Probit Model - Maximum Likelihood Estimation

Maximum Likelihood Estimation

Suppose data set contains n independent statistical units corresponding to the model above. Then their joint log-likelihood function is

The estimator which maximizes this function will be consistent, asymptotically normal and efficient provided that E exists and is not singular. It can be shown that this log-likelihood function is globally concave in β, and therefore standard numerical algorithms for optimization will converge rapidly to the unique maximum.

Asymptotic distribution for is given by

where

$\Omega = \operatorname{E}\bigg, \qquad \hat\Omega = \frac{1}{n}\sum_{i=1}^n \frac{\varphi^2(x'_i\hat\beta)}{\Phi(x'_i\hat\beta)(1-\Phi(x'_i\hat\beta))}x_ix'_i$

and φ = Φ' is the Probability Density Function (PDF) of standard normal distribution.

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