Page 370 - Special Topic Session (STS) - Volume 3

STS550 Kyle Hood et al.
and so in the literature sometimes do not cover the entire period of time being
nowcasted. In this case, however, we have access to all three months of data
that cover our sample period. We convert these data to a quarterly value and
use the quarterly data in the GB model. The regression specification for a
single detailed component is

= + ∑ + ∑ ∑ ̅ + . (1)

−1
, ,−
=1 =1 =0
Here, t is time (quarter) and j(1, ... , ) indexes indicators if there are more than
one. yt is the dependent variable (the nowcast target), while ̅ j,t-1 represents
the quarterly average of a monthly indicator, and ∈t is the error term. j,i and
i are collections of parameters on the lag polynomials, and c is a constant.
Note that this regression is estimated separately for each detailed component
(i.e., there is no pooling of observations across cross-sections). p and k are the
number of lags used for the dependent variable and for the indicators and are
assumed to be equal to each other in most of the specifications below.
Because of its use of indicators, we sometimes refer to this as an “indicator
model.”
Our BF model is very similar in its basic specification to the GB model, except
that the indicators (x’s) in Equation (1) are replaced with common factors,
denoted with an ƒ (and of which there are r), namely,

̅
= + ∑ + ∑ ∑ ̅ + . (2)
−1
, ,−

=1 =1 =0
These common factors, meant to capture movements in the entire collection
of (monthly) indicators, are assumed to be related to the (monthly) indicators
via
xm = ⋀ ƒm + ζm (3)
Note that we have omitted the bars above these variables (which indicate
quarterly averages of monthly variables) and replaced the time subscript with
an m ∈ {1, ... ,M}, indicating that these variables are expressed at a monthly
frequency. Here, the Λfm is known as the “common component” of xm while the
ζm represents the “idiosyncratic component” of xm. Λ is an n×r matrix of “factor
loadings” which is constant over time (and where n is the number of indicators)
and fm is an r×1 vector of common factors for month m. The factor model is
estimated via principal components analysis (PCA), which is appropriate if any
cross-sectional dependence in xm coming from the idiosyncratic component
(ζm) is transitory. The PCA procedure produces a collection of min{M,n}
mutually-orthogonal monthly factors which can be fed into Equation (2). The
Bai-Ng criterion (Bai and Ng, 2002) is used to select the number of factors that

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