Exchange Rate Economics: Theories and Evidence

Empirical evidence on the monetary approach 161
law of iterated mathematical projections,and using the past three equations,we
can obtain:
∞∑
( ) i
g ′ β1
z t =
h ′ A i z t ,
1 + β 1
i=1
= h ′ φA(I − φA) −1 z t ,(6.47)
where φ = (β 1 /1 + β 1 ). If equation (6.47) is to hold non-trivially,the following
2p parameter restrictions can be imposed on the VAR:
g ′ − h ′ φA(I − φA) −1 = 0,(6.48)
and post-multiplying (6.48) by (I − φA) we can get a set of non-linear restrictions
on the VAR for (ζ t , x t ) ′ :
H 0 : g ′ (I − φA) − h ′ φA = 0. (6.49)
The non-linear restrictions test is the third formal first test of the forward-looking
monetary model. Fourth,a variance bounds test,popularised by Shiller (1980) in
the context of the stock market literature,may be calculated using this framework.
For example,if expectations are rational then it must follow that:
ˆζ t = ζ t + u t ,(6.50)
where u t is a random error and with rational expectations must be uncorrelated
with ζ t . Given that variances cannot be negative,it must further follow that:
Var(ζ t ) ≤ Var(ˆζ t ). (6.51)
Such a test is seen as an improvement over the basic variance bounds tests used
by,for example,Huang (1981) as it addresses the issue of non-stationarity in the
presence of cointegration. A final test of the model can be obtained by noting that
the perfect foresight spread, ˆζ t ,may be calculated as:
ˆζ t = h ′ φA(I − φA) −1 z t .
Campbell and Shiller argue that a simple graphical comparison of the calculated
perfect foresight spread with the actual spread will give a qualitative measure of
how well the model fits.
In terms of the first test of the model,we saw in the previous section that a number
of studies support cointegration of the exchange rate with monetary fundamentals
particularly when the methods of Johansen or panel methods are used. Consider
again the results of MacDonald and Taylor (1993),who use a data period spanning
January 1976–December 1990 for the DM–USD. As we noted in Section 6.3.1,
clear evidence of cointegration was found for the general system which included
relative interest rates. The evidence of cointegration also existed for the base-line