# Econ 703 Doctoral Program

1

Quantitative Methods I
Mid-Term Exam
1. Consider the linear regression model:
y  X  with  ~ N(0, 2I)
where X is an n  k matrix, and I is an n  n identity matrix. The log-likelihood function of this
model with a multivariate normal density for  is:
( ) ( )
2
1
ln
2
ln 2
2
( , 2 ) 2 2  

l    n  n   y  X  y  X
Use the exponential of the estimated log-likelihood function to show that the maximum
likelihood (ML) estimation of this model yields similar results to least squares estimation. Note
that the estimated variance is given by ˆML 2 ˆˆ / n .
(10 Marks)
2. In maximum likelihood estimation, the necessary condition for maximizing the log of the
likelihood function ln L( | X ) is:
( , , ) ln  0

m y X  L
where y denotes the dependent variable, X denotes the sample data of the explanatory variables,
and  is the unknown parameter vector. This condition is, however, a moment condition since
m(y, X ,)  E  lnL    0
where E is the expectation operator. Rewrite this maximization condition using a generalized
method of moments (GMM) approach, and show that the maximum likelihood estimator can be
viewed as a GMM estimator.
(10 Marks)
3. Consider the general non-linear regression model with non-spherical disturbances:
y  f (X,β)  ε ; ε ~ N(0, 2)
2
where y is an n1 vector of n observations on the dependent variable, X is an n k matrix of n
observations on the k regressors, β is a k 1 vector of the regression coefficients,  is a
positive definite matrix of order n, and ε is an n1 vector of the n error terms.
Show how this model parameters can be estimated using the method of generalized method of
moments (GMM).
(10 Marks)
4. Show how to carry out a test of over-identifying restrictions within a GMM estimation
methodology for the non-linear model:
y  f (X,β)  ε ; ε ~ N(0, 2I)
where the notation is similar to the previous question, I is the identity matrix, and potentially
E(X )  0
(10 Marks)

Pages (550 words)
Approximate price: -

Why Choose HelpHub

Quality Researched Papers

We always make sure that writers follow all your instructions precisely. You can choose your academic level: high school, college/university or professional, and we will assign a writer who has a respective degree.

Qualified Writers

We have hired a team of professional writers experienced in academic and business writing. Most of them are native speakers and PhD holders able to take care of any assignment you need help with.

Unlimited Revisions

If you think we missed something, send your order for a free revision. You have 10 days to submit the order for review after you have received the final document. You can do this yourself after logging into your personal account.

On Time Delivery

All papers are always delivered on time. In case we need more time to master your paper, we may contact you regarding the deadline extension. We will always strive to deliver on time.

Original & Confidential

We use several writing tools checks to ensure that all documents you receive are free from plagiarism. Our editors carefully review all quotations in the text.

Our support agents are available 24 hours a day 7 days a week and committed to providing you with the best customer experience. Get in touch whenever you need any assistance.

Try it now!

## Calculate the price of your order

Total price:
\$0.00

How it works?

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

HelpHub Writing Services

No need to work on essay at night. Sleep tight, we will cover your back. We offer all kinds of essay writing services.

## Essay Writing Service

No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.

Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.

Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied.