1042 The Econometrics of Exchange Rates

also, from (22.37):

plim(̂θ− 1 )=θ− 1 =

Cov(−rpt+ (^) t+n,Etst+n−st+rpt)
Var(ft−st)

−Var(rpt)−Cov(rpt,Est+n−st)
Var(ft−st)

. (22.42)

We note from (22.41) that a negative estimate ofθimplies thatCov(rpt,Est+n−st)<

0. This is the important point made by Fama (1984), that negativity of estimates
   ofθrequire a negative covariation between the risk premium and the expected
   rate of depreciation. In addition, this covariation has greater absolute magnitude
   thanVar(Etst+n−st). From (22.42), a negative estimated coefficient implies that
   Var(rpt)has greater absolute magnitude thanCov(rpt,Est+n−st).
   A time-varying risk premium is well motivated. For example, in a consumption
   capital asset pricing model (CAPM) framework, assuming logarithmic utility and
   that all variables are jointly lognormally distributed, we can derive that:

ft−Etst+ 1 =0.5Vart(st+ 1 /st)−covt([st+ 1 /st].pt+ 1 /pt)

−δcovt([st+ 1 /st]. log(ct+ 1 /ct)), (22.43)

whereδis the coefficient of relative risk-aversion.^41 In general, optimizing models
built on microtheoretic underpinnings will imply that the risk-premium depends
on the variance of the exchange rate. As with other asset prices, there is consider-
able evidence of time-varying volatility in spot exchange rates at high frequencies.
Various authors have employed extensions of the work of Engle (1982) and esti-
mated multivariate GARCH models and included the own conditional variance
in the mean equation (see, e.g., Baillie and Bollerslev, 1990; Bekaert and Hodrick,
1993). For instance, the basic idea (without the multivariate generalization) is to
estimate:

st+ 1 −st=α 0 +α 1 (ft−st)+α 2 ht+ 1 + (^) t+ 1 , (22.44)
ht+ 1 =δ 0 +δ 1 ht+δ 2
t^2 +δ 3
∣
∣ft−st
∣
∣, (22.45)
where variables are defined as above andhtis the conditional variance of the error
term. The absolute difference
∣∣
ft−st
∣∣
is included in the variance equation based
on empirical observation by Hodrick (1989).
GARCH effects disappear with aggregation (Drost and Nijman, 1993) and so are
not usually statistically significant in low-frequency data such as monthly or quar-
terly. Also consumption and price data are not available at weekly or daily levels.
Consequently, some of the terms in the risk-premium have to be assumed constant
in empirical work. Nevertheless, it would appear that a time-varying risk-premium
does not rationalize the forward premium anomaly.^42 However, it is interesting
in this context to note the results reported by Flood and Rose (1996). They found
that estimates ofθwere positive in credible periods in the EMS target zone when
risk-premia might,a priori, be expected to be smaller in magnitude.^43