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RISK PREMIA IN FOREIGN EXCHANGE MARKETS


BY

WEN-HE LU





















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY




UNIVERSITY OF FLORIDA


1986















I dedicate this dissertation to Mr. Li You-chai, Mr. Li You-fu, Mr.

Li You-qien and all of my fellow kinsmen in Xi-go-men Village,

Dui-jiu-go District, Feng-zhen County, Inner Mongolia, China, whose

experience in the last half a century has led me to comprehend the

significance of economic theory.















ACKNOWLEDGEMENTS


I would like to thank my dissertation committee members, Dr. G.S.

Maddala, Dr. S.R. Cosslett, Dr. E. Zabel, and Dr. R.D. Emerson for their

guidance and comments on earlier drafts of this dissertation.

This dissertation would not have been started if this topic had not

been suggested by my committee chairman Professor G.S. Maddala, to whom

I owe all the intuition I have developed in econometrics. I am deeply

grateful that I have had this opportunity of being exposed to his mind.

Gratitude is accorded to Professor Stephen R. Cosslett, whose

patience and guidance have led me through all the tides in my empirical

experimentation. The sacrifice of his time in disciplining this study

into the form it now takes can never be adequately repaid no. His

expertise in econometrics and high standard of academic studies will be

enlightment to me in the years to come.

Dr. Roger Huang provided me with some of the data used in this

dissertation. Dr. Mark Rush and Dr. Leonard Cheng provided some

references. Thanks to them are due.

Special thanks go to Debra Hunter and DeLayne Redding for efficient

and accurate typing of many drafts of this dissertation.

I would not have started this long process of self-education and

eventually formal training in the U.S. but for the encouragement,

guidance and discipline from my parents and my uncles. I would like to

think them for their kindness.

Last but not least I thank Lyndall for all the support she has shown

over these four years of graduate study here at the University of Florida.















TABLE OF CONTENTS


ACKNOWLEDGEMENTS . . . . . . . . . .

ABSTRACT . . . . . . . . . . . .

CHAPTER

I INTRODUCTION AND SURVEY OF THE LITERATURE. . . .

II LUCAS MODEL AND ITS EXTENSIONS: AN EMPIRICAL STUDY.


Lucas Model and its Extension .


White's Adjustment and Empirical Results..
White's test . . . . . . .
Engle's ARCH test . . . . .

III A MONETARY APPROACH TO THE FOREIGN EXCHANGE RATE
DETERMINATION . . . . . . . .

The Model . . . . . . . . .
Estimation and Testing . . . . .
ARCH Method of Estimation . . .
GMM Estimation . . . . . .
Variation on the theme . . . .

IV SUMMARY AND CONCLUSIONS . . . . . .

APPENDIX A . . . . . . . . . .

APPENDIX B . . . . . . . . . ..

APPENDIX C . . . . . . . . . ..

APPENDIX D . . . . . . . . . ..


BIBLIOGRAPHY . . . . . . . . . .

BIOGRAPHICAL SKETCH. . . . . . ... . .


PAGE

iii

vi



1

7

8
20
21
23


II I


*
.















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


RISK PREMIA IN FOREIGN EXCHANGE MARKETS

By

Wen-he Lu

August, 1986


Chairman: Dr. G.S. Maddala
Major Department: Economics


We have attempted to test the existence of time-varying risk premia

in foreign exchange markets under two models that we have developed in

this dissertation. This first one is an extension to Lucas's general

equilibrium model of international finance. By assumption of the Cobb-

Douglas utility function of the consumers we are able to derive a closed

form for the risk premia in the foreign exchange markets on the setting

of a two-country economy model. We used White's test and Engle's test

for homoscedasticity and used White's heteroscedasticity-consistent

variance-covariance matrix to derive the correct standard errors. The

time varying risk premium is tested jointly with the efficiency of the

foreign exchange market, i.e., whether the forward exchange rates are

unbiased predictors of the future spot exchange rates. The empirical

findings indicate that the notion of market efficiency is rejected and

there is no risk premium for any of the three cases we studied.

In the monetary approach, however, we test the existence of time-

varying risk premia alone. By PPP and an extension to the uncovered

v















interest parity we introduced the risk premia into our monetary approach

to foreign exchange rate determination. The forward premium is used as

a driving force of the risk premium. A rational expectation hypothesis

is made and the forward solution derived.

Since it is a non-linear single equation model and there is

evidence of heteroscedasticity we used GMM estimators and the

corresponding variance-covariance matrix and found that there is

constant risk premia in the case of Germany and Japan but not in the

case of Canada.

We also did an empirical study of monetary model with the formation

of risk premium derived before. The findings we have is that there is

time-varying risk premium in the case of Germany but not in the cases of

Japan and Canada.

Since our monetary model relaxes the restriction imposed on the

semi-elasticity of interest rate the empirical results are based on a

more general setting than most of the monetary models of foreign

exchange rates. The conflicting empirical results from the two attempts

are attributed to the different setting of the models.

Extensions to the current data will test whether the conclusion we

have drawn is valid.















CHAPTER I
INTRODUCTION AND SURVEY OF THE LITERATURE


Since the early 1970s, the foreign exchange rates among all the

major currencies of the industrialized world have been determined

largely by private market forces within a floating exchange rate system.

The enormous volatility of the spot and forward foreign exchange rates

as predictors of the future spot rates indicates the unexpected amount

of speculation in exchange markets and causes concern for policy makers,

financial associations and research economists. One feature of this

volatility of the exchange rates is that it far exceeds the variation in

the price levels and in the inflation rates in all of the developed

countries over the same period. The search for a rational explanation

for this exchange rate behavior in the setting of integrated

international financial markets has been the central thrust of empirical

as well as theoretical research over the last decade.

A huge literature already exists on this topic. R.M. Levich (1983)

gives a comprehensive survey of the existing models and empirical

findings. These empirical studies convey mixed messages about the

validity of the different models proposed. Most of these models fit the

data for the countries and for the length of time the authors proposed,

but they do not fit the data for the others. There is still no simple

consensus on this topic. See Levich (1983).

Many researchers suspect that the failure of the models over-all is

due to the simplicity of the models. The relaxation of the restrictions









imposed on the existing models may well provide us with a more realistic

picture and a closer fit to the data.

There arises a question whether the foreign exchange market is

efficient. If the foreign exchange market is efficient and if the

exchange rate is determined in a fashion similar to the determination of

other asset prices, we should expect that current rates reflect all

current and available information. Expectations concerning future

exchange rates should be incorporated and reflected in forward exchange

rates, i.e., in the regression


lnet = a + blnFt_1 + ut (1.1)


where et denotes the spot exchange rate, Ft denotes the forward exchange

rate at time t, a = 0, b = 1 and residual u should contain no

systematic information and should have no serial correlation. Frankel

(1982) does the above regression and a number of other regressions for

the period June 1973 to August 1980 and concludes that the market is

efficient. However, when the data are extended to the 1980s the same

conclusion can not be drawn. Due to difficulty in collecting the

relevant data, we restrain our preliminary investigation to monthly data

of Dollar/Mark, Dollar/Yen and Dollar/Canada from September 1973 to

December 1984. The results are reported in Table 1.

The hypothesis of market efficiency under the context of (1.1)

amounts to the joint hypothesis H0: a0 = 0, al = 1. We observe that

all the OLS parameter estimates are far off the targets. We do a

conventional F-test, and the null hypotheses are all rejected.

However, we can not reject the hypothesis of the market being

efficient by the results reported, since we implicitly assume that there

exists no risk premium. It is probable that the market efficiency









Table 1
Preliminary test of the hypothesis of market efficiency
Ine = a + a nFt + ut
t+1 0 1 t t


2
a0 a R URSS RSS F2,131


German 0.815 0.180 0.0170 2.733 73.398 1693.5812
(0.018) (0.120)

Canada 0.193 -0.088 0.0196 1.149 69.967 3923.0453
(0.046) (0.055)

Japan 2.097 0.618 0.76 0.574 1.295 257.4
(0.167) (0.030)


Note: The numbers in parentheses are standard errors from
the estimation.


notion is rejected because we do not include the risk-premium term,

preferably a time-varying risk premium term. Therefore, we need to

investigate the risk premia's existence to test for market efficiency.

The purpose of this dissertation is to investigate the existence of the

time-varying risk premium by different approaches in the foreign

exchange rate determination. Before my own studies are introduced, a

survey of the established research efforts is in order.

The approaches taken in this area reflect the different viewpoints

of the exchange rates themselves. As Dornbusch (1980) notes, there are

basically three views of foreign exchange rates: that of the exchange

rates being the relative prices of monies, the relative prices of goods,

and the relative prices of different financial assets. Each of these

three views can not help being partial on its own.

Those who view the exchange rates as the relative prices of monies

concentrate their investigations on the foreign exchange markets per se,

i.e., they study only the relationship between the exchange rates of









different sorts and assume that the other economic factors (price

levels, interest rates, money supplies, etc.) are not arguments for the

determination of the foreign exchange rates. Stochastic properties of

the exchange rates under rational expectations constitute the main focus

of studies falling into this category. In the context of rational

expectation, all of the economic factors other than the exchange rates

are excluded in the information sets of the agents in the economy

considered. See Driskill and McCafferty (1982), Hansen and Hodrick

(1980), (1983), Hodrick and Srivastava (1983), M. Kawai (1984) and Huang

(1984) among others. Compared with these theoretical attempts, the

empirical works are relatively scarce. Hansen and Hodrick (1983) treat

the risk premia as not being rejected for some pairs of countries and

rejected for the others.

The theory of purchasing power parity (PPP) views the exchange

rates as the relative prices of goods in the two countries involved.

Although PPP is theoretically appealing, its restrictions reflect

themselves in the empirical studies. Rush and Husted (1985) is referred

to for detailed empirical studies. The major theoretical drawbacks of

PPP are that PPP ignores the notions of governments intervening in the

foreign exchange markets (tariff), and that not all commodities are

involved in the transactions of foreign trade and the existing

transportation costs. This may be one of the reasons why the PPP

approach to the foreign exchange rate determination is not appealing to

research economists. However, we would expect that PPP would fare

better in the long run than in the short run, since in the long run the

intervention and other obstacles to the flow of trade are not

consequential. Woo (1985) proposes a monetary model with lagged










exchange rates as well as a lagged inflation rate, a real income index

and interest rates. He found that the forecasting power of this model

is better than that of random walk model. However, in his model he

implicitly assumed the absence of the risk premium in the foreign

exchange markets.

More recent research follows the popular portfolio-balance model,

which views the exchange rates mainly as relative prices of different

assets for the two countries. See Frankel (1982a), Dooley and Isard

(1983) and Ueda (1983), among others for theoretical models of this type.

Frankel (1982a) does empirical studies in testing the existence of

the risk-premia for the major six currencies. He assumes that the risk

premia depend specifically upon asset supplies and variance-covariance

of the asset supplies. His conclusion is that empirical results fail to

support the existence of the risk premia in foreign exchange markets.

Dooley and Isard (1983) also find from their investigation that the

assumption of risk premia is a weak one.

Among all the theoretical models Lucas's (1982) is the most general.

His model encompasses all the economic factors (besides news). Domowitz

and Hakkio (1985) extend it by a simplifying specification of the utility

function and derive the risk premium as a function of the difference of

variances of the future money supplies of the two countries. However,

they depart from the model by applying Engle's (1982) ARCH (autoregressive

conditional heteroscedasticity) method which changes the specification of

the risk premia into a function of the realized forecasting errors and

therefore undercuts the theoretical base of their attempt. The results

are quite misleading. If interpreted correctly, the hypotheses that risk

premia exist and are time varying are all rejected, contrary to their

conclusions.









We plan to investigate the existence of time-varying risk premia

along two lines. In Chapter II we set out with a derivation of Lucas's

model and an extension is made with some extra assumptions. An

empirical study follows. Since it is a linear model, we introduce

White's (1980) covariance adjustment robust to heteroscedasticity. We

then carry out a series of tests on market efficiency and the existence

of time-varying risk premia. In Chapter III we pursue a monetary

approach to the foreign exchange determination. We also make a general

assumption of time varying risk premium with the forward premium as a

proxy, as suggested by Hansen and Hodrick (1984). Since it is a

non-linear model, we use the non-linear least square method of estimation

to get consistent estimates of the parameters and we introduce the

generalized method of moments by Hansen (1982) to correct the covariance

matrix which is designed to be robust against heteroscedasticity and it

is proved to be asymptotically efficient. The reason that we lay so much

emphasis on the robustness of the estimates against heteroscedasticity is

that its presence causes the variances of the estimates to be inefficient

and makes the statistical inference invalid. A summary and concluding

remarks are collected in Chapter IV.















CHAPTER II
LUCAS MODEL AND ITS EXTENSION: AN EMPIRICAL STUDY


Lucas (1982) developed a general two-country model of international

trade and finance. Since then it has been extended in a number of

empirical studies of foreign exchange models. Notably, Hodrick and

Srivastava (1983) derive a model with risk premia as a linear function

of the forward premia, and Domowitz and Hakkio (1985) derive an

extension by assuming a specific form of utility function for the agent.

The risk premium in that context is a linear function of the difference

between the variances of the forecasting errors of the money supplies

of the two countries involved. However, they departed from their own

model and used Engle's (1982) ARCH method in their estimation.

Although Lucas (1982) is in close parallel with Lucas (1978), it

has not generated as much response in the field of international finance

as the latter. The purpose of this chapter is to pursue where Domowitz

and Hakkio (1985) left off in their extension and to introduce the

rational expectation hypothesis and White's (1980) heteroscedasticity-

consistent covariance adjustment to estimate the original Lucas' model

and also to test the null hypothesis of risk premia being time-varying

in the foreign exchange rate determination and the notion of foreign

exchange market efficiency. The derivation of the model is contained in

Section 1. White's adjustment and the empirical results are reported in

Section 2. Some detailed derivations are collected in the Appendix A.









Section 1 Lucas Model and its Extension

Consider a world economy with two countries. These countries have

identical constant populations; all variables will then be expressed in

per (own country) capital terms. Each citizen of country 0 is endowed

each period with E units of a freely transportable, non-storable

consumption good X. Each citizen of country 1 is endowed with n units

of consumption good Y. These endowments E and n are stochastic

following a Markov process with transition given by


Pr{(+ < '|I = E, n, = ) = F{}',n',=,n}


Assume that the process 9tn t} has a unique stationary distribution

9((,n). The realizations of ,,n are taken to be known at the beginning

of the period, prior to any trading, but no information (other than full

knowledge of F) is available earlier. Each agent in country i wishes to

maximize an intertemporal utility function of the form



E{ ZE t U(xit,yit )}, 0 < B < 1 i = 0,1 (2.1)
t=0


where xit is consumption in country i in the period t of the good X and

yit is consumption of the good Y. The utility function U and discount

factor 8 are common to both countries. In (2.1) U is assumed to be

continuously differentiable, increasing in both arguments, and concave.

Here we specify a Cobb-Douglas utility function


U(x,y) = Axyl-a (2.2)


We assume that agents trade in both goods, spot and in advance

contingent on all possible realizations of the shock process { ,nt}.









The preferences of agents have been assumed to be independent of their

nationalities, so that agents differ only in their endowments. Agents

are assumed risk-averse so that in the face of stochastically varying

endowments, one would expect them to use available securities markets to

pool their risks. In this context, pooling must come down to an exchange

of claims on 'home' endowment for claims on 'foreign' endowment in return.

Perfectly pooling in this sense would involve agents of each country

owning half of the claim to 'home' endowment and half of the foreign.

Let s = {,n} be the current state of the system. Take the prices

of all the goods, current and future, to be functions of the current

state s, with the understanding that prices are assumed stationary in

the sense that the same set of prices is established at s independent of

when s is realized. Then the knowledge of the equilibrium price

functions together with knowledge of the transition functions F(s',s) =

F{(',n',~,n} amount to knowledge of the probability distributions of all

the future prices. Therefore, the agents are assumed to have rational

expectations formulated as such.

For a system in any current state s, let the current spot price of

good X be unity, so that all the other prices will be in terms of

current X-units. Let p (s), qx(s), and q (s) be the spot price of good

Y at state s, the current price of a claim to the entire future (from

the next period on) stream { t} of the endowment of good X, and the

current price of a claim to the future stream {nt}, respectively, all in

X-units. Therefore p y(s), q(s) and qy(s) are all relative prices.

Consider an agent entering a period endowed with 0 units of wealth,

consisting of claims to current and future goods, valued in X-units. He

wants to maximize his utility by choosing the current consumption (x,y)

at spot prices (l,p (s)), equity shares x y in future endowments {Et}









and {nt} at the prices per share qx(s) and q (s) respectively. The

budget constraint is


x + py(s)y + qx(s)Ox + qy(s)y = e (2.3)


The corresponding endowed wealth e' valued in X-units at the

beginning of the next period will be


6' = 6x [' + q (s')] + 6 [p (s')n' + q (s')] (2.4)


since X,Y are assumed to be non-storable and V' and n' are new transfer

endowments.

The use of money is motivated by two constraints imposed upon all

traders to the effect that goods can be purchased only with currency

accumulated in advance. The idea is that under current circumstances

currency can serve as an inexpensive bookkeeping device for

decentralized transactions, thus enabling a decentralized system to

imitate closely a centralized Arrow-Debreu system. See Lucas (1982).

The timing of trading is taken to be the following. At the

beginning of a period, traders from both countries meet in a centralized

marketplace, bring securities and currency holdings previously

accumulated, and engage in perfectly competitive securities trading.

Before the trading opens, the current period's real state s = {(,n} and

current money supply state {M,N} are both known to all. We note here

that the current money supply shocks are also known at this stage. At

the conclusion of securities trading, agents disperse to trade in goods

and currencies. Since any security earns a positive nominal return in

some currency, it is evident that traders will hold non-interest-bearing

currency in exactly the amount they need to cover their perfectly

predictable current-period goods purchase. In this context there is









only transaction demand for money of the current period. Also, the

consumers are assumed to have no speculative demand for money.

Let Mt,Nt denote nominal dollars per capital for both countries.

Prior to any trading in period t, let each trader's money holdings be

increased by a lump-sum wtMt_ and vtNt-1 so that money supply evolves

according to


Mt+ = (1 + wt+l)Mt,


Nt+1 = (1 + vt+ )Nt (2.5)


Let (w } follow a Markov process, possibly related to the real

process (st with transition function


H{w,w',v,v',s,s'} =


P t+1 w' t+t t= ,t+1 = s',st = s) (2.6)

We then have the following financial constraint


M(t) = Px(s,M)


N(t) = p (s,N)n (2.7)


with the equality sign when equilibria are reached. Here px(s,M) and

p (s,N) are nominal prices.

The objective value function for this agent, therefore, is


V(O,s) = max {U(x,y) + 6 f V(O',s') f(s',s)ds'} (2.8)
x,y, x,9
x y

for the consumer in state s with wealth 6. He is assumed to maximize

(2.8) with 6' denoted in (2.4) subject to the constraints (2.3) and

(2.7). We now have Lagrangian function






12


L(x,y,x' y, X,,1'2,3) = {U(x,y) + 8 / V[{ x[' + qx(s')]


+ e [p (s')n' + q (s')]},s] f(s',s)ds'


+ Al[e x py(s)y qx(s) Ox qy(s) y]


+ 2[Mt Px(s,M)S]


+ x3[Nt py(s,N)n]} (2.9)


The first order conditions for the Lagrangian problem are

L = U (x,y) 1 = 0 (2.10.1)


L = U (x,y) A1p (s) = 0 (2.10.2)


L = B f V,,(e',s')[E' + qx(s')] f(s',s)ds'
x

X1q (s) = 0 (2.10.3)


Le = f V ,(6',s')[p (s')n'
y

+ q (s')] f(s',s)ds' Xq y(s) = 0 (2.10.4)


We also know L = 1 as 0 is given and

L = 6 x p(s)y qx(s) yq (s) (2.10.5)
A1 y x A y y


L = Mt p (s,M)E (2.10.6)


L = N p y(s,N)n (2.10.7)


Reorganizing the above conditions, we have

U (x,y) = Xp (S)


Using L, = 1 = Ux(x,y)









U (x,y) U (x,y) U (i,itn)
p(s) = = TTTn (2.11.1)
y x U (x,y) U (i ,I,)
X x


S/f V (e',s')[v' + qx(s')] f(s',s)ds' = 1q x(s)


q (s) = B[U (x,y)]I 1 U (s')[E' + q (s')] f(s',s)ds' (2.11.2)


B / Ve(6',s')[p (s')n' + q (s')] f(s',s)ds' = X1qy(s)


q (s) = B[U (s)]-1 / U (s')[p (s')n'


+ q (s')] f(s',s)ds' (2.11.3)


and

M
px(s,M) = t (2.11.4)

N-r
N
p (sN) = (2.11.5)
y n


Equations (2.8) and (2.9) can be solved forward as in Lucas (1978).

However, it is (2.11.4) and (2.11.5) with which we proceed to derive the

time-varying risk-premia.

Equation (2.11.1) means at such an equilibrium, the relative price

of Y in terms of X, p (st) depends only on the real state of the system,

and is given by the ratio of the marginal utility of Y to the marginal

utility of X.

Given the specification of our utility function, we have


U (,,n) A(1-a)xYa 1-a x a
U x( ), l - A x y y x- ( 2 1
Sy = in

y = In









Here the perfect pooling assumption seems to be restrictive. As we see

later, this will not affect our derivation of the time-varying-risk-

premia.

Assuming purchasing power parity we have


p (s ) Mn M.
xM(stNMt) M ttt t 1-a M (
e(st'MtNt) ps y(s) = P p(s ( (2.13)


by (2.11.4), (2.11.5) and (2.12).

The imperfect pooling would mean an additional coefficient of the

above. It should be noted that the exchange rate in (2.13) does not

incorporate expectations of the future contrary to many rational

expectations models of exchange rate [see Levich (1983)]. This result

follows from the set-up of the model: all decisions are made after the

resolution of uncertainty, so there is no speculation component to the

money demand. In our paper, endowments and money supplies are assumed

to follow conditional Gaussian AR(1) stochastic process


Int = P1lnt-1 + ult


Innt = P2nnt-l + u2t


nt -lnM1 + 3t


nNt = Y2nNt-1 + ut


ut' = (Ult,u2t,u3t,u4t)


ut It-1 I N(0,Ht)

where I indicates the information set at the period t-1.
t-diag(h) (2.14)
t= diag(hll,th22,t h33,t'h44,t) (2.14)









Here we assume zero covariance between the stochastic processes. It is

possible to generalize the model to allow for non-zero covariance. This

would lead to the risk premia also depending upon the conditional

covariances of the money supply.

To derive the forward rate we need interest rate parity. First,

consider the home-currency price in period t of a claim to one unit of

home-currency with certainty in period t+1. Such a claim is equivalent to



tM 1 t+l- (2.15)
t+1 M
Px(St+lMt+1) Mt+l


units of X in period t+l. Similarly



HN t+l (2.16)
t+l Nt+
t+l


Since l and tl are both functions of the future real output and

nominal money supply, there is uncertainty as to their exact values.

The I units of X will be valued by agents in period t+l at the
t+1
marginal utility of X, Ux(st+ ), which must be discounted back to period

t by multiplication by the discount factor. The x-unit price of the

claim to one unit of home currency is therefore Et[Ux(st+) n + Ux(St)-1]

which is obtained by taking the conditional expectation of the marginal

value of the payoff on the asset and dividing it by the marginal utility

of X in period t, since the opportunity cost of the investment is its

x-unit price times the marginal utility of X in period t.

The home currency price of the investment is then obtained by

multiplication of the x-unit price by p (st,M) or division by It.

Therefore, the period t discount bill paying one unit of home currency

in period t+l in terms of the home currency price is









Mt+ M
U (s )1M
b(s ) = E[ s 1Mt+1 EQ (2.17)
x t U (s )M t+1
x t t

Similarly, by replacing x with y in the above argument, the period t

foreign currency price of a claim to one unit of foreign currency in t+1

is


BU (s )H+
b (stt) = E[ t+ E N (2.18)
Y t t U (s )N = t+
y t >"t


where U (s ) is the marginal utility of y in period t and HN is the
yt t
purchasing power of one unit of foreign currency in terms of Y. The

intertemporal marginal rate of substitution of money is an index that

weights the change in the purchasing power of one currency by the inter-

temporal marginal rate of substitution of goods between the two periods.

Since the exchange rate is the relative price of two monies, each of the

rates of substitution is important in determining the risk premium in

the forward foreign exchange market.

In order to determine the nature of the risk premium in one forward

foreign exchange market, we must derive the forward price of foreign

exchange, that is, the contract price set in period t at which one can

buy and sell foreign exchange in period t+1.

If there is no default risk on either nominal investment discussed

above or on the forward contracts, investors must be indifferent between

investing in the riskless home currency dominated asset, in which case the

return is 1/b (s ,6 ) per unit of home currency invested, and the alterna-

tive covered interest arbitrage strategy of converting the home currency

into foreign currency and selling the proceeds in today's forward market

at price F(stOt,Mt,Nt) of home currency in terms of foreign currency.






17


The covered investment in foreign currency yields the return

1
[e(stlMN)111/by(st,6t)]f(st,9tMtNt)



per home currency invested e(st,M,N) is the exchange rate, i.e., the

foreign currency per one unit of the home currency. Equating the two

strategies,


1 1 [ /
bx(stt) e(stM,N) [i/by(St'et)]F(st'et'Mt'Nt)
x t, t t


we have

b (s ,0 )
F(st,et,Mt,N ) = e(st,M,N) b(s (2.19)
x t t


The forward rate formulation is


EN
EtQ t+1
F =e
t = et M
EtQt+1

From (2.14) we have

U (s')(' M
Et+ = B x t f(ul u u )du du du
St+ U (s) M 1,t+l' U2,t+l' U3,t+l) dUlt+1 du2t+1 du3t+1
x t+l1


BU (s ) M
E x t+1 t+1
=E
t U (s )fl
x t t

ca-1
t+1 t
t t+l

a 1-(
= E t+1 t+1
E B (-) (t -)
t t nt


M <
t t+1
t+1 t


M
t
M
t+1


(2.20)











u (s')n' N
EQ f (u l du du du
Ety B (sn N t- f(ult+l,u2,t+l 3,t+l) dltd2tdu4t
y t+1


BU (s )1N
E y t+1 t+1









t 't+1 "t+1


It is shown in Appendix D that
E NNN







InEtQ InEtQ+ = (l-Y)lnNt + h


(1-Y )lnM h3
1 t 33,t

Substituting the above into (2.19) and taking the logarithm, we have


InFt = Inet + [(1-Y2)lnNt (1-Y )lnMt h33,t + h44,t


= ln[(l-a)/a] + InMt nNt + (1-Y2)lnNt (1-Y1)lnMt h33,t


+ 3h
44,t

= ln[(l-a)/a] + YllnMt Y21nNt 2h33,t + h44,t


Also from (2.13) we have


E tne = ln[(l-a)/a] + InMt+1 InNt


= ln[(l-a)/a] + yllnM Y2lnNt + Etu3t Etu4t









The last two terms are zero, hence we have


EInet nFt = (h 33,+ h44+) (2.23)


Equation (2.23) indicates that the risk premium depends upon the condi-

tional variances of the forecasting errors of the domestic and foreign

money supplies. An increase in the conditional variance of domestic

money, h33,t+, increases the conditional variance of domestic prices and,

therefore, increases the expected future purchasing power of the foreign

currency. Consequently, the expected return on a long position in foreign

exchange (the risk premium) must rise to compensate investors. Similarly,

there is an increase in the demand for future foreign currency which

drives down the forward rate, leading to an increase in the risk premium.

The model of (2.23) has the term representing relative conditional

variances of the future money supply shock h33,t+ h44,t+ which can

not be observed at time t. We here make a naive rational expectation

assumption, i.e., that the economic agents use the relative variance of

current money supply shocks h33,t h44,t a proxy to that of the future

money supply shocks. There are other ways of formulating the rational

expectation hypothesis, but they are just as, if not more, arbitrary;

hence (2.23) becomes


Etlne nFt = (h33, h44,t) + [i(ht+ 33t
t t 2 33,t 44,t h33,t

(h44, h44,t)] (2.24)

We approximate Et net and InFt by


et et-l Ft et
t t-1 t t
and
et-1 et


and estimate the equation









et etl Ft I- etl
et-= a0 + aI e + (h33t h44,t) + t (2.25)
et-1 t-1


There are two implications of equation (2.25) for the behavior of

the risk premium. First, the only time series movement of the risk

premium is due to the movement of the 'relative variance of the money

supply shock.' Second, the risk premium can take both signs and can

switch signs, depending upon the values of a,, a2. This is important

since, for example, Stockman (1978, p. 172) found that 'the point

estimates of the risk premium change signs,' when he estimated a

hypothesized constant risk premium over different subsamples. Suppose

that a, < 0, a2 > 0. Then for small relative variance of money supply

shocks the risk premium will be negative, while for large relative

variance of money supply shocks, the risk premium may change signs.

Equation (2.25) allows tests of several hypotheses concerning the

time-series behavior of the risk premium. According to the model, al

should be 1 if the market efficiency hypothesis is to hold, and t+l

should be white noise, independent of whether or not a risk premium

exists. A test of the hypothesis that a2 = 0 is a test of whether the

risk premium is time-varying or not, in determining the deviation of the

forward rate from the expected future spot rate.

Maintaining ia = 1 and t+l white noise, a0 = 0, a2 = 0 means a zero

risk premium. a0 e 0, a2 = 0 implies a constant risk premium. a0 z 0,

a2 = 0 means that data dictate that the risk premium is a time varying one.


Section 2 White's Adjustment and Empirical Results

The model of (2.25) is the one we proceed to estimate. From (2.24)

and (2.25) it is clear that t = vt + a2[I(h33,t+l h33,t) + (h44,t -

h44,t)] and is clearly heteroscedastic. The consequence of this approxi-









nation is that the error terms are heteroscedastic; hence OLS estimates of

parameters are consistent and unbiased but not efficient. The OLS

estimates are reported in Table 2. The DW statistics show that the first

degree serial correlation is absent for all cases. Therefore we can treat

the heteroscedasticity as being 'cross-sectional' in nature. Before we

proceed to estimate the consistent variance-covariance matrix, we do two

tests for heteroscedasticity and the test scores are reported in Table 2.

White's test

The information matrix test of White (1980) is a test of the validity

of the model against any alternative which renders the usual maximum

likelihood inference techniques invalid. When the model is correctly

specified, the information matrix may be expressed either in Hessian form,

- E[2 L/UDE'] or in the outer product form, E[8L/85 WL/W'] where E is

the vector of parameters being estimated. The White procedure tests the

equality of these alternative expressions. In the present context, the

information matrix test for normality is sensitive to skewness or kurtosis

[White (1980)], and can also be interpreted as a test for parameter

constancy [Chesher (1983)]. White (1980) also proves that the test score
2 2 2
is equivalent to TR x k(k+1)/2 where T is the sample size; R is the

constant-adjusted squared multiple correlation coefficient from the

regression

^2 t t 2
=2 = I )2 2
ut + 1 e +2 (h33,t 44,t



+ 3 [(h33,t h44,) t 1]



where again ut is the OLS residual term of model (2.25). The results

are contained in Table 2. The test scores indicate that there is










Tests for Serial
with

e -et
t t-1
e 0 + 1
t-I


Table 2
Correlation and Heteroscedasticity
ARCH as a special case

Ft l-et-)
[ e ] + a2[(h33 h44 )] + E


OLS Estimation

critical
score
Germany Japan Canada at 5%
significance
level


a0 0.0313 -0.0022 0.0038
(0.0170) (0.0033) (0.0035)

a 0.0560 0.0426 -0.0016
(0.0329) (0.0218) (0.0031)

a2 -3.425 1.3865 0.2706
(2.612) (2.2926) (0.3503)

R2 0.0329 0.0316 0.0063

DW 2.007 1.863 2.218

ARCH-1 8.619 2.769 2.392 5.02

ARCH-2 8.656 3.548 2.3607 7.38

ARCH-3 8.9984 3.571 2.5344 9.35

ARCH-4 9.3345 5.779 2.794 11.1

White's test 9.694 4.1003 0.5371 9.35


Note: The score for ARCH-P is equivalent to TR2 with T as sample size and
2 2 P 2
R from regression e = Y + E .. The numbers in parentheses
Rt from regression Yit-i.
i=l
are standard errors from the estimation.



heteroscedasticity in the German/U.S. case but not in the Canada/U.S.

and the Japan/U.S. cases.









Engle's ARCH test

We also carry out Engle's test for the presence of ARCH (auto

regressive conditional heteroscedasticity). In a general model of Engle's


yt = f(Xt,B) + ut


where X is a vector of explanatory variables and 6 the parameter vector

with the distribution of the disturbance term assumed conditional normal

as following


ut t-1 N(0,o)
where
2 P 2 (2.26)
t 0 + Yiut-i26)
i=1


The highlight of Engle's ARCH formulation is to capture the severe

volatility of the error terms in a model with time series data. It is

particularly attractive in the research involving the foreign exchange

rate modeling, since we observe immense volatility of both spot and

foreign exchange rates. The conditional variance of the current time

period is assumed to be a (linear) function of the squared past errors;

therefore, one larger past error tends to be followed by errors with

larger dispersion. One weak point of ARCH is that p in (2.26) is

arbitrary and has no strict economic theoretical backing.
2
Engle (1982) shows that X score for testing the null hypotheses
2
H Y = Y = ... = Y = 0 is equivalent to TR where T is the sample

size, R2 is the constant-adjusted squared multiple correlation

coefficient from the regression


^2 P ^2
ut = Y + E Yiut-i (2.27)
i=1










The test scores reported in Table 2 indicate that there is no ARCH for

the cases of Canada and Japan. However, ARCH effects of 1st and 2nd

order are significant in the Germany/U.S. case, although longer lags are

not significant.

Since (2.25) is linear, ARCH estimates can be obtained from the OLS

regression (2.27). See Engle (1982). The results are reported in Table
A ^
3b. The estimates show that y0 and yl are significant in the case of

Germany for all the lag lengths p which we tried. This partly confirms the

results we had in White's test and Engle's test, i.e., there is hetero-

scedasticity in the case of Germany case and it is of ARCH type.


Table 3a
ARCH Estimates and White's heteroscedasticity-consistent
variance estimates for Mark/$

^2 P ^2
t = Y0 + Yiu t-i
i=l
White's heteroscedasticity-consistent estimates and standard errors


a0 1 a2


Germany 0.0313 0.0560 -3.425
(0.215) (0.0399) (2.447)

Japan -0.00225 0.04256 1.3865
(0.00356) (0.02448) (2.10544)

Canada 0.0038 -0.0016 0.2706
(0.00356) (0.00313) (0.1195)



Note: The figures in the parenthesis are standard errors.
indicates significance at 5% level.









Table 3b
ARCH Estimates and White's heteroscedasticity-consistent
variance estimates for Mark/$

^2 P ^2
ut = Y + 1 Yiut-i
i=1
ARCH parameter estimates


YO Y1 Y2 Y3 Y


Germany 0.00074
(0.00018)
0.00076
(0.00019)
0.00081
(0.00021)
0.00082
(0.00022)


0.00087
(0.00019)
0.00096
(0.00021)
0.00092
(0.00023)
0.00076
(0.00024)

0.00015
(0.00041)
0.00016
(0.000043)
0.00015
(0.000046)
0.00016
(0.000048)


0.2574
(0.0851)
0.2651
(0.0887)
0.2633
(0.0893)
0.2634
0.0897

0.14597
(0.08711)
0.15694
(0.08850)
0.15945
(0.08890)
0.14616
(0.08876)

0.1356
(0.0872)
0.13638
(0.08874)
0.13766
(0.08931)
0.14121
(0.09004)


-0.0289
(0.0888)
-0.0126
(0.0925)
-0.0132
(0.0931)


-0.08101
(0.08851)
-0.07780
(0.08974)
-0.06193
(0.08937)



-0.00993
(0.08875)
-0.01317
(0.09015)
-0.01484
(0.09076)


-0.9583
(0.0895)
-0.0511
(0.0931)


0.0097
(0.08893)
-0.0097
(0.08921)






0.03463
(0.08930)
0.03921
(-0.0474)


Note: The figures in the parenthesis are standard errors.
indicates significance at 5% level.


Following White (1980), we calculate the heteroscedasticity-

consistent covariance matrix. The procedure is to save residuals ut


from regression (2.25) and calculate the following


Japan










Canada


-0.0375
(0.0901)


0.14662
(0.08824)









-0.04215
(0.08992)









I -1 ^2 -1
0 = (X'X/n) (X'(diag(u ))X'/n)(X'X/n) (2.28)


where X is the matrix of the explanatory variables.

Comparing the heteroscedasticity adjusted standard errors with

those from OLS in Table 2, we find that all but one of them are greater

than the OLS estimates. The coefficient of the risk premium (h33,t

h44,t) for Canada is significant.

Maintaining al = 1, we first test the joint hypothesis a0 = 2 = 0.

The results in Table 4 (in the column RP1) indicate that this null hypothe-

sis can not be rejected for any currency. We next test the expanded

hypothesis H0: a0 = a2 = 0, al = 1. The results of this test are reported

in Table 4 (in the column RP2). The results indicate that the null hy-

pothesis of no risk premium are rejected for Germany, France and Japan.

The third hypothesis we test is a, = 1. This is a test of market

efficiency when the time-varying risk premium is present in the model.

The null hypothesis are rejected for all the cases.


Table 4
Testing scores for existence of the Risk premia


RP1 RP2 RP3

Germany 2.6427 431.9083 409.25

Japan 6.6010 971.5318 1001.98

Canada 1.1785 53017.746 50922.34


Critical value 99.5 26.2 99.5


Note: The numbers reported for RP1 and RP2 are F-statistics
at 1% significance level.
RP1 is an F-test of the hypothesis a0 = a2 = 0;
RP2 is an F-test of the hypothesis a0 = a2 = 0, a = 1.
RP3 is an F-test of the hypothesis al = 1.









The above tests suggest that there is no empirical evidence in our

model that the foreign exchange market is 'efficient', i.e., the forward

exchange rates is not an unbiased predictors of the future spot rates.

This finding coincides with the results of the models of the other

authors (see Park (1984), Frankel (1982), Domowitz and Hakkio (1985)

Murfin and Ormerod (1984), Baillie, Lippens and MacMahon (1983)).

However, the tests of the other authors for the market efficiency are

based upon one assumption of risk premia being constant. Although they

also rejected the hypothesis that the forward rates are unbiased

predictors for the spot rates, they can not preclude the possibility

that with time-varying risk premia the market is efficient. The test of

our model does not have this shortcoming and the rejection of the market

efficiency is made upon a more general setting than most of the

empirical studies in the literature.















CHAPTER III
A MONETARY APPROACH TO THE FOREIGN EXCHANGE RATE DETERMINATION


In this chapter we derive a model based on a monetary approach to

foreign exchange rate determination with time varying risk premium.

Except for a time-varying risk premium term, this model is a straight-

forward extension of Dornbusch (1980) and Woo (1985).

The model is partial in that money supplies, real incomes, price

levels and interest rates in both countries are assumed to be exogenous

to the foreign exchange market. The rational hypothesis is made when

the revised uncovered interest rate parity is assumed. All the

exogenous variables are assumed to be represented by a finite order

autoregressive process. However, the semi-elasticity of money demand

with respect to real income levels is not assumed to be the same for

both countries. We hope that this relaxation can lend support to the

validity of the model. We use the forward solution to handle the

expectation of the future endogenous variable and the generalized

method of moments to estimate the model. The model is derived in

Section 1. Section 2 contains the estimation and testing procedure.

Some details are provided in Appendix B.


Section 1 The Model

Consider an economy of two countries. The standard money market

equilibrium condition is


mt Pt =








mt Pt = y r + u tl,..., (3.2)



T is the sample size. Here mt pt' Yt stand for the logarithm of

money supply, price level and real income level at time t for the home

country; rt is the nominal interest rate and ut is a white noise

disturbance term, 4 and 3 are unknown coefficients of the real income

and interest rate and are both positive. Similarly, mt, pt, yt, rt, ut

and t are the corresponding variables and parameters for the foreign

country in the model. Note that we here relax the constraint that most

of the models impose upon the coefficients of the real income levels

being the same for both countries. We hope that this relaxation makes

this model more general than the conventional practice.

Subtracting (3.2) from (3.1) we have


(mt-mt (Pt-)t = t t B(rt-rt) + (ut-ut) (3.3)


The cornerstone of the monetary approach, PPP (purchasing power

parity) is assumed, i.e.,


Pt = pt + et (3.4)


where et is the home currency price of the foreign currency. The PPP

says that arbitrage of commodities will make the exchange rate equal to

the ratio of the price levels of the two countries involved. The

underlying assumptions are that all commodities are traded and that

there is neither government intervention nor transportation cost. These

assumptions are of course not realistic in the real world. However,

these distortions in the long-run are not consequental. Officer (1981)









and Rush and Husted (1985) provide evidence in support of this

assumption. See Levich (1983) for further discussion.

Departing from the risk-neutrality assumption in Dornbusch (1980)

and Woo (1985), we modify the uncovered interest rate parity by

introducing the risk premium term


S- r = E(et+ e RtI)= E(et+) e R (3.5)


where It is the information set the agents in the economy have up to

time t, R(t) is the risk premium term at period t. E(.I t) denotes the

rational expectation of the agents at time t.

Under the assumption of perfect capital mobility, the equation

looks exactly the same as the uncovered interest rate parity with

investors being risk-neutral.

The risk premium we introduced shows our assumption of investors in

general being risk averse. Admittedly, there is difficulty in

differentiating the risk-neutrality from perfect capital mobility, since

both arrive at the same form of expression of (3.5) without Rt. Here we

choose the former interpretation.



This can also be interpreted as deriving from the perfect state of
capital mobility. Papell (1985) defines this equation as


r rt = E(el e a(e qt)) + v


where et and qt are deviations from the steady state of the exchange rates
and relative price levels. His formulation implies that imperfect capital
mobility is caused by the short-run deviation from the long-run PPP.
We can interpret the risk premium along the same lines, i.e., risk
premium reflects the short-run deviation from PPP relating to the
uncovered interest rate parity in a particular way and this deviation is a
function of the past forecasting error on the spot exchange rates.









Substituting (3.4), (3.5) into (3.3) and rearranging, we have

1 4 8 8
e I (m -m ) y + y R + E e +v (3.6)
t 1+8 t 1+7 t 1+ t 1+ Rt 1+ Et t+l (3


(ut-ut)
where vt vt is white noise too.
t 1+p t
Now we proceed to formulate the risk premium and the distribution

of the error term. We assume that R has the general form

2 2 2
Rt = f(d ,d ...,d ) (3.7)
t t-1t-2 t-p

where dt is the forward premium, the difference between the realized

future spot rate and the forward rate divided by the future spot rate
f -e
te Hodrick and Srivastava (1983) developed an extension to Lucas's
e
(1982) model and derived that the risk-premium is a function of forward

premium. We take this as given and work from there. The equation

(2.23), the risk premium we derived in Chapter II is developed also from

another extension of Lucas's (1982) model. Since the difference between

these two formulations do not affect the rest of the derivations, we

keep our formulation of the risk premium as the former. In a later

section empirical study with (2.23) is done.

This formation of the risk premium departs from the constant risk-

premium assumed in most of the empirical and theoretical studies of

exchange rate models. In a period of rampant forward speculation

constant risk premium is less likely to be the case.

In our investigation, we specifically assume the following:


R = a0 + adt-2 + (3.8)


where a0 and a are constant unknowns and dt is the forward premium at

time t, i.e., the difference of logarithms of the forward exchange rate









at time t-1 and the spot rate at time t; wt is white noise. Although we

observe asymmetry in the above formulation of the risk premium, we do

allow the risk premium to take either sign as the data dictate.

We here assume that the agents have a naive rational expectation,

i.e., use the current forecasting error as estimate of the future

forecasting errors. Therefore we have


ER = a0 + ad
tt 0 1 t-1

ERt+i = + a d i=1,2,... (3.9)
t t+i 0 1 t-1

Substituting (3.8) into (3.6), we finally have the model



1t 1+- t +B t 1+6 Y 1+8 tt+1 1+R


( + a d _) + v (3.10)
0 1 t-1 t

There are usually two ways of handling the rational expectation of

future endogenous variables. The first is Wickens's (1982) error-in-

variable method, which replaces the expectation of the future endogenous

variables with the true observation of that variable. One restriction of

this method is that it is not applicable when the realized and expected

variable appears in the same equation, which is the case with our model.

Therefore, we are left with the method of substitution due to Shiller

(1978) and Wallis (1980). In this model B must take a positive value;

therefore, the backward-solution method is explosive. See Shiller (1978)

and Wallis (1980) for a discussion of the forward and backward solution

methods. Hence, we proceed to simplify the model using the forward

solution method.










Since Etet+1 involves the future values of all the exogenous
*
variables, we assume at the outset that mt, mt, t, yt are all from a

first order autoregressive process.


Inmt = O + Pl nm- + t

*
Inmt = PO + llnmt- + t


Inyt = k0 + kllnyt- + nt

*
Inyt = ko + k lnyt-1 + nt (3.11)

*
where Et, Et' qt' qt are all white noises.

The stationary conditions require that


PO + P" 1

*
PO + PI 1


k0 + k1 1

*
k0 + k < 1


The forward solution method leads us to


1 1 *
et = +(1-l mt ( ) mt it l +
-+B l-k Yt Yt



BO 110 1O k o 0 k
B +1 [1-l l- 1- ]

0 +I --l ) -11 1- 1 1-k
1~**
011 110111









w w
kOkl1 koklp
[1+k(1- k)I(1-k ) +
1 1-k [1+8(1-kl)](1-k)



a1 2 a1 2
Bd d- + v (3.12)
1+8 t-1 +I t t


The detailed derivations are collected in Appendix B. Equation (3.12)

is the model we proceed to estimate.


Section 2 Estimation and Testing

In this section we use both the ARCH (autoregressive conditional

heteroscedasticity) method introduced by Engle (1982) and GMM

(generalized method of moments) introduced by Hansen (1982) to estimate

the model developed in the last section.

When we address the empirical estimation and testing of our model,

we have to consider the plausibility of the assumption that the distur-

bance term is homoscedastic. The volatility of the spot exchange rates

in all years since September 1973 suggests that this assumption may not

be valid, and consideration of this possibility should be given in

estimation and testing to prevent the estimates from being inefficient

if heteroscedasticity is truly the case. There are two ways now

available to tackle this problem. One is the ARCH method of estimation

and the other is GMM. ARCH is more suitable if the residuals show signs

of having a distribution that has a fatter tail than the normal

distribution, whereas GMM has the advantage of not having to be this

specific. We shall use both methods. Unfortunately, the ARCH method

fails to produce convergent maximum likelihood estimates. So after the

derivation of both the methods, only the results of GMM are reported.









ARCH Method of Estimation

McNee (1979) says that "large and small errors tend to cluster

together" in the international finance models. This would suggest that

forecasting variance may change over time, and is predicted by past fore-

casting errors. Engle's (1982) ARCH method is intended to help solve this

problem in empirical studies. The basic idea is that the conditional

variance of the error term is nonconstant and moves in an autoregressive

fashion with the past forecasting error. This implies that large errors

tend to be followed by large errors and small errors tend to be followed

by small errors, in either direction. We start with the model


Yt = f(xt ) + v (3.13)


where Y is a vector of observations of the endogenous variable and 6 is

a vector of parameters to be estimated; X is a matrix of explanatory

variables; v is the vector of error terms. The conditional distribution

of the error term is assumed to be of the form


vt Itl N(O,o2) (3.14)


where It- is the information set at time t-1 and var vt is generally

defined by


2 = f(v v2 v2 ) (3.15)
at = f(t t-2 Vt (3 5t-

Specifically, we formulate a linear function as follows


o2 = Zy (3.16)
t









where
2 2 2
Z= 1, v ,v ,...,v _} (3.17)
t t-1 t-21 t-p t

a = {y0,Y1,..., p} (3.18)


To restrict the variance to be non-negative, we impose


Yi > 0 i=0,1,2,...,p (3.19)


and to prevent the variance from being explosive we impose


p
E Yi < 1 (3.20)
i=l1


We note that (3.18) is sufficient but not necessary.

The likelihood function is

2
1 2 1 vt
InL =const. In (3.21)
t 2 t 2 2
at

2
where v = Y f(x ,B) and oa is defined as (3.16). This is maximized
t tt
to obtain the MLE of . The first and second order conditions can be

easily derived. However, in our empirical study likelihood functions

fail to converge.

GMM Estimation

The generalized method of moments (GMM) estimator introduced by

Hansen (1982) consists of finding the element of the parameter space

that sets linear combinations of the sample cross products as close to

zero as possible. The GMM estimator is consistent and is asymptotically

normal if the observable variables are stationary and ergodic. Since

many linear and non-linear econometric estimators reside within this









category, GMM is a good alternative to the conventional maximum likeli-

hood method. One advantage of the GMM estimator over MLE is that the

variance-covariance matrix obtained by GMM is robust with respect to the

specification of the disturbance distribution; hence, we can expect to

produce a heteroscedasticity-robust variance-covariance matrix of the

estimated parameters when the researchers are not confident about the

specification of the disturbance distribution and/or when the specifica-

tion of one distribution is too complex.

Consider a model similar to (3.12)


y = Xf(5) + e (3.22)


where y is a Txl dependent variable and X a matrix of explanatory varia-

bles of dimension TxM; 6 is the unknown hxl parameter vector; f(6) is

MxN a one-to-one function (not necessarily linear) taking element 6 of

the parameter space into a space of greater or equal dimension, i.e., M'h.

It is assumed that there exists an integer P having the property that


E{ tlEtp ,t_p_l...XtXt-l*** .) = 0 (3.23)


This formulation admits the possibility of serially correlated residuals.

p=l is equivalent to the absence of serial correlation, since (3.23) implies


E{stle t,St-2,...,Xt,Xt_,...} = 0 (3.24)


The GMM estimator is the vector 6 that minimizes the quadratic form


$(6) = (y Xf(6))'X -1X'(y Xf(6)) (3.25)


where 0 is a consistent estimator of the positive-definite matrix


R = lim (- E(X'ce'X)). (3.26)
T+cT









Hansen (1982) proves that


p-1 1 T (
= iX'E X ttqXtq} (3.27)



is a consistent estimator of n.

The variance-covariance matrix of the GMM estimator of 6 is

v'X ^-1 X'v -1
plim [( T TX 1 (3.28)


af af f
where v = X X [ ... (3.29)
a6 1 36 62 6h *


where 6 is the GMM estimate.

In our model (3.12)


ft {f f2 f3 f4 f5 f6 f7
where
1
1


f 2
f
1 1+B-Bp1



I+B-Bv1


f =



*
6I+B-Bk1



1+B-Bk1


2
-a, 8
f5 -
1+B


oilg
6 = 1+









*
^ 0 B0 0 Bk0 O 0k0
f = + + (3.30)
f7 -0 + 1+B-By 1++ + _k-Bk


2 2
and X = (m m y y d2 d 1.)
and Xt = (m t t Yt t t-d t

6 = (B 4 ) 0 a 1)

^* af
The derivatives evaluated at 6 7, for the model (3.12) are derived in

Appendix C.

Before we estimate (3.12) the rational expectation hypothesis allows

us to impose the first order autoregressive process on the money supply,

and on the income index for the models of two countries (Germany vs. U.S.

Japan vs. U.S. and Canada vs. U.S.) defined in (3.11). The estimates of

P0' il, k0, k1 are reported in Table 5. The U.S. is treated as the

foreign country in all cases, as usually the case in empirical studies of

exchange rate models.


Table 5
Estimations of the first-order autoregression process
*
on the exogenous variables mt, mt, Yt' Yt.


U.S. Germany Japan Canada

V0 0.00115077 0.09747710 0.30776482 0.00467322

1 1.0078970 0.98287155 0.972608783 1.00037203

k0 0.05466299 0.37540327 0.01169507 0.41615339

k. 0.98914601 0.92667755 0.99814542 0.91531332


Note: The above estimates are from the regressions

mt = + ~imt-1 + lt

and Yt = k0 + kYt- + v2t

where mtYt are logarithms of money supply and income index
respectively.









Substitute the estimates from the autoregressive processes into

equation (3.12), and then estimate it with NLLS.

Before we set out to adjust the variance-covariance matrix of the

GMM estimator, we first do some diagnostic checking on the residuals we

obtained from the NLLS estimation.


Table 6
Tests of ARCH in equation (3.12)


2 p
2t =0 +
i=1


Germany







Japan







Canada


2-
Bi t-1


Note: The numbers in parentheses are all t scores
indicates significance at the 5% level.


for H :


i = 0.
1


B0 1 2 B3

0.0035 0.7321
(1.787) (12.255),
0.0037 0.7910 -0.0804
(1.889) (8.97) (-0.911)
0.0025 0.8101 -0.2671 0.2364
(1.341) (9.464) (-2.449) (2.759)


0.0043 0.8372
(0.046) (17.708),
0.0027 0.5649 0.3278
(1.342) (6.792), (3.96) ,
0.0022 0.5495 0.2207 0.1233
(1.113) (6.557) (2.396) (1.479)


0.0010 0.8219
(1.719) (16.969),
0.0007 0.5844 0.2981
(1.099) (6.988) (3.605),
0.0008 0.62988 0.2561 -0.0221
(1.139) (7.329) (2.688) (-0.263)









We first test the existence of serial correlation by the standard

regression


2 P 2
s = 0 + Bt-i (3.31)
i=l


The results are reported in Table 6. Engle's X2 test of ARCH is

conveniently equivalent to TR2 of (3.31), with T as sample size and R2

as the squared multiple correlation coefficient from (3.12). Table 6

shows that p in (3.31) should be picked as 3 for Germany, 2 for Canada

and Japan. Table 7 contains the GMM estimates.

In Table 7 the estimates of the semi-elasticity of money with

respect to interest rates B show that they are not significant from zero

for all the cases. The estimates of the semi-elasticity of money with

respect to the income index for Germany, Japan and Canada are all

significantly different from that of U.S. These findings have confirmed

that the restrictions imposed in the conventional monetary models are

too rigid and the corresponding restrictions should be relaxed.

Therefore the results we have obtained are based on a more general

setting then the conventional practice. Unfortunately, the estimates of

are not the same for the three cases. We concede here that further

research should be made along these lines.

Woo (1985) develops a monetary model for the dollar/deutschmark

rate with a partial adjustment mechanism assumed in the money demand

function. There he also relaxes the restrictions as we do but he

assumes a zero risk premium. His estimate of B is significantly

different from zero but (, i are not. Since he concentrates on the

dollar/deutschmark rate only, we do not know how his model would fare in









the other cases. In contrast, our model is built up with a time-varying

risk premium. The results from our empirical study should have a better

reflection of the reality than Woo's.

The other monetary models are mostly built using the data up until

1980, therefore their results can not be compared in the same light with

ours. See Frankel (1983) for the other empirical studies.

Turning to the terms of time-varying risk premium, GMM estimates

indicate that there are constant risk premia in the case of Germany and

Japan.


Table 7
GMM estimation of (3.12)



B f f 0 a1


Germany 3.9920 2.4402 1.05637, -2.0918 0.4884
(S.E.) (3.6176) (1.4070) (0.2444) (0.9269) (0.4415)


Canada 0.0307 0.9484 7.2775 -35.7719 1.8947
(S.E.) (0.1039) (0.2824) (0.2116) (135.921) (5.5671)


Japan 0.0056 0.1257 0.1723 -40.0190 -0.1697
(S.E.) (0.5483) (0.6316) (0.3458) (0.3958) (0.1413)


Note: S.E. stands for the standard error according to the generalized
method of moments.
indicates significance at 5% level.


In the case of Canada, GMM estimates indicate that risk premia do

not exist. There have been few attempts on modeling time-varying risk

premium into the monetary approach of foreign exchange determination.

One reason is that the monetary approach did not fare well in empirical

studies. Therefore there are not many papers of monetary approach in










the recent five or six years. Another reason is that the source of the

time-varying risk premium is usually outside the system of the monetary

models. It is my belief that incorporating the time-varying risk

premium into the monetary approach may prove to revive the interests of

researchers in this area. The current paper is one attempt.

Variation on the Theme

We proceed to do another empirical study with model (3.12).

Instead of using forward premium as argument of the time-varying risk

premium so far we use (2.23), the risk premium derived in Chapter II,

i.e., the difference of the variances of the money supplies as the

argument of the risk premium.

Parallel to the preceding procedure, NLLS of (3.12) is run with new

formulation of risk premium. Residuals are saved and ARCH test is

taken. The result is reported in Table 8.

The results show evidence of ARCH effect. We choose 2, 2, 1 as

values of p in (3.30) for Japan, Canada and Germany respectively. ARCH

estimation failed to converge again. GMM is used to estimate (3.12).

The results are reported in Table 9.

We note that in the case of Japan the estimate of coefficient of
*
U.S. income index ( hit the bound. As in the preceding empirical

study, the estimates of < ) are significant for the case of Canada

while risk premium is non-existent. Again, there is empirical evidence

that in the case of Germany there is time-varying risk premium but not

in the case of Japan and Canada. These results partly conform to those

we obtained in the first empirical study of (3.12).









Table 8
Tests of ARCH of equation (3.12) with
risk premium defined as (2.23)


2 P
Et 0 + E
t=1


E2-i
i t-i


80 81 62 83 TR2

0.0061 0.8280 88.805
(0.0027) (0.0501)
0.0054 0.7288 0.1190 59.706
(0.0027) (0.0877) (0.0889)
0.0049 0.6910 -0.0027 0.1441 92.648
(0.0024) (0.0770) (0.095) (0.0779)

0.0029 0.8649 97.595
(0.0016) (0.0444)
0.0021 0.5977 0.3123 100.256
(0.0016) (0.0837) (0.0840)
0.0018 0.5990 0.2647 0.0498 100.736
(0.0016) (0.0867) (0.0977) (0.0875)

0.0014 0.8140 60.892
(0.0008) (0.0487)
0.0009 0.6022 0.2719 92.248
(0.0007) (0.0841) (0.0828)
0.0010 0.6575 0.2478 -0.0561 92.288
(0.0007) (0.0862) (0.0969) (-0.0838)


Note: The figures in the parentheses are the standard errors. The
2
critical X score is 53.7 at 0.5% significance level.
indicates that the coefficient is significant at 5% level.


Germany







Japan







Canada









Table 9
GMM estimation of (3.12) with (2.23) as risk premium


S0 "1

Germany 0.7083 2.1196 0.6534 -11.8076 8.0253
(S.E.) (15.5247 (4.8193) (0.7628) (3.54133) (3.19257)


Canada 0.3728 1.2322 1.2743 -6.6495 0.6425
(S.E.) (0.9642) (0.3085) (0.3272) (1134.7) (165.869)


Japan 4.5285 0.2833 0.0 -0.4098 -0.1427
(S.E.) (1.2580) (0.7046) (0.3293) (0.4405) (3.3414)


Note: S.E. stands for the standard error.
indicates significance at 5% level.















CHAPTER IV
SUMMARY AND CONCLUSIONS


We have attempted to test the existence of time-varying risk premia

in the foreign exchange markets under two models we have developed in

this dissertation. This first one is an extension of the Lucas (1982)

general equilibrium model of international finance. By the assumption

of a Cobb-Douglas utility function for consumers we are able to derive a

closed form for the risk premia in the foreign exchange markets in the

setting of a two-country economy model. The time varying risk premium

is tested jointly with the efficiency of the foreign exchange markets,

i.e., whether the forward exchange rates are unbiased predictors of the

future spot exchange rates.

We note that the model we developed in Chapter 2 is an extension to

Lucas (1982) under the strict assumption of Cobb-Douglas utility

functions. This assumption immensely simplifies our derivation of the

source for the risk premia in our model. We also made the assumptions

of independent Markov processes for the incomes and the money supplies

of both countries. Finally, we used the variance of current money

supply shocks as estimates of the variance of the future money supply

shocks for both countries. Any of the three assumptions can not avoid

being arbitrary and there are other alternatives for each of them.

Further investigations along these lines are topics for future research.

We used White's test and Engle's test for homoscedasticity and used

White's heteroscedasticity-consistent variance-covariance matrix to









derive the correct standard errors. The joint hypothesis of market

efficiency and non-existence of risk premia in the foreign exchange

market are rejected for Germany, Japan and Canada. Market efficiency is

rejected by the data in our model. Although market efficiency is also

rejected in the models of Park (1984), Frankel (1982), Murfin and

Ormerod (1984), Baillie, Lippens and MacMahon (1983), their models

assumed zero or constant risk premium. Therefore their results are

based on a less general setting than the one we have. Domowitz and

Hakkio (1985), however, did develop an exchange rate model with time-

varying risk premium and also tested the market efficiency and the time-

varying risk premium. Unfortunately their conclusions were wrongly

drawn. Interpreted correctly, their findings are parallel to what we

have in our model.

In the monetary approach, we tested the existence of time-varying

risk premia alone. The semi-elasticities with respect to interest rates

in the money demand equation are assumed to be identical. However, the

parameters of the income index term are not assumed identical.

Therefore, our model is estimated on a more general setting than most of

the monetary models of risk premium in the foreign exchange markets. By

PPP and an extension to the uncovered interest parity we introduced the

risk premia into our monetary approach to foreign exchange rate

determination. The forward premium is used as a proxy for the source of

the risk premium. Rational expectation hypotheses are made and the

forward solution derived.

In the first stage of investigation we impose the first-degree

autoregressive process for all the exogenous variables and we also

impose independence of the disturbance terms. Although these are










conventional practices in the literature with the rational expectation

hypothesis, we must concede that there is arbitrariness in these

formulations. Other rational expectation hypotheses can be incorporated

into our model in future research.

Since it is a non-linear single equation model, and we found

evidence of the existence of heteroscedasticity in the error term we

used GMM estimators and corresponding variance-covariance matrices and

found that for the case of Germany there is risk premium.

Redefining the risk premium as we derived in the Lucas's model, we

repeat the empirical studies in the monetary approach and find that

there is risk premium in the case of Germany and Japan and no risk

premium in the case of Canada.

There have been few attempts on modeling time-varying risk premium

into the monetary approach of foreign exchange determination. One

reason is that the monetary approach did not fare well in empirical

studies. Therefore there are not many papers of monetary approach in

the recent five or six years. Another reason is that the source of the

time-varying risk premium is usually outside the system of the monetary

models. It is my belief that incorporating the time-varying risk

premium into the monetary approach may prove to revive the interests of

researchers in this area. The current paper is one attempt.

The conflicting empirical results from the two attempts are

attributed to the different settings of the models. However, there is

enough empirical support to the notion that German mark/U.S. Dollar

exchange market generates risk premia.

Extension to the current data will test whether the conclusion we

have drawn is valid.















APPENDIX A


The monthly data used in Chapter II and III are obtained from the

following resources.


Canada


Japan


U.S.


Germany


spot IFS IFS IFS
rate ae ae ae


73.10 78.12 73.10 78.12 73.10 78.12 ,
forward Harris Bank Weekly Harris Bank Weekly Harris Bank Weekly
rate 79.1 84.12 79.1 84.12 79.1 84.12
Financial Times Financial Times Financial Times


price IFS IFS IFS IFS
index 64 64 64 64


income IFS IFS IFS IFS
index 66.c 66.c 66.c 66.c


interest IFS IFS IFS IFS
rates 60c 60b 61 60c


money IFS IFS IFS IFS
supply 34.b 34.b 34.b 34.b


Note: The spot exchange rates and forward exchange rates are all relative
to U.S. Dollars. Interest rates and money supply figures are all
reported figures. Price indices and income indices are all with
1967 as the base year. Forward rates are collected for the last
Friday of each month from the Financial Times from January 1979 to
December 1984.
*
The forward data are obtained from the Harris Bank Weekly Review
for October 1973 to December 1978. Weekly data are available, and
the last week's data of every month is used for tests and
estimations involving 1-month forward rate.










International Financial Statistics
Exchange market rate
consumer prices
Industrial Production, Seasonally adjusted
call money rate
treasury bill rate
public authority bond yield
Money, Seasonally adjusted


IFS:
ae:
64:
66.c:
60.b:
60.c:

61:
34.b:















APPENDIX B


We first make the assumption that all the exogenous variables are

from afirst-order autoregressive process, which is the conventional

practice as well as the simpliest assumption in models of rational

expectations. The other types of assumptions of the stochastic process

are certainly just as arbitrary.


mt = 10 + 1mt-1 + t

** *
m = 0 + mt-l + t


S= k0 + klt-1 +t

* *
Yt = k + klYt1 + w t=2,...,T (Bl)

*
where vt, vt, wt, wt are white noise; T is the sample size. Equation

(Bl) and the rational expectation hypothesis lead to




t t+i 1-11 mt
1

*i
0(-1 ) *i *
Em = + mt
t t+i 1 t


i *i
k0(1-kl) i k0(l-kl ) *i
E y 0 + kly Ey -0 + k y (B2)
tt+i 1-k t t t+i k k Yt



The expectations of the e's projected on It are










1
Etet+I 1+ O P"O+ mt


*
1+6


*
- o 1+8t (k ( + klYt


** 2 +
(k0 + klYt) ~T (a0 + aldt) + -- Etet+2


1 O0(-u ) i
Eet+i + pm
t t+i 1+ 1-1 t
1


*1 *i
0-pl) *i *
1-p* 1 t
1


k0(1-k )
+ 4 ( 1-k +
1 K


*i
i k0(l-k1 )
klt) + ( +
l-k1


6
+ Etet+i+
1+6 t t+i+1


018 2
Sd2
1+6 t


Substituting (B3) into (3.10) recursively and collecting terms we have


1I
e=
t 1+6-B6i


62
a1 2
1+B t


1 4 ,
m TTm Y +
t t 1+B-k t *
1 +B-Bk


a16 2
d t-
1+8 t-1


+6 0 ~ 0 k k0
1+B 1-P 1-k
I 1-- 1
1-


*
4 kO
+ ) a 0
1-k
1


**

17- (1-1PP)(1+6-1) (- P*)(1+6_ 1
11 )


4kok1 4 k0k1
k1 + k)0k +vt
(1-k )(1+6-6k ) (1-k )(1+*-6k t
1 (l (lkl)(l+ -kl)


which we use in our empirical study of this chapter.


*i *
k Yt)]


(B3)


(B4)










To make the derivation clear, we repeat (B4) and have


1
vt = e +-B
t t 1+a-6p



y *
SYt
1+g-6k
1


1( P"
1+B l-uI


+ *
(l-k )(l+6-f k )


The likelihood function is


1 *
m + m + y
t 1+6-^ t + 1+8-BkI Yt


2
1 B 2 a2 2
1+r t 1+T t-l


*0 k0 O k0
S -k + +
l-v 1 l-k
1 1


0 _)1 k 0k1
*l(1-k 1)(i++-k


1 l 2 1
2 ogat 2-


2
v
t
2 + const.
t


we impose here

2 2
t = 2 + avt-
and the first order conditions are
and the first order conditions are


af 1
an 2
a
t


St-1 t vt
(av) +--
(3t-I a- 2 +
a
t


2
vt
2 )2
(a )
t


tt-1
vt-1 an 3


2
v tv v av v
t t t-1 t-1 it
+ 7 (- 1)
2 3n 3 2 an 2
t t t


f(v ) = -










af 1 1
a2 2 2
t


af
3f
3o 3


2
1 vt
2 (2 )2
t


3f 2
- 2* v
aa t-1
2


*
where n = {6,<,( ,C0,a }.


We need


1-plI
= 2 m
(1+6-6Ul)


* (1-k1)
+* 2
(1+6-Ok)I )


*
t


*
1-pl
--
*2
(1+8--8pl)


alB(B+2) 2

(1+B)2 t


* 4(1-kI)
m -- y
t 2 Yt
(1+8-8k)


al 2
+ d
(+)2 t-1
(1+B)


1 0 0 0ko k 0
+- +
SO2 1 1- 1-k -
(1+6) I 1 0 1 1-k1


1 kO l 1Ol
+ ( -


kokI 0 kok1
(1-k )(1+B-Bk ) + )
1 1 (1-kl)(1+B-8k)


**
+ -0 + (- +
1+ (l+2 -B )2
(1+B-BUl) (1+B-8pl)


S kk1
(1+6-+Bk )


1 B ko kkl1
+-Bk t 1+ 1 -k1 1+B (l-k )(1+B-Bk)
1+-Rkl t + T+ + k1


1
1+B-Bk1 Yt


k k
S0k 1
+ (1 -
1+B 1-k 1+6-Bk
1 1


av
t
w8


av
t


* *
4 k0k1
1- )
2
(1+S-Bk1)









**
1 k __ k0k1
t 1+e 1+6 -
1+B-kI l-k1 (1l-kl) (l+B-ak)


1
1+B-Bk1


*
ko k
S k(1 -
l -k1 1+B-Bk1


2
I a 2 d2 + d2
1+a t 1+6 t-1


av

a300


avt
ml


Dv














Appendix C


8f
For (3.28) we derive 2


fi1 -(1-u)
1 1 1
a8 [ i+f3(l-i)]2




[1+BC1-k) 2

af3 (1-k )



^ 2
af4 (l-kI)


af5 _l_(2+_ )


S (1+)2


af6
as (1+S)2
Jr6


as following


af7 1 10 "
aB (1+)2 1- 1-[l
Da [-P -

*


(1i-1)[ l+B(l-1) ]2


+ ^--- 2
1+P [I1+e(1-1l) ]


^
Ik 1
l-kl


^* *
+ ko 1__ O1
--- -0 [2
1-k (1+6) (1- 1[1+8(1-u)]


k0k1 +
0-k

(1-kl)[l+B(1-kl)]


^* *
Skk1k
^ ^
(1-k1 (1+8-Sk1)


* ^ ^ *
110k O0k 1 k k k

[1+(1-p)] [1+(1-k)]2 1+6(1-k) ]2
^ ]2 [ l+ (llkl









af af2
1 2
af -


af3




af7




afl



af 4

3({


af7

03f




0


af4 af af6
4 = = T- = 6
p -( -(( -0


-1

1+B(1-k1)


A
ak
= ^ (1
(l+8)(l-k1)


af2 af3
2 3
---w-*



1

1+8(1-kI)


af
~---*
*<(


^ *
Sk
(1 -
(1+8)(l-kI)


af2
2
T0
0


8f 3 f
3 4
0 0


k
^ -)
1+B(1-k1)


af
--- 0
*>


k




af5 af6
- 5 6T
0 0


af7
7
5 -a_ B
0


af2
1


af af 9f
3 4 6
1 1 1


8f
7
= -- = 0
1


af
5 6
1 1+8


afl

1















APPENDIX D


From equation (2.19) we know


b (s ,6 )
F(st'tM Nt) = e(stM,N) b t t
x tt


The forward rate formulation is



EtQt+

t t where

where


U (s') '
EtQti + U (s)
x


M
t
Nt+l


6U (s )t+
x t+1 t+l
Et U(s)TM
x t t


t+l1
t E t
t


t
(-)
t+1


M
t
M
t+1


t+l
Et


a 1-a
t+1 t+l t
Et E n M
t t t+1


Et Q =


U (s')n' N
y ) f(u lt tu t) dutdu dut
U (s) f Itu2t' 4t it 2t 4t
y t+1


(DI)


f(ultu2t'u3t) dultdU2tdu3t


(D2)









0 (s ) I
=E y t+1 t+1
- USE
U (s )v
y tt


(t+1)
= E ( t
5t


nt)
nt+l


t+l1
= E5 (-)
Et


a a-1
n, N
t t
>(-)
t+ N
t+l t+1


From (2.14)


Ult = Int+l tln t


Ult N(0,hlt)


In t+1 = Ult + P11nt N(Pllnt,hllt)

Hence the probability density function of lnSt+ given hlt is
t~l gvenlilt


(D3)


f(inEt+',hlit)


= 1
1 t


exp[- 2h (ln
lit


- P1lnt)2]


Similarly


f(innt+lh22t




f(1nMt+1,h33t)




f(InNt+ ,h44t )


1
= -- exp[-
22t


exp[-
33t


S 1

44t


2h (1nnt+1
22t



(InMll
2h t t+
2h33t (nMt+l


exp[- (inNt+1
44t


- P21nnt )2




- Yl1nMt)2]




- Y21nNt)2]


N t+l t+l
Et t+1 t Et -
t t


N
t
t+I


nt+
ntt


N
t
t+i






60


S a cn i-a N -1
t+1 t+1 t+1
=E ( ) (-) (-)
Et ( N
t t t t


t a a e 1-a N
(t+1 t+- 2t+1u
= / ( ) ) ( N ) f(ult,u2tu) dultdu2tdu4t
t t t

Since ult, u2t and u4t are assumed to be uncorrelated, we have


E a t+ 1-a
N t+l (t+1 -
EQ = 8 (t) f(u t)dult f )
t t+1 t It It nt


N -1
t+1
f(u2t) dut I ( N)
t


f(u ) dut
4t 4t


( t+ l) 1 exp{- n- PI-nE- '2} d(lnt )
B -t r27 i 2hllt t+i


( t + )I 1 0 e x pt +- 2 t d ( in n + 1)
S-t /2-h22t 2h22tl t+
2






Nt+ 1 (InNt+l Y21nNt)2
= (-) -- exp{- t 1 d(lnNI )
t 2h t+1
t 2th 1tt+
44t



44t


Since


E[E(' =
t+1


exp(ap lnEt + a2 hllt)


E(naI ) = exp{(a-1)pl2nnt + 1(a-l)2h22

-1
E(Nt+I) = exp{-l21nNt + h44t

(M-1) = exp{-Yn + h }
E(Mt+I) = expf-YllnMt + -tht
t+I I 33t}


In{EtQ 4
t t+1


2
=In8 alnt + ap lnt + a 2hlt


+ alnnt + (a-l)p21nnt + i(a-) 2h22t


+ iNt Y2nNt + h44t






61


1n{E Q m} = -n + aP lnEt + laa 2 h


+ alnfl + (ct-l)p2lnflt + (Ca-1)2 h22t


+ lriiM Y21' t + jh44t


Therefore


InE Q
t t+1


- lnEtQtil = ('2)'nNt + 44t


(l-yl)lntt 1h33t















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BIOGRAPHICAL SKETCH


Wen-he Lu was born in Shanghai, China, January 2nd 1953. When the

Cultural Revolution was launched, he was a first-year student in

Nan-yang Middle School in Shanghai.

In May 1969 he volunteered to settle down as an educated youth in

Xi-go-men Village, Diu-jiu-go Commune, Feng Zhen County, Inner Mongolia,

where he spent four and a half hard years both as a farmer and as a

young revolutionary.

After passing a primitive entrance-exam, he was assigned to study

English for three years in Inner Mongolia Teacher's College in Huhehot,

Inner Mongolia, China. Then he was assigned as a teacher of English and

Physics to No. 2 Senior and Middle School in Feng Zhen County, Inner

Mongolia.

In July, 1978 he took and passed the nation-wide entrance-exam for

the Graduate School of the Academy of Sciences in Peking and studied

English and Education there until August 1980.

He came to the United States as a special student in the Department

of Economics, University of Pittsburgh and was awarded his M.A. in

Economics in August, 1982. Since then he has been a Ph.D. student in

the Department of Economics, University of Florida.










I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.


G.S. Maddala, Chairman
Graduate Research Professor of
Economics


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.


Edward Zabel
Professor of Economics


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.


Stephen R. Cosslett
Associate Professor of Economics


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dis ion for degree of
Doctor of Philosophy.


Robert D. Emerson
Associate Professor of Food and
Resource Economics


This dissertation was submitted to the Graduate Faculty of the Department
of Economics in the College of Business Administration and to the Graduate
School and was accepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.


August, 1986
Dean, Graduate School




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