Page 304 - Special Topic Session (STS) - Volume 3

STS544 M. Camachoa et al.
and typically refer to either growth rates of hard indicators or the level of soft
indicators. Without any loss of generality, we will consider a set of only two
economic indicators, which implies = ( , )′. The single-index dynamic
2

1
factor approach attempts to forecasting a targeted time series, , by using its

own dynamics and the dynamics of a set of indicators, that capture

aggregate economic activity.
Let us assume that the variables used in the model admit a dynamic factor
representation. In this case, the -th indicator of the model can be written as
the sum of two stochastic unobserved components: a common component,
, which represents the overall business cycle conditions, and an idiosyncratic

component, , which refers to the particular dynamics of the time series. Both

the unobserved index and the idiosyncratic component are modeled as having
linear stochastic structures. In particular, the underlying business cycle
conditions are assumed to evolve with ( ) dynamics

= + ⋯ + − + , (1)

1 −1

where ~. . . (0, ).

2

Apart from constructing an index of the business cycle conditions, we are
interested in computing accurate short-term forecasts of . To compute these

forecasts, we start by assuming that the evolution this time series depends
linearly on and on their idiosyncratic dynamics , which evolves as an

( ),

= + (2)

= −1 + ⋯ + + . (3)

1
−
where ~. . . (0, ).

2

In the same way, the economic indicators also admit a common factor
representation, in the sense that they depend contemporaneously on and

on their idiosyncratic dynamics, . Again, the idiosyncratic components of the

n monthly indicators can be expressed in terms of autoregressive processes of
orders

= + (4)

= + ⋯ + + , (5)
1 −1

−
where ~. . . (0, ) = 1,2.

2

The main identifying assumption is that the co-movements of the multiple
time series arise from the single source of the common factor . This is made

precise by assuming that ( , , , ) are mutually and serially uncorrelated
1
2

at all leads and lags. In addition, the scale of is identified by setting the

normalization restriction that = 1.
2

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