Exchange Rate Economics: Theories and Evidence

144 Empirical evidence on the monetary approach
at the rate T 1/2 ,in the case of stationary processes,least squares estimates of
non-stationary but cointegrated processes converge at a rate T . This means that,
asymptotically,endogeneity will have a negligible effect on the coefficient estimates
(in finite samples,however,endogeneity biases can still be significant – see Banerjee
et al. 1986).
Again,paralleling the cointegration-based studies of PPP discussed in Chapter 2,
the first set of cointegration studies of the monetary model relied on the
Engle–Granger two-step method,while later studies used fully modified estimators
such as the Johansen (1995) full information maximum likelihood method. A
summary of a selection of the studies which have used these methods is contained
in Table 6.2. There are a couple of key results generated by this table. First,when
the Engle–Granger two-step method is used,the null of no cointegration is usually
not rejected. However,when the Johansen estimator or other estimators which
include a correction for endogeneity and/or serial correlation of the error term
are used the null of no cointegration is rejected. Indeed,notice that when the
Johansen method is used there is clear evidence of multiple cointegrating vectors.
To illustrate the results obtained for the monetary model using cointegration
methods we take as an example MacDonald and Taylor (1991),who used the
cointegration methods of Johansen (1995) to test model for the German mark–US
dollar exchange rate. In particular,define the monetary vector:
x ′ t =[s t , m t , m ∗ t , y t, y ∗ t , i t, i ∗ t ],(6.17)
and assume it has a VAR representation of the form:
x t = η +
p∑
x t + ε t ,(6.18)
i=1
where η is a (n × 1) vector of deterministic variables,and ε is a (n × 1) vector
of white noise disturbances,with mean zero and covariance matrix . Expression
(6.18) may be reparameterised into the vector error correction mechanism
(VECM) as:
p−1
∑
x t = η + i x t−i + x t−1 + ε t ,(6.19)
i=1
where denotes the first difference operator, i is a (n × n) coefficient matrix
(equal to − ∑ p
j=i+1 j), is a (n × n) matrix (equal to ∑ p
i=1 i − I ) whose
rank determines the number of cointegrating vectors. If is of either full rank,
n,or zero rank, = 0,there will be no cointegration amongst the elements in
the long-run relationship (in these instances it will be appropriate to estimate the
model in,respectively,levels or first differences).
If,however, is of reduced rank, r (where r < n),then there will exist (n × r)
matrices α and β such that = αβ ′ where β is the matrix whose columns are