#maximum likelihood estimation

Fret: Is this a good idea?

Minamimoto: Is assuming a normal distribution a good idea when you could easily do an ML estimation?

Purpose: Testing multiple null hypotheses using MLE\(^{1}\) estimates. When testing for the joint hypothesis, test that all slope coefficients are equal to zero except the intercept.

Consider the following model:

\[Y_{i} = \beta_{0} + \beta_{1}X_{1i} +\beta_{2}X_{2i} + \epsilon_{i}\]

The test: 

\(H_{0}\): \( \beta_{1} = \beta_{2} = 0\)

\(H_{1}\): One of more \(\beta_{i}\)’s \(\neq 0\)

Test statistic:

\[LR = -2 (log L^{R} - log L^{U}) \sim^{a} \chi^{2}_{number\: of\: restrictions}\:\:\:\:^{2}\]

Rejection Rule:

Reject the \(H_{0}\) if \(LR\) > \(\chi^{2}_{\alpha,\:number\:of\:restrictions}\)

where, \(\alpha\) is the significance level, e.g. 5% or 1% and the degrees of freedom is the number restrictions.

Assumptions:

The MLE estimators must: 1. Consistent 2. Asymptotically normal (i.e. there must be a large sample) 3. Asymptotically efficient

Notes: \(^{1}\). Maximum Likelihood Estimators

\(^{2}\). \(\sim^{a}\) means that the LR test statistic asymptotically follows a \(\chi^{2}\) when the sample is large