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)