-13-
The dynamic factor model in VAR form (FAVAR) can be obtained by combining
factor analysis (equation 1) with a VAR model (equation 2):
(1) Xt = AFt + D ( L ) Xt _1 + vt
(2) Ft = Φ( L ) Ft _i + Gηt
where q ≤ r ≤ qp with r static factors Ft and q dynamic factors. A is a n × r
matrix, D(L) is a n × n matrix lag polynomial of order p , Φ(L) is a r × r matrix lag
polynomial of order p , G is r× q , ηt is a r -variate vector of global shocks driving the
common factors, vt equals a n -variate vector of idiosyncratic shocks. Substituting the
factor evolution equation (2) into equation (1) and collecting terms yields the complete
FAVAR form:
(3)
Ft
Xt
Φ(L) 0
AΦ(L) D(L)
F-
Xt-1
SFt
εxt
where
SFt
ε Xt
Gηt +
0
vt
The FAVAR contains the exclusion restriction implied by factor analysis, i.e. Xt
does not predict Ft given Φ(L)Ft-1 . Restrictions of this form closely resemble the
assumption of exogenous world variables which are used to identify global shocks in
open economy VARs. The FAVAR can be estimated via a two-step principal component
approach. In the first step, the common components Ft are estimated using the first r
principal components of Xt . In the second step, a VAR is estimated on these common
components. Bernanke, Boivin and Eliasz (2005) point out that this two-step approach
implies the presence of “generated regressors” in the second step. However, the
uncertainty in the factor estimates should be negligible when N is large relative to T.
In the first step, estimates of the common factors are obtained by dividing the data
set Xt into categories of variables. These categories are capturing different dimensions of
the economy across countries (Xt1, Xt2,..., XtI) : economic activity as reflected by real
GDP; inflation which include consumer prices, producer prices, import prices and the
GDP deflator; commodity prices which include the HWWI and the CRB commodity
price index in domestic currency; house prices; monetary liquidity which include broad
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