Exchange Rate Economics: Theories and Evidence

380 Spot and forward exchange rates
Hansen and Hodrick (1983) test (15.26) by assuming that the conditional
covariances between returns and the marginal rate of substitution move in proportion
across assets over time. With this assumption the ratios of covariances in
(15.24) or β in (15.26) are constant. Their approach may be viewed as a latent
variable approach because R b is unobservable and it implies a set of overidentifying
restrictions. With β assumed constant,and forward forecast errors used as
instruments,Hansen and Hodrick find that the overidentifying restrictions are not
rejected. However,in an alternative set of tests Hodrick and Srivastava (1984) find
that the overidentifying restrictions are rejected when f −s,the forward premium,
is used as an instrument. There are a large number of other tests of the latent
variable model and these produce mixed results (see also Hodrick and Srivastava
1986; Campbell and Clarida 1987; Giovannini and Jorion 1987). However,even
tests which do not reject the overidentifying restrictions are unable to provide a
measure of how much of the forward premium puzzle is explained by the risk
premium term. Cumby (1990) and Lewis (1991) argue that one reason for the
rejections of the model could be due to the auxiliary assumption that covariances
move in proportion to each other. They note that this condition only seems to hold
at longer horizons and when long horizon returns are used there is less evidence
of rejection; however,this failure to reject at long horizons could simply reflect the
low power of the test.
15.3.2.3 Hansen–Jaganathan bounds
The Hansen–Jaganathan approach uses combinations of excess returns to provide
a lower bound on the volatility of the inter-temporal marginal rate of substitution in
consumption – the Q t term. This lower bound is seen as a useful empirical tool for
comparing excess returns – here the excess return from taking a forward position –
with the implications of a particular model,in this case the general equilibrium
model of Lucas. The derivation of these bounds may be illustrated in the following
way (see,for example,Lewis 1995). Consider again (15.27):
E t {Q t+1 ex t+1 }=0,
where the j superscript has been dropped.
If it is assumed that Q t can be written in terms of a simple linear projection as:
Q t+1 = δ 0 + δ ′ ex t+1 + e t+1 ,(15.31)
where e is the error term. Using the standard OLS formulae the parameter vector
can be written as:
δ = ∑ −1
[E(Q t+1 ex t+1 ) − E(Q t+1 )E(ex t+1 )],(15.32)
which given (15.27) implies
=− ∑ −1
E(Q t+1 )E(ex t+1 ),
where ∑ is the variance,or variance covariance matrix (if ex is a vector) of ex.