244 The Basics of financial economeTrics
Estimation of Factor Models
After defining the two fundamental classical factor models—that is, strict
and scalar factor models—we can now ask three important questions:
- Given the data, how do we determine the number of factors?
- Given the data, how do we estimate parameters and factors?
- Are parameters and factors uniquely determined by our data?
problem of Factor Indeterminacy
Suppose for a moment we know the number of factors. Can we estimate
parameters and factors? The somewhat surprising conclusion is that we can
determine the model’s parameters but we cannot uniquely determine fac-
tors. This is the well-known problem of factor indeterminacy.
Historically, factor indeterminacy provoked much debate in the litera-
ture on statistics. The reason is that factor models were initially proposed in
the area of psychometrics, with factors being the determinant of personal-
ity. Based on factor analysis, psychologists claimed that personality can be
explained almost deterministically in terms of a number of basic factors.
The discovery of factor indeterminacy weakened this proposition.
Finite and Infinite Factor Models In practice, every quantity we deal with in
our personal and professional lives is finite, that is, it can be measured with
an ordinary number. For example, the universe of potential stocks from
which a portfolio can be constructed consists of a finite number of stocks.
The number of candidate stocks can be very large, say in the range of thou-
sands, but still it is finite. The number of dates, or even instants of trading,
is finite.
However, many mathematical properties can be better stated in the limit
of an infinite number of time series or an infinite number of dates and time
points. Mathematicians distinguish many different types of infinity, and
many different types of infinite processes. We cannot go into the details of
the mathematics of infinite quantities. However, the intuition behind infinite
numbers can be stated as follows. To be concrete, let’s define what is meant
by an infinite market formed by infinitely many stock return processes.
Essentially an infinite market means that whatever large number we
choose, as large as we want, the market will have more stocks. A market is
infinite if no ordinary number, regardless of how big we choose it, will be
able to count the market. The same concept can be applied to the concept
of an infinitely long time series: whatever number we choose, the series will
have more points.