Chapter 3 ML estimation

Question 1

Identification condition in ML estimation

Answer 1

There is no set of parameters observationally equivalent to the true set of parameters.

Inconsistency of the ML estimator can arise if the likelihood is flat arount the true value => curvature of the likelihood can be interpreted as a measure of the precision of the ML estimate

Question 2

Asymptotic distribution of ML

Answer 2

Random sampling is assumed.

• When the likelihood is well specified the information equality applies, as a result the asymptotic variance is simplified.
• The information matrix just measures the curvature of the likelihood function and as such it is a measure of the precision of the estimates

Question 3

Consistency of ML

Answer 3

• Identification condition is required.
• Random sampling also, so that LLN applies.

Question 4

Definition of MLE

Answer 4

Question 5

Wald test

Answer 5

H0: h(theta)=0

H1: h(theta)<>0

• We use the unrestricted estimator.
• In the case of the ML we replace the estimator of the variance by the -1*hessian of the log-likelihood
• The result is true even for mispecified likelihoods
• Notice we just need to compute the unrestricted model

Question 6

LM test

Answer 6

Delta tilde is the maximizer of the log likelihood subject to the restriction

• Under the null, its distribution is given by the one below
• Notice for the LM test we just need to compute the restricted estimator

Question 7

LR test

Answer 7

Againt theta tilde is the restricted estimator

• Notice that to perform this test we have to compute both the restricted and unrestricted models
• Asymtotic distribution requires a well specified likelihood

Question 8

Nominal size

Answer 8

Supposed size of an asymptotic test different of true size of the thest