Exchange Rate Economics: Theories and Evidence

Real exchange rate determination 215
and C 0 is an n × n matrix which describes the contemporaneous correlations
among the disturbances:
x t = C 0 e t + C 1 e t−1 + C 2 e t−2 +··· (8.33)
Identifying an estimated reduced form VAR system,such as (8.31),becomes a
matter of choosing a unique value for C 0 . Since C 0 contains n × n elements,
identification requires imposing n 2 restrictions. The structural disturbances and
the reduced form residuals are related as e t = C0 −1 ε t,and so the choice of C 0 also
implies the choice of the covariance matrix:
= C0 −1 −1′
C0 ,
where is the variance covariance matrix of structural disturbances. Maximum
likelihood estimates of and C 0 can be obtained through sample estimates of .
As we have said,identification requires choosing n 2 elements of C 0 . Almost all
approaches to identifying VAR models start with restricting the n(n +1)/2 parameters
of the covariance matrix . The first n of these usually come from normalising
the n diagonal elements to be unity,while the remaining n(n − 1)/2 restrictions
come from assuming that the structural shocks are mutually uncorrelated
or orthogonal. Taken together,these restrictions on C 0 imply:
C0 −1 −1′
C0 = I ,
where I is the identity matrix. This leaves a further n(n −1)/2 restrictions on C 0 to
fully identify the model. The approach adopted in all of the papers in the SVAR
literature is to assume that C 0 is equal to C(1),the long-run coefficient matrix on
the structural shocks,and to impose the remaining restrictions by assuming that
this is lower triangular. This means that the long-run coefficient matrix has a Wold
representation and the particular ordering of variables is achieved by appealing to
economic theory.
Latrapes (1992) was the first to apply an SVAR decomposition to real exchange
rate behaviour. In particular,he estimated bivariate VAR models consisting of
real and nominal exchange for five US dollar bilateral rates over the period March
1973–December 1989. Using the identification methods described earlier,he
extracted two shocks – a real and nominal – and he restricted the nominal shock to
have a zero long-run impact on the level of the real exchange rate. A set of variance
decompositions showed that real shocks were the predominant source of both real
and nominal exchange rate behaviour for the sample,and he interprets this as
evidence favouring the Stockman–Lucas view of exchange rate determination (see
Chapter 4). However,as Lastrapes recognises,his results may be a reflection of
the two dimensional nature of his system: with multiple real and nominal shocks
the results could turn out to be different. Clarida and Gali (1994) use the SVAR
approach to decompose real exchange rate behaviour into three shocks and since
there work has become something of a benchmark in this literature,we consider
it in some detail here.