3 Our Approach
3.1 Methodology
Consider a vector autoregression in reduced form,
Yt = B(L)Yt-I + ut, E[utut] = Σ
for some vector of variables Yt, coefficient matrices B(L) and a variance-
covariance matrix for the one-step ahead prediction error Σ. The key to
identification is to represent the one-step ahead prediction error ut as a linear
combination of orthogonalized “structural” shocks,
ut = Avt , E [utu′t] = I
Traditional identification strategies impose a recursive ordering or structural
restrictions on A or A-1. Here, we use the methodology of sign restrictions
as in Uhlig (2005).
As a consequence, it is not necessary to identify all structural shocks.
Identifying a single shock is equivalent to identifying an impulse vector:
Definition 1 The vector a ∈ Rm is called an impulse vector, iff there is
some matrix A, so that AA = Σ and so that a is a column vector of A.
Simple matrix algebra shows that any impulse vector a can be character-
ized by
a = Aa, (5)
where AA' = Σ is some decomposition of Σ and a is an m-dimensional vector
of unit length. Let ri(k) ∈ Rm be the vector response at horizon k to the i-th
shock in a Cholesky-decomposition of Σ. Then, the impulse response ra(k)
for a is given by
∞
ra(k) = ai ri(k). (6)
i=1
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