E4708. Financial Data Analysis. Professor S. Kou.
Final Solution, Dec. 22, 2011. 1:10pm-4:00pm.
Closed Book Exam. Total 100 pts.
1. (a) (2 pt) Market returns, individual stock returns, and the CPI. We use nonlinear SUR
because the Black CAPM implies a non
Financial Data Analysis
Professor S. Kou, Department of IEOR, Columbia University
HWK 8 Solution
1. (a) Since
Ri
2
N ( i;
Xi jRi
);
N (Ri ; 1);
we have the posterior density
f (Ri jXi ) / f (Xi jRi )f (Ri )
1
p exp
2
=
/ exp
= exp
= exp
Therefore,
(
(
1
(
Financial Data Analysis Professor S. Kou, Department of IEOR, Columbia University Lecture 1.b Review of Mean Variance Analysis
1
Review of The Mean Variance Analysis
Consider a one period economy as before with N risky assets, and a money market account w
Data Analysis for Financial Engineers Professor S. Kou Department of IEOR, Columbia University Lecture 5. The second attempt to model the dependent structures: MA and ARMA models
1
1.1
The Moving Average Model
Introduction to the moving average model
In t
Financial Data Analysis. Mathematical Formula Sheet. -distribution 1 2 = 1 and 2 , respectively. 1 = 2 1 1 where 2 1 is a 2 distribution with d.f. 1 . Leptokurtic 4 3 Distributions. The kurtosis is dened as = (4) Skewness. = (3) .
2 . 2 1 1 2 2 2
where 2
E4709. Data Analysis for Financial Engineering. Professor S. Kou.
Midterm, Oct 20, 2011. 2:20pm-4:20pm.
Closed Book Exam. Total 100 pts.
1. (26 pts) Basic Facts from Empirical Finance.
(a) (2 pts) What is the continuously compounded return if the correspo
Financial Data Analysis Professor S. Kou Department of IEOR, Columbia University Lecture 6. Seasonality, Unit Root Test, and Tests of Stationary
1
Seasonality
Last time we learned MA, ARMA, and ARIMA models, which works for stationary time series without
E4708. Statistical Inference for FE. Professor S. Kou.
Final Exam, Aug 25, 2011, 2pm to 4:30pm.
Closed Book Exam. Total 45 pts.
1. (6 pts) Basic Facts.
(a) (1 pt) Describe the analogies of the null hypothesis and type I error in a murder trial.
(b) (1 pt)
E4702. Statistical Inference for Financial Engineering. Professor S. Kou.
Midterm, August 11, 2012. 2pm-4:30pm.
Closed Book Exam. Total 40 pts.
x
1. a. (4 pts) If x < 0, then P ( n( Mn ) x) = P (
Mn
Mn > 0:
n ) = 0, because
If x 0, then
P(
n(
Mn )
x)
x
+
Data Analysis for Financial Engineering Professor S. Kou, Department of IEOR, Columbia University Lecture 3. Fitting Stock Return Distributions and Introduction to Dependent Structures of Stock Returns
1
1.1
Preliminary Denitions of Returns
Simple Returns
Columbia University Department of IEOR Data Analysis for Financial Engineers E4709, Fall 2010 R 2:40pm-5:10pm, 303 Mudd Columbia Course Work Web Page Prof. Steven Kou 312 Mudd Building [email protected] Tel: 212-854-4334 Office Hours: TA: e-mail: TA Offic
Columbia University M.S. Program in Financial Engineering
IEOR 4709:
Data analysis for Financial
Engineering
Tuesday 4:10 PM- 6.40 PM
Instructor: Rama CONT
Teaching Assistant: Ka Chun [email protected]
Lecture 2: Probabilistic tools.
1
Lecture 2: Probab
Bayesian Statistics
1
Bayesian Hypothesis Testing
In Bayesian hypothesis testing, we use the posterior probabilities of the null and alternative hypothesis. More specifically, consider the null hypothesis H0 : 0 versus H1 :
/ 0 . The Bayesian
computes P(
Columbia University
Statistical Analysis and Time Series
IEOR-4709
A. Capponi
Spring 2017
Problem Set #2
Issued:
February 7, 2017
Due: BEFORE CLASS February 22, 2017
Note: Please put the number of hours that you spent on this homework set on top of
the fi
Lecture 4: Consistency, Unbiasedness, Method of Moment
Estimator, and Maximum Likelihood Estimator (MLE)
1
Recapt from Last Time
An estimator of a parameter is a function f (X1 , X2 , . . . , Xn ) of the samples X1 , . . . , Xn . It is
unbiased if E [f (X
Likelihood Ratio Statistics, and Asymptotic Distribution
1
Likelihood Ratio Tests
Last time we discussed hypothesis testing. Consider two simple hypothesis, that is H0 : = 0
versus H1 : = 1 . Define = P(accept H0 |H1 ). The power of the test is 1 = P(acce
Likelihood Ratio Statistics, and Asymptotic Distribution
1
Likelihood Ratio Tests
Last time we discussed hypothesis testing. Consider two simple hypothesis, that is H0 : = 0
versus H1 : = 1 . Define = P(accept H0 |H1 ). The power of the test is 1
= P(acce
Statistical Tests and Bayesian Statistics
1
Goodness of Fit Tests
Last time, we have seen goodness of fit tests. These are used to test the hypothesis: H0 : pi = i
for all i, versus H1 : pi 6= i for some i. The probability distribution is what we expect t
Bayesian Statistics
1
Introduction
The Bayes formula states:
P (H|D) =
P (D|H) P (H)
P (D)
If the prior and likelihood are known for all hypotheses, then Bayes formula computes the posterior
exactly.
Example: Consider two urns, A and B: urn A contains con
Goodness of Fit Tests
1
Goodness of Fit Tests
Example: In practise, it is always very difficult to estimate the probability distribution of default
(bankruptcy events). Moodys KMV has defined the concept of distance to default, to capture the
likelihood o
Lecture 9: Hypothesis Testing
1
Hypothesis Testing
Consider independent and identically distributed samples X1 , X2 , . . . , Xn coming from a Gaussian
distribution N (, 2 ) with unknown mean and unknown standard deviation . We want to know
whether = 0 fo
Asset Allocation
1
Asset Allocation
Classical portfolio theory, as it existed in about 1975, has two main parts. The first is mean variance
analysis and constitutes a way to allocate assets in a world of risk and return. The investor is
assumed to know th
Midterm Sample Exam
Note: This exam is closed book, closed notes, calculators allowed. Show all
your work to receive full credit. Please, clearly write your first and last name on
the first page of the exam. The exam time is from 10:05-11:25. Please, retu
Columbia University
Statistical Analysis and Time Series
IEOR-4709
A. Capponi
Spring 2017
Problem Set #3
Issued:
March 1, 2017
Due: BEFORE CLASS March 20, 2017
Note: Please put the number of hours that you spent on this homework set on top of
the first pa
Lecture 8: The EM Algorithm
1
The Expectation Minimization (EM) Algorithm
Suppose we have an estimation problem in which we have m independent samples of a random
variable X, cfw_x1 , x2 , . . . , xm . We wish to fit the parameters of a model which depend
Estimation in Asset Allocation
1
Announcements
The midterm exam will be on Wednesday, March 22, from 10:05am to 11:25am, Location
Mudd 1121.
Midterm review sheet has been posted. It contains all information related to the midterm
exam.
Midterm sample e
COLUMBIA UNIVERSITY
Midterm Review Sheet
IEOR E4709: Statistical Analysis and Time Series
Spring 2017
Instructor: A. Capponi
Information
The midterm exam will be held on Thursday, March 22, from 10:05am to 11:25am,
Location Mudd 1121.
The exam will be clo
Columbia University M.S. in Financial Engineering
IEOR 4709:
Data analysis for Financial
Engineering
Lecture 10: Tails and extreme returns
Instructor: Rama CONT
Teaching Assistants:
MA Ka Chun [email protected]
KIM Sang Won [email protected]
1
Tails a
Lecture 3: Consistency, Unbiasedness, Method of Moment
Estimator, and Maximum Likelihood Estimator (MLE)
1
Recapt from Last Time
Last time, we have seen the different notions of convergence of a sequence of random variables. Let
cfw_Xn n1 be a sequence of
function [f,axe]=kernelregression(x,y)
% function [f,axis]=kernelregression(x,y)
% Computes the Nadaraya Watson estimator for a nonlinear regression of y(t)
% on x(t). Uses the kernel defined in kernelfunction.m
% Rama Cont, 2006.
dx=(max(x)-min(x)/length