Flashcards in Chapter 3 ML estimation Deck (8)

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1

Q

Identification condition in ML estimation

A

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

2

Q

Asymptotic distribution of ML

A

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

3

Q

Consistency of ML

A

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

4

Q

Definition of MLE

A

5

Q

Wald test

A

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

6

Q

LM test

A

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

7

Q

LR test

A

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

8

Q

Nominal size

A

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