ft = Dft-1 +εt (2)
where D is the r × r autoregressive coefficient matrix and εt is an r × 1 vector of (reduced
form) innovations. The coefficient matrix D and the residuals εt of the VAR(1) model in
(??) are estimated by OLS (once the r factors ft have been retrieved in the first stage of the
analysis). Then, an r × q matrix R is obtained using the following eigenvalue-eigenvector
decomposition of Σ (which is the sample covariance matrix for the innovations, εt, in ??):
R=KM (3)
In particular, M is a diagonal matrix having the square roots of the q largest eigenval-
ues of Σ on the main diagonal; K is an r × q matrix whose columns are the eigenvectors
corresponding to the q largest eigenvalues of Σ. The matrix R measures the relationship be-
tween the r dimensional vector of reduced form innovations, εt , and a q dimensional vector
of common shocks ut (with q < r):
εt = Rut (4)
From equations (??) and (??) we can observe that the matrix R measures the impact
of the common shocks ut on each factor ft and it is crucial in retrieving the impact of the
common shocks ut on each series of the dataset x (via equation ??).
3.2 Density forecast from the principal components model
In this section we describe how we obtain the density forecasts of the EMP index associated,
with the unobservable common shock ut (interpreted as a regional vulnerability indicator).
Given that crisis events are related to the distribution tail of the EM P index, the focus of the
forecasting exercise in this paper is not on average scenarios, but on the adverse realisations
of shocks either common or specific to each variable in the dataset x. For this purpose, in
this section we show how to obtain predictions (corresponding to adverse scenarios) from the
Dynamic Factor model described. The first model we consider is the augmented Dynamic
Factor model (see Stock and Watson, 2002) which gives the following projection of the
unstandardised EMP index of the j-th country:
EMPj,t+ι = a о + XX αk EMPit+1 -k + X βift +1 + j+1 (5)
k=1 i=1
In this model, fft+1is the one step ahead prediction for the ith component of the r di-
mensional vector of factors, given by Dift, where Di is the i-th row of D and ft is the last