The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 438 Wednesday, February 4, 2004 1:08 PM

438 The Mathematics of Financial Modeling and Investment Management

T

∑ztz' Kt

Bˆ = --------------------------t =^1 -

T

∑zKtz' Kt

t = 1

The APT restriction can be tested with Likelihood Ratio methods which

compare the likelihood of the constrained and unconstrained model.

Testing and Estimating APT When Factors are not Portfolios

If factors are not portfolios and if they are given exogenous processes,

multifactor models are multivariate regressions on the factors. If the

regression innovations are assumed to be jointly normally distributed and

no restriction is imposed, models can be estimated with MLE methods

that are, in this case, equivalent to OLS estimates. Writing the multifactor

model in real returns, OLS estimation yields the following results:

aˆ = μμμμˆ – Bˆμˆ

K

T

∑(rt – μˆ^ )(fKt–^ μˆ^ K)

Bˆ = -----------------------------------------------------------t =^1 -

T

∑(fKt – μˆK^ )(fKt–^ μˆ^ K)

t = 1

T

∑

ˆ^1

μ = ----

T

rt

t = 1

T

∑

ˆ^1

μ (^) K = ---- fKt
Tt = 1
Testing the zero intercept restriction from the above estimates can be
performed using MLE methods. Note that in this case only one model is
estimated because factors are given. Should factors be portfolios, the
constrained and unconstrained models yield different factors.