..., rel

the double sum in the far right expression can be
rewritten in terms of the y5"s as

* _ y-l j j+l r-j
Yo =f J
3 7 A Oy) Oy Asay ty Asae tees tony TAL)
for j=1,..., r-l. This can be seen by writing out the double sum
r-l or kej
E(t uy 7A) explicitly.
jel k=j+l

27/ In practice it is possible to impose these restrictions by inverting the
‘polynomial matrix A(L) in the frequency domain by using procedures suggested by
Sargent [1979]. The moving average representation thus obtained is then truncated at
values for the moving average parameters which are extremely small and the inverted
polynomial matrix is evaluated at the discount factor of interest (n,) or (n2), Having

done this, the restrictions embodied in the ¥,'s j=0,.++ 5 r-1 can be built up

forming the criterion function to be minimized.

28/ These ‘lag length specification tests are done for Canada and the
U.S. over the period March 1973 to August 1980 for monthly data. See
Appendix 3 for a description of the data and Appendix 2 for a more

detailed explanation of the tests for finite lag length.

29/ More formally, for the theoretical model to be identified Hansen
and Singleton [1982] argue” that the QxR partial derivative matrix
oF,
Dp = Elg=(éq)] must be of full rank and the nonlinear £ function
0
RxQ

characterizing the orthogonality restrictions must be differentiable.
Moreover. in implicit exclusion restriction in the: current specification

of the flex-price model concerns the fact that the variables in the vector
autoregression are not allowed to be dependent on contemporaneous values
of the spot exchange rate or domestic prices. In addition, the overidenti~
fying restrictions implied by the theorétical ‘model can be tested by con-
structing fairly straightforward statistics which only involve estimation
of the model subject to the nonlinear restrictions in Y*(-) and ¥**(-)

unlike likelihood ratio tests. See Glaessner [1982A] Chapter IV, or Hansen
and Singleton [1982].


results should be considered preliminary, however, due to several technical
problems discussed in section IV.

The organization of this paper is as follows: Section II describes
the equations in the flex-price model and derives closed form solutions for
the spot exchange rate and domestic prices, which embody the cross-equation
restrictions implied by the assumption of rational expectations. These
solutions are obtained by using methods from prediction theory .9/

Section III presents the orthogonality restrictions implied by the
theoretical model which can be used in order to form the GMM estimator.
Section IV presents and discusses the empirical results obtained in
estimating and testing the theoretical restrictions implied by the flex-
price model for Canada and the U.S. over the current floating exchange rate
period from March 1973 to August 1980. Finally Section V summarizes the

conclusions of this study and suggests topics for future research.

II. The Flex Price Model:

The model presented below is a slightly modified version of the
models developed by Flood [1981] and Mussa [1982A]. The country under
consideration is assumed to be large only in the markets for its owa money
and its own output. It is assumed to be small in the world (or foreign)
goods and capital markets, so that the foreign price of foreign goods
(P_*) and the foreign interest rate (i,*) are exogenous variables.
Equations (1 - 12) characterize the flex-price model of exchange rate
determination. The model presented below is not yet complete in that

specific stochastic processes generating the error terms and the


The CAT criterion attributable to Parzen [1975] and the more
* standard likelihood ratio tests for choosing the finite unknown erder

of a vector autoregressive process, presented in Table A2 have been
described by Meese [1980]. ~

To limit computational costs the maximum value of lag length (r)
was chosen to be 4, Thus in Table A2 the multivariate Cat criterion
suggests a lag length of r=l. This is because the minimum vlaue of
cat(c*) occurs at r* = 1 and thereafter increases monitonically for
r= 2,3, 4. In contrast the likelihood ratio tests tend to suggest

a lag length of three since the restrictions imposed by a one or two lag model

are rejected at the .05 significance level. Thus, the results in Table A2

suggest that it is not unreasonable to assume that the order of the vector

autoregression generating the forcing variables is unity.


- 20 -

eight instruments contained only predetermined values of the forcing variables
Xt-25 Xe-3, where the vector X¢ is defined as before.

Heuristically the application of Hansen's [1982] GMM estimation
procedures within the present multi-equation context, involves the
minimization of a criterion function which is a weighted average of the small
sample estimates of the population orthogonality conditions in equation (26).
The weighting scheme is chosen so as to obtain the minimum asymptotic variance
covariance matrix for estimators that can be viewed as exploiting the same
fixed set of orthogonality restrictions. A formal derivation of the GMM
estimator and an explanation of the actual procedures used to obtain the
parameter estimates and standard errors in Tables I and II below can be found
in Glaessner [1982A].

Tables I and II reveal that the autoregressive parameters in the spot
exchange rate (py) and price (94) equations are statistically significant.
Both of the coefficients Py and Po have a value close to unity. The other
six structural parameters in the model a, (the semi-interest-elasticity of the
demand for money), 42 (the income elasticity of the demand for money), by
(the relative price elasticity of the demand for domestic goods), bg (tne
elasticity of the demand for domestic goods with respect to the real interest
rate), b3 (the real income elasticity of the demand for domestic goods) and
o (the domestic consumption share of the domestic good) are all found to be
insignificant in Table I. However, these first stage estimates of the
additional structural parameters do seem to have the correct signs and be of
the proper magnitudes. The coefficients on the first own lags of the left
hand side variables (me, yt, it, Pe*) in the vector autoregression

(see Table I) are usually close to unity and statistically significant.


-2%-

Vv. Conclusions and Possible Extensions

The results presented in this study suggest that the
theoretical restrictions implied by the Flex-price model of exchange
rate determination are rejected soundly. This is the case even though
the restrictions implied by the assumption of rational expectations
are imposed in estimating the model, and the assumption of purchasing
power parity is not a foundation of the flex-price model estimated.
In addition the rejection of the flex-price model cannot simply be
attributed to an incorrect exogéneity specification since the use of
ford procedures in estimation allows one not to have to make strong
assumptions about the exogeneity of "forcing" variables. This is in
contrast to all previous work aimed at estimating and testing the
rational expectations restrictions embodied in exchange rate
determination models.

In light of the above conclusions one might ask two questions.
First, what might explain the poor performance of the flex-price model
for Canada and the U.S. over the current floating exchange rate
period, and what alternative specifications for exchange rate deter-
mination models would be interesting to estimate? Second, to what
extent should future research be devoted to solving some of the
technical problems mentioned at the end of section IV.?

It is difficult to answer the first question, because the test of
the theoretical restrictions conducted is a joint test of rational
expectations and the structure of the model. Thus it is difficult to

determine whether it is the assumption of (1) rational expectations


Table II
Final GMM Estimates of the Constrained Flex-price Model

Using a Consistent Estimate of the Optimal Weighting ‘Matrix for the

Orthogonality Restrictions in Eq. (26) A

Parameter Estimate x7)
rey "1164 (0966)
; a, .936L = (1.239)* “-
Soon | Ps =.6270  (-.0864) - 22,473
by -.3919  - (-.1003)
by “7055 (21949)
a «9223 (14,575) **
» 1.0884  (16.232)**
Lo

-9153 (2.5816) **

A) Statistics in parentheses are ratios of coefficient estimates to their
asymptotic standard errors.
** Indicates significance at both the .05 and .01 significance levels.

* Indicates significance at the .1 significance level.


- A6 =

Appendix 3

The data for this study were obtained from the following

sOUrCcES?
[e>) Mo Canadian money supply, Ml (currency + demand deposits)

IFS DATA TAPES, not seasonally adjusted.

(2) Y, = Canadian Industrial production index from the OECD (MEI),

mot seasonally adjusted.

(3) int U.S. eurodollar interest rate, obtained from DRI. These

are London Interbank offered rates (LIBOR) for Eurodollars.

Not seasonally adjusted.

*
(4) Ps Wholesale U.S. price index, obtained from Federal Reserve

Board Data Base. Not seasonally adjusted.

(5) Ss. The spot exchange rate expressed in units of .the domestic
currency (i.e. Canadian dollars) per unit of foreign currency
(i.e. the U.S. Dollar). This series was obtained from DRI.

These are interbank rates (bid prices) - Not seasonally adjusted.

(6) PB = the industrial selling price index for Canada from the
Canadian Statistical Review or the STAT Canada data base at the

Federal Reserve Board. Not seasonally adjusted.


-17-

(25) wr = Ata)

yt = Aten,

j jel -§
5 2) (mp7 As, tng?” tj

Asi5 +... +

j+2

for j = 1, ... , r-l.
These expressions for the coefficients %5 and ia for j = 0, —-, r-l can be
derived from the closed form solutions given in equations (13)' and (14)'
above.26/ Thus, equations (24) and (25) provide one with an explicit
representation for the rational expectations restrictions across the parameters
in equations (13) - (15), which can be imposed when constructing the GMM
estimator .27/ Equations (24) and (25) also focus attention on the
importance of the choice of r (the order of the vector autoregression) in
determining the number of parameters to be estimated (Q) and the complexity of
the cross equations restrictions. Appendix 2 below explains and implements
procedures for choosing finite lag lengths based on work by Parzen [1975].
These procedures suggest a lag order of unity (or r = 1) for the vector
autoregression generating the "forcing" variables in X_, for Canada and the
u.s.28/ . Thus, if r = 1, only contemporaneous values of X¢ would appear in
the closed form solutions. In addition only twenty-four structural parameters
would have to be estimated (i.e., sixteen structural parameters in the vector
autoregression and eight additional structural parameters {a,, o> g by,
by» b3, Py Po}.

Equation (21), also suggests that the total number of orthogonality
conditions present in the model will depend upon (1) the number of instruments,

denoted by h (2) the number of error terms denoted by N and (3) how many lagged

values (i.e. lag order) of each "instrument" is chosen,


(14) P. = (d58)"+ d285)

woo

k 1 eo k
(n.)” EX. 1%.) + 4.8" of (nm)
o 2 tek! 't Ml yl)

= ys
(15) ACL)X, = ve
4x4 4x1
where
'
8) = (0, 1, “Bo, (B(1 - o)F - Bo (1-0) - BL]
82 = (2, -8), a), ((o-1) + aC 0))],
' 1 oon
Xe [mys ¥y aD, PE
_ Boo
my * By + Be
2
ng = Tea)
= 1
a, - By + Boe * d,
(e,(o- 1) - 9)
d, = ——
2 By + Bo) (i+ 0)
=i
ds dra,
be
1 iz bz
poe 2. W/
2 l-b


The results presented in Table Al test whether the sixteen co-
efficients on the hypothesized endogenous variables in ‘the four equa-
tion system are significantly different from zero. For both the raw
data (only detrended using a linear trend) and the deseasonalized and
detrended series the joint exogeneity specification of the model was
rejected at both the .05 and .01 levels using the likelihood ratio test
of the restrictions described above. Applying "sims" correction still
results in test statistics which are significantly different from

zero at the .05 level for the flex-price model.


Appendix 1

Exogeneity Specification Tests of the Flex-Price Exchange Rate
Determination Model for Canada and the U.S.
March 1973 - August 1980

This appendix presents the results of performing joint exogeneity
tests for the flex-price model of exchange rate determination using

the procedures suggested by Geweke [1978] or Geweke and Dent [1978].

Specifically, the joint exogeneity of the variables in the vector
autoregression x = Im, Ye is, Px] is tested with respect to the
hypothesized endogenous variables in the flex-price model, Pr the price of
domestic (i.e. Canadian) goods and S_ the Canadian U.S. dollar exchange
rate, The variables mt and Yt are defined as Canadian money and real

income while if and P# are the U.S. eurodollar interest rate and the U.S.
price level respectively. The data series used are described in more detail
in appendix 3. The results of conducting these tests for the detrended data
alone versus data which has been deseasonalized are presented in Table Al
below. The tests were conducted by estimating a restricted and unrestricted

version of the following four equation system, written in vector notation as

w £
(Al.1) = X = £ FRX + = Gry +
. wt s=l Ss Lt-s s=l s it-s &
4x1 4x4 2x2 4x1
where x, is defined as above, ve - (s, PJ, ry and Gk are matrices of

coefficients and COov( ft X,_,) = 0 forall tands=1, ..., w,

cov & Ye = 0 for allt ands=1, ..., & Also


Definitions of V

and Notation

is the natural log of the general price level.

the domestic consumption share of the domestic good.

is the logarithm of the domestic price of the domestic good.
is the logarithm of the foreign price of the foreign good.

is the logarithm of the spot exchange rate, which is expressed
in wits of domestic currency per mit of foreign currency.

defines the relative price of domestic'goods in logs.

is the logarithm of the quantity of money.

is the level or ln(1+ iD) of the domestic interest rate.
is the log of real output.

is the level of the foreign interest rate.

is the log of the forward exchange rate.

the log of the foreign price of the domestic good.

the log of the domestic price of the foreign good.

= EX, 12,] where @, contains all of the variables and dis-
turbances in the model dated t or earlier and the values of

the models parameters.

The flex-price model (eq. (1) - (12)) is identical to that in Flood

[1981] except that an additive disturbance term has been introduced into

the money market (eq. (3)) and specific stochastic processes for the

exogenous variables have not been assumed .10/ i/


hypothesized exogenous variables mp, Ye, Pe*, and i,* (defined

below) are not assumed at the outset. The necessary assumptions are made

below.
The Model
qy m, = oP, + (1-0) (S, + PI)
*
(2) a = Pe - G+ PDD
(3) mt eg =a, - ai + a5(PL+Yo-m)+u a,, a, a, > 0
t t 0 vt 2. t ot ot t’ 0’ “1? ~2 .
s _
4) me Me
s_id
(5) m= my
. .*
(6) Art As,
@ FL FSi
8 S
(8) YY,
(9) dye - ba (i. - -
Ye = Bg- bya bo. - CEM) Te) + bs + ep, bys dy, by > 0
Ss _ vd
(10) Yea Ye
(11) P. = Q + Ss.
(12) * = p*
Q Pe + Ss.


- 18-

denoted by P. Equation (21) defines fp Gy) as an N((P-1) + h) x 1 = Rxl
vector of random variables where BLE, (¢5)] = Q. Thus in the present context
given N=6 and h=4, the number of orthogonality restrictions is determined by
P. Hansen [1982] and Hansen and Singleton [1982] have shown that models
estimated using GMM procedures will be overidentified if R > Q and if certain
regularity conditions hold.29/ In the present context R > Q if P> 3.

Note that if P=3, R=48 which is a large number of orthogonality restrictions
relative to the number of observations 89 in the model. Hansen and Singleton
[1982] have pointed out that as the number of orthogonality restrictions gets
large relative to the number of observations there is a loss of precision in
both the estimated coefficients and asymptotic standard errors.

To avoid the problems associated with using a representation for the
orthogonality restrictions like that in (21) an alternative, less efficient
estimation procedure, which conserves on (1) the number of orthogonality
restrictions and (2) upon the number of simultaneously estimated parameters
was pursued, The structural parameters in the vector autoregression were
estimated using OLS and were held fixed when computing the GMM estimator for
the eight additional structural parameters fa), Wy 9 by, bys bg, e1> P5}- In
this case the orthogonality restrictions appeared as

(26)
pt 1 1
ELE,(Ep)] = EL (Ls got, @ |X, o1] = 0

C2C(P- 1) +4) x1

R Xee3


Equation (7) states that the log of the forward exchange rate (ft)
equals the expected future spot exchange rate (eESe+1). Empirical
evidence regarding this relationship has been mixed.14/ The weight of
the evidence does seem to suggest that this relationship may not hold
exactly, and that a time varying risk premium may be present. The present
paper follows Flood [1981], Dornbusch [1976], Mussa [1982A], and Rogoff
[1979] by asserting that equation (7) can be regarded as "a good
approximation" in the present context.

Equation (8) states that the supply of output is exogenous. This
simplifying assumption may be unrealistic; in future work it will be
relaxed.15/

Equation (9) says that the demand for domestic goods depends
negatively upon the log of the relative price of domestic goods (q.) and
the real interest rate (it - CEt e417 7 ) and positively upon the log
of real output (or income) ¥p 16/ Finally, equation (10) is simply
the equilibrium condition for the goods market.

Equations (11) and (12) state that goods market arbitrage holds for
each of the goods (domestic and foreign) in the model.

The flex-price model (eqs. 1 - 12) can be readily solved using
difference equation techniques .17/ Using vector notation the solut:on
is:

a3) S$. = dei K ery . ' nok
t 7 S25) co? (Xe, 19) + (4,8) + 4582) F (ng)

ELE, 44 12,1 , Vie


-22-
technical nature suggest that the results obtained above. should be considered
preliminary.

First, the chi-square statistic (see Glaessner [1982A]) will tend to
be biased toward rejection of the theoretical restrictions implicit in the
flex-price model to the extent that the true minimum value of the second stage
criterion function has not been-obtained. This consideration is important in
the present context since the convergence criterion for minimizing the
criterion functions used to obtain the parameter estimates in Table I or Table
II, had to be relaxed substantially, to obtain convergence. Specifically, the
convergence criterion was 1.0E-2 or 1.0E-3 for changes in the coefficients.
Often these convergence criterions for changes in the coefficients were not
met after two hundred iterations even though the change in the function at
each iteration was only occurring out to five places (e.g. 1.0E-6). Thus
convergence for each criterion function was achieved by relaxing the
convergence criterion from 1.0E-2 to 1.0E-l1, after seeing that very little
change was occurring in the actual value of the function being
minimized .33/

Second, the fact that in the final (GMM) estimates reported in Table
II the value of the autoregressive parameter in the price equation Co, ) is
greater than unity (1.0884)is quite troubling in light of the theoretical
restriction noted in equation (16) that 0 < | Py | <1 for i= 1,2. Such a
finding could be due to either (1) problems in minimizing the second stage
criterion function or (2) possible problems in characterizing the

restrictions. Thus, it may be necessary to not allow ey i=l, 2 to take


International Finance Discussion Paper
Number 208

April 1982

FORMULATION AND ESTIMATION OF A DYNAMIC MODEL OF
EXCHANGE RATE DETERMINATION: AN APPLICATION OF GENERAL
METHOD OF MOMENTS TECHNIQUES

by

Thomas C. Glaessner

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


-b5-

Obstfeld, M., 1981, "Relative Prices, Employment, and the Exchange Rate in an
Economy with Foresight," unpublished paper, Columbia University.

Papell, D. H., 1981, "An Econometric Portfolio Balance Model of an Open Economy with
Rational Expectations, Ph.D. Thesis, Columbia University.

Parzen, E., 1975, "Multiple Time Series: Determining the Order of Approximating

Autoregressive Schemes," Technical Report No. 23 (State University of New York,
Buffalo, NY).

Rehm, D., 1981, "Staggered Contracts and Capital Markets in the Open Economy,"
Chapter III of unpublished Ph.D. dissertation.

Rogoff, K., 1978, "Essays on Exchange Rate Volatility," Ph.D. Thesis, M.I.T.,
Cambridge, MA.

Salemi, M.K., 1979, "A Dynamic Stochastic Model of Inflation and Exchange
Depreciation," unpublished manuscript, University of North Carolina.

Sargent, T. J., 1978, "Rational Expectations, Econometric Exogeneity and
Consumption," Journal of Political Economy.

Sargent, T. J., 1979, Macroeconomic Theory, Academic Press.

Shiller, R. J., 1972, "Rat ional Expectations and the Structure of Interest Rates,"
Ph.D. Dissertation (Massachusetts Institute of Technology, Cambridge, MA).

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

Taylor, J. B., 1979, "Estimation and Control of a Macroeconomic Model with Rational
Expectations," Econometrica, Vol. 47, No. 5.

Tryon, R., 1979, "Testing for Rational Expectations in Foreign Exchange Markets,"
Intéraational Finance Discussion Paper No. 139.


- dl -

' - rl or .
G4)" PEs (4) Bla tony) (I+ 2 cr m3, y05y
1x4 4x4 j= kej+1

- r-l oor
* (4589+ d,g1)A tng) [1+ =o E

K-54 55
j=l kejer 2 AJL Nx, *V

2t

1 =
a) AM RE

Equations (13)', (14)' and (15) represent a compact notation for a six
equation system in Po Si and the "forcing" variables Mes Yeo ri and Pr.
The restrictions delivered by the assumption of rational expectations are
embodied in equations (13)' and (14)'. To actually express S_ and Py
as functions of all the structural parameters in the model [contained in
qd, ao, 8» 8) d3, Tye Nos A(L) ] it is necessary to determine the
order of the vector autoregression (r) in (15). For example, if r=2 we
would have to invert the 4x4 2nd order polynominal matrix A7!(L), and
evaluate it at L= Ty or Ls No: Thus, the order of the vector
autoregression will be critical in determining the number of structural
parameters to be estimated and the degree of complexity of the cross-
equation restrictions in the flex-price model. The important advantage of
the representation above is that the structure of the vector autoregression
is very general so that each element of X; depends on its own past values
and upon past values of the other varibles within Xy.

Equations (13)', (14)' and (15) almost provide one with enough
information to form an estimator for the flex-price model using GMM
procedures; however, it is still necessary to give a specific

parameterization to the error terms Vjr, i=l, 2 in the spot exchange


-10-

The polynomial matrix A(L) is assumed to be of finite order r so that
Ae) eI- At - A,L? =e. ATL, The innovat ions in the vector auto-
regression are defined as ve = x - ELK, | Xo 1? Xi_2> .+] where e(eve'] =
and eLVive! J we for all j # 0. These assumptions imply that it is
possible for there to be contemporaneous correlation across the equations
in the vector autoregression (a nondiagonal V matrix), but serial
correlation in the individual components wD of vu are not permitted
to be serially correlated. Finally, define F as FEZ, |2,] = ELZ 44 jo.)
and 1e(2, [2] = E(Z,_,10,_,] = some Zp, where E[-l] is
interpreted as the linear least squares projection operator, whereas E(-)
is defined to be the unconditional projection operator.

Finally as in the work of Mussa 19824] the price of domestic goods,
(P,_) and the exchange rate (S_) depend upon the discounted expected
future values of the real and nominal variables included within the vector
Xe. In contrast to the work of Frenkel [1976] and Bilson [1978], there

are two discount factors y present in the solutions for Py and

Ny
St in equations (13) and (14). The discount factor n, is a function of
only the parameters in the goods market whereas the discount factor 4 is a
function of the parameters in the money market .19/

Closed form solutions for the spot exchange rate and prices which

express each of these variables as functions of contemporaneous and >ast

kok
values of the "forcing" variables in x to (@m,, te ipePD are20/
t-l of .
aay sp @ggatinptr + 2 Cr Sa Ly
4x4 jel ksje+l
r-l or kj j
+ (4,8) +4, 8))A7 lLinytle £2 ns jL°J) xX, + V.
2 ee ca gegen? OED EE Mae

4x.


Bibliography

Amemiya, T., 1974, "The Nonlinear Two-Stage Least-Squares Estimator,"
Journal of Econometrics, Vol. 2, pp. 105-110.

Barro, R. J., Grossman, H. I., 1976, Money, Employment and Inflation,
Cambridge; Cambridge University Press.

Bilson, J., 1978, "Rational Expectations and the Exchange Rate," in
Jacob Frenkel and Harry Johnson (eds.) The Economics of Exchange
Rates, Addison-Wesley, Reading, MA.

Calvo, G. A., 1981, "Staggered Contracts and Exchange Rate Policy,"
Columbia University, N. Y., Unpublished Manuscript. |

Canzoneri, M., 1980, "Exchange Intervention Policy in a Multiple
Country World," Federal Reserve Board, November, Unpublished
manuscript.

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

Cumby, R. E., Obstfeld, M., 1983, "International Interest Rate and
Price-Level Linkages under Flexible Exchange Rates: A Review of
Recent Evidence." Forthcoming 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.

Dornousch, R., 1976, “Expectations and Exchange Rate Dynamics,"
Journal of Political Economy, December.

Dooley, M., Isard, P., 1980, "The Portfolio - Balance Model of
Exchange Rates," International Finance Discussion Paper, No. 141,
Federal Reserve Board.

Driskill, R. A., Sheffrin, S. M., 1981, "On the Mark: Comment,
American Economic Review, Vol. 71 No. 5, P. 1068-1074.

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

Flood, R. P., 1981, "Explanations of Exchange Rate Volatility and
Other Empirical Regularities in Some Popular Models of the
Foreign Exchange Market," Carnagie-Rochester Conference Series on
Public Policy 15, Autumn.

Flood, R. P., 1982, "Sticky Prices and Inventory Adjustment"
Unpublished Manuscript, Federal Reserve Board.

Flood, R. P., Garber, P.M., 1980A, "A Pitfall in Estimation of Models
with Rational Expectations," Journal of Monetary Economics,
July.

Flood, R. P., Garber, P.M., 1980B, "Market Fundamentals vs. Price
Level Bubbles: A First Test," Journal of Political Economy,
August.



Test of the Joint Exogeneity Specification of

- AQ -
Table Al

the Flex-Price Model

for Canada and the U.S.

1973 - 1980
Flex Price Model * Likelihood Ratio* © Sims Correction®
x71). x7(16)
Data
. ak *
Series detrended 43.145 28.596
only .
Data series
ek
detrended and . 53.336 28 .528*
deseasonalized
* indicates significance at the .05 significance level
** «indicates significance at the .05 and .01 significance levels.
A

The likelihood ratio is defined as L = Tin Det Ty

- &£n Det P| where pry is the restricted model variance co-

variance matrix of residuals (i.e. the case where all
hypothesized endogenous variables are assumed to have zero

coefficients) and z is the unrestricted variance covariance
matrix of residuals.

The sims correction takes the form (T-K)/AV-L

given the
definition of L in A. Also T is the number of observations
(86 in the present case) and K = the number of regression

coefficients per equation. Thus k = 29 in the case where

only the detrended series were used and k = 40 when the data

series were deseasonalized and detrended.


- Al -

ek will be serially uncorrelated if w is chosen to be large enough.

-In choosing w and % properly the investigator 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, The results presented below employ a value of

w= 6 and & = 2as the choice of lag length for the hypothesized
exogenous and endogenous variables respectively. The results in Table
Al test whether ce = 0 through the use of the likelihood ratio stat:.stic

(a1.2) L = T | #n Det (Z,) - tn Det(Z,) [~x2¢9 where I, is the

variance covariance matrix of residuals for the restricted model where fed =0,

Ly is the unrestricted variance covariance matrix of residuals obtained by
estimating (Al.1) with no restrictions imposed, T is the number of

observations, and q is the number of restrictions imposed. Table Al also

presents the results of applying Sim's [1980] correction to these likelihood
ratio statistics since the chi square statistics computed can be biased toward
rejecting the exogeneity specification when the number of degrees of freedom

left in the data is not a different order of magnitude than the number of

restrictions being tested.


-7-

Equation (1) defines the log of the general price level (or index
(mp); a is the domestic consumption share of the domestic good, (Pt) is
the logarithm of the domestic price of the domestic good and (p*) is the
logarithm of the foreign price of the foreign good. (S_) is the log of
the spot exchange rate which is expressed in units of domestic currency per
unit of foreign currency.

Equation (2) defines the terms of trade (qt) or the relative price
of domestic goods in logs.

Equation (3) describes the money demand equation. The demand for
real money balances (mp - 1 ) depends negatively upon the level of the
domestic nominal interest rate (it) and positively upon the natural
logarithm of real domestic income (Pe + Ye - Te). In contrast to the
work of Dooley and Isard [1979] and others on portfolio balance models,
wealth does not enter the money demand function explicitly. However, the
specification adopted here only neglects the net foreign asset component of
wealth due to the presence of real income in the money demand equation.
This specification would seem reasonable in countries where net foreign
asset holdings do not account for a large proportion of total
wealth .12/

Equation (4) states that the natural logarithm of the quantity of
money (mz) is an exogenous variable determined by the author-
ities,13/ and equation (5) expresses the equilibrium condition for the
money market.

Equation (6) expresses the covered interest rate parity condition,

where if is the level of the foreign interest rate.


Table I
First Staye Nonlinear Instrumental Variable (or GNI)
Estimates of the Constrained Flex Price Model (73:3 - 30:8)A in Eq. (26)

r a : "2.1533 (-.2359)
a : 1.0944 € 3903)
c by -.5020 | (-. 0885)
-1,T by / : =.4314 | (-.1112)
bs 5857 ( 1402)
+9304 (6.173) **
2 .9273 (4.222) **
Lo ; +8199 ( £36979
M. PS any .8086 (13.1593) **
equation ay) -.13909 (-2.2909)*
ayy 0022 ( .3462)
ay -.1034 (-1.9251)*
Po. ayy .0993 (1.2328)
zy, ayy +7129 . (8.9568) ** ;
equation 205 +0153 (1.8598)
| 254 -.1969 (-2.794) **
r a5) 1.0849 : (1.9506)*
. .
Pe a5 9540 (1.7361
equation
a .9051 (15.9926) **
3
L834 6248 (1.2844)
ave -.0704 (-1.4862)
: ay .0360 _ ©7699)
. .
t a 0121 (2.1213)*
43 .
equation

L 244 : +9239 (22.317) **

A) Statistics in ‘parenthesis are ratios of coefficient estimates to their
asymptotic standard errors.

*k Indicates significance at the .01 and .05 level.
* Indicates’ significance at the .05 level.


- F3-

8/ It should be observed that the "flex-price" model derived below can also be viewed
as a special case of a more general "sticky-price" model (see Glaessner [1982A]);
thereby permitting empirical tests of different price adjustment specifications
posited by various authors. This model is solved in Chapter III of Glaessner [19824]

where closed form solutions for the exchange rate and prices are derived which can be
used to estimate the model.

9/ For a treatment of prediction theory, see Sargent [1979] or Hansen and Sargent
T1980).

10/The error term in the model ‘could also have been introduced on the supply side of
the market once specific stochastic processes have been specified for the "forcing"
variables me, Yt, if and Pr. (See Glaessner [1982A] Chapter III).

11/ Specifically using difference equation techniques to solve the
model allows one to make assumptions about the stochastic processes
generating the exogenous variables later in the analysis. More

importantly, following such a procedure allows for assuming that the
exogenous variables follow a general vector autoregressive process.

12/ Also allowing for wealth to enter the model explicitly would
result in the need to model savings behavior which will increase the

order of the dynamics in the system. This point has also been made by
Mussa [1982A].

13/ This assumption is, of course, a major simplification. The work .
“of Flood [1980] suggests that intervention can lead to more volatility
in exchange rates. Amore detailed treatment of the effects of
intervention are provided by Henderson [1979, 1980] who handles the
two country case and Canzoneri [1980] who discusses the multiple
country case. Allowing for intervention in the present model might be
an interesting extension (see Henderson [1980], p. 41).

14/ Meese and Singleton [1980B] Hansen and Hodrick [1980, 1981], Stockman [1978],
Tryon [1979] and Geweke [1979] find that this relationship does not tend to hold
while Hakkio [1980] and Frenkel [1979] find that it does.

15/ As mentioned in Chapter III of Glaessner [1981A
endogenous without changing the order of
system may be useful in efforts to deriv
restrictions explicitly. In addition,
have pointed out that a richer specific
model (with less exogenous variables) m
volatility of the exchange rate without
processes that have infinite variances.

J letting output he
the difference equation

e the rational expectations
Meese and Singleton [19804]
ation for an exchange rate

ay explain a good deal of the
appealing to exogenous driving


-F4-

16/ Note that there is an asymmetry here between ‘the definition of
real income used in the money demand equation vs. goods demand. In
future work real income will be defined similarly for each equation,
however, the correct specification does not affect the results. In
addition, the specification of the demand for domestic goods equation
does not allow for the impact of government on the demand for domestic
goods. Incorporating government into the model in the demand function
for domestic goods would be a further extension of previous work. For
example, see Flood [1981], Mussa [1982A], and Dornbusch [1976].

17/ Specifically, the model can be written as a pair of simultaneous first order
difference equations in Pe and S_ with the roots

Ay = Ap = By 4 BF 31
1 By o

as shown in Glaessner [1982A] Chapter III. The model is solved by following the
conventional practice of choosing terminal conditions which make the general rational

expectations solution and the particular solution coincide (i.e.bubbles in the sense
of Flood and Garber [1980B] are excluded).

18/ Note that constant terms do not appear in these solutions. This is because each
series and various lagged values of each series are detrended and deseasonalized so
that the resulting series are assumed to be mean zero, linearly indeterministic,
covariance stationary processes.

19/ When output is allowed to be an endogenous variable, this property of the model no
longer holds. Thus, authors who have emphasized this point (see Mussa [1982A]) have
been drawing conclusions which are too strong given the specific structure of the
model.

20/ See Glaessner [1981] Chapter III, (Section 3.4(a) and Appendix 4) for a detailed
derivation of the equations presented below. Note also that to be able to apply the
prediction theory discussed in Hansen and Sargent [1980, 1981] or Sargent [1979]
requires that the variables in the vector X¢ be jointly covariance stationary. To
insure that this was the case it was assumed that the roots of det A(Z) = 0, for Z
complex, were outside the unit circle. This insured that the polynomial matrix A(L)
was invertible. Weaker conditions which suffice for applying these prediction
formulas are discussed in Hansen and Sargent [1980].

21/ The conditions under which Vj_ and V2_ will follow ARMA(1,1) processes are
‘explored in Section 3.4(a) of Chapter III in Glaessner [1982A]. Specifically, this
will be the case if one of the structural errors is white noise (e.g., Ut) and the
other error (e.g., e¢) follows an AR(1) process. In this case, the error terms
(Wye V5 2) appearing in the price and spot exchange rate equations will each be

sums of. tne correlated white noise and AR(1) processes which Granger and Morris [1976]
show to be ARMA(1,1) processes. If both ut and € are assumed to follow

independent AR(1) processes, the V, i=1,2 will be an ARMA(2,1). The assumption of
an ARMA(2,1) process is not adopteaé because the higher order of the autoregressive
component of the error term will complicate the nature of the restrictions greatly and
will increase the number of structural parameters to be estimated. An additional


-25-

(2) a flexible domestic price of domestic goods (3) the money demand
or goods demand specification (4) the possible instability of the
vector autoregression generating the forcing variables or (5)
assumptions made about the serial correlation properties of error

terms which result in the rejection of the model. One possible method

of attacking this problem is to look at these foundations explicitly

(see Meese and Rogoff [198], 1982]. ‘two other implications of these

findings are (1) to make more variables in the model endogenous (e.g.
allow for an intervention function), and (2) assume that domestic
prices adjust slowly (as in Glaéssner [1982A] Chapter III). This
latter assumption in conjunction with the assumption of rational
expectations leads to price and exchange rate equations with different
nonlinear restrictions than those in the flex-price model which are
testable using the procedures developed in this paper.

Ic. answer to.the second question posed above it would seen useful
to pursue several approaches to remedying the technical problems
suggested at the end of section IV. First, experimentation with
diffe

rent minimization and derivative approximation routines to see

how sensitive the results are would seen proper before describing the

results above as conclusive. Second, constraining the autoregressive

parameters in the price and exchange rate equations to be less than
unity in estimating the model would be an important improvement in the
preliminary procedures followed. Third, it would also seen useful to
try to estimate all twenty-four parameters jointly by reducing the

number of orthognality conditions to twenty-five, (e.g. this is


The estimation techniques employed in this paper are applied to a
“flex-price" model based on the work of Flood [1981] and Mussa [1982A]
which has several unique features. First, in contrast to the so called
"monetary approach" models developed and estimated by Frenkel [1976, 1979],
Hartley [1982] and Bilson [1978] there is more than one good in the model
so that purchasing power parity is not assumed to hold at all points in
time. Second, in contrast to the monetary models both the money and goods
market are modelled explicitly so that real factors are allowed to affect
the exchange rate through channels other than the income term in the money
demand function, Finally, in contrast, to the "sticky price” models
develcped by Frankel [1979], Hooper and Morton [1980], Salemi [1979],
Dornbusch [1976] Driskill and Sheffrin [1981], Flood [1981, 1982] and Mussa
[1982A]7/ the domestic price of domestic goods is not assumed to adjust
slowly according to some explicit price adjustment specification.8/

Thus, the flex-price model can be viewed as lying in between the two types
of models which have previously been estimated.

Applying the GMM estimation techniques to the "flex-price" model
results in a strong rejection of the joint hypothesis of rational
expectations and the structure of the model for the Canadian U.S. floating
exchange rate period of 1973:3 - 1980:8. In fact, these results indicate
that only the autoregressive parameters in both the exchange rate and price
equations are statistically significant. These autoregressive parameters
are close to unity, a finding consistent with the results of Meese and
Singleton [1981] that the univariate process for the log of the

Canadian/dollar spot exchange rate may have a unit root. These empirical


- b4 -

Hoffman, D. L., Schmidt, P., 1978, "Testing the Restrictions Implied by the Rational
Expectations Hypothesis" Econometrics Workshop Paper, No. 7804.

Isard, P., 1977, "How Far Can We Push the Law of One Price?", American Economic
Review.

McCallum, B.T., 1979, "A Macroeconomic Model with Predetermined Wages, Mark-Up
Pricing and Classical Properties," University of Virginia, Unpublished
Manuscript.

McCallum, B. T., 1979, "A Macroeconomic Model with Predetermined Wages, Mark-Up
Pricing and Classical Properties," University of Virginia, Unpublished
Manuscript.

McCallum, B. T., 1980, "Monetary and Fiscal Policy in a Disequilibrium Model of the
Barro Grossman Type," Presented at the Southern Economic Association Meetings.

Meese, R., Rogoff, K., 1981, "Empirical Exchange Rate Models of the Seventies: Do
They Fit out of Sample?" International Finance Discussion Paper No. 184.

Meese R., Rogoff K., 1982, "The Out-of-Sample Failure of Empirical Exchange Rate
Models: Sampling Error or Misspecification," International Finance Discussion
Paper No. 204.

Meese, R., Singleton, K., 1980A, "Rational Expectations and the Volatility of
Floating Exchange Rate," Unpublished manuscript, Federal Reserve Board.

Meese, R., Singleton, K., 1980B, "Rational Expectations, Risk Premia and the Market
for Spot and Forward Exchange," Working Paper, Board of Governors of the
Federal Reserve System, Wash., D.C.

Meese, R., R., Singleton, K., 1981, "A Note on Unit Roots and the Empirical
Modelling of Exchange Rates," Journal of Finance, (forthcoming)

Mussa, M., 1982A., "A Model of Exchange Rate Dynamics," Journal of Political
Economy, February.

Mussa, M., 1982B, "Sticky Prices and Disequilibrium Adjustment in a Rational Model
of the Inflationary Process" American Economic Review, Vol. 71, No. 5.

Obstfeld, M., 1983, "Rational Expectations and the Foreign Exchange Market: A
Comment" Forthcoming in Jacob A. Frenkel (ed.) Exchange Rates and International
Macroeconomics. Chicago: University of Chicago Press for the National Bureau
of Economic Research, 1983.



-b2-

Frankel, J., 1979, "On the Mark: The Theory of Floating Exchange Rates
Based on Real Interest Differentials," American Economic Review,
69.

Frankel, Jeffrey A., 1981, "On the Mark: Reply", American Economic
Review, pp. 1075-1082.

Frenkel, J., 1976, "The Monetary Approach to the Exchange Rate:
Doctrinal Aspects and Empirical Evidence," Scandinavian Journal
of Economics, 78, No. 2.

teeta , 1979, "Flexible Exchange Rates in the 1970's" National Bureau
of Economic Research Working Paper 450, Cambridge, MA.

Geweke, 1979, "A Note on the Testable Implications of Expectations
Models," December, University of Wisconsin, Unpublished
Manuscript.

--------- 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 Equetion
Models," SSRI,University of Wisconsin, Madison.

Geweke, J., Meese, R., 1981, "Estimating Regression Models of Finite
but Unknown Order," International Economic Review,Vol. 22, No. 1,
February.

Glaessner, T. C., 1982A, "Theoretical and Empirical Essays on the
Determination of Spot and Forward Exchange Rates," Unpublished
Ph.D. dissertation, University of Virginia.

Glaessner, T. C., 1982B, "The Modern Theory of Forward Foreign
Exchange: Some New Consistent Estimates under Rational
Expectations," International Finance Discussion Paper No. 206.

Granger, C.W.J., Morris, M.J., 1976, "Time Series Modelling and
Interpretation," Journal of the Royal Statistical Society, A.
139, Part 2, pp. 246 - 257.

Hakkio, C., 1980, "Expectations and the Forward Exchange Rate,
"National Bureau of Economic Research, Working Paper 439,
Cambridge, Mass.

Hakkio, C.S., 1981, "Exchange Rate Determination and the Demand for
Money." Working Paper No. 766, National Bureau of Economic
Research.

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


as), «
Vo. = (1l- . ' ' k
ar * (ho e2h) (Py (428) * 4585), F (ng) EX 19]

1

. k
- dB. =
171 oP ELK, 41% 1)

where

*
Vig = G- 8b),

ea.
Vor = G- 821)Q_-

. .
it 7d Vo,

Hence, it can be shown that E(VT, 12 9] = 0 and E(V3,12,_] =

are first order moving average error processes.

/

Notice that V

Thus to estimate the model given the assumption that Vie i=1,2 are
generated by an ARMA (1,1) process involves choosing instruments from

the set 2 {X }. In the present context we express

t-2 ~ t-2? Xa? ue
the orthogonality conditions for the first order moving average errors

* =
Vit i=1,2 as

* :
(20) ELVit X 7] =0 for t>2, i=1,2.

1xl 4x1 4x1

The orthogonality conditions have been expressed in terms of the vt is 1,2
because the closed form solutions for the spot rate (S_) and price.
equations (P_) (13)' and (14)' can then be used to express these errors

*
Wie V5) in terms of observable variables. Expressing the orthogonality

restrictions in this fashion allows the investigator to ignore the moving


-16-

is a vector of instruments (predetermined values of the forcing variables).

Finally

ACL} Eo) = AplEy) + AY(Eglh + «+ * a Co

where

(22)

(1-9), 0, -Q- oy1)[4, B,¢ olEy) + (58) + 48 syyes (i9)1 2

xd

AolSq) = 10. Q-eh, -(i- p5L) [4,8) + 4585) ¥9** (Eq) + 4,8)¥ “(Eg ]s 9

6x10 _

Oo, I
x2 4x4 4x4

1x4 : 1x4

' -Ay

0, 0, -(1= o,L)[dy8i¥5 (Eq) + (dy8y # 4582) (Eq) 1» 0!

(23 Q
4 1x4
y*
d, 50) = 10, 0, -(l- oot) (4,8) + a. 382)¥5 “(&q) + 4,8 8) ¥5 (Eq) 1 y
6x10 1x4
0 0 “A
2) , jel
4x2 4x 4x4
* * kk kk
for j = 1, —-, tl, and the coefficients Yo % or Yo 5 can be expressed
as
(24) vy © An)
0 2

and

* +1 .
Yo J
yj A Cty A;


-2-
typically have been assumed to be either serially uncorrelated or follow a
first order autoregressive process (AR(1)). Both Glaessner [1982A] and
Hakkio [1981] have noted that very stringent conditions on structural error
terms must be made in order to claim that the reduced-form error term in
the spot exchange rate equation will take such a simple form. Hakkio
[1981] has found evidence of serial correlation (of a greater order than
AR(1)) for several different bilateral spot exchange rate equations.
Finally Obstfeld [1983] has pointed out that spot exchange rate
determination models have been estimated using techniques which presume
that equation disturbances are conditionally homoskedastic, an assumption
which may have been untenable.4/

In light of these considerations, this paper aspires to improve upon
the estimation techniques previously brought to bear upon these types of
models. A nonlinear instrumental variables or general method of moments
(GMM) estimator developed by Hansen [1979, 1982], Hansen and Sargent [1981]
and Hansen and Singleton [1982] is used in order to estimate and test the
theoretical restrictions implied by the structure of the model and the
assumption of rational expectations. These procedures do Rot require that
(a) the forcing variables be exogenous (b) that equation disturbances be
characterized by a specific distribution function or (c) that equation
distrubances be conditionally homoskedastic.5/ Also in contrast to
previous work restrictive assumptions are not made concerning the structure
of the vector autoregression generating the forcing variables.
Specifically, the polynomial matrix multiplied by the vector of forcing
variables is not assumed to be diagonal, upper triangular, or contain any
exclusion restrictions with regard to the other variables within the vector

autoregression.6/


- F2-

5/ As will be noted below (Section III) and as is discussed in

more detail in Glaessner [19824] Chapter IV, the GMM estimator as
applied in the present context does require that the investigator make
some assumption regarding the order of the serial correlation in
quasi-reduced form error terms. Such an assumption is necessary when
predetermined values of the forcing variables are used as instruments.
This assumption, is weaker than that typically assumed, i.e. that the
error term is characterized by a specific distribution function. It
should also be observed however, that to the extent that the
distribution function’ assumed for the error terms in using maximum
likelihood procedures is the true one, the estimates obtained will be
more efficient than those obtained using GMM procedures.

6/ These assumptions about the lack of interaction between "forcing"
variables are frequently made so as to obtain closed form solutions
more easily. For example Driskill and Sheffrin follow Frankel [1979]
by assuning that the log of the growth rate of the money supply
follows a random walk. Papell [1981] assumes either an independent
first order autoregressive process on each exogenous variable or an
upper traingular vector autoregressive process on the exogenous
variables.- Finally Hartley [1982] assumes that foreign income and
money do not effect domestic income and money in obtaining closed form
soluticns so that he need only invert 2x2 polynomial matrices in
obtaining closed form-solutions. However, given that in Hartley's
model agents see both foreign and domestic variables it is not clear
why they would use only a~subset of this information in order to form
optimal. predictors of domestic vs. foreign variables. This dichotomy
does not seem reasonable.

7/ The price adjustment specifications adopted by these authors are

‘hot all alike. Dornbusch [1976] assumes that the rate of price change
depends only upon excess demand in the goods market. Flood [

contrast assumes that pricing is completely anticipatory so that the

rate of price change depends only upon the expected rate of change in

the price which would clear the domestic goods market. The price
adjustment specifications adopted by Salemi [1979], Hooper and Morton
[1980], Frenkel [1979] and Driskill and Sheffrin are similar to that

in Dorrbusch [1976] with some minor modifications (see Glaessner

[19824] chapter III. The price adjustment specifications of Mussa [1982A, 1982B]
and Flood [1982] tend to subsume both of the above approaches to price adjustment.
See Glazssner [1982A] for a more detailed explanation of these different price
adjustment rules and for a critical review of some of these different specifications
The excnange rate overshooting properties of these models can also be obtained by
assuming that wages are sticky. (See Obstfeld [1981] or Rehm [1981].


- 23-

on a value greater than unity in the process of minimizing the criterion
function. This would simply be a way of imposing the inequality constraint on
the p,'s in equation (16).

Third, even though the sixteen parameters in the vector
autoregression were treated as fixed in obtaining estimates of the eight

additional structural parameters it was extremely difficult to calculate the
3£,00) la

aq) | fayr
to the appearance of the terms ato and ee) within the orthogonality

partial derivative matrix Dp ~ analytically. This was due
restrictions in (26). The asymptotic standard errors calculated above in
Tables I and II were sensitive to how the derivatives were approximated, since
if the perturbations to £(Eq)> were too small (e.g. out to the eighth or
ninth decimal place) the Dr matrix tended to not be of full rank, so that

the Dr! Dy matrix used in the calculation of standard errors, was

singular. This numerical problem was solved by perturbing £, (bp) by

somewhat larger amounts in approximating derivatives so as to ensure that the
partial derivative matrix Dp was of full rank 34/

Fourth, attempts to estimate all of the parameters in the model
jointly using the orthogonality restrictions in equation (21) were not
successful. because the large number of orthogonality restrictions (R = 48 in
this case resulted in severely ill conditioned estimates of the weighting
matrix which made it impossible to calculate asymptotic standard errors, or

calculate the second stage criterion funct ion.35/


Formulation and Estimation of a Dynamic Model of Exchange Rate

Determination: An Application of General Method of Moments Techniques.

by

Thomas C. Glaessner*

I. Introduction

The advent of floating exchange rates in 1973 has resulted in a
proliferation of papers aimed at formulating and estimating models of spot
exchange rate determination. Few of these models have been estimated with
techniques that explicitly utilize the identifying restrictions imposed by
the assumption of rational expectations. Moreover, those authors who have
estimated their models subject to these restrictions (including Driskill
and Sheffrin [1981], Salemi [1979] and Hartley [1982])1/ have
explicitly or implicitly made strong assumptions about (a) the exogeneity
of the "forcing" variables and (b) the serial correlation properties of
reduced form error terms.2/

In virtually all of this work the "forcing" variables are assumed to
be strictly econometrically exogenous. Recently Meese and Rogoff [1981]
and Glaessner [1982A] have found that the necessary conditions for the
joint exogeneity of the variables in many of these models are

rejected.3/ Also, in estimation, the error terms in these models


- AG

Appendix 2
Tests for the Order of the Vector Autoregression you xs ve
This appendix presents the results of conducting tests for the
finite lag length of the vector autoregression generating the forcing
variables in the vector x, - {m,, Yeo f, oa J where the elements in
x, are defined in Appendix 3. Table A2 below presents the results of
applying two different multivariate criterions for choosing lag length

for the vector autoregression of "forcing" variables described in the

text.
Table* 42
lag length. CAT(r)° Likelihood Ratio? -
1 -31,947.2 75.25% x7 (48)
2 ~30,885.9 50.328% %2(32)
3 -29,886.0 25.953% (16)
4 -29,155.4
* Indicates rejection at the -05 significance level.

a) The author wishes to thank Richard Meese for providing soft—
ware which was modified in order to obtain the results in
‘this table.

“b) The likelihood ratio statistics represent tests of four vs.

one lag distributed x2(48), 4 vs.2, and 4 vs, 3 lags
respectively,
c) The CAT criterion attributable to Parzen [1975] chooses lag
length n to minimize the quantity.
trace [4/T p vi wy n=1, ---,M
ge. 4 hw? ’ aM,
where T is sample size, M is the maximal lag considered (eg. M=4)

and Vy is an estimate of the contemporaneous covariance matrix of

distrubances for the model with j lags.


-F7-

30/ This is primarily due to the fact that joint estimation of all equations using the
Orthogonality restrictions in equation (21) would take into account the nonlinear

restrictions across the vector autoregression and the price and exchange rate .
An additional implication of this less efficient estimation procedure is

equations. 0
that the vector of innovations wre is assumed to be contemporaneously uncorrelated
1
with the error terms Ve i=l, 2 within d_ so that
'
Bd, l= [zt 0
6x1 1x6 “|2x2 0 2x4
Vv where Z and V are as defined
ao 4x4 in the text.
x

Also it is necessary that the innovations in not be correlated with the

“t
fundamental white noise components Qjr¢ i=1,2 at all leads and lags, a cmdition
which holds given the assumptions about the Qit i=l,and © disturbance terms
made in the text. “t

31/ A more formal discussion of this point can be found in Glaessner [1982A] Chapter
TV. Also see Hansen [1982], Hansen and Sargent [1981] or Hansen and Singleton [1982]
for a discussion of how to form GMM estimators within a variety of different
contexts.

32/ A formal discussion of this test statistic is given in Glaessner [192A], Hansen
and Sargent [1981] and Hansen and Hodrick [1981].

33/ Use of a different algorithm (other than DFP) to minimize the criterion functions
Zn question would seem useful in this context. Also using different procedures to
approximate derivatives would be useful since the search procedure used to find a
minimum depends critically on the calculation of these derivatives, Moreover meeting
the convergence criteria on each coefficient can be difficult when one is minimizing a
function over a fairly large number of parameters (eight or twenty-four).

34/ In practice perturbations of about ((.0001)* coefficient value) or larger were
sufficient to ensure that the Dy matrix would be of full rank. Also the asymptotic
standard errors tended to get smaller as the perturbation became larger (i.e. as the
approximation to the "true" derivatives became worse.) More experimentation with
regard to developing some way of optimally choosing the amount by which to shock

£, (tp) in approximating Dp would seem useful.

35/ This weighting or distance matrix and the criterion functions referred to are
discussed in detail in Glaessner [1982A] Hansen [1982] and Hansen and Singleton
[1982].

36/ This is due to the fact that as the number of orthogonality conditions grows
relative to sample size, the estimates of the parameters, and asymptotic standard
errors tend to become very imprecise (see Hansen and Singleton [1982] anc. Section IV
above).


- 26 -

equivalent to reducing the number of instruments used). This would result in
more efficient parameter estimates and more importantly might provide a way of
obtaining a consistently estimated weighting matrix which is not
illconditioned. However, given a sample of only eighty-nine observations,
obtaining the joint estimates of twenty-four structural parameters will tend
to reduce the precision of both parameter estimates and asymptotic st.andard
errors.36/ Fourth, further experimentation regarding the optimal choice

of the value by which one perburbs parameters in approximating derivatives
should be explored. Finally, Hansen and Singleton [1982] have suggested that
the GMM estimator formed above is often sensitive to the choice of maximal lag
length chosen for the instruments. Thus it would be interesting to conduct
the estimation of the orthognality restrictions in (26) under different

assumptions about the maximum lag length (P) chosen for the instruments.


»7-12-

rate and price equations. It is shown in (Glaessner [1982A] Chapter III)
that in general the error terms in the exchange rate and price equation
will follow some finite order ARMA (p,q) process depending upon the
properties of the structural error terms a and &. Thus in the

section which follows orthogonality restrictions between instruments and
error terms are presented which can be used in order to form the GMM or
instrumental variables estimator developed by Hansen [1982], Hanser. and

Sargent [1980, 1981] and Hansen and Singleton [1982].

III. The Orthognality Conditions for the Flex-Price Model

In this section two sets of orthogonality restrictions between the
errors Vj, i=1,2, ve and the instruments (i.e. predetermined: values of
the forcing variables) are presented based on the more detailed derivation
in Glaessner [1982A]. It should be observed at the outset that this
methodology is adopted because (1) the forcing variables in the flex-price
model were not found to meet the necessary conditions for joint exogeneity
with respect to St and Pt (see appendix 1), (2) the error terms Vj¢t
i=1,2 are probably serially correlated for the reasons mentioned above and
(3) the disturbance terms in the spot exchange rate price equations may not
be conditionally homoskedastic.

To derive the orthogonality conditions suggested by the flex-price
model we need to give a precise parmeterization to the error terms Vj¢
and Vo¢. This is due to the fact that predetermined values of the
"forcing variables" are used as instruments in estimation, For the sake of
simplicity we assume that Vj_ and V2_ follow ARMA(1,1)

processes 21/


-15-

average parameters gj, g2. In fact, these parameters will not need to be
estimated when the GMM estimator (see Hansen [1982]) is formed below. Thus
this estimator has computational advantages vis a vis maximum likelihood
techniques.

Equations (13)', (14)' and (15) can be used in conjunction with
equations (17) and (20) above to express the orthogonality: conditions25/

in compact notation as

(21) Elf, (q) } = Eldt g Xoo ]1=0

bxl
Rel 4
xl Rxl
*te3
“teP
where *.
S$. = [Mag] = ACLS Sy) e,
6x1 * 6x10 10xl *
Vor
Xx
Ye

The symbol @ indicates the Kronecker product, X; is defined as above, the

vector of observable variables (St, Py, Xs Xp-1)' is denoted

by tee b> is the (Qxl) vector of true structural parameters

- =, AL), and [X.',,  -—-8' j

(a, G5, 9, bys bys bys Py» Pas Ay» Ara cP