Formulation and Estimation of a Dynamic Model of Exchange Rate Determination: An Application of General Method of Moments Techniques

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to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

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

-?2typically 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 hot 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.3/ 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/

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 [19824]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

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 flexprice 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 themodels 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

hypothesized exogenous variables me, Ye, Pe*, and i-* (defined

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

below. (1) ™| = oP, + (1-0)(S,+ Ps) (2) a, = PL - (S,+ PD) d ; . . (3) Me TE FA - Syd, + oo(PL + Y.-L) tu, ap, G1, a, > 0. (4) a “. (5) m, = me (6) i,t ipt tps, (2). f= LS, (8) . ey, aan ia = Bo- Pit ~P2Ge- GE MED) + SY, +e, bys dL, b, > 0 ay ved | (11) P= Q. + S. (12) Qs Pie S.

Tv

Definitions of Variables 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.

var

the logarithm of the spot exchange rate, which is expressed wits of domestic currency per wit 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 + i.) 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.

= EIX, 19,1 where Q. contains all of the variables and disturbances 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/

- 7 ~-

Equation (1) defines the log of the general price level (or index (we); o is the domestic consumption share of the domestic good, (PE) 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 - 7 ) depends negatively upon the level of the domestic nominal interest rate (it) and positively upon the natural logarithm of real domestic income (Pr + Yt - ™t). 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 (mp) is an exogenous variable determined by the authorities ,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.

Equation (7) states that the log of the forward exchange rate (fr) equals the expected future spot exchange rate (cESt+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 - (En tc+]7 % t) and positively upon the log of real output (or income) Y_ 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 solution is:

(13) S, = dog! og (n,)* efx Co aety eee yk p28) Cyd” Ee yO.) + (458) + 4523) 2 (ny)

ELS. 12] * Vie

- 9 -

COD Py = (2p) 4585) E lag)* BOE Ia) + a8! a E tn)"

EX 4,1 } + V5,

_ yx (5) A(L)XV = VE

4x4 4x1

where

8) = (0, 1, -B,, (B)(1-o)F - BL (1- o) - B,)]

to * «

Bog

n = — 2 __ 1 BL? Boo Ng * l+a, 1 dj = ———__.+q l BL + Boo 2 (a, (¢- 1) -9) dj = —— 2 (B, + Bo) (1+a,) I d. ~ lta, poe lL l l- bs ae 2 2 1l-b

- 10 -

The polynomial matrix A(L) is assumed to be of finite order r so that

a 2 ACL) =I - A,L - AjL — see ATL’, The innovations in the vector auto- 4x4 :

‘= t. rezression are defined as vr - x. E[X, | Xp X92» J where ele ] = 4

and elVive! J ne tor all j # 0. These assumptions imply that it is possible for there to be contemporaneous correlation across the equat ions in the vector autoregression (a nondiagonal V matrix), but serial correlation in the individual components (TD of ve are not permitted to be serially correlated. Finally, define F as FE(Z, |2.] = E(Z.., |9,] and LEZ, [2] = E(Z,_,19,_,] for some Z_, where E[-] 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 [1982a] 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 ny Ny present in the solutions for Py and 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 y 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 Xo (m, »¥ i, »Pp are2Z0/

t?

T-l Tr . . (13)" Ss (458147 (m1) [ I+ £ (f ny PAL] 4x4 jel kejel

' roel tel r kj j + (d58,+d,8,)A (no) [I+ © (2 no “AJL y) yy

= =j t j=l ksjel 4x1

4

- fl -

' : tie r-] r

k-j j . n )L 1x4 4x4 j=l k=j+l 1 “Ax J + (834 diB'ya7)¢ are

k-j j n L

2t

(15 =

Equations (13)', (14)' and (15) represent a compact notation for a six equation system in Pye Sis and the "forcing" variables Mer Yeo The restrictions delivered by the assumption of rational expectations are

* * i. and Pee

embodied in equations (13)' and (14)'. To actually express S_ and Pt

as functions of all the structural parameters in the model [contained in

a, d,, By> 85, 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= ™ or L= Nos 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 crossequation 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 X_.

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 Vjt, i=1, 2 in the spot exchange

»-.12 -

rate and price equations. It is shown in (Glaessner [19824] 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 ue 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], Hansen. 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 S_ and Py (see appendix 1), (2) the error terms Vit. 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 Vjt 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 Vo_ follow ARMA(1,1)

processes. 2!/

- 13 -

Specifically, write

(16), (1- g.L). V.. = ———~ Q, it (1 - pb) it

where i=1,2,0< lo. | <1, O< ig. | <1, and £(Q;,) = 0 for allt,

EIQ .&-_5] = 0 for all j#0, i=1,2. Also, Ef{q

qi] = = where ~TTo 2x2 qa. = (Q4> Q,) so that the reduced form error terms in the spot ex-

change rate and price equations are allowedto be contemporaneously

correlated.

To derive the orthogonality restrictions present in the flex-price

model, let us define the information set Qo = Xp Xeope ces } which

is composed of predetermined values of the forcing variables.22/ The definition of the innovations in the vector autoregression ( Vi) stated in section II suggests that E(VEl2, 4] = 0. 23/ this observation implies the following set of orthogonality conditions for the four equation vector

autoregression represented by equation (15)

(17) ry! =

for all v 21, 4xl 4x4

The orthogonality restrictions for the price and exchange trate

equations can be obtained by multiplying equations (13) and (14) by (1-p,L)

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

(18) * _ ‘ . k Vie = Go eh (8, = 48) F (my) EEX, 1%]

oe k ~ (4,8) * 8582) Fe? ELM 11)

(19)

Vv: = - - ’ ' ' . k | at = (- poh) (P, (298) * 4585) (ng) ELK, 4, 1%)

ao

' k - dy 8) Saat E(X. 4,121)

where

in > G- 8,

‘o< l

ay Vee (1 - 851)Q,,-

* and V2

Notice that Vit ot

are first order moving average error processes. 24/ Hence, it can be shown that E(V},12,_9] = 0 and E[V5,12,_5] = 0.-— 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 2.2 * {X,_>> Xpoge cee }. In the present context we express

the orthogonality conditions for the first order moving average errors

kos 8 Vee i=l,2 as

* = fo > 2 i=i1,2. (20) E{V.. X od 0 rtzés lxl 4xl 4xl

The orthogonality conditions have been expressed in terms of the vy"

1t because the closed form solutions for the spot rate (St) and price

i=1,2

equations (P_) (13)' and (14)' can then be used to express these errors (Vy

* it? Vo4) in terms of observable variables. Expressing the orthegonality

restrictions in this fashion allows the investigator to ignore the moving

-15-

average parameters g], 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: conditions2>/

in compact notation as

(21) = Eld = ELE, (q) | bet 4 kee] | @ Rxl 4 x1 Rxl “tea3 *t=P where * GF Vag] = ACLS EQ) fk, 6x1 * 6x10 10x1 Vor x Ve

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

vector of observable variables (S;, Pr, Xe Xe-1)! is denoted

by ue bo is the (Qxl) vector of true structural parameters

--, AL), and [X,' -—-x' J

(a, Qo» Oo, by, bos bas Pir P2 Ay» “4-3? t-P

- 16 -

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

Finally MCLs £9) = AplEg) + Ay(Egdh t vee + Ap yEQdey

where

(1-9), 0, -(.-e,L) [48% e(E9) + (dyBt + dshy¥** (ET, 2 (22) 1x4 AolEo) = [0, (1-e,L), ~~ eb) [4, B+ 2% (59) * + dBi 9959) ), 2 6x10 _ 1x4 Oo, I, -Ay jaxz 4x4 4x4

e z¢« ‘ ’ 03) 0, 0, -(1- pyL) [dy81¥5 (Sq) + (428y + 4582)"; (Eq)I» 9 1x4 = -fle- ' "y ytt toe ' A (Gp) = [0+ 0 ~(1~ opt) (C48) + d582)¥5" (Eq) * 418,45 CEQ)I> 2 6x10 1x4 0, 0, “Asay 4x2 Axa 4x4 k & kk kk for j = 1, —-, r-l, and the coefficients Yo ¥5 or Yo 45 can be expressed as | . (24) y* = Alen) ; , 0 ) * -] : . Y. =A J j+l j (mp (MPAs) # MP TAR tee # MT FAD

and

-17-

25 ee . - 9) Y= A Cay)

5S = A7*(n,) (197454) + ng) TA 42 +... 4 td for j = 1, ... , r-l. These expressions for the coefficients ¥F and " 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 representat:ion for the rational expectations restrictions across the parameters in equations (13) - (15), which can be imposed when constructing the GMM estimator .2// 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 Xt, for Canada and the u.s.28/ . Thus, if r = 1, only contemporaneous values of Xt 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 {c,, Go. G by bos b,; a> 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,

- 18 -

denoted by P, Equation (21) defines fr (Sq) as an N((P-1) + h) x 1 = Rxl vector of random variables where e[f (tJ = 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,, 45, 9 bj), by, by, Py» Po} In

this case the orthogonality restrictions appeared as

(26) El (59) = ELT; godt? @ [x .]) = 0 (2((P- 1) +4) x1, R [Rees

- 19 -

where

1 ‘y* 2x6 ' -, |* 9281 * 4582) 8** (Eq)

0, (L- pl), -(1- eyL) (458) + d585) ¥**(g,)

lee 1 - pL) {d,8'¥%(¢,,) + (do8. +.d yet" (c.)) A5(Eq) = 10, 0, -(L- pL) [4 8)¥5 (eo) + (428) *.4582)%5 (Ep 2x6

' te te w 0, 0, -(1- pol) [(d,8, + 458)¥; (Eq) + 448) %5 (Eq)

for j = 1, ... , r-l and Po = [S., P X.]. Observe that N=2 so that

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 flexprice 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

- 20 -

eight instruments contained only predetermined values of the forcing variables Xe-25 Xt-3; where the vector Xy 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 (p,) and price (95) 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 ay (the semi-interest-elasticity of the demand for money), 9 (the income elasticity of the demand for money), by (the relative price elasticity of the demand for domestic goods), bg (tae 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.

Table I First Staye Nonlinear Instrumental Variable (or GMM) Estimates of the Constrained Flex Price Model (73:3 - 30:8)A in Eq. (26) eee a ben Trex emice mode! |

ALL EA

CO

a) -.1533. (~.2359) a, - 39944 ( .3903) c by -.5020 | — (+,0885) -1,T bo 31 (41122) bp 5857 | ( .1402) P -9304 (6.173) %* .9273 - (4,222) ** Co : .8199 ( 36979 M_ ay ; 8086 _ (43.1593) ** equation 415 | -.13909 (-2.2909)* ay .0022 ( 3462) a | 0993 | (1.2328) Y, = ao. +7129 | (8.9568) ** | equation 255 0153 | (1.8598) 254 | -.1969 (-2.794) #8 | a3) | 1.0849 - (1..9506)* i | a3, £9540 (1.7361 equation an. 9051 (15.9926) ** 33 ; a5, 6248 (1.2844) ay -.0704 _ (41.4862) : 249 | 0360: — '¢ .7699) ; a JF 8A5 0121 (2.1213)*. equation 7 ay, 9239 (22.317) **

ne

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

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

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

Using a Consistent Estimate of the Optimal Weighting Matrix for the

Orthogonality Restricticns in Eq. (26) A

Parameter Estimate x7(8)

ay "41164 (0966) im a, 9361 (1.239) _ Sor | Py -.6270 (=.0864) «22.473

bs -.3919 ~~ (=.1003)

bs 7055 (1949)

PL 69223 (14.575) **

2 1.0884 (16.232)**

S

69153 (2.5816) **

A) Statistics in parentheses are ratios of coefficient estimates to their

asymptotic standard errors.

*k Indicates significance at both the .05 and .01 significance levels.

* Indicates significance at the .1 significance level.

- 21 -

The parameter estimates presentred in Table II were obtained by using the initial estimates S17 te 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,.3!/

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, Go (the income elasticity of money demand) is now significant at the .2 level and o (the domestic consumption share of the domest ic 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 x2 (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

- 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-1, 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 Cp, ) is greater than unity (1.0884)is quite troubling in light of the theoretical restriction noted in equation (16) that 0 < | 5 | <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 Py i=l, 2 to take

- 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 - %f,00) ..

partial derivative matrix Dr~ —— t G9) -1i,T

to the appearance of the terms AvP (ny) and aly) within the orthogonality

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 £ op)» 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 £. (op 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 wh:ch made it impossible to calculate asymptotic standard errors, or

calculate the second stage criterion funct ion.39/

- 24 -

V. 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 exogeneity specification since the use of om procedures in estimation allows one not to have to make strong assumpt ions 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 per formance 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 determination 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

~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 [1981, 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 Glaessner [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.

Ir. answer to. the second quest ion 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 different 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 im estimating the model would be an important improvement in the preliminary procedures followed. Third, it would also seem useful to

try to estimate all twenty-four parameters jointly by reducing the

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

- 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 standard 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 instrumerits.

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 = (m,, Ye it, Px } is tested with respect to the

~ ; hypothesized endogenous variables in the flex-price model, Py 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 L (Al.1) = X= £ FRX + £ Gey + Wt s=1 ° ~t-s sxl § ~t-s & 4x1 4x4 2x2 4x1

where x, is defined as above, Ye = (s. Pid Fx and Gs are matrices of

coefficients and COV( ct X,_.) =O forall tands=1, ..., y,

Cov ex Y.2 = 0 for allt and s=1, ..., 2% Also

- Al -

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

-In choosing wand &% 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= 6and 2% = 2 as 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 (Al.2) L=T | 2£n Det (4) - in Det (Z,) Iv x09) where i, is the

variance covariance matrix of residuals for the restricted model where G = 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.

- A2 - Table Al

Test of the Joint Exogeneity Specification of the Flex-Price Model

for Canada and the U.S.

1973 - 1980 Flex Price Model ° Likelihood Ratio” Sims Correction® 2 x7(16) x? (16) Data - ak * Series detrended 43.145 28 .596 only Data series kk detrended and 53.336 . 28.528% - deseasonalized * indicates significance at the .05 significance level kk

indicates significance at the .05 and .01 significance levels.

A The likelihood ratio is defined as L = T| 2n Det ty

- &£n Det Z| where t 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 ey is the unrestricted variance covariance

matrix of residuals.

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

B The sims correction takes the form (T-K)A-L

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

series were deseasonalized and detrended.

The results presented in Table Al test whether the sixteen coefficients on the hypothesized endogenous variables in the four equation 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.

- AG -

Appendix 2

Tests for the Order of the Vector Autoregression A(L) X= vy,

4xg “7b -t

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,, Ye i

* t? Pe ] 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(z)& Likelihood Ratio? » l -31,947.2 75.25% x7 (48) 2 -30,885.9 50.328% y%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 seft=

“b)

c)

ware which was modified in order to obtain the results in

this table.

The likelihood ratio statistics represent tests of four vs. one lag distributed x7 (48), 4 vs.2, and 4 vs. 3 lags respectively,

The CAT criterion attributable to Parzen [1975] chooses lag length n to minimize the quantity.

trace [4/T : vie v7} n=, ---,M j=1 j h > bd > 9

where T is sample size, M is the maximal lag considered (eg. M=4)

and V5 is an estimate of the contemporaneous covariance matrix of

distrubances for the model with j lags.

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 viaue of cAT(r#*) occurs at r* = l 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.

- A6 -

Appendix 3

The data for this study were obtained from the following SOUTCES:

(1) M, = Canadian money supply, Ml (currency + demand deposits)

IFS DATA TAPES, not seasonally adjusted.

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

mot seasonally adjusted.

*& . . (3) i. U.S. eurodollar interest rate, obtained from DRI. These are London Interbank offered rates (LIBOR) for Eurodollars.

Not seasonally adjusted.

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

a - Board Data Base. Not seasonally adjusted.

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

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

(6) PL = 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.

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 11979, 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 AR(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.

- 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 asstmption 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 ig 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 optima]. predictors of domestic vs. foreign variables. This dichotomy does not seem reasonable.

7/ The price adjustment specifications adopted by these authors are

fot 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 Glaessner [1982A] for a more detailed explanation of these different price adjustment rules and for a critical review of some of these different specifications. The exchange rate overshooting properties of these models can also be obtained by assuming that wages are sticky. (See Obstfeld [1981] or Rehm [1981].

-~ 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 cf prediction theory, see Sargent [1979] or Hansen and Sargent [1980].

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 m, Yr, if and Pee (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

endogenous without changing the order of the d System may be useful in efforts to derive t ‘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.

[1981A] letting output be

ifference equation

he 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 Pt and S_ with the roots

pid gy An = By 4 B29 1 ia +t 1 Boo

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 st ac ionary « 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 Vz¢ 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 (V2 V 5) appearing in the price and spot exchange rate equations will each be sums of uncorrelated 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

- Ff -

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 that, = {X,, X jo0777 5 where,both economic agents and the econometrician useQ; to form optimal predictors of x - Assuning that agents observe a different information set than the econometrician, (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 Py 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/ E[vela, 4] = E[X, - E(X,1@, 53194) = 0. 4x]

24/ ™ = - = ri 24/ Note that E(v5,1% 5] EIQ, £:Q4.,1% 9] = 0 for i=1,2 if it is assumed that ink «=

um at the elements in Q. {X.> X.u3e +++ } are uncorrelated with Q:_ and Q:4-) for i=1,2. This will be the case

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

is not projected on oy 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

E[Vext 4] = 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 parameter estimates.

r-1 . 26/ For example, to see how the coefficients (¥5) in ¥¢L) = <r yw*yJ were ; j=0 derived, note that from equation (13)' and (14)' we can derive expressions of the form r-1 . -] r-l or k-4 . Ye(L) = I ¥*pJ sa (nj) C1 + ©ct uy AL] joo ? j=l k=j+l

- F6 -

. 8 -1 * -1 Note thet when j= 0 we have A (",) so *0 =A (7), however, for j=1,..., r-1 the double sum in the far right expression can be

rewritten in terms of the ¥*'s as ij.

* ,-l j j+l . r-j

yr os | . J

j A (ny) (ny Asay try Aso tee. ty TAL) for j=1,..., r-l. This can be seen by writing out the double sum r-l or k-4 Z(t ny 7A) explicitly.

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 (no), Having

done this, the restrictions embodied in the ¥;'s j =O, ++ » 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. ove: 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

Do = Els=(t,)] must be of full rank and the nonlinear f function "0 RxQ

characterizing the orthogonality restrictions must be differentiable. Moreover. an 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 constructing fairly straightforward statistics which only involve estimation of the model subject to the nonlinear restrictions in Y*(+) and Y¥**(.-)

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

-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. " is assumed to be contemporaneously uncorrelated

that the vector of innovations ve

with the error terms re i=l, 2 within d~ so that E[d, d }= {12 0 ey “12x2 2x4 6x1 1x6 x . Vv where Z and V are as defined te 4x4 in the text. x

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

~t fundamental white noise components Qjt i=1,2 at all leads and lags, a cmdition which holds given the assumptions about the Qjt i=l,and Vv’ disturbance terms made in the text. ~t

31/ A more formal discussion of this point can be found in Glaessner [1982A] Chapter IV. 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 [1982A], Hansen and Sargent [1981] and Hansen and Hodrick [1981].

33/ Use of a different algorithm (other than DFP) to minimize the criterion functions ‘in 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 Dp 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 £03) in approximating Dy 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).

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.

Dorndusch, R., 1976, "Expectations and Exchange Rate Dynamics,"

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.

- 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.

“------ , 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.

aaacinienananaias 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.

- b3 -

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

Hansen ],., 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, Mass.).

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

- 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.

- b5 -

Obstfeld, M., 1981, "Relative Prices, Employment, and the Exchange Rate in an Econony 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, "Rational 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.