262 The Basics of financial economeTrics
If we look at the fundamental relationship ΣΨ=+BB' , it is easy to
see that there is no simple solution to this question. In fact, if Ψ is not a
diagonal matrix, we could nest another factor structure that might yield
ΨΩ=+HH'. But nesting another factor structure would increase the num-
ber of factors. That is, if we accept that residuals be correlated, then the
original factors cannot be the only factors given that residuals will also
exhibit a factor structure.
Intuitively, we could split the eigenvalues into two groups, “large” and
“small” eigenvalues so that ΣΨ=+BB' is the sum of two parts, one BB'
due to large eigenvalues and the other Ψ due to small eigenvalues. However,
this is not a theoretically satisfactory solution because the splitting into large
and small eigenvalues is ultimately arbitrary.
In order to find a theoretically rigorous solution, the setting of factor
models was modified by assuming that both the number of time points T
and the number of time series are infinite. This solution was developed by
Stephen Ross, whose arbitrage pricing theory (APT) published in the mid
1970s makes use of infinite factor models.^3 Factor models with correlated
residuals are called approximate factor models. The theory of approximate
factor models was developed in the early 1980s, again in the setting of an
infinite number of both time points and time series.^4 Approximate factor
models allow residuals to be correlated but they are defined for infinite
markets.
The assumption of infinite markets is essential for defining approximate
factor models. This is because the assumption of infinite markets allows for
the distinction between those eigenvalues that grow without bounds from
those eigenvalues that remain bounded. Roughly speaking, in an infinite
market, true global factors are those that correspond to infinite eigenvalues
while local factors are those that correspond to bounded eigenvalues.
Of course this distinction requires a carefully defined limit structure. In
order to define approximate factor models, Chamberlain and Rothschild
first defined an infinite sequence of factor models with an increasing num-
ber of series and data points.^5 It is assumed that as the size of the model
increases, only a finite and fixed number of eigenvalues grow without lim-
its while the others remain bounded. This assumption allows one to define
an approximate factor model as a sequence of factor models such that a
(^3) Stephen Ross, “The Arbitrage Theory of Capital Asset Pricing,” Journal of Eco-
nomic Theory 13 (1976): 341–360.
(^4) Gary Chamberlain and Michael Rothschild, “Arbitrage, Factor Structure, and Mean-
Variance Analysis in Large Asset Markets,” Econometrica 51 (1983): 1305–1324.
(^5) See Chamberlain and Rothschild, “Arbitrage, Factor Structure, and Mean-Variance
Analysis.”