|Department of Economics
Faculty of Economics and Political Science
|Econ 703 – 2020/21
Quantitative Methods I
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 ) 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 ˆˆ / n .
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 lnL 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.
3. Consider the general non-linear regression model with non-spherical disturbances:
y f (X,β) ε ; ε ~ N(0, 2)
where y is an n1 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 n1 vector of the n error terms.
Show how this model parameters can be estimated using the method of generalized method of
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
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