Exchange Rate Economics: Theories and Evidence

Spot and forward exchange rates 389
[ ]
σss σ
and let sf
be the var–covar of (ε
σ fs σ st ε ft ). It follows then that the parameters
ff
in (15.4) are related to (15.3) as follows:
b 0 = σ sf σ −1
ff
, b i = b si − σ sf σff
−1 bf i , c i = c si − σ sf σff −1 c fi ,
and the noise term ε t = ε st − σ sf σff
−1 ε ft . That equation (15.4) is a special case of
(15.53) can be seen by rearranging (15.53) as:
s t =−α s β 0 − α s (s t−1 − f t−1 ) + b 0 f 1 + α s (1 − β 1 )f t−1
k−1
∑ ∑
+ b i s t−i + c i f t−1 + ε t ,(15.55)
i=1
k−1
i=1
which will degenerate to (15.1) when b 0 = 0, β 1 = 1, b i = 0, c i = 0, i =
1, ..., (k −1). b 0 = 0 means that the contemporaneous cross-equation covariance
in the VAR of the spot and forward rate, σ sf ,in the vector autoregression of the
spot and forward rate must be zero. And the lag length in the VAR,given b i = 0,
c i = 0,must be unity. So for equation (15.1) to be valid the following conditions
must hold:
1 The spot and forward rate must be cointegrated.
2 The slope of the cointegrating vector must be unity.
3 The forward rate must be weakly exogenous – the derivative market must
drive the underlying market.
4 The cross-equation residual covariance in the VECM must be zero.
5 The lag length of the VECM must be exactly 1.
MacDonald and Moore consider to what extent these conditions are met for
10 currencies against the USD and DM,for the period 1978–1994. In the vast
majority of cases it turns out that it is in fact the spot rate change that is weakly
exogenous and not the forward rate and so the forward rate puzzle may simply be
a function of the chosen regression equation rather than saying anything else.
The MacDonald and Moore finding is consistent with McCallum’s (1994) interpretation
of the forward premium puzzle. McCallum argues that if central banks
target the interest rate according to the function:
i t − i ∗ t
= λ(s t − s t−1 ) + σ(i t−1 − i ∗ t−1 ) + ξ t,(15.56)
and if the risk premium follows a first-order autoregressive process,with coefficient
ρ,the coefficient in the basic forward premium representation will be equal to
(ρ − σ) −1 λ,and if ρ is small relative to σ this would give a negative coefficient in
the standard forward premium unbiasedness equation.
Abadir and Talmain (2005) demonstrate that the forward premium puzzle may
be explained because of a failure of researchers to model the non-linear long memory
properties of the series used to test the unbiasedness of the forward premium.