The Importance of Global Shocks for National Policymakers: Rising Challenges for Central Banks

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specify latent factors which capture the overall dynamic features of the international
economies (Fry, 2004).

Recent work with Factor-Augmented Vector Autoregression (FAVAR) suggests
that standard VAR analysis can be improved by incorporating the information in a large
number of macroeconomic series. A general formulation of the dynamic factor model is
X_it = Λ_i(L) f + ε_it with X_it as the observed data for the macroeconomic time series i at
it i t it it

time t for i = 1,...,N and t = 1,...,T . If the lag polynomials Д.(L) are assumed to have
finite orders, the equation can be written in static form: X_t = AF_t + ε_t. Hence, the factor
F_t can be thought of as a weighted average of the variables in a data set. The factor
loadings A , i.e. the weights, can be either positive or negative and reflect how correlated
each variable is with the factor. F_t, A and ε_t are not directly observable and have to be
estimated. To separate factors from idiosyncratic disturbances, the following identifying
assumptions are made (Justiniano, 2003):

■ Orthogonality of idiosyncratic errors, i.e. εt 1 ε¹_t (t)Vi, j = 1,..., N and i ≠ j . Usually,
the assumption of no cross-correlation is relaxed. The model is then said to have an
approximate factor structure.

■ Orthogonality of factors, i.e. f_t^j 1 f_t^kVk, j = 1,..., K and k ≠ j . However, factors can
be correlated in time.

■ Idiosyncratic errors are orthogonal to factors, i.e. ε_tⁱ 1 f_t^jVi = 1,..., N and j = 1,..., K .

These assumptions imply that all co-movements across variables are attributed
exclusively to a set of orthogonal factors. Stock and Watson (1998, 2002) show that the
factors in a model of the form X_t = AF_t + v_t can be consistently estimated by principal
component analysis when the time series dimension (T) and the cross-section dimension
(N) are large. The factors are extracted in a sequential fashion, with the first factor
explaining the most variation in the data set, the second factor explaining the next most
variation (not explained by the first factor), and so on.

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