2.9 (a), (b)
(a) Since f is a smooth function of , we can switch the order of dierentiation and expectation,
E( log f (X1 , . . . , Xn ) = E
=
=
f (X1 , . . . , Xn )
f (X1 , . . . , Xn )
f (X1 , . . . , Xn )
f (X1 , . . . , Xn )dx1 dxn
f (X1 , . . . , Xn
#Although results are not given in the solution, you should provide necessary values,
#plots or tables in your assignment.
#Unless the formula you used can be easily read from the code, e.g. mu <- mean(x)
#otherwise you have to write out the formula you u
Hints to Assignment 2
October 31, 2012
Problem 2.9
No need to decompose the joint density as log f (Xn , . . . , X1 ) = log f (Xt |Xt1 . . .),
instead use the joint density directly in your proof. Just remember all it matters is f being a density and the
Stat240: Homework 1 - due at beginning of class on Friday October 21, 2011
LX = Lai and Xing, Statistical Models and Methods for Financial Markets
1. Problem 1.7 in LX.
2. Problem 2.3 in LX.
3. Problem 2.5 in LX.
4. Problem 3.2 in LX.
5. (Optional) Proble
#Codes and results must be provided for full credits
#Unless the formula you used can be easily read from the code, e.g. mu <- mean(x)
#otherwise you have to write out the formula you used but not just provide the codes.
#5.9
library(forecast)
library(tim
STATS 240 Team Project
Due Dec 5, 2014 by 5pm
Each team can consist of 1 to 4 students. All members names should be included in the
project; each member of the same team will get the same grade.
Hand in to the head TA Milan, or slip it into her o ce.
1. (
Black-Litterman Asset Allocation and
Mean-Variance Portfolio Optimization when
Means and Covariances of Asset Returns are
Unkown
Tze Leung Lai
Stanford University
2014
1 / 31
Outline
Review of Markowitzs portfolio optimization theory and Black-Litterman
a
Black-Litterman Asset Allocation in a Bayesian
Framework
Black and Litterman start with a normal assumption for the asset return rt at period t with
expected return :
rt |, N (, ).
(1)
To simplify the problem, they implicitly assumed that is known ( is es
Tze Leung Lai Haipeng Xing
Statistical Models and
Methods for Financial
Markets
123
Haipeng Xing
Department of Statistics
Columbia University
New York, NY 10027
USA
[email protected]
Tze Leung Lai
Department of Statistics
Stanford University
Stanford
Introduction to Black-Litterman Asset Allocation in a
Bayesian Framework
October 30, 2012
Although Markowitzs mean-variance portfolio optimization is sound in theory, in practice
it has many issues. The main problem, a.k.a. the Markowitz enigma, is that t
Part II
6
Method of maximum likelihood (Sect. 2.4.1)
Suppose X1 , . . . , Xn have joint density
function f (x1 , . . . , xn ) where is an
unknown parameter vector written as a
column vector of dimension p.
The likelihood function based on the
observatio
An Introduction to R
Notes on R: A Programming Environment for Data Analysis and Graphics
Version 2.13.0 (2011-04-13)
W. N. Venables, D. M. Smith
and the R Development Core Team
Copyright
Copyright
Copyright
Copyright
Copyright
c
c
c
c
c
1990 W. N. Venabl
Part I
9
Linear regression models and
ordinary least squares OLS
(Sect. 1.1)
A linear regression model relates output
(or response) yt to q input (or predictor) variables xt1 , . . . , xtq , also called
regressors, via
yt = a + b1 xt1 + + bq xtq +
t
(10)