Math 185 Final Projects
Project 1. Consider the multiple-response regression model of the form
X = (X1;:::;Xm)| = + Bf +";
so that for each observation (Xi;fi) satis es
Xi = (Xi1;:::;Xim)| = + Bfi +"i; i = 1;:::;n; (0.1)
where is a m-dimensional unknown mean vector, B = (b1;:::;bd)| 2Rm d is the
factor loading matrix, fi 2Rd is a vector of common factors to the ith observation
and is independent of the noise "i. An iconic example of model (0.1) is the factor
pricing model in nancial economics, where Xik is the excess return of fund/asset k at
time i, fi’s are the systematic risk factors related to some speci c linear pricing model,
such as the capital asset pricing model (CAPM) (Sharpe, 1964), and the Fama-French
three-factor model (Fama and French, 1993).
Consider the problem of simultaneously testing the hypotheses
H0k : k = 0 versus H1k : k 6= 0; for k = 1;:::;m: (0.2)
Although the key implication from the multi-factor pricing theory is that the intercept
k should be zero, known as the \mean-variance e ciency" pricing, for any asset k,
an important question is whether such a pricing theory can be validated by empirical
data. In fact, a very small proportion of k’s might be nonzero according to the Berk
and Green equilibrium (Berk and Green, 2004).
The goal of this project is to investigate the stock market data (monthly data for 500
S&P 500 constituents from 2000 to 2016, available on TritonED) via Fama-French
three factor model of the form. (0.1). In this dataset, stock return is an n m matrix
and factor return is an n d matrix as follows: