%% This document created by Scientific Word (R) Version 2.5
%% Starting shell: article


\documentclass[12pt,thmsa]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{sw20lart}

%TCIDATA{TCIstyle=article/art4.lat,lart,article}

%TCIDATA{Created=Fri Feb 27 17:19:02 1998}
%TCIDATA{LastRevised=Tue Mar 24 21:06:48 1998}

\input{tcilatex}
\input tcilatex
\begin{document}

\title{Parameterizing Currency Risk in the EMS: The Irish Pound and Spanish Peseta
against the German Mark\thanks{%
Prepared for ESEM98, Berlin. } }
\author{G. C. Lim\thanks{%
Department of Economics, University of Melbourne, Parkville, Victoria,
Australia 3052. Email: g.lim@ecomfac.unimelb.edu.au} and Paul D. McNelis%
\thanks{%
Department of Economics, Georgetown University, Washington, D.C., USA
20057-1036. Visiting Professor, Spring 1998, Universitat Pompeu Fabra,
Barcelona, Spain.Email: McNelisP@gunet.georgetown.edu}}
\date{March 1998}
\maketitle

\begin{abstract}
This paper compares alternative estimates of systemic time-varying excess
returns for the Irish pound and the Spanish peseta, against the German mark,
since 1985. We make use of progressively more complex models, going from the
GARCH in Mean specification, to the International Capital Asset Pricing
model (ICAPM) with a time-varying ''beta'', to a general equiblirum Constant
Relative Risk Aversion model (CRRA), with trivariate GARCH-M esimation. The
results show significant relative risk aversion as well as significant
volatility effects on redictable excess returns. The time-varying ''beta''
has also declined in the past five years for both Ireland and Spain.
\end{abstract}

\section{\protect\bigskip \ Introduction}

This paper compares alternative estimates of systemic time-varying excess
returns for the Irish pound and the Spanish peseta, against the German mark,
since 1985, a period of financial openness and adjustment to low inflation,
but also a period of turbulence, with major swings in the value of the
exchange rates during the European currency crises of 1992.

From a policy perspective, it is helpful to know if the underlying or
systemic components of excess returns have been increasing or decreasing
during the past five years. One of the goals of monetary unification as well
as domestic policy reform programs has been the reduction of risk through
greater market transparency, lower inflation rates and liberalized capital
flows. How have these programs worked?

Casual empiricism suggests that there may be some decline in these excess
returns. The interest rates on similar assets, such as the Irish, German,
and Spanish call rates are slowly converging (see Figure 1). Certainly, the
volatilities of the spot rates of the Irish pound and Spanish peseta
relative to the German mark seem less in the late 1990's compared to the
mid-1990's and the mid-1980's (see Figure 2).

The question we ask in this paper is straightforward: Is there any
quantitative evidence of a decline in systemic currency risk in the sample
period examined?

The literature contains many studies of the role of the excess returns or
risk premia in foreign exchange markets; for survey see Lewis (1995). Models
of the risk premia developed include: models which relate the risk premia
directly to spot rate volatility;\footnote{%
See Bollerslev (1990), Baillie and Bollerslev (1990), Dukas, Fatemi and Lai
(1993).} models which explain risk in terms of the international capital
asset-pricing framework;\footnote{%
For examples, see McCurdy and Morgan (1991), Malliaropulos (1997).} and
models which include explicitly a role for the coefficient of relative risk
aversion.\footnote{%
For example, see Ayuso and Restoy (1996).}

There is also the calibration approach. One recent study tries to account
for both the predictability and the volatility of risk premia in currency,
bond, and equity markets between the United States and Japan.\footnote{%
See Baekert, Hodrick, and Marshall (1997).}

In this paper we employ the multivariate generalized autoregressive
conditional heteroskedastic in mean (M-GARCH-M) framework to estimate
progressively more complicated versions of risk premia models. The models
include: the spot volatility in mean model (SPOT-M), the international
capital asset pricing model (ICAP-M), and the intertemporal model with
explicit reference to the coefficient of relative risk aversion parameter in
the utility function (CRRA-M).

The paper is structured as follows. Section 2 contains a brief discussion of
the theoretical background to the models estimated. It shows that the three
models examined - the model which focuses on the role of exchange rate
volatility as an explanator of currency risk; the model which focuses on
time-varying betas, and the model which focuses on the role of the risk
aversion parameter - can all be viewed as variations of the Euler condition
derived from the dynamic general equilibrium intertemporal consumption
portfolio model.

Section 3 presents results for all three approaches estimated within the
M-GARCH-M framework. While the SPOT-M and ICAP-M have been popularly
estimated as univariate and bivariate GARCH systems respectively, the
intertemporal model has been usually estimated by generalized methods of
moments (GMM). We show how the CRRA-M can also be set up as a trivariate
GARCH system. Hence we present a unified framework for estimating all three
models. Section 3 applies these methods to the Irish pount/German mark and
Spanish peseta/German mark exchange rates. Results are presented for the
in-mean variance, the time-varying beta and the risk aversion coefficient
approaches to the parameterization of exchange-rate risk. The last section 4
concludes.

\section{Theoretical Framework}

The general equilibrium approach for explaining risk in currency markets is
inspired by the two-country complete markets model of Lucas (1982). In this
framework, asset prices are determined from the Euler condition for an
intertemporal choice problem of an investor who can trade freely in asset $%
j, $ and who maximizes the expectation of a time-separable utility function.
The Euler condition is:

\begin{eqnarray}
&&  \nonumber \\
\frac{U^{\prime }(C_{t})}{P_{t}} &=&\delta E_{t}\left[ R_{t}^{j}\frac{%
U^{\prime }(C_{t+1})}{P_{t+1}}\right]  \label{g1} \\
&&  \nonumber
\end{eqnarray}
where $\delta $ is the time preference parameter and $C_{t}$ is real
consumption, $U^{\prime }(C)$ denotes the marginal utility of consumption, $%
P_{t}$ is the price index. $R_{t}^{j}$ is the one-period gross nominal
return on asset $j;$ $R_{t}^{j}=(1+r_{t}^{j}).$ This expression is more
often presented as:

\begin{eqnarray}
&&  \nonumber \\
E_{t}\left[ R_{t}^{j}Q_{t+1}\right] &=&1  \label{g2} \\
&&  \nonumber
\end{eqnarray}
where $Q_{t+1}$ is the intertemporal marginal rate of substitution: 
\begin{eqnarray}
&&  \nonumber \\
Q_{t+1} &=&\delta \frac{U^{\prime }(C_{t+1})}{U^{\prime }(C_{t})}\frac{P_{t}%
}{P_{t+1}}  \label{g3} \\
&&  \nonumber
\end{eqnarray}

To derive the measure of excess returns, first recognise that the Euler
equation for the riskless asset is:

\begin{eqnarray}
&&  \nonumber \\
R_{t}^{f} &=&\frac{1}{E_{t}\left[ Q_{t+1}\right] }  \label{g4} \\
&&  \nonumber
\end{eqnarray}
and the equivalent first-order condition for asset $j$ is:

\begin{eqnarray}
&&  \nonumber \\
E_{t}\left[ Q_{t+1},R_{t}^{j}\right] &=&1  \label{g5} \\
&&  \nonumber
\end{eqnarray}
From equation (\ref{g4}) and the definition of covariance applied to (\ref
{g5}) we obtain the standard excess returns result:

\begin{eqnarray}
&&  \nonumber \\
E_{t}\left[ R_{t}^{j}-R_{t}^{f}\right] &=&-R_{t}^{f}Cov_{t}\left[
Q_{t+1},R_{t}^{j}\right]  \label{g6} \\
&&  \nonumber
\end{eqnarray}

When applying the model to international asset pricing, we define the the
risk-free rate as the return to a domestic asset, while the return to asset $%
j$ includes the currency risk:

\begin{eqnarray*}
&& \\
R_{t}^{j} &=&(1+r_{t}^{*})\frac{S_{t}}{S_{t+1}} \\
&&
\end{eqnarray*}
where $S_{t}$ is the spot rate, defined as the units of foreign currency per
domestic currency. Hence equation (6) shows that the conditionally expected
excess return associated with an uncovered position in the foreign currency
is proportional to the conditional covariance of the spot price with the
intertemporal marginal rate of substitution of domestic currency.

Since $Q$ is not observable, additional assumptions are needed to obtain a
testable version. A typical assumption is to re-express equation (6) in
terms of a benchmark portfolio on the conditional mean-variance frontier.
The benchmark portfolio has gross return $R^{m},$ which can be written as a
linear combination of the minimum variance portfolio nominal return (which
is perfectly correlated with $Q)$ and the risk-free return, $R^{f}$. The
equilibrium return on asset $j$ can then be expressed as the conditional
beta CAPM model examined by, for example, McCurdy and Morgan (1991):

\begin{eqnarray}
&&  \nonumber \\
E_{t}(R_{t}^{j}-R_{t}^{f}) &=&\frac{Cov_{t}\left[ R_{t}^{j},R_{t}^{m}\right] 
}{Var_{t}\left[ R_{t}^{m}\right] }E_{t}(R_{t}^{m}-R_{t}^{f})  \label{g7} \\
&&  \nonumber
\end{eqnarray}

Another version of the first order condition is based on the assumption that 
$q_{t+1}=\log (Q_{t+1})$ and $r_t^j=\log (R_t^j{})$ are conditional joint
lognormally distributed.\footnote{%
See Campbell, Lo, and MacKinlay (1997), various sections, for a discussion
of the following substitutions.} Hence an alternative expression for excess
returns is:

\begin{eqnarray}
&&  \nonumber  \label{g8} \\
E_t\left[ r_t^j-r_t^f\right] &=&-\frac{Var_t(r_t^j)}2-Cov_t(r_t^j,q_{t+1})
\label{g8} \\
&&  \nonumber
\end{eqnarray}
where $Var_t(r_t^j)$ is the conditional variance of $r_t^j,$ and $%
Cov_t(r_t^j,q_{t+1})$ is the covariance between $r_t^j$ and $q_{t+1}.$ If
the covariances are assumed to be negligible, we have the spot volatility in
mean model studied by for example, Baillie and Bollerslev (1990):

\begin{eqnarray*}
&& \\
E_{t}\left[ r_{t}^{j}-r_{t}^{f}\right] &=&-\frac{Var_{t}(s_{t+1})}{2} \\
&&
\end{eqnarray*}

To render (8) empirically tractable, it is common to assume a time-separable
power utility function:

\begin{eqnarray*}
&& \\
U(C_{t}) &=&\frac{C_{t}^{1-\gamma }-1}{1-\gamma } \\
&&
\end{eqnarray*}
where $\gamma $ is the coefficient of relative risk aversion.\footnote{%
In this setting, risk premia critically depend on the specification of the
underlying utility function. Mark (1985) has found that the parameter of
risk aversion in a constant relative risk aversion specification of the
utility function has to be quite large (in the range 12 to 20) to explain
the variability of excess returns. Cumby (1988) suggested that the
non-separability over time of the utility function in $t$ may account for
the failure of the general equilibrium models to explain speculative returns.%
}

This implies a first-order condition of form:

\begin{eqnarray}
&&  \nonumber \\
E_{t}\left[ \delta R_{t}^{j}\left( \frac{C_{t+1}}{C_{t}}\right) ^{-\gamma
}\left( \frac{P_{t}}{P_{t+1}}\right) \right] &=&1  \label{g9} \\
&&  \nonumber
\end{eqnarray}
which gives the general volatility-in-mean expression:

\begin{eqnarray}
&&  \nonumber \\
E_{t}\left[ r_{t}^{j}-r^{f}\right] &=&-\frac{Var_{t}(r_{t}^{j})}{2}%
-Cov_{t}(r_{t}^{j},\pi _{t+1})+\gamma Cov_{t}(r_{t}^{j},c_{t+1})  \label{g10}
\\
&&  \nonumber
\end{eqnarray}
where $\pi _{t+1}=\log (P_{t+1}/P_{t})$, $c_{t+1}=\log (C_{t+1}/C_{t})$ and $%
Cov_{t}(r_{t}^{j},\pi _{t+1}),$ $Cov_{t}(r_{t}^{j},c_{t+1})$ are the
conditional covariances. An estimate of $\gamma $ can be obtained if
consumption data is available.

However, high frequency data on consumption are generally unavailable. Ayuso
and Restoy (1996), suggest approximating nominal consumption growth with the
return on the equilibrium portfolio. We thus have:

\begin{eqnarray}
&&  \nonumber \\
E_{t}\left[ R_{t}^{j}\left( R_{t}^{m}\right) ^{-\gamma }\left( \frac{P_{t}}{%
P_{t+1}}\right) ^{1-\gamma }\right] &=&1  \label{g11} \\
&&  \nonumber
\end{eqnarray}
Again assuming joint log normality, the expression for excess returns
becomes a function of second moments of asset returns:

\begin{eqnarray}
&&  \nonumber \\
E_{t}\left[ r_{t}^{j}-r_{t}^{f}\right] &=&-\frac{Var_{t}(r_{t}^{j})}{2}%
+(1-\gamma )Cov_{t}(r_{t}^{j},\pi _{t+1})+\gamma Cov_{t}(r_{t}^{j},r_{t}^{m})
\label{g12} \\
&&  \nonumber
\end{eqnarray}
where $Cov_{t}(r_{t}^{j},r_{t}^{m})$ is the conditional covariance of the
return on asset $j$ with the return on the market portfolio.

The advantage of equation (12) is that it allows direct estimation of the
CRRA parameter $\gamma $. It is apealing as a model of risk premium because
it includes the three main explanators of currency risk: spot volatility,
co-variability of the spot rate with inflation and co-variability of the
spot rate with the world return. The major weakness is that it is derived
from a specific type of utility function.

\section{Empirical Analysis}

\subsection{Data}

Figure 3 shows the evolution of the excess returns ($\log (R^{j}/R^{f})_{t})$
from investing in a foreign asset, in Ireland or Spain, for a German
investor. The frequency is monthly covering the period 1985:1 to 1997:04.
The interest rates are monthly returns on comparable assets - the
three-month bill rates for the three countries.

From Figure 3, it appears that the two series are very similar, with the
exception of the large jump in the Irish series between 1992 and 1993. Table
I presents some descriptive statistics. The mean excess return is non-zero
and highly volatile for both countries. The series are clearly non-normal,
but stationary. Ireland satisfies the Engle and Ng (1993) test for symmetry
in the volatility, but Spain does not.

The aim of the paper is to extract information from the two series about
systemic risk. Did the non-random components of these series decline as the
economy adjusted to an era of lower inflation\thinspace and increased
globalisation of financial markets?

\subsection{Estimating the M-GARCH-M Framework}

The ARCH\ model proposed by Engle (1982), and generalised by Bollerslev
(1986), GARCH, have been used extensively to model the behaviour of
volatility over time.\footnote{%
For a review of the numerous applications of the GARCH framework for
estimating volatility, see Bollerslev, Chou and Kroner (1992).} In
particular, the GARCH-M model of Engle, Lilien and Robins (1987) explicitly
links the conditional variance to the conditional mean of the returns and
hence is the ideal framework to study the relationship between measures of
currency risk and volatility. This technique has been used extensively in
this literature, to estimate the SPOTM and the ICAPM analysis of risk
premia. In this paper, the framework is also applied to the CRRAM. All
models were estimated for the GARCH (1,1) case. Alternative orders of the
GARCH process will be examined at a later stage.

All data are demeaned, and deseasonalized. We estimate all three non-linear
systems by maximum likelihood methods. Following Dorsey and Mayer (1995), we
initialized the parameter search with a genetic algorithm, and then used a
gradient-descent optimization algorithm to reach the reported estiamtes.

Normally, one would report heteroscedastic-consistent estimates of the
standard error based on the Hessian matrices. . Bollerslev and Wooldridge
(1988), for example, have shown that asymptotically valid inference may be
based on a quasi maximum likelihood procedure, when a robust covariance
matrix for the parameters is calculated from $H^{-1}(GG^{\prime })H^{-1},$
where $H$ is the Hessian, and $G$ is the outer product of the gradients.

However, the high degree of non-linearity in the multivariate GARCH systems
made inversion of the Hessian matrices impossible. We thus made use of the
boostrapping technique for obtaining the standard errors and the
corresponding t-statistics. \footnote{%
We are indebted to Andreas Weigand for suggesting the use of bootstrapping
methods for calculating the standard errors.}\footnote{%
For an exposition of boostrapping techniques, see Mooney and Duval (1993).}

\subsection{Spot Rate Volatility}

Baillie and Bollerslev (1990) argue that the covariances in equation (\ref
{g12}) are negligible. So the simplest version of the currency risk model
relates the excess returns to the square root of the conditional variance of
the spot exchange rate, and may be set up as a univariate GARCH-M model as
follows:\footnote{%
In a related model by Dukas, Fatemi, and Lai (1993), the variable $s_t$ is
modeled as a random walk process and the currency risk is hypothesied to be
related to the conditional variance of the exchange rate, $h_t^s$ which
follows the GARCH process.} 
\begin{eqnarray*}
&& \\
y_{t} &=&\alpha y_{t-1}+\sqrt{h_{t}}+\varepsilon _{t} \\
&& \\
&&\varepsilon _{t}|I_{t-1}\symbol{126}N(0,h_{t}) \\
&& \\
h_{t} &=&c+a\varepsilon _{t-1}^{2}+gh_{t-1}+\zeta _{t} \\
&&
\end{eqnarray*}
where $y_{t}$ is either $(f_{t-1}-s_{t})$ or $(\log (R^{j}/R^{f})_{t}).$
With these models, the conditional mean and volatility of the series are
assumed to be predictable using past available information on returns and
volatility measures.\footnote{%
See McCurdy and Stengos (1992) for a comparison of parametric versus
non-parametric conditional mean estimators. Their results show that a
parametric specification of the GARCH process avoids the problem of
over-fiting.}

Table II presents the results of the GARCH-in-Mean model for the two
countries. The results show that there are significant GARCH processes, and
that volatility in the underlying spot exchange rates has significant
effects on the excess returns, for mean and volatility. The diagnostics
indicate consistency under the Pagan-Sabau tests, but also show evidence of
autocorrelation and lack of symmetry.

\subsection{The ICAP-M Method}

In the partial equilibrium models of asset pricing, Adler and Dumas (1983)
were the first to show that the ``market return'' in the international CAPM
model should generalize across many countries. The international capital
asset pricing model evaluates the risk and return of an asset $j$ relative
to a benchmark return and risk; specifically the ICAPM postulates that the
equilibrium excess return on any asset $j,$ is related to the excess returns
from a benchmark portfolio as follows:

\begin{eqnarray*}
&& \\
&& \\
r_{t}^{m}-r_{t}^{f} &=&\delta [r_{t-1}^{m}-r_{t-1}^{f}]+\varepsilon _{1t} \\
y_{t} &=&\frac{h_{t}^{12}}{h_{t}^{22}}E_{t-1}[r_{t}^{m}-r_{t}^{f}]+%
\varepsilon _{2t} \\
&& \\
\varepsilon _{t}^{\prime } &=&[\varepsilon _{1t},\varepsilon _{2t}];\text{ }%
\varepsilon _{t}^{\prime }|I_{t-1}\symbol{126}N(0,H_{t}) \\
&& \\
H_{t} &=&C^{\prime }C+A^{\prime }\varepsilon _{t-1}\varepsilon
_{t-1}^{\prime }A+G^{\prime }H_{t-1}G \\
&&
\end{eqnarray*}
\begin{eqnarray*}
H &=&\left[ 
\begin{array}{cc}
h^{11} & h^{12} \\ 
h^{12} & h^{22}
\end{array}
\right] ;C=\left[ 
\begin{array}{cc}
c_{11} & c_{12} \\ 
0 & c_{22}
\end{array}
\right] ;\text{ } \\
A &=&\left[ 
\begin{array}{cc}
a_{11} & a_{12} \\ 
a_{12} & a_{22}
\end{array}
\right] ;\text{ }G=\left[ 
\begin{array}{cc}
g_{11} & g_{12} \\ 
g_{12} & g_{22}
\end{array}
\right] \\
&&
\end{eqnarray*}
where $h_{t}^{22}$ is the conditional variance of the benchmark portfolio
and $h_{t}^{12}$ is the conditional covariance between the excess return of
the foreign asset and the benchmark portfolio. Following McCurdy and Morgan
(1991), we have used the Morgan Stanley Capital International world equity
index as the appropriate benchmark portfolio to compute the market return, $%
R_{t}^{m}$.\footnote{%
For the benchmark world portfolio, McCurdy and Morgan (1991) included a
moving average component, to reflect the effects of non-synchronized trades
of the components of the world index. Since this study is based on monthly
data, non-synchronicity is not an issue.} In this model, beta is not a
constant but is conditional on the covariance between the asset and market
returns relative to the variance of the market return.

The ICAP-M time-varying beta model makes use of GARCH-in-mean estimation,
and the vector of errors $\varepsilon _{t}^{\prime }$ is assumed to have a
conditional bivariate normal distribution with zero mean and conditional
covariance matrix defined by the Engle and Kroner (1993) BEKK form, where $%
C, $ is an upper triangular matrix and $A$ and $G$ are symmetric matrices of
parameters.\footnote{%
For an alternative parametrization, see Malliaropulos (1997) application of
Bollerslev (1990) model which assumes a constant conditional correlation
while allowing the conditional variances and covariances to vary over time.}

The results from estimating the time-varying beta for the excess returns
model appear in Table III.

The first point is that few of the coefficient estimates are significant.
The second point, based on the Pagan-Sabau tests for error consistency, is
that the error structure for the excess returns formulation is better
captured by the GARCH-M specification than by the ICAP-M one. The error
diagnostic for the systemic risk premia suggests the need for a more complex
error structure.

Concentrating on the excess returns, Figure 4 shows that systemic
time-varying beta, given by $\frac{h_{t}^{12}}{h_{t}^{22}}$ has become
relatively more stable in the 1990's for both countries.

But as can be seen from Table III, there is clearly room for improvement,
possibly due to an omission of shocks associated with consumption.

\subsection{CRRA Approach}

The generalized method of moments (GMM) has been used to estimate the ``deep
parameters'' such as the coefficient of risk aversion in the representative
agent's utility function, which ultimately determine excess returns under
efficient markets.\footnote{%
The GMM method is due to Hansen (1982). For applictions in this area see
Bodurtha and Mark (1991), Ayuso and Restoy (1996).} An alternative procedure
is to estimate the CRRAM embodied in equation (\ref{g12}) as a trivariate
GARCH-M system:\footnote{%
For an earlier work on a tri-variate CAPM model, see Bollerslev, Engle and
Wooldridge (1988), their work also suggest ptential role for consumption
shocks. See Bekaert (1995) for a combined VAR-GARCH model.}

\begin{eqnarray*}
r_{t}^{m}{} &=&\alpha _{1}r_{t-1}^{m}{}+\varepsilon _{1t} \\
&& \\
\pi _{t} &=&\alpha _{2}\pi _{t-1}+\varepsilon _{2t} \\
&& \\
y_{t} &=&-\frac{1}{2}h_{t}^{33}+(1-\gamma )h_{t}^{32}+\gamma
h_{t}^{31}+\varepsilon _{3t} \\
&& \\
\varepsilon _{t}^{\prime } &=&[\varepsilon _{1t},\varepsilon
_{2t},\varepsilon _{3t}];\text{ }\varepsilon _{t}^{\prime }|I_{t-1}\symbol{%
126}N(0,H_{t}) \\
&&
\end{eqnarray*}
\begin{eqnarray*}
H_{t} &=&C^{\prime }C+A^{\prime }\varepsilon _{t-1}\varepsilon
_{t-1}^{\prime }A+G^{\prime }H_{t-1}G \\
&&
\end{eqnarray*}
\begin{eqnarray*}
H &=&\left[ 
\begin{array}{c}
h^{11}h^{12}h^{13} \\ 
h^{12}h^{22}h^{23} \\ 
h^{13}h^{23}h^{33}
\end{array}
\right] ;C=\left[ 
\begin{array}{ll}
c^{11}0 & 0 \\ 
c^{21}c^{22} & 0 \\ 
c^{31}c^{32} & c^{33}
\end{array}
\right] ; \\
\text{ }A &=&\left[ 
\begin{array}{c}
a^{11}a^{12}a^{13} \\ 
a^{12}a^{22}a^{23} \\ 
a^{13}a^{23}a^{33}
\end{array}
\right] ;\text{ }G=\left[ 
\begin{array}{c}
g^{11}g^{12}g^{13} \\ 
g^{12}g^{22}g^{23} \\ 
g^{13}g^{23}g^{33}
\end{array}
\right] \\
&&
\end{eqnarray*}

This framework has the same structure as previous models and has the
advantage of allowing for a time-varying risk premia.\footnote{%
In many other empirical studies, such as those of Lewis (1988) and Engel and
Rodriguez (1989), the relative risk aversion coefficient is not
significantly different from zero. A larger class of dynamic asset pricing
models has recently been studied by Bakshi and Naka (1997). They note that
stochastic discount factors that incorporate habit forming behaviour are
better at explaining the empirically observed asset prices.}

Results are presented in Table IV. The main point is that the coefficient of
relative risk aversion is significant for Ireland, there are strong and
significant persistence effects in the trivariate GARCH processes for both
countries, and the errors appear to be adequately parametrized by the CRRA
framework for Ireland, somewhat less so for Spain. There is evidence of
autocorrelation in the residuals for the excess returns for both countries.

\section{Conclusion}

Understanding the determinants of excess returns is important for
appropriate policy responses. The results show that the excess returns
cannot be attributable solely to spot exchange rate volatility, nor can they
be explained by the risk aversion behaviour embodied in the time separable
power utility function. But, results from the ICAPM framework shows that
beta has declined in volatility, which in turn suggest a decline in foreign
currency risk against the German mark, for both Ireland and Spain.

Although the coefficient of relative risk aversion was not significant for
Spain, it must be recalled that the linear form of the intertemporal asset
pricing condition estimated was based on the power utility function. In
recent years, more complex utility functions allowing for habit persistence
behaviour are being developed. Future research incorporating more complex
risk behaviour in the utility function into the M-GARCH-M modeling
framework, with its emphasis on the second moments of asset returns, could
potentially lead to better models of currency risk.

Comparing the last two models, the CRRA model is more complex in terms of
parameter space. The ICAP-M model, on the other hand, is more complex
insofar as its functional form is highly nonlinear in parameters. In this
model, a time-varying conditional covariance is divided by a time-varying
conditional variance. Given the insignificant coefficient estimates and
significant tests for consistency produced by the ICAP-M,our estimation
results show that the ''curse of dimensionality'' may not be as severe as
the ''curse of complexity''.

\newpage \ 

{\LARGE \ }

\begin{thebibliography}{99}
\bibitem{}  Adler, M. and Dumas, B. (1983), ``International Portfolio Choice
and Corporate Finance: A Synthesis'', \textit{Journal of Finance,} 38,
925-984.

\bibitem{}  Ayuso, J. and Restoy, F. (1996), `` Interest Rate Parity and
Foreign Exchange Risk Premia in the ERM'', \textit{Journal of International
Money and Finance,} 15, 369-382.

\bibitem{}  Baillie, R.T. and Bollerslev, T. (1990), ``A Multivariate
Generalized ARCH Approach to Modelling Risk Premia in Forward Foreign
Exchange Rate Markets'' \textit{Journal of International Money and Finance,}
9, 309-324.

\bibitem{}  Bakshi, G. and Naka, A. (1997) ``An Empirical Investigation of
Asset Pricing Models Using Japanese Stock Market Data'', \textit{Journal of
International Money and Finance}, 16, 81-112.

\bibitem{}  Bekaert, G. (1995), ``The Time Variation of Expected Returns and
Volatility in Foreign-Exchange Markets, \textit{Journal of Business \&
Economic Statistics}, 13, 397-408.

\bibitem{}  Bekaert, G and Hodrick, R.J. (1993), ``On biases in the
measurement of foreign exchange risk premiums'', \textit{Journal of
International Money and Finance}, 12, 115-138.

\bibitem{}  Bekaert, G, Hodrick, R.J. and Marshall, D.A. (1997), ''The
Implications of First Order Risk Aversion for Asset Market Risk Premiums'', 
\textit{Journal of Monetary Economics}, 40, 3-40.

\bibitem{}  Bera, A. and Jarque, C. (1980), ``Efficient Tests for Normality,
Heteroscedasticity, and Serial Dependence of Regression Residuals'', \textit{%
Economic Letters}, 6, 255-259.

\bibitem{}  Bodurtha, J.N. Jr, and Mark, N.C. (1991), ``Testing the CAPM
with Time-Varying Risks and Returns'', \textit{Journal of Finance}, 46,
1485-1505.

\bibitem{}  Bollerslev, T. (1986), ``Generalized Autoregressive Conditional
Heteroskedasticity'', \textit{Journal of Econometrics} 31, 307-327.

\bibitem{}  Bollerslev, T. (1990), ``Modelling the Coherence in Short-Run
Nominal Exchange Rates: A Multivariate Generalized ARCH Model'', \textit{%
Review of Economics and Statistics}, 72, 498-505.

\bibitem{}  Bollerslev, T., Engle, R.F. and Wooldridge, J.M. (1988), ``A
Capital Asset Pricing Model with Time-varying Covariances'', \textit{Journal
of Political Economy}, 96, 116-131.

\bibitem{}  Bollerslev, T., Chou, R. and Kroner, K. (1992) ``ARCH Modelling
in Finance:\ A Review of the Theory and Empirical Evidence'', \textit{%
Journal of Econometrics}, 52, 5-59.

\bibitem{}  Bollerslev, T. and Wooldridge, J.M. (1988) ``Quasi-maximum
Likelihood Estimation and Inference in Models with Time Varying
Covariances'', \textit{Econometric Reviews}, 11, 143-172.

\bibitem{}  Campbell, J.Y., Lo, A.W. and Craig, A. (1997), \textit{The
Econometrics of Financial Markets}, Princeton University Press, Princeton.

\bibitem{}  Cumby, R.E. (1988), ``Explaining Deviations from Uncovered
Interest Parity'', \textit{Journal of Monetary Economics,} 16, 139-172.

\bibitem{}  Dorsey, R.R. and Mayer, W.J. (1995), ''Genetic Algorithms for
Estimation Problems With Multiple Equilibria, Nondifferentiability, and
Other Irregular Features'',\textit{\ Journal of Business and Economic
Statistics}, 13,53-66.

\bibitem{}  Dukas, S., Fatemi, A., and Lai, T.Y. (1993), ``An Examination of
the Relationship Between the Forward Exchange Risk Premium and Spot Rate
Volatility''. \textit{The International Journal of Finance}, 5, 441-452.

\bibitem{}  Engle, R.F. (1982), ``Autoregressive Conditional
Heteroskedasticity with Estimates of the Variance of UK Inflation'', \textit{%
Econometrica}, 50, 987-1008.

\bibitem{}  Engle, R.F. and K.F. Kroner (1995), ``Multivariate Simultaneous
Generalized ARCH'' \textit{Econometric Theory}, 122-150.

\bibitem{}  Engle, R.F., Lilien, D. and Robins, R. (1987) ``Estimating
Time-Varying Risk Premia in the Term Structure: The ARCH-M Model'', \textit{%
Econometrica}, 55, 391-407.

\bibitem{}  Engle, R.F. and Ng,V.K. (1993), ``Measuring and Testing the
Impact of News on Volatility'', \textit{Journal of Finance}, 48, 1749-1778.

\bibitem{}  Engel, C.M. and A.P. Rodrigues (1993), ``Tests of International
CAPM with Time-Varying Covariances'', \textit{Journal of Applied Econometrics%
}, 4, 119-138.

\bibitem{}  Fama, E.F. (1984) ``Forward and Spot Exchange Rates'', \textit{%
Journal of Monetary Economics}, 14, 319-338.

\bibitem{}  Hansen, L.P. (1982), ``Large Sample Properties of Generalized
Method of Moments Estimators'', \textit{Econometrica,} 50, 1029-1054.

\bibitem{}  Hodrick, R.J. and Srivastava, S. (1984) ``An Investigation of
Risk and Return in Forward Foreign Exchange'', \textit{Journal of
International Money and Finance}, 3, 5-29.

\bibitem{}  Lewis, K.K. (1988), ``Inflation Risk and Asset Market
Disturbances: The Mean-Variance Model Revisited'', \textit{Journal of
International Money and Finance}, 7, 273-288.

\bibitem{}  Lewis, K.K. (1995), ``Puzzles in International Financial
Markets''. Chapter 37 in \textit{Handbook of International Economics},
vol.III, edited by Grossman, G. and Rogoff, K.

\bibitem{}  Lucas, R.E. Jr. (1982), ``Interest Rates and Currency Prices in
a Two-Country World'', \textit{Journal of Monetary Economics, }10, 336-360.

\bibitem{}  Malliaropulos, D. (1997), ``A Multivariate GARCH Model of Risk
Premia in Foreign Exchange Markets'', \textit{Economic Modelling} 14, 61-79.

\bibitem{}  Mark, N. (1985), ``On Time-Varying Risk Premia in the Foreign
Exchange Market'', \textit{Journal of Monetary Economics,} 16, 3-18.

\bibitem{}  Marston, R.C. (1997), ``Tests of three parity conditions:
Distinguishing Risk Premia and Systematic Forecast Errors'', \textit{Journal
of International Money and Finance}, 16, 285-303.

\bibitem{}  McCurdy, T.H. and Morgan, I.G. (1991), ``Tests for a Systematic
Risk Component in Deviations from Uncovered Interest Parity'', \textit{%
Review of Economic Studies,} 58, 587-602.

\bibitem{}  McCurdy, T.H. and Stengos, T. (1992), ``A Comparison of Risk
Premia Forecasts Implied by Parametric versus Nonparametric Conditional Mean
Estimators, \textit{Journal of Econometrics}, 52, 225-244.

\bibitem{}  Mooney, C.Z. and Duval, R.D. (1993), \textit{Boostrapping:A
Nonparametric Approach to Statistical Inference}, Calif: Sage Publications,
Inc.

\bibitem{}  Pagan, A.R. and Sabau, G.W. (1992), ``Consistency Tests for
Heteroskedasticity and Risk Models'', \textit{Estudios Economicos}, 7, 3-30.
\end{thebibliography}

\end{document}