Page 62 - Special Topic Session (STS) - Volume 2

STS459 Gan C.P. et al.
the -th row represents the value of macroeconomic variables in the -th

quarter. We form the -th sub-table using the first to − 1 + rows of

the table.
A factor model with its static representation given by
= ᴧ + ,
∗
(2.1)
may be used to describe the vector . In Equation (2.1), ᴧ is an × matrix
∗

of factor loadings, is an × 1 vector of common latent factors underlying
∗
and is a × 1 vector of random errors.

We perform a principal component analysis of the columns of the

observations in the -th sub-table. Suppose the principal component with

∗
the -th largest variance is . We obtain the first principal components

( < ( + )) , , … , ∗ . Suppose is the value of extracted from
∗

2

1
the -th row of the sub-table. The row vector = ( , , … , ∗ ) then
2

1

∗
represents the values of important latent factors in the -th quarter.
Whenever the first value of the vector represents the value of the
macroeconomic variable in the -th quarter, we replace this first value by the
value of which represents the values of important latent factors in the -
∗

th quarter. In this way we can obtain the -th window of × ( − 1 + )

rows, each of which represents of an updated value of .
∗
The data for in the -th window is fitted with an [ + ( − 1) +

3( − 1)] -dimensional MPN distribution. From the fitted MPN distribution, a
large number of the values of are generated. The components of are

∗
transformed to the vector (1) of which the first components gives the
values of the latent factors, the ( + )-th component represents the
∗
∗
index of the company, while the last 3 components are the ratings in the
previous, present and future quarters.
From the large number of the (1) generated, we form a table consisting
of the values of (1) which correspond to a chosen company and the chosen
ratings in the previous and present quarters. We next form a sub-table by
deleting the + 1 to + columns of the original table. A row in the sub-
∗
∗
∗
table then gives the value of a vector (2) of which the first component are
∗
the value of the latent factors and the ( + 1)-th component is the rating
∗
in the next quarter for the selected company with the specified rating ( ) in
the previous quarter and the rating () in the present quarter.
∗
∗
When the first values of (2) are given by the first values in the -th
∗
row of the sub-table, a conditional distribution is obtained for the + 1 value
of (2) . From the conditional distribution, we obtain the probability that the

∗
( + 1)-th component of (2) lies in the interval . We may investigate the

∗
dependence of the probability on the values of the latent variables given

∗
by the first components of (2) .
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