
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/UF00102773/00001
Material Information
 Title:
 Risk premia in foreign exchange markets
 Creator:
 Lu, Wenhe, 1953 ( Dissertant )
Maddala, G. S. ( Thesis advisor )
Zabel, Edward ( Reviewer )
Cosslett, Stephen R. ( Reviewer )
Emerson, Robert D. ( Reviewer )
 Place of Publication:
 Gainesville, Fla.
 Publisher:
 University of Florida
 Publication Date:
 1986
 Copyright Date:
 1986
 Language:
 English
 Physical Description:
 vi, 65 leaves : ill. ; 28 cm
Subjects
 Subjects / Keywords:
 Currency ( jstor )
Economic models ( jstor ) Efficient markets ( jstor ) Exchange rates ( jstor ) Foreign exchange markets ( jstor ) Interest rates ( jstor ) Money supply ( jstor ) Observational studies ( jstor ) Prices ( jstor ) Risk premiums ( jstor ) Dissertations, Academic  Economics  UF Foreign exchange Foreign exchange futures
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Abstract:
 We have attempted to test the existence of timevarying 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 twocountry economy model. We used White's test and Engle's test
for homoscedasticity and used White's heteroscedasticityconsistent
variancecovariance 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 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 nonlinear single equation model and there is
evidence of heteroscedasticity we used GMM estimators and the
corresponding variancecovariance 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
timevarying 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
semielasticity 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.
 Thesis:
 Thesis (Ph.D.)University of Florida, 1986.
 Bibliography:
 Includes bibliographical references (leaves 6264).
 General Note:
 Vita.
 General Note:
 Typescript.
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 University of Florida
 Holding Location:
 University of Florida
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Full Text 
RISK PREMIA IN FOREIGN EXCHANGE MARKETS
BY
WENHE 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 Youchai, Mr. Li Youfu, Mr. Li Youqien and all of my fellow kinsmen in Xigomen Village, Duijiugo District, Fengzhen 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. Gosslett, 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 selfeducation 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.
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . .
TABLE OF CONTENTS
PAGE
iii vi
1 7 8
20 21 23
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 . . . . . . . . . . . . . . . . . . . . . .
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
Wenhe Lu
August, 1986
Chairman: Dr. G.S. Maddala
Major Department: Economics
We have attempted to test the existence of timevarying 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 CobbDouglas 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 twocountry economy model. We used White's test and Engle's test for homoscedasticity and used White's heteroscedasticityconsistent variancecovariance 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 timevarying 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 nonlinear single equation model and there is evidence of heteroscedasticity we used GMM estimators and the corresponding variancecovariance 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 timevarying 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 semielasticity 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 overall 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 + blnF t1 + u t 0.1)
where e t denotes the spot exchange rate, F t 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 H 0 : OL 0 = 0, a, = 1. We observe that all the OLS parameter estimates are far off the targets. We do a conventional Ftest, 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
a 0 a I R 2 URSS RSS F 2,131
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 riskpremium term, preferably a timevarying 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 timevarying 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
Table 1
Preliminary test of the hypothesis of market efficiency
lne t+I C O + a 1 lnF t + u t
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 portfoliobalance 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 riskpremia for the major six currencies. He assumes that the risk premia depend specifically upon asset supplies and variancecovariance 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 timevarying 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 timevarying 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 nonlinear model, we use the nonlinear 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 twocountry 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) heteroscedasticityconsistent covariance adjustment to estimate the original Lucas' model and also to test the null hypothesis of risk premia being timevarying 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) capita terms. Each citizen of country 0 is endowed each period with E units of a freely transportable, nonstorable 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{' tl =< ' t E' nt = F} F{ ', ',E,
Assume that the process {t~nt } has a unique stationary distribution ( 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{ Z at U(x itYit )}, 0 < a < 1 , i = 0,1 (2.1)
t=O
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 a 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 CobbDouglas utility function
U(x,y) = Axaya (2.2)
We assume that agents trade in both goods, spot and in advance contingent on all possible realizations of the shock process {E t,t}.
The preferences of agents have been assumed to be independent of their nationalities, so that agents differ only in their endowments. Agents are assumed riskaverse 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 JEn 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) FJ'n'En 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 Xunits. Let p y(s), q x(s), and q y(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 {E t of the endowment of good X, and the current price of a claim to the future stream In } , respectively, all in Xunits. Therefore p y(s), q x(s) and q y(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 Xunits. He wants to maximize his utility by choosing the current consumption (x,y) at spot prices (1,p y(s)), equity shares 8 , e yin future endowments J
and {T at the prices per share q x(s) and q y(s) respectively. The budget constraint is
x + p y (s)y + q x(s)O x+ q y(s)Oy e (2.3)
The corresponding endowed wealth e' valued in Xunits at the beginning of the next period will be
6' = 6 [E' + q x(s')] + 8 [ p (s')q' + q y(s')] (2.4)
since XY are assumed to be nonstorable 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 ArrowDebreu 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 = {E,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 noninterestbearing currency in exactly the amount they need to cover their perfectly predictable currentperiod 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 capita for both countries. Prior to any trading in period t, let each trader's money holdings be increased by a lumpsum wtMt_ and vt N t so that money supply evolves according to
Mt+I = (I + wt+i)Mt,
Nt+1 ( + vt+l)Nt (2.5)
Let {w t} follow a Markov process, possibly related to the real process {st I with transition function
H{w,w',v,v',s,s'} =
P r ~ w t + < w ' = v ' w t = v ' s t + s
+ w',vt+ = w v = s = s} (2.6)
We then have the following financial constraint
M(t) = px(S,M)
N(t) = p y(s,N)n (2.7)
with the equality sign when equilibria are reached. Here px(s,M) and py (s,N) are nominal prices.
The objective value function for this agent, therefore, is
V(O,s) = max {U(x,y) + f V(O',s') f(s',s)ds'} (2.8)
x,y,Ox,Oy
for the consumer in state s with wealth 0. He is assumed to maximize (2.8) with 0' denoted in (2.4) subject to the constraints (2.3) and (2.7). We now have Lagrangian function
12
L(x,y,O , 'xlX2,x3) = {U(x,y) + B f V[{6x [' + qx(s')]
+ e [p (s')n' + qy(s')]},s] f(s',s)ds'
+ xl[1  x  py(s)y  qx(s) Ox  q(S)ey I
+ x2[Mt  Px(s,M)E]
+ x3[Nt  py(s,N)n]} (2.9)
The first order conditions for the Lagrangian problem are
L = U (x,y)  X1 = 0 (2.10.1)
Ly = Uy(x,y)  1py(S) = 0 (2.10.2)
LO = B f V ,(e',s')[E' + qx(s')] f(s',s)ds'
x
 X1q(s) = 0 (2.10.3)
Le = f V,(6',s')[py(s')n'
+ qy(s')] f(s',s)ds'  Xlqy(s) = 0 (2.10.4)
We also know L, = 1 as 0 is given and
L = a x p(s)y  q We  q (s) (2.10.5)
L x2 = Mt  P(S,M)E (2.10.6)
L x = Nt  p y(s,N)n (2.10.7)
Reorganizing the above conditions, we have
U y(x,y) = py (S)
Using L, =X = Ux(x,y)
U (x,y) U (x,y) U ( ,in)
Uy(S(=  = (2.11.1)
yx U x(x,y) U ( iE, O
X X
a f V (6',sq)[' + qx(s')] f(s',s)ds' = xlqx(s)
q(S) = B[U (x,y)I f Ux (s')[E' + qx (s')] f(s',s)ds' (2.11.2)
B f V0(e',s')[py(s')n' + qy(s')] f(s',s)ds' = Xqy(s)
q (S) = B[Ux(S)]I 1 Ux(s')[py (s')'
+ qy (s')] f(s',s)ds' (2.11.3)
and
M
Px(s,M) = E (2.11.4)
Nt
py(sN) = (2.11.5)
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 timevarying riskpremia.
Equation (2.11.1) means at such an equilibrium, the relative price of Y in terms of X, py (s t) 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
py(S U (,1 () )xy _ 1a x _ a  (2.12)
y a2
yUt AxT, n) A~xlla a y x 2 y = i
Here the perfect pooling assumption seems to be restrictive. As we see later, this will not affect our derivation of the timevaryingriskpremia.
Assuming purchasing power parity we have
P(StMt) Mtnt 1 Mt
e(st'MtNt) x t t t t 1a t (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 setup 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 lnnt = 21nt_1 + u2t 1imt = YlnMt_ + ut
t 2 t1 2t
lnNt = Y21nNt1 + u4t
u t'= (ult,u2t,u3t,u4t)
u t Ilt_1 "I N(0,H t)
where I indicates the information set at the period t1.
t I
t dig 1l,th22,t h33,t'h44,t)(.4
Here we assume zero covariance between the stochastic processes. It is possible to generalize the model to allow for nonzero 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 homecurrency price in period t of a claim to one unit of homecurrency with certainty in period t+1. Such a claim is equivalent to
I 1 ~l (2.15)
+1 pXt+1Mt+1 M l
units of X in period t+1. Similarly
IN 71t+l (.6
~t+1 N t+i2.6
Since 11 and ? are both functions of the future real output and
nominal money supply, there is uncertainty as to their exact values. The 4 units of X will be valued by agents in period t+1 at the
t+1
marginal utility of X, U X(s t1), which must be discounted back to period t by multiplication by the discount factor. The xunit price of the claim to one unit of home currency is therefore E WS (s ) IT U (S )t X t+1 t+l x t
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 xunit price times the marginal utility of X in period t.
The home currency price of the investment is then obtained by multiplication of the xunit price by p (stM) or division by TIm. Therefore, the period t discount bill paying one unit of home currency in period t+1 in terms of the home currency price is
b (sEt Xt+1t+1 EQM (.7
bx t )=t U (S )HlM t Et+i 2.7
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
b (s ) E[ YU (t+i t+ E QN (2.18)
Y t, t U (s )flN = +
where U (s )is the marginal utility of y in period t and HNis the
y t 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 intertemporal 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 x(s t,O6 ) per unit of home currency invested, and the alternative 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
Ie I,,N][i/by(St,6t)]f(stStPMt,$Nt
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,
b (s I a e(s ,M,N) [i/by(Stet )]F(st'et'Mt'Nt)
x t, t C
we have
by(St,0t) F(stet,Mt,Nt) = e(st,M,N) b(S (2.19)
t ~x stlt)
The forward rate formulation is
EN
Ft = e t t+1
t t+1
From (2.14) we have
_U (s)v' M
EQ t l f d,t+l'U2dt+lU3,t+) dudt+Idu2t+Idu3t+I
x Mt+l
OU(S ) M =E x t+1 t+1 t Ux(St)RM
x t t
cii
t+1 ) 1t
t Tt+,
t+ nIt+11E () () t  t t
Mt Et+1 t+1 Et
Mt
Mt+1
(2.20)
N u (s')fl' N
EQ f (u l du du du
y W  N ,t+l,u2,t+l 3,t+l it 2t 4t
y t+
OU (s 1N
=E By(St+I) t+1
t U(S t)TI
E It+, St N n t+i
t 1t+i t+i t
a iN
(t+l) T1t N t
Ea(ti) (lt () (2.21)
t T1t+1 Nt+1
It is shown in Appendix D that
inEtQ +1 lnEtQt+1 = (1Y)lnNt + 1h
 ( )lnMt  h3
1 t 2h33,t Substituting the above into (2.19) and taking the logarithm, we have
lnF = ine + [(Iy2)lnN  (iY )lnM  h3t +h
t t 2 t 1 t 1h33, h4,
= ln[(la)/a] + InMt  lnNt + (1Y2)lnNt  (Y1 )lnMt  2h33,t
+ 1h 44,t
= ln[(Wa)/a] + Y llnMt  Y2lnNt  h33,t + jh44,t Also from (2.13) we have
Etlnet = ln[(ia)/al] + inMt+1  inNt+I
= ln[(ia)/a] + YilnMt  Y2lnNt + Etu3t  Etu4t
The last two terms are zero, hence we have
E tlne t  lnF t = J(h 33,t+  h 44,t+l) (2.23)
Equation (2.23) indicates that the risk premium depends upon the conditional variances of the forecasting errors of the domestic and foreign money supplies. An increase in the conditional variance of domestic money, h 33,t 11 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 h 33tl h 44tlwhich 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
curentmony uppy sock 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
E lne lnFt = (h h + [i(hj+i 3t
t t t 33,t 4, 3tl h3,
 J(h 4t h )4, (2.24)
We approximate E t me tand lnF tby
e t  t  at t  e t
e t1 an F
and estimate the equation
e~ te tl F t _I  e l
e ti act0 +a I e ti + a 2(h33t  h 44,t) + Ct (2.25)
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 aW~ 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 ao < 0, a 2 > 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 timeseries behavior of the risk premium. According to the model, a1 should be 1 if the market efficiency hypothesis is to hold, and C t+1 should be white noise, independent of whether or not a risk premium exists. A test of the hypothesis that a 2 =0 is a test of whether the risk premium is timevarying or not, in determining the deviation of the forward rate from the expected future spot rate.
Maintaining a, = 1 and Ct white noise, a0 = 0, a 2 = 0 means a zero
risk premium. a 0 ;e 0, a 2 = 0 implies a constant risk premium. 0o 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 et= v t + a2 [2(h 33,t+l  h 33t + Jh4,
h 4,t)]and is clearly heteroscedastic. The consequence of this approxi
mation 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 'crosssectional' in nature. Before we proceed to estimate the consistent variancecovariance 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[D2 L/E' or in the outer product form, E[aL/$  L/ ' 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 is equivalent to TR2 x2 k(k+1)/2 where T is the sample size; R2 is the constantadjusted squared multiple correlation coefficient from the regression
= I 22
ut 0 + I t + a2 (h33,t  44,t
+ 83 [(h33,t  h44,t) tet]
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_1 e t1
Table 2
Correlation and Heteroscedasticity ARCH as a special case
Ft 1 ]et[ I 2[ (h33  h t4)]+s
OLS Estimation
critical
score
Germany Japan Canada at 5%
significance
level
a0 0.0313 0.0022 0.0038
(0.0170) (0.0033) (0.0035)
a1 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
ARCHi 8.619 2.769 2.392 5.02
ARCH2 8.656 3.548 2.3607 7.38
ARCH3 8.9984 3.571 2.5344 9.35
ARCH4 9.3345 5.779 2.794 11.1
White's test 9.694 4.1003 0.5371 9.35
Note: The score for ARCHP is equivalent to TR2 with T as sample size and
RP from regression 2 = + p 2 The numbers in parentheses
i=1l t
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
Y= f(Xt)6) +u
where Xtis a vector of explanatory variables and the parameter vector with the distribution of the disturbance term assumed conditional normal as following
u II " N(Q,a 2
t t1 t
where
~2 p 2u(.6
at = iuti 2.6
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.
Engle (1982) shows that X2 score for testing the null hypotheses H: 0 0 YO=Y . = Y = 0 is equivalent to TR where T is the sample size, R 2is the constantadjusted squared multiple correlation coefficient from the regression
ut Y0+ E 'u'.u. (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 3b. The estimates show that yo and y1 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 heteroscedasticity in the case of Germany case and it is of ARCH type.
Table 3a
ARCH Estimates and White's heteroscedasticityconsistent variance estimates for Mark/$
A2 P ^2
t = Y0 + i i Uti
White's heteroscedasticityconsistent estimates and standard errors a0 a1 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 heteroscedasticityconsistent
variance estimates for Mark/$ ut = YO+ Y Yu
ARCH parameter estimates
YO Y1 Y2"Y3
Germany 0.00074
(0.000 18) 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)
0.000 16
(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) 0.0392 1
(0.0474)
Note: The figures in the parenthesis are standard errors.
* indicates significance at 5% level.
Following White (1980), we calculate the heteroscedasticityconsistent covariance matrix. The procedure is to save residuals t
from regression (2.25) and calculate the following
Japan Canada
0. 0375 (0. 0901)
0. 14662 (0. 08824)
0.042 15 (0. 08992)
= (X'X/n) 1(X'(diag(u2 ))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 h 44,t) for Canada is significant.
Maintaining aI = 1, we first test the joint hypothesis m0 = a2 =0.
The results in Table 4 (in the column RPI) indicate that this null hypothesis can not be rejected for any currency. We next test the expanded hypothesis H0: a0 = a2 = 0, a, = 1. The results of this test are reported in Table 4 (in the column RP2). The results indicate that the null hypothesis 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 timevarying 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
RPI 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 Fstatistics
at 1% significance level.
RPI is an Ftest of the hypothesis a0 = a2 = 0;
RP2 is an Ftest of the hypothesis a0 = a2 = 0, i = 1.
RP3 is an Ftest of the hypothesis a, = 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) Murf in 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 timevarying 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 timevarying risk premium term, this model is a straightforward 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 semielasticity 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
m t  p t = y t  r t + u t (3.1)
m~ Pt = y r + u t,.,T (3.2)
T is the sample size. Here m t9 pt, y stand for the logarithm of money supply, price level and real income level at time t for the home country; r t is the nominal interest rate and utis a white noise disturbance term, 4, and 0 are unknown coefficients of the real income and interest rate and are both positive. Similarly, m. t9pt3 yt, r t, u t
and 4,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
tm t t (pt t yt  rtr t ) utu t(3)
The cornerstone of the monetary approach, PPP (purchasing power parity) is assumed, i.e.,
t= pt+ e t(3.4) where e t 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 longrun 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 riskneutrality assumption in Dornbusch (1980) and Woo (1985), we modify the uncovered interest rate parity by introducing the risk premium term
r t  rt E(et+  etR 1t E(e t+1 e~ t 35
where 1 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(.jII ) 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 riskneutral.
The risk premium we introduced shows our assumption of investors in general being risk averse. 1Admittedly, there is difficulty in differentiating the riskneutrality from perfect capital mobility, since both arrive at the same form of expression of (3.5) without R t' Here we choose the former interpretation.
1 This can also be interpreted as deriving from the perfect state of capital mobility. Papell (1985) defines this equation as
rt  t =Et (t+1 t aet qt ))+vt
where e t and qtare deviations from the steady state of the exchange rates and relative price levels. His formulation implies that imperfect capital mobility is caused by the shortrun deviation from the longrun PPP.
We can interpret the risk premium along the same lines, i.e., risk premium reflects the shortrun 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
e~ I ~(m m) y ~ Rt + i. E e~ v (3.6)
1+8 T t t 7 T7 t TT t 1 + +v
where N~ (u  ~ .V is white noise too.
Now we proceed to formulate the risk premium and the distribution of the error term. We assume that R thas the general form
22 2
Rt=f(d d,. ._ 2 ,'d ) (3.7)
where d t is the forward premium, the difference between the realized future spot rate and the forward rate divided by the future spot rate f te
te . Hodrick and Srivastava (1983) developed an extension to Lucas's (1982) model and derived that the riskpremium 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 riskpremium 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 t= a + a 1d t12 +wt (3.8)
where cx0and a1are constant unknowns and d tis the forward premium at time t, i.e., the difference of logarithms of the forward exchange rate
at time t1 and the spot rate at time t; w t 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
E R + a0 d 2
E R a + ad 2 i12.(3.9)
t t+i 0 1iti
Substituting (3.8) into (3.6), we finally have the model
t 1+8 t t 1+0 t + t 1+8 t t+1 1+R
(a +ad 2) +v (3.10)
0 i ti t
There are usually two ways of handling the rational expectation of future endogenous variables. The first is Wickens's (1982) errorinvariable 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 8 must take a positive value; therefore, the backwardsolution 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 Ete t+1 involves the future values of all the exogenous
variables, we assume at the outset that mt, mt, Yt, Yt are all from a first order autoregressive process.
inmt = PO + Pllnmti + t
* * * * *
inmt = PO + Pllnm tI + t inyt = ko + k1lnYti + t
lnyt = ko + k1lnYt1 + Tt (3.11)
where Et' Et' qt' qt are all white noises.
The stationary conditions require that
PO + Pl 1 PO + Pl1 1 k0 + k1 < 1 k0 + k1 < 1
The forward solution method leads us to
_* * *
0 + 6 [ 7 , lk lk
 601 110111
k 0(1 k)1k + + k1
[l+ (lk)1 lkl) [1+(lk )](1k1)
2 a1 2
d  d + v (3.12)
1+ t1 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 disturbance 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 forecasting 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
Y t= f(x t ) + v t(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
v 1I 1 N(O, cr) (3.14)
where It1is the information set at time t1 and var v t is generally defined by
a2 =fv2 v2 v2 (3.15)
at t1 t2 v. ) )
Specifically, we formulate a linear function as follows
at2 = Zy (3.16)
where
= f ,Vt ,Vt_,.,Vt2_p (3.17)
a {y0,y1,. ,p} (3.18)
To restrict the variance to be nonnegative, we impose
Yi > 0 i=0,1,2,.,p (3.19)
and to prevent the variance from being explosive we impose
p
Yi < 1 (3.20)
We note that (3.18) is sufficient but not necessary.
The likelihood function is
2
1 2 1 vt
lnLt =const.  f intt  2 (3.21)
at
2
where vt = Y  f(xt',) and ot is defined as (3.16). This is maximized
t t
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 nonlinear econometric estimators reside within this
category, GMM is a good alternative to the conventional maximum likelihood method. One advantage of the GMM estimator over MLE is that the variancecovariance matrix obtained by GMM is robust with respect to the specification of the disturbance distribution; hence, we can expect to produce a heteroscedasticityrobust variancecovariance matrix of the estimated parameters when the researchers are not confident about the specification of the disturbance distribution and/or when the specification of one distribution is too complex.
Consider a model similar to (3.12)
y = Xf(S) + 6 (3.22)
where y is a Txl dependent variable and X a matrix of explanatory variables of dimension TxM; 6 is the unknown hxl parameter vector; f() is MxN a onetoone 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 t jt_p,6t_p_l,.XtXtel,.) = 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{stletil,5t_2,.,Xt,Xtil,.} = 0 (3.24)
The GMM estimator is the vector 6 that minimizes the quadratic form
i(D) = (y  Xf(6))'XQ X'(y  Xf(M)) (3.25)
where 0 is a consistent estimator of the positivedefinite matrix
1
= lim (1 E(X'ce'X)). (3.26)
T o T
Hansen (1982) proves that
1 I T (3.27)
q=p+l t=1ttt
is a consistent estimator of n.
The variancecovariance matrix of the GMM estimator of 6 is
rv'x '1 x'_v 1
plim [(   X  1 (3.28)
where v = X X [ 'f (3.29)
a6 * 36i 1 * 62 6 6h *
where 6 is the GMN estimate.
In our model (3.12)
ft {f 1f2 f3 f4 f5 f6 f7} where
1
1
1
f2
f
2
f5
f6 = 1+
^ 1 80 110 Ok 0 k
_7 =kI+ + 1 1 + , * (3.30)
* * d2 d2 1.)
andXt = (mt mtt Yti t
6= (B 4) 4) ci0 c )
^* 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' PiV 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 firstorder autoregression process
on the exogenous variables mt, m, Yt' Yt.
U.S. Germany Japan Canada
V 0.00115077 0.09747710 0.30776482 0.00467322
1.0078970 0.98287155 0.972608783 1.00037203
k0 0.05466299 0.37540327 0.01169507 0.41615339
k 1 0.98914601 0.92667755 0.99814542 0.91531332
1
Note: The above estimates are from the regressions mt P0 + Pimti + vIt
and Yt k 0 + klyti + v2t
where mt,Yt 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 variancecovariance 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 t= + p
i=1
Germany Japan Canada
1i2_l
Note: The numbers in parentheses are all t scores
* indicates significance at the 5% level.
for H0:
0 1 2 63
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)
6. = 0.
We first test the existence of serial correlation by the standard regression
2 = p 2
6 t O + ti (3.31)
The results are reported in Table 6. Engle's X 2 test of ARCH is conveniently equivalent to TR 2 of (3.31), with T as sample size and R 2 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 semielasticity of money with
respect to interest rates show that they are not significant from zero for all the cases. The estimates of the semielasticity 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 a is significantly different from zero but 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 timevarying 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 timevarying risk premium, GMN estimates
indicate that there are constant risk premia in the case of Germany and Japan.
Table 7
GNN estimation of (3.12)
a 0a
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 timevarying 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 timevarying risk premium is usually outside the system of the monetary models. It is my belief that incorporating the timevarying 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 timevarying 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 nonexistent. Again, there is empirical evidence that in the case of Germany there is timevarying 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).
0 2 3 TR 2
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)
Table 8
Tests of ARCH of equation (3.12) with
risk premium defined as (2.23)
p
O + E
t=1
E: 2
Germany Japan Canada
Note: The figures in the parentheses are the standard errors. The
critical X 2 score is 53.7 at 0.5% significance level.
* indicates that the coefficient is significant at 5% level.
Table 9
GMM estimation of (3.12) with (2.23) as risk premium
01
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
SUMMTARY AND CONCLUSIONS
We have attempted to test the existence of timevarying 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 CobbDouglas 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 twocountry 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 CobbDouglas 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 heteroscedasticityconsistent variancecovariance matrix to
derive the correct standard errors. The joint hypothesis of market efficiency and nonexistence 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), Murf in 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 timevarying risk premium and also tested the market efficiency and the timevarying 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 timevarying risk premia alone. The semielasticities 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 firstdegree 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 nonlinear single equation model, and we found
evidence of the existence of heteroscedasticity in the error term we used GMM estimators and corresponding variancecovariance 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 timevarying 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 timevarying risk premium is usually outside the system of the monetary models. It is my belief that incorporating the timevarying 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.
The monthly data used in Chapter II and III are obtained from the following resources.
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 1month forward rate.
APPENDIX A
U. S.
Germanv
Japan
Canada
IFS: ae: 64: 66. c: 60. b:
60.c: 61:
34.b:
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
APPENDIX B
We first make the assumption that all the exogenous variables are from afirstorder 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.
m= 110 + 1 Im t + vt
* * ** *
mt =0 + Imti + vt
= k0 + klYt + w
Yt= k0 + klyt1 + w t=2,.,T (BI)
where vt, vt, wt, wt are white noise; T is the sample size. Equation
(BI) and the rational expectation hypothesis lead to
10(1J1)
tmt+i 1 1 1mt
1
* 110(I1I ) *
Etmt = * + i m
Et t+i +1
4 * *i
k0(1kl) * ko(1kl *i
E t, 0 + kly 2 E y *0 + k* I(B2
t 1lk lt , t+i 1k 1 ik t
The expectations of the e's projected on It are
1
E tet+l = + PO0 + U Im t
+
 10  Pilmt]  i (ko + klYt)
(k* 0 + 2
(k0 + klY) 1+6 (a0 + ad) +   Etet2
tet+i 1+6 111 t
P 0 I  l ) * i * lu*___  l Mt
1
k0(1k )
+4, (, k1 +
* *i
k * k0(1k1 )
kly)+4, ( * +
IL 1k1
6
+ t Ett+i+1
1 + d2 ITT t
Substituting (B3) into (3.10) recursively and collecting terms we have
e t _1 2
1+P t
1 * 4, 4
t *mt Yt+ *
1++k11t +66k1
a1 2
+ Tdt1
+ 1I0 vi0 4,k0
(lP 1 k
4 k0
+  )  a a
Ik1
 11011l 10111
 s + (1TZ1)(1+6 1i) (,P*)(1+ _6 1 )
0ko1 1 k0k1
(1k )(l+6k 1) (1kl)(1+ k) t
which we use in our empirical study of this chapter.
kI Yt)]
(B3)
(B4)
To make the derivation clear, we repeat (B4) and have
1 1 *
vt e  m + m +
* + 2 d2 +ca1 a2
ï¿½ t + t 1+ dti
P 10 10 1k 0 k
1+ 1111 1 1k10
+ ((i110
+ k 0
(TkI )unc+ I ) The likelihood function is
f(v t) = 
2L logo2 _2 1 2 1
SO_1 kk1 (1k*1(l+8B 1)
2 vt
2 + const.
t
we impose here
2 2
ay a 2 +av
t 2 3 t1
and the first order conditions are
af 1
a
t
(vt1 vt Ivt
( 3vtI q 2 3Ti
t
2 v t
2)2
t
v1
vtI TI 3
2
v v v v v2
t t+ tI t it
77 (1
2 3n 3 2 no a a a
t t t
2
af 1 1 +1 v t
a 2 2 G2 2 (ao2) 2
ï¿½t
af af 2
=D 2 vt1
3 ={2
where nj = 04 ,to0a I
(I+BB 1)2 mt
* (1k1) + * 2
( l+6OkI )
Yt +
*2
1B( 1 +2) 2 (1+B)2 t
, ï¿½(1kI)
m  )2 Yt
+ 1 2 + d
(i+8)2 dti
1 1O 0 0 k1
+ eO 1 (* lk + " )
(1+6)2 Ip I 1 k
1 111
+ B1 2 (0 1 *1
ko0k I ï¿½kok1
(1k )(l+B_k ) + * * )
1 1 (1k1 )(l+ fkI)
2 2
+ 0k 1
+(1i+BBk I )2
Sko k0k1
Y+ l+1 k1 T7+ (lk )(lBkI)
1+ Rl t T+  1
1
I+BBk1Yt
+0 k 1o
1+B 1kI +BOkI
We need
av
t
wB
av
t_
k k k 1
 , 2)
(I+BBkI)
* 0 __ kkl
+* k I 1k (1k )(l+ R Bk )
__ 1
*+k
Sk0 k 1
1k1 1+6S
I a 2 d2 + I_  d2 1+a t 1+6 t1
avt
300
avt lI
Dv t
Appendix C
af
For (3.28) we derive 2f
fi (1u i)
2
8 [i+s(iwi)]2
Sf2 i_I 1
af3 _ (ikI)
[1+5(1k)2
af 4 +(1kI)
3f 5 1iB(2+ )
 a (1+5)2
af6
D (1+S)2
as following
af7 1 0 P
DalPl
)OPl
+  '2
w +8 [ 12 [ )
1k
+ kol _ _1 1
 1 0 1kI1 (i+ )2 ii [ + i )]
i k 0 k 1 +
( * *
(1lk1 (1i+SSkI1)
10"l Ok0k 1 k0k
[2+(lp1I) [1+S(k)]2 1+(1k) ]2
af1 af2 T1 = T_
af3
af7 afl
w
acf4
Df7
0
af 4 af 5 af6 f4 f5 T f 6
1
1+~(1k1)
S0 (1
af 2 af3 1f2 13
1
Df 5
A *
(1+)(k1)
af2
0
af 3 f4
0 0
 ^ k )
1+B(1k1)
af6
_ _ 0
k1
af5 af6
0 0
af7
0
f2
af 3 af 4 f 6
1 1 1
f7
 .Tw = 0
af 5 ' f5 ^ 1i 1+8
afl
APPENDIX D From equation (2.19) we know
F(st,t,Mt,N) e(st,M,N) Y t t t b x(s t)
x t t
The forward rate formulation is
EtQN
F et t+1
t t
EtQ t+1
where
Ux(s') ' EtQ ti =  U (s)
x
M
t t+l
6U (s t+1i
E x t+1+l Et Ux(st)1
( E t + ) = E t
nt t1
M
(Mt
Mt+1
(t+l
(t+l) nt+l (1a M Et Et (t t+1
EtQ N = 5 f
+t t+1
U (s')n' N
y N f (u U ) du du du
U (S)TIft uitu2t' u4t it 2t 4t
(D)
f(uit,U2tu3t) du tdu2tdu3t
(D2)
01 (S )I
=E y t+i t+i
t U (st)IN
yt t
(t+i)
= E
(nt)
(t)
= E5 ()
( )
It+l t+1
From (2.14)
Ult = t+i  tlnEt
u1 t %N(O,hlt )
in t+i = 'it + PIln t n N(pIlnt,hlit)
Hence the probability density function of lnt+1 given hilt is
(D3)
f(in ht+hlit )
1
Ii
 1 inE t+1 exp[ 2hlit
 Pllnt) 2]
Similarly
f(inn t+l h 22t ) f(1nMt+i ,h33t) f(1nNt+l ,h44t)
1
 exp[22t
 exp[33t
_ 1
2 44t
h1 (int+1
2 2t
2h 1t+ 33t
exp[ h nNt+1
44t
 P2lnfnt) 2
 Y ll nrM t )2 ]
 Y2lnNt)2
N ( t+l)i(Tt+l
EtQt+1 = E t ( (t nt
Mt
Nt+I
nt+i Tt
N
Nt
Nt+i
60
znt+1 ia(Nt+1 1
E  ( ((Et+l)a(ft+l)lo(Nt+l) f(udu du du
E t Tit N t
Since uIt' u2t and u4t are assumed to be uncorrelated, we have
EQ N)f(U )du f (t+I)Itt+t It It nt
N 1 f(u t) dut f (,")
2t 2t Nt
f(u 4t) du4t
aint+1 l l t)
(t+l)a 1 exp{ (nE ' nE t2} d(in t+
t 27 i 2h llt t+
(innt+  P2innt)2
( t + I 1  0 e x pt + 2 t d ( n n + ) f t )h22t exp{ 2h22t t+1
Nt+1 i 1(InN  )2
() exp{ 2h 4 t  d(linN t+)
Nt 2 4t244tti
Since
E[E I =
t+l
exp(aPllnEt + a 2hllt)
E( nt1) = expt(al)P21nnt + 1(al)2h22t1
E(Nt1I) = exp{Y21nNt + 1h
E ~ 2 t e + 44t}
E(MtII) = exp{YllMt + lh
t+I I 33t}
in{EtQ }
t t+l
In8  alnt + a P1lnt + a hllt
+ anit + (al)P21nnt + (al)2 h22t
+ inN  Y2nN + h
61
1n{EtQ+I} = 1n  lnt + aPllnEt + a2h
+ alnlt + (ctl)p2lnflt + 1(a1)2 h22t
+ riMt  Y21nMt + h44t
Therefore
inE Q N
t t+1
 lnEtQt+ = (ly2)'nNt + 44t
 (lyl)lnMt  h33t
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B IOGRAPHI CAL SKETCH
Wenhe Lu was born in Shanghai, China, January 2nd 1953. When the Cultural Revolution was launched, he was a firstyear student in Nanyang Middle School in Shanghai.
In May 1969 he volunteered to settle down as an educated youth in Xigomen Village, Diujiugo 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 entranceexam, 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 nationwide entranceexam 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.
Ttephe' 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 d' s ion for 'degree of
Doctor of Philosophy.
or
Kobert 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
UNIVERSIIN OF ILORIDA
082 52731

Full Text 
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R ISK PREMIA IN FOREIGN EXCHANGE MARKETS BY WENHE 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
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I dedicate this dissertation to Mr. Li Youchai, Mr . Li Youfu, Mr . Li Youqien and all of my fellow kinsmen in Xigomen Village, Duijiugo District, Fengzhen County, Inner Mongolia, China, whose experience in the last half a century has led me to comprehend the significance of economic theory.
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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 enlightm ent 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 referenc es . 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 selfeducation 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. iii
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TABLE OF CONTENTS ACKNOWLEDGEMENTS. ABSTRACT. CHAPTER I II INTRODUCTION AND SURVEY OF THE LITERATURE ..â€¢... LUCAS MODEL AND ITS EXTENSIONS: AN EMPIRICAL STUDY. Lucas Model and its Extension ..... . White's Adjustment and Empirical Results .. Whi te's test Engle's ARCH test .......â€¢.. III A MONETARY APPROACH TO THE FOREIGN EXCHANGE RATE DETERMINATION. . . . . The Model . Estimation and Testing .â€¢. ARCH Method of Estimation .. GMM Estimation. Variation on the theme IV SUMMARY AND CONCLUSIONS .â€¢ APPENDIX A. APPENDIX B. APPENDIX C. APPENDIX D .. BIBLIOGRAPHY. PAGE iii vi 1 7 8 20 21 23 28 28 34 35 36 42 45 48 so 55 57 61 BIOGRAPHICAL SKETCH. . . . . . . . . . 64 iv
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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 I N FOREIGN EXCHANGE MARKETS By Chairman: Dr. G.S. Maddala Major Department: Economics Wenhe Lu August, 1986 We have attempted to test the existence of timevarying 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 twocountry economy model. We used White's test and Engle's test for homoscedasticity and us ed White's heteroscedasticityconsistent variancecovariance 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
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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 nonlinear single equation model and there is eviden ce of heteroscedasticity we used GMM estimators and the corresponding variancecovariance 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 monetar y model with the formation of risk premium derived before. The findings we have is that there is timevarying 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 semielasticity of interest rate the empirical results are based on a more general setting than most of the mon e t a ry 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 t est whether the conclusion we have drawn is valid. v i
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CHA PTER I INTRODUCTION AND SURVEY OF THE LITERATURE Since th e early 197Os, 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 f orward foreign exchange rates as predictors o f the future spot rates indicates the unexpected amount of speculation in ex change 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 i nflation r a tes 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 fina n cial markets has been the central thrust of empirical a s well as theoretical research over the last decade. A huge literature alre a d y exists on this topic. R .M. Levich (1983) gives a comprehensive survey of the e x isting models and empirical findings. These empirical studi es convey mixed messages about the validit y of the different models proposed. Mo st of these models fit the data for the countries and for the length of time the authors proposed, but they do not fit the dat a for the oth e rs. There is still no simple consensus on this topic. See Levich (1983). Many res ea rchers suspect that the failure of the models overall is due to the simplicity of the models . The relaxation of the restrictions 1
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2 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 lne =a+ blnF 1 + ut t t(1.1) where et denotes the spot exchange rate, Ft denotes the forward exchange rate at time t, a= 0, b = 1 and residual ut 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 H 0 : a 0 = O, ~ 1 = 1. We observe that all the OLS parameter estimates are far off the targets. We do a conventional Ftest, 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
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3 Table 1 Preliminary test of the hypothesis of market efficiency lnet+l = a 0 + a 1 lnFt + ut ao al R2 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 riskpremium term, preferably a timevarying 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 timevarying 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
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4 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 e x pect 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
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5 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 portfoliobalance 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 riskpremia for the major six currencies . He assumes that the risk premia depend specifically upon asset supplies and variancecovariance 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.
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6 We plan to investigate the existence of timevarying 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 timevarying 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 nonlinear model, we use the nonlinear 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.
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CHAPTER II LUCAS MODEL AND ITS EXTENSION: AN EMPIRICAL STUDY Lucas (1982) developed a general twocountry 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 timevarying 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. 7
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8 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) capita terms. Each citizen of country O is endowed each period with~ units of a freely transportable, nonstorable consumption good X. Each citizen of country 1 is endowed with n units of consumption good Y . These endowments ~ and n are stochastic following a Markov process with transition given by Assume that the process {~t,nt} has a unique stationary distribution ~(~,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 CX) E{ E St U(xit'yit)}, 0 < S < 1 , i = 0,1 t=O (2.1) where x. is consumption in country i in the period t of the good X and it y. is consumption of the good Y . The utility function U and discount it factor Sare common to both countries. In (2.1) U is assumed to be continuously differentiable, increasing in both arguments, and concave. Here we specify a CobbDouglas utility function U(x,y) a 1a = Ax y (2.2) We assume that agents trade in both goods, spot and in advance contingent on all possible realizations of the shock process {~t,nt}.
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9 The preferences of agents have been assumed to be independent of their nationalities, so that agents differ only in their endowments. Agents are assumed riskaverse 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. Lets= {~,n} be the current state of the system. Take the prices of all the goods, current and future, to be functions of the current stat e s, with the understanding that prices are assumed stationary in the sense that the same set o f prices is established at s independent of whens is realized. Then the knowledge of the equilibrium price functions together with kn o wledge 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 ar e assumed to have rational expectations formulated as such. For a system in any current states, let the current spot price of good X be unity, so that all the other prices will be in terms of current Xunits. Let p (s), q (s), and q (s) be the spot price of good y X y Y at states, the current price of a claim to the entire future (from the next period on) stre am {~t} of the endowment of good X, and the current price of a claim to the future stream {nt}' respectively, all in Xunits. Therefore p (s), q (s) and q (s) are all relative prices. y X y Consider an agent entering a period endowed with 0 units of wealth, consisting of claims to current and future goods, valued in Xunits. He wants to maximize his utility by choosing the current consumption (x,y) at spot prices (l,p (s)), equity shares 8 , 8 in future endowments{~} y X y t
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10 and {nt} at the prices per share q (s) and q (s) respectively . The X y budget constraint is x + p (s)y + q (s)8 + q (s)8 = e y X X y y The corresponding endowed wealth 8' valued in Xunits at the beginning of the next period will be 6' = e [~â€¢ + q (s')] + e [p (s')n' + q (s')] X X y y y (2.3) (2.4) since X,Y are assumed to be nonstorable and~â€¢ 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 ArrowDebreu 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 states= {~,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 noninterestbearing currency in exactly the amount they need to cover their perfectly predictable currentperiod goods purchase. In this context there is
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11 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 capita for both countries. Prior to any trading in period t, let each trader's money holdings be increased by a lumpsum w M and v N 1 so that money supply evolves t t1 t taccording to (2.5) Let {wt} follow a Markov process, possibly related to the real process {st} with transition function H{w,w' ,v,v' ,s,s'} = Pr{wt+l w' v v' lw = w v ' t+l t ' t We then have the following financial constraint M(t) p (s,M)~ X N(t) = p (s,N)n y = s' s = s} ' t (2 . 6) (2 . 7) with the equality sign when equilibria are reached. p (s,N) are nominal prices. Here p (s,M) and X y The objective value function for this agent, therefore, is V(8,s) = max {U(x,y) +Bf V(8' ,s') f(s' ,s)ds'} (2.8) x,y,e ,8 X y for the consumer in states with wealth 8. He is assumed to maximize (2.8) with 8' denoted in (2.4) subject to the constraints (2.3) and (2.7). We now have Lagrangian function
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12 + 8 [p (s')n' + q (s'))},s] f(s' ,s)ds' y y y + A3[N p (s,N)n)} t y The first order conditions for the Lagrangian problem are L = U (x,y) X X ).. = 0 1 L y U ( x ,y) Alp (s) = 0 y y 1 8 =BJ v 8 ,(8' ,s')[~â€¢ + qx(s')] f(s' ,s)ds' X A q (s) = 0 1 X L =BJ V ( 8 ' s')[p (s')n' 8 8' ' y y + q (s')] f (s' ,s)ds' A 1 q (s) = 0 y y We also know 1 8 =Alas 8 is given and = 8 x p (s)y q (s) e q (s)e y X X y y LA 2 Reorganizing the above conditions, we have U (x,y) = >..p (s) y y (2.9) (2.10.1) (2.10.2) (2.10.3) (2.10.4) (2.10.5) (2.10.6) (2.10.7)
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and 13 U (x,y) U (x,y) U (~,n) p ( s ) = a...y ....,.= y = a...Y ..,...,___,.__....... y A U (x,y) U (~,n) X X 8 JV (8' s')[~â€¢ + q (s')] f(s' s)ds' = A q (s) 8 ' X ' 1 X q (s) = 8[U (x,y)]l JU (s')[~â€¢ + q (s')] f(s' ,s)ds' X X X X 8 JV (8' s')[p (s')n' + q (s')] f(s',s)ds' = 8 ' y y q (s) = 8 [U (s)]l JU (s')[p (s')n' y X X y p (s,N) y + q (s')] f(s' ,s)ds' y n (2.11.1) ( 2 .11.2) (2.11.3) (2.11.4) (2.11.5) Eq uati ons (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 timevarying riskpremia. Equation (2.11.1) means at such an equilibri um, the relative pric e of Yin terms of X, p (s) depends only on the real state of the system, y t 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 CH,n) = y u CH,n) X A(l a)x'y a. et1 1et Aetx y 1Ct X Ct y _ 1Ct X = H Ct n Y = n (2.12)
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14 Here the perfect pooling assumption seems to be restrictive . As we see later, this will not affect our derivation of the timevarying risk premia. Assuming purchasing power parity we have (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 setup 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 a ssumed to follow conditional Gaussian AR(l) stochastic process where I 1 indicates the information set at the period t1. tH = diag(hll ,h22 ,h33 ,h44 ) t ,t ,t ,t ,t (2.14)
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15 Here we assume zero covaria n ce between the stochastic processes. It is possible to generalize the model to allow for nonzero 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 homecurrency price in period t of a claim to one unit of homecurrency with certainty in period t+l. Such a claim is equivalent to ~+l 1 = ~t+l I Mt+l px(st+l'Mt+l) (2.15) units of X in period t+l. Similarly N nt+l Ilt+l = Nt+l (2.16) Since ~+land ~+l are both functions of the future real output and nominal money supply, there is uncertainty as to their exact values. The ~+l units of X will be valued by agents in period t+l at the marginal utility of X, Ux(st+l), which must be discounted back to period t by multiplication by the discount factor. The xunit price of the M 1 claim to one unit of home currency is therefore E [BU Cs +l) IT +lU Cs) ] t X t t X t 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 xunit price times the marginal utility of X in period t. The home currency price of the investment is then obtained by multiplication of the xunit price by px(st,M) or division by~ Therefore, the period t discount bill paying one unit of home currency in period t+l in terms of the home currency price is
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16 (2.17) Similarly, by replacing x with yin the above argument, the period t foreign currency price of a claim to one unit of foreign currency in t+l is (2.18) where U Cs ) is the marginal utility of yin period t and TIN is the y t t purchasing power of one unit o f 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 f oreign exchange market, we must derive the forward price of foreign exchange, that is, the contract price set in period tat which one can buy and sell foreign exchange in period t+l. 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 Cs ,e ) per unit of home currency invested, and the alternax t t tive covered interest arbitrage strategy of converting the home currency into foreign currency and selling the proceeds in today's forward market at price FCst, e t,Mt,Nt) of home currency in terms of foreign currency.
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17 The covered investment in foreign currency yields the return per home currency invested e(st,M,N) is the exchange rate, i.e., the foreign currency per one u n it of th e home curre ncy . Equating the two strategies, we have The forward rate formulation is F t From (2.14) we have E QM t t+l = E t a1 a 1 n M B C t+ 1 ) (t) (t) = Et (t nt+l Mt+l (2.19) (2.20)
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18 And E QN U (s ' )n ' N 8 f y t dultdu2tdu4t = Nf(ul t+l'u2 t+ l 'u3 t+l) t t+l U (s ) n y t+ 1 ' ' ' = E 8U/st+l)II~+l t U/st)II~ s a a n N n = ES ( t+l) ( t) ( t) ( t+l) st 1 \+1 Mt+l nt s a n a1 N = EB ( t+ l ) ( t ) ( t ) nt+l Nt+l (2.21) I t is shown in Appendix D that (1y )lnM ~h 1 t 2 33 ' t Substituting the above into (2.19) and taking the logarithm, we have Also from (2 . 13) we have
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19 The last two terms are zero, hence we have (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, h 33 ,t+l' increases the conditional variance of domestic prices and, therefore, increases the expected fut ur e 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 h 33 ,t+l h 44 ,t+l which can not be observed at time t. We here make a naive rational ex pect ation assumption, i.e., that the economic agents use the relative variance of current money supply shocks h 33 ,t h 44 ,t a proxy to that of the future money supply shocks . There are othe r ways of formulating the rational expectation hypothesis, but they are just as, if not more, arbitrary; hence (2.23) becomes We approxim a te E lne and lnF by t t t et et1 and F e t t et and estimate the equation (h44 t+l h44 t)] ' ' (2.24)
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20 (2.25) 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 0 , a 2 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 < 0, a 2 > 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 timeseries behavior of the risk premium. According to the model, a 1 should be 1 if the market efficiency hypothesis is to hold, and Et+l should be white noise, independent of whether or not a risk premium exists. A test of the hypothesis that a 2 = 0 is a test of whether the risk premium is timevarying or not, in determining the deviation of the forward rate from the expected future spot rate. Maintaining a 1 = 1 and Et+l white noise, a 0 = 0, a 2 = 0 means a zero risk premium. ao = 0, Uz 0 implies a constant risk premium. a 0 = 0, a 2 = 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 Et= vt + a 2 [(h 33 ,t+l h 33 ,t) + 44 ,t h 44 ,t)] and is clearly heteroscedastic. The consequence of this approxi
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21 mation is that the error terms are heteroscedastic; hence 01S estimates of parameters are consistent and unbiased but not efficient. The 01S 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 'crosssectional' in nature. Before we proceed to estimate the consistent variancecovariance 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[a 2 1/asas 1 ] or in the outer product form, E[a1/as a1/asâ€¢] wheres is the vector o f parameters being estimat e d. The White procedure tests the e quality 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 is equivalent to TR 2 x 2 k(k+l)/Z where Tis the sample size; R 2 is the constantadjusted squared multiple correlation coefficient from the regression F e t t et where agai n ut is the 01S residual term of model (2.25). The results are contained in Table 2. The test sc ore s indic ate that there is
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22 Table 2 Tes ts for Serial Correlation and Heteroscedasticity with ARCH as a special case DW ARCH1 ARCH2 ARCH3 ARCH4 e e t t1 et1 White's test Germany 0.0313 (0.0170) 0.0560 (0.0329) 3.425 (2.612) 0.0329 2.007 * 8.619 * 8.656 8.9984 9.3345 * 9.694 OLS Estimation Japan 0.0022 (0.0033) 0.0426 (0.0218) 1.3865 (2.2926) 0.0316 1.8 63 2.769 3.548 3.571 5. 779 4.1003 Canada 0.0038 (0.0035) 0.0016 (0.0031) 0.2706 (0.3503) 0.0063 2.218 2.392 2.3607 2.5344 2.79 4 0. 5371 critical score a t 5% significance level 5.02 7 .38 9.35 11.1 9.35 Note: The score for ARCH Pis equivalent to TR 2 with T as sample size and R 2 from regression!= y 0 + i y. 2 .. The numbers in parentheses i=l l t1 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.
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23 Engle's ARCH test We also carry out Engle's test for the pres e nce of ARCH (auto regressive conditional heteroscedasticity). In a general model of Engle's where X is a vector of explanatory variables and 8 the parameter vector t with the distribution of the disturbance term assumed conditional normal as following where 2 p 2 a t= YO+ L Y.ut . i=l l i (2.26) 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 pin (2.26) is arbitrary and has no strict economic theoretical backing. 2 Engle (1982) shows that X score for testing the null hypotheses H 0 : Yo= Y 1 = = Yp = 0 is equivalent to TR 2 where Tis the sample size , R 2 is the constantadjusted squared multiple correlation coefficient from the regression A2 p A2 ut =YO+ L Y.ut . i=l l 1. (2.27)
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24 The test scores reported in Table 2 indic a te that ther e is no ARCH for the cases of Can a d a and Japan. However, ARCH ef f ects of 1st and 2nd order are significant in the Germany/U.S . case, a l though longer lags are not sign i ficant. Since (2.25) is linear, ARCH estimates c a n be obtained from the OLS regression (2 . 2 7). See Engle (1982). T he results are reported in Table 3b. The estimat e s show that Yo and y 1 are significant in the case of German y for al l the lag lengths p which we tried . This p a rtly confirms the results we had in White's test and Engle's test, i.e. , there is heteroscedasticity in the case of Germany c a s e and it is of ARCH type. Table 3 a ARCH E stimates and White's heteroscedasticityconsistent variance estim a tes for Mark/$ A 2 u = y + t 0 White's het e roscedasticityconsistent estimates and standard errors ao a l Germany 0 . 0313 0 . 0560 (0.215) (0.0399) Japan 0.00225 0 . 04256 (0.00356) (0 . 0 2 448) Canada 0.0038 0.0016 (0.00356) (0.00313 ) Note: The figures in the parenthesis are standard errors. * indicates significa n ce at 5 % level . a2 3 . 425 (2. 447) 1. 3 865 (2.10544) * 0.2706 (0 . 1195)
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25 Table 3b ARCH Estimates and White's heter o scedasticityconsistent variance estimates for Mark/$ A2 p AZ ut = Yo+ [ y iuti i=l ARCH parameter estimates Yo yl Y2 Y3 * * Germany 0.00074 0.2574 (0.00018) (0 . 0851) * * 0.00076 0.2651 0.0289 (0 . 00019) (0 . 0887) (0 . 0888) * * 0.00081 0.2633 0.0126 0.9583 (0.00021) (0.0893) (0.0925) (0.0895) * * 0.00082 0.2634 0. 0132 0 . 0511 (0.00022) 0 . 0897 (0 . 0931) (0.0931) * Japan 0.00087 0.14597 (0 . 00019) ( 0 . 08711) * 0.00096 0.15694 0.08101 (0.00021) (0.08850) (0.08851) * 0.00092 0.15945 0.07780 0.0097 (0 . 00023) (0.08890) (0.08974) (0.08893) * 0.00076 0.14616 0.06193 0.0097 (0 . 00024) (0.08876) (0.08937) (0.08921) * Canada 0 . 00015 0.1356 (0.00041) (0.0872) * 0.00016 0. 13638 0.00993 (0.000043) (0.08874) (0.08875) * 0 . 00015 0.13766 0.01317 0.03463 (0.000046) (0 . 08931) (0.09015) (0 . 08930) * 0.00016 0.14121 0.01484 0 . 03921 (0.000048) (0.09004) (0.09076) ( 0.0474) Note: The figures in the parenthesis are standard errors. * indicates significance at 5 % level. Y4 0.0375 (0.0901) 0 . 14662 (0.08824) 0.04215 (0.08992) Following White (1980), we calculate the heteroscedasticityconsistent covariance matrix. The procedure is to save residuals ut from regression (2.25) and calculate the following
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26 1 A2 1 = (X'X/n) (X'(diag(ut))X'/n)(X' X /n) (2.28) where Xis the matrix of the explanatory variables. Comparin g the heteroscedast ici ty adjusted standard errors with those from 01S in Table 2, we find that all but one of them are greater than the 01S estimates. The coefficient of the risk premium (h 33 ,t h 44 ,t) for C anada is significant. Maintaining a 1 = 1, we first test the joint hypothesis a 0 = a 2 = 0. The results in Table 4 (in the column RPl) indicate that this null hypothe sis can not be rejected for any currency. We next test the e x panded hypothesis H 0 : a 0 = a 2 = 0, a 1 = 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, Fr ance and Japan. The third hypothesis we test is a 1 = 1. This is a test of market efficiency when the timevarying 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 RPl 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 No te: The numbers reported for RPl and RP2 are Fstatistics at 1% significance level. RPl is an Ftest of the hypothesis a 0 = a 2 = O; RP2 is an Ftest of the hypothesis a 0 = a 2 = 0, a 1 = 1. RP3 is an Ftest of the hypothesis a 1 = 1.
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27 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 timevarying 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.
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CHAPTER III A MONETARY APPROACH TO THE FOREIGN EXCHANGE RATE DETERMINATION In this chap t er 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 semielasticity 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 (3.1) 28
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* * * * ct> y Br + ut t t 29 t=l, ,T (3.2) Tis 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 count ry; rt is the nominal interest rate and ut is a white noise disturbance term, ct> and Bare unknown coefficients of the real income * * * * * and interest rate and are both positive. Similarly, mt' pt' yt' rt, ut * and
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30 and Rush and Husted (1985) provide evidence in support of this assumption. See Levich (1983) for further discussion. Departing from the riskneutrality assumption in Dornbusch (1980) and Woo (1985), we modify the uncovered interest rate parity by introducing the risk premium term (3.5) where It is the information set th e agents in the economy have up to time t, R(t) is the risk premium term at period t. j it) 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 riskneutral. The risk premium we introduced shows our assumption of investors in general being risk averse . 1 Admittedly, there is difficulty in differentiating the riskneutrality from perfect capital mobility, since both arrive at the same form of expression of (3.5) without R. Here we t choose the former interpretation. 1 This can also be interpreted as deriving from the perfect state of capital mobility. Papell (1985) defines this equation as 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 shortrun deviation from the longrun PPP. We can interpret the risk premium along the same lines, i.e., risk premium reflects the shortrun deviation from PPP relating to the unc overed interest rate parity in a particular way and this deviation is a function of the past forecasting error on the spot exchange rates.
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31 Substituting (3.4), (3.5) into (3.3) and rearranging, we have (3.6) vt is white noise too. Now we proceed to formulate the risk premium and the distribution of the error term. We assume that R has the general form t (3.7) 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 t t et Hodrick and Srivastava (1983) developed an extension to Lucas's (1982) model and derived that the riskpremium 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 a nother e xte nsion of Lucas's (198 2) model. Since the difference between these two formu lations 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 fo rmation of the risk premium departs from the constant risk premium assumed in most of the empirical an d 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: (3.8) where a 0 and a 1 are constant unknowns and dt is the forward premium at time t, i.e., the difference of logarithms of the forward exchange rate
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32 at time t1 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 i=l,2, ... (3. 9) Substituting (3.8) into (3.6), we finally have the model (3.10) There are usually two ways of handling the rational expectation of future endogenous variables. The first is Wickens's (1982) errorin 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 backwardsolution 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.
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33 Since Etet+l involves the future values of all the exogenous * * variables, we assume at the outset that mt, mt' yt, yt are all from a first order autoregressive process. lnmt = llo + llrunt1 + e: t * * * * * lnmt = llo + llnmt1 + Et lnyt = kO + kllnyt1 + nt * * * * * lnyt = kO + kllnyt1 + n (3.11) t * * where e:t, e: t' nt' nt are all white noises. The stationary conditions require that The forward solution method leads us to * 1 1 * t * et l+S01 ) m * m l+ s 0k 1 ) yt + * yt t l+sCl1 ) t l+s0k 1 ) * * Bo llo llo
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34 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 norm a l 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.
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35 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 where Y is a vector of observations of the endogenous variable and Bis a vector of parameters to be estimated; Xis a matri x of explanatory variables; vis the vector of error terms. The conditional distribution of the error term is assumed to be of the form (3.14) where I 1 is the information set at time t1 and var v is generally tt defined by 2 2 2 f(v 1 ,v 2 , ... ,v ) tttp (3.15) Specifically, we formulate a linear function as follows (3.16)
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36 where 2 2 2 zt = {l, v l'vt } ttp To restrict the variance to be nonnegative, we impose y. > 0 l i=0,1,2, ... ,p and to prevent the variance from being explosive we impose p E i=l y. < 1 l We note that (3.18) is sufficient but not necessary. The likelihood function is 1 2 = const. 2 lnat (3.17) (3 .18) (3.19) (3.20) (3.21) 2 where vt = Yt f(xt,B) and at is defined as (3.16). This is maximized to obtain the MLE of s. 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 nonlinear econometric estimators reside within this
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37 category, GMM is a good alternative to the conventional maximum likeli hood method. One advantage of the GMM esti mator over MLE is that the variancecovariance matrix obtained by GMM is robust with respect to the specification of the disturbance distribution; hence, we can expect to produce a heteroscedastici tyr obust variancecovariance matrix o f 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(o) + e: (3.22) where y is a Txl dependent variable and X a matrix of explanatory varia bles of dimension TxM; o is the unknown hxl parameter vector; f(o) is MxN a one too ne function (not necessarily linear) taking element o 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{e: le:t ,e:t l, .â€¢. ,x ,x = o t p pt t(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 (3.24) The GMM estimator is the vector o that minimizes the quadratic form ~(o) = (y Xf(o))'Xn1 X'(y Xf(o)) (3.25) where~ is a consistent estimator of the positivedefinite matrix = lim E(X'e:e:'X)). T+oo (3.26)
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Hansen (1982) proves that p1 St = I q=p+l is a consistent estimator of St, 38 The variancecovariance matrix of the GMM estimator of o is [( v'X) A lcX'v)] 1 plim T Q T where v = X .!_ = X rlL af a ao ao 1 ' ao 2 ' . .. aoh * * * * 0 0 0 0 * where 0 is the GMM estimate. In our model (3.12) ft = {f l f2 f 4 } where fl 1 1+ss 1 1 = * l+BB 1 f3 = ~ l+Ssk 1 * f4 = * 1+ssk 1 2 o: B = 1 l+B f6 0:1 s l+ B (3.27) (3 . 28) (3.29)
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39 * * * f7 = a. s + S O SO Sk 0 + S k 0 (3 . 3 0 ) 0 1 +BS 1 * l+S Sk 1 * l+ S S l l+SSk 1 The deriv a tives evaluated at 6* !~ for the mode l (3.12) are derive d in Appendix C. Before we estimate (3 . 12) the ration a l ex p ectation hypo t hesi s allows us to impos e the first order a u toregressive process on the money s up p l y, and on the income index for the models of t wo countries (Germany v s . U . S . Japan v s. U . S. an d Canada vs. U.S . ) defined in (3 . 11) . The estima t es of 0 , 1 , k 0 , k 1 are reported in T a ble 5. The U.S . is t reated a s th e foreign country in all cases, as usually the cas e in empirical studies of ex c hange rate models. Table 5 Estim a tions of the firstorder autoregression process * * on the e xo g eno u s varia b les mt, mt' Yt' Y tâ€¢ u. s . Germany Japan C anada o 0.00115077 0 . 09747710 o . 30776482 0.00467322 1.00 7 8970 0.98287155 0 . 972608783 1 .0 0037203 kO 0.05466299 0.37540327 0 . 01169507 0 . 41615339 kl 0.989146 0 1 0 . 926677 5 5 0 . 99814542 0 . 91531332 Note: The above estimates ar e from t he regression s mt O + lm t1 + vlt a n
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40 S ub s t i t ute t h e estimates from the autoregressive pr oc e s s es i n t o equation (3 . 12), and then estimate it with NLLS . B efore we se t out to adjust the variance cova r ia n ce m a tr i x o f t he GMM estimator, we first do s ome diagnostic checking on the residuals we obtain e d from the NLL S estimation . Germany Japan Canada Note: Table 6 Tes t s of ARCH in equation (3 . 12) 8 0 0.0035 ( 1. 787) 0 . 0037 ( 1. 889) 0.0025 ( 1. 3 41) 0.0043 (0.046) 0.0027 ( 1. 342) 0.0022 (1.113) 0.0010 ( 1. 719) 0 . 00 0 7 ( 1.099) 0 . 0008 (1.139) The numbers * indicates p 2 8 0 + I 6. l l i=l 8 1 8 2 * 0. 7321 (12.255)* 0.7910 0 . 0804 (8 . 97) * (0 . 911) 0.8101 0. 2671 ( 9 .464) (2.449) * 0 . 8372 (17 . 708)* 0 . 564 9 0.3278 * (6 . 792)* (3 . 96) * 0 . 5495 0 . 2207 (6.557) (2.396) * 0. 82 19 (16.969)* * 0.5844 0.2981 (6.988) * (3.605)* 0.62988 0 . 2561 (7 . 329) (2.688) in parentheses a re a ll t scores significance at the 5 % level . 83 * 0.2364 (2 .7 59) 0 . 1233 ( 1.4 79) 0.0221 ( 0.263) for H 0 : 8. o. l
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41 We first test the existence of serial correlation by the standard regression (3.31) The results are reported in Table 6. 2 Engle's x test of ARCH is conveniently equivalent to TR 2 of (3.31), with T as sample size and R 2 as the squared multiple correlation coefficient from (3.12). Table 6 shows that pin (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 semielasticity of money with respect to interest rates S show that they are not significant from zero for all the cases. The estimates of the semielasticity 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 Sis significantly * different from zero but~'~ are not. Since he concentrates on the dollar/deutschmark rate only, we do not know how his model would fare in
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42 the other cases. In contrast, our model is built up with a timevarying risk premium. The results from our empirical study should have a better reflection of the reality than Woo's. The o ther 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 timevarying 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) * s
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43 the recent five or six years. Another reason is that the source of the timevarying risk premium is usually outside the system of the monetary models. It is my belief that incorporating the timevarying 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 timevarying 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 pin (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 are significant for the case of Canada while risk premium is nonexistent. Again, there is empirical evidence that in the case of Germany there is timevarying 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).
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Germany Japan Canada 44 Table 8 Tests of ARCH of equation (3.12) with risk premium defined as (2.23) * 0 .0061 (0.0027) * 0.0054 (0.0027) * 0.0049 (0.0024) 0.0029 (0.0016) 0.0021 (0.0016) 0.0018 (0.0016) 0.0014 (0.0008) 0 .0009 (0.0007) 0.0010 (0.0007) * 0.8280 (0.0501) * 0. 7288 (0.0877) * 0.6910 (0.0770) * 0.8649 (0.0444) * 0.5977 (0.0837) * 0.5990 (0.0867) * 0.8140 (0.0487) * 0.6022 (0.0841) * 0.6575 (0.0862) 0.1190 (0.0889) 0.0027 (0.095) 0.3123 * (0.0840) * 0.2647 (0. 0977) * 0.2719 (0.0828) * 0.2478 (0.0969) 88.805 59.706 0.1441 92.648 (0.0779) 97.595 100.256 0.0498 100.736 (0.0875) 0.0561 (0.0838) 60. 892 92.248 92.288 Note: The figures in the parentheses are the standard errors. The critical x 2 score is 53.7 at 0.5 % significa nce level. * indicates that the coefficient is significant at 5% level.
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45 Table 9 GMM estimation of (3.12) with (2.23) as risk premium * B a.a a.l 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.
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CHAPTER IV SUMMARY AND CONCLUSIONS We have attempted to test the existence of timevarying 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 CobbDouglas 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 twocountry economy model. The time varying risk premium is tested jointly with the ef f iciency 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 CobbDouglas 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 countri e s. 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 heteroscedasticityconsistent variancecovariance matrix to 46
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47 derive the correct standard errors. The joint hypothesis of market efficienc y and nonexistence 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 timevarying risk premia alone. The semielasticities 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 firstdegree autoregressive proc e ss for all the exogenous variables and we also impose independence of the disturbance terms. Although these are
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48 conventional practices in the literature with the rational expectation hypothesis, we must concede that there is arbitrariness in these formulations. Other rational expec tatio n hypotheses can be incorporated into our model in future research. Since it is a nonlinear single equation model, and we found evidence of the existence of heteroscedasticity in the error term we used GMM estimators and corresponding variancecovariance 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 timevarying 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 man y papers of monetary approach in the recent five or six years. Another reason is that the source of the timevarying risk premium is usually outside the system of the monetary models. It is my belief that incorporating the timevarying 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.
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APPENDIX A The monthly data used in Chapter II and III are obtained from the following resources. U.S. Germany Japan Canada 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 Imonth forward rate. 49
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IFS: ae: 64: 66.c: 60.b: 60.c: 61: 34.b: 50 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
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APPENDIX B We first make the assumption that all the exogenous variables are from afirstorder 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 o + 1 mt1 + Vt * * * * * mt = o + lmt1 + V t yt = kO + klyt1 + w t * * * * * yt = ko + klyt1 + wt t=2, ... , T (Bl) * * where vt' V t' w t' w t are white noise; Tis the sample size. Equation (Bl) and the rational expectation hypothesis lead to i Etmt+i (1) i = + lmt 11 * *i * o(l) *i * Em+ * + mt t t l 11 kO(lki) * *i i * ko(lkl ) *i Etyt+i = 1k + klyt Etyt+i = + kl yt (B2) * 1 1k 1 The expectations of the e's projected on I t are 51
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52 (B3) Substituting (B3) into (3.10) recursively and collecting terms we have 1 et= 1+BBl mt 1 * m * t l+ BB 1 y + l+ S Bk 1 t a B 0 + V t * * * yt 1 +BBk 1 which we use in our empirical study of this chapter. (B4)
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53 To make the derivation clear, we repeat (B4) and have 1 1 * e m + :::::,m + ~::::~ y t l+S8 1 t l+ S 8 1 t l+SSk 1 t * * * o o
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* 2 V t1 54 where n = {/3,~ , ~ ,a 0 ,a 1 }. We need * av 11* q> (1kl) t 1 1 asm m 2 t * 2 t (l+f3Skl )2 (l+BB 1 ) ( l+/3/3) * * (1kl) * a 1 f3(B+2) i+ al 2 + (1 +Bf3k;) 2 yt + 0+8) 2 (1+(3)2 dt1 t * * * 1 llo o ~ko ko + a (+) O (1 +8 )2 l 1 1* l k 1 1k * 1 1 1 yt
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1 55 * kl (1 ,*) l+ s s kl
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For (3 . 28 ) we derive:~ as ,H 1 _ (1 ) 1 "a""B [l+S(l 2 * * at 2 = 1 1 as r 1 +so~ ) 1 2 A a = ~(1kl) as [ l+S(l kl) ]2 A* * at 4 (1 kl) as [ l+BClk~)] 2 A af 5 = c. 1 S(2+S) as o+s)2 at 6 = _ c. __ as Cl+s)2 Appendix C following ~kOkl A + (lk 1 )[l+S(lk 1 )] 56
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57 af 1 af 2 af 4 at 5 at 6 = = = = = 0 kl (1 ,.) 1 +B(lk 1 ) af 4 1 * * a ct> l+ S (lk 1 ) * * af 7 S k 0 (1 kl * ) * * a ct> (l+ S ) (lk 1 ) l+ S (lk 1 ) af 1 af 2 af 3 af 4 a t 6 a t 7 = = = = = aa l = o
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APPEND IX D From equation (2.19) we know The forward rate formulation is where And E QM t t+l E QN t t+l f3 f = f3 f 58 (Dl) (D2)
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59 s a n a1 N = EB ( t+l) (t ) ( t ) st nt+l Nt+l (D3) From (2.14) Hence the probability density function of lnst+l given hll t is f ( lns t+ l, h 11 t) 1 exp[1 (lnst+l 2 2h11 t pllnst) ] 12h11 t Similarly f(lrn7t+l , h 22 t) 1 exp[1 (lnnt+l 2 = p 2 lnnt) ] 2 h22t f(lnM t +l'h33t) 1 exp[1 (lnMt+l 2 = yllnMt) ] 2 h33t f(lnNt+l'h44t) 1 exp[1 (lnNt+l 2 2 h44t y 2 lnNt) ] E QN s a n a1 N = EtB ( t+ 1 ) ( t+ 1) ( t) t t+l st nt Nt+l
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60 Since ult' u 2 t and u 4 t are assumed to be uncorrelated, we have E QN (t+l a n 1a N 1 = B f (~) f(ult)dult f ( t+l) f(u ) du 2 t f ( ~+l) f( u4t) du4t t t+l nt 2t t t 2 ( a 1 (ln(t+l plln(t) = B f ( t+l) exp{} d(ln't+l) ~t Zhllt 2 n 1a 1 (lnnt+l P 2 lnnt) } f ( t+l) exp{d(lnnt+l) nt 2 h22t N 1 2 1 (lnNt+l y 2 lnN t) f ( t+l) exp{2h } d(lnNt+l) N t 44t Since a exp(ap 1 ln~t + 1 2 ) E[(t+l] = 2a hllt a1 exp{(al)p 2 lnnt + a1) 2h22t} E(nt+l) = 1 exp{y 2 lnNt + h 44t} E(Nt+l) = 1 E(Mt+l) exp{y 1 lnMt + h 33t}
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6 1 1 {E Q M } = 1 B 1 + 1 + 1 2 h n t t+l n a Il s t ap 1 Il s t 2a llt Therefore (1y )lnM h 1 t 33t
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BIBLIOGRAPHY Baillie, R.T., Lippens, R.E. and MacMahon, P.C. "Testing Rational Expectations and Efficiency in the Foreign Exchange Market," Econometrica 51 1983(2): 553563. Blanchard, O.J. "Backward and Forward Solutions for Economics with Rational Expectations," American Economic Review 69, Papers and Proceedings (1979), 114118. Chesher, A. "The Information Matrix Test: Simplied Calculation via a Score Test Interpretation," Economic Letters 13 (1983), 4548. Diebold, F.X., and Pauly, P. "Endogenous Risk in a Portfolio Balance Rational Expectation Model of the DeutschemarkDollar Rate," Discussion paper in Department of Economics, University of Pennsylvania (Dec. 1983). Domowitz, Ian, and Hakkio, Craig S. "Conditional Variance and the Risk Return in the Foreign Exchange Market," Journal of International Economics 19 (1985), 4766. Dooley, M.P., and Isard, P. "The Portfoliobalance Model of Exchange Rates and Some Structural Estimations of the Risk Premium," I.M.F. Staff Paper 30 (1983), 683702. Dornbusch, R.D. "Exchange Rates Economics: Where Do We Stand?" Brookings Papers on Economic Activity 1980(1): 143194. Driskill, R.A. "Some Evidence in Favor of a Monetary Model of Exchange Rate Determination," Working Paper, Department of Economics, University of California, Davis (Aug. 1982). Driskill, R.A., and McCafferty, S. "Spot and Forward Rates in a Stochastic Model of the Foreign Exchange Market," Journal of International Economics 12 (1982), 313331. Engle, R.F. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of the U.K. Inflation," Econometrica (July 1982), 9871007. Frankel, J.A. "In Search of the Exchange Risk Premium: A Sixcurrency Testing Assuming Meanvariance Optimization," Journal of International Money and Finance 1 (1982a) 255274. Frankel, J.A. "The Mystery of the Multiplying Marks: A Modification of the Monetary Model," Review of Economics and Statistics 64 (1982b), 515519. 62
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63 Frankel, J.A. "Monetary and Portfoliobalance Models of Exchange Rate Determination," in J. Bhandari and B. Putnam (eds.), Economic Interdependence and Flexible Exchange Rates. Cambridge: MIT Press, 1983, 84115. Hansen, L.P. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica 50 (1982), 1029 1054. Hansen, L. P., and Hodrick, R. J. "Forward Exchange Rates as Optimal Predictors of Future Spot Rates: An Econometric Analysis," Journal of Political Economy 88 (1980), 829. Hansen, L.P., and Hodrick, R.J. "Risk Averse Speculation in the Forward Foreign Exchange Market: An Econometric Analysis of Linear Model," Ch. 4 in J.A. Frankel (ed.), Exchange Rates and International Macroeconomics. Chicago: University of Chicago Press, 1984. Hodrick, R.J., and Srivastava, Sanjay "An Investigation of Risk and Return in Forward Foreign Exchange Market," Discussion Paper WP 1160 Graduate School of Industrial Administration, CarnegieMellon University, Pittsburgh (1983). Huang, R.D. "Some Alternative Tests of Forward Exchange Rates as Predictors of Future Spot Rates," Journal of International Money and Finance (1984), 153167. Kawai, Masahiro . "The Effect of Forward Exchange on Spot Rate Volatility Under Risk and Rational Expectations," Journal of International Economics 16 (1984), 155172. Levich, R.M. "Empirical Studies of Exchange Rates: Price Behavior, Rate Determination and Market Efficiency," NBER Working Paper, No. 1112, (1983). Lucas, R.E., Jr. "Asset Pricing in an Exchange Economy," Econometrica 46 (November 1978), 14291445. Lucas, R.E. "Interest Rates and Currency Prices in a TwoCountry World," Journal of Monetary Economics 10 (1982), 335360. McNees, S.S. "The Forecasting Record for the 1970s," New England Economics Review (September 1979), 3353. Murfin, A. and Ormerod P. "The Forward rate for the U.S. Dollar and the Efficient Markets Hypothesis, 19781983," Manchester School of Economics Studies (1984), 293297. O f ficer, L. "Effective Exchange Rate and Price Ratio Over the Longrun: A Test of Purchasing Power Parity," Canadian Journal of Economics _!2, (1981), 206230. Papell, D.H. "Activist Monetary Policy, Imperfect Capital Mobility and the Overshooting Hypothesis," Journal of International Economics (1985), 219240.
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64 Park, Keehwan. "Tests of the Hypothesis of the Existence of Risk Premium in the Foreign Exchange Market," Journal of International Money and Finance 3 (1984), 169178. Rush, M., and Husted, S. "Purchasing Power Parity in the LongRun," Canadian Journal of Economics (1985), 137145. Shiller, R.J. "Rational Expectations and the Dynamic Structure of Macroeconomics Models," Journal of Monetary Economics 4 (1978), 144. Stockman, A.C. "Risk Information and Forward Exchange Rates," in J.A. Frankel, and H.G. Johnson, (eds), The Economics of Exchange Rates. Reading, Mass.: AddisonWesley Publishing Co., 1978, inclusive pages. Ueda, K. "Permanent and Temporary Changes in the Exchange Rate and Trade Balance Dynamics," Journal of International Economics 15 (1983), inclusive pages. Wallis, K.F. "Economic Implications of the Rational Expectation Hypothesis," Econometrica 48 (1980), 4974. White, H. "A HeteroscedasticityConsistent Covariance Matrix Estimator and a Direct Test for Heteroscedasticity," Econometrica 48 (1980), 817838. Wickens, M.R. "The Efficient Estimation of Econometric Models with Rational Expectations," Review of Economic Studies 69 (1982), 5567. Woo, W.T. "The Monetary Approach to Exchange Rate Determination Under Rational Expectations," Journal of International Economics 18 (1985), 116.
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BIOGRAPHICAL SKETCH Wenhe Lu was born in Shanghai, China, January 2nd 1953. When the Cultural Revolution was launched, he was a firstyear student in Nanyang Middle School in Shanghai. In May 1969 he volunteered to settle down as an educated youth in Xigomen Village, Diujiugo 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 entranceexam, 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 nationwide entranceexam 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. 65
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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. 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. 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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