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.

Read more about this topic:  Probit Model

Famous quotes containing the words maximum, likelihood and/or estimation:

    “Only at his maximum does an individual surpass all his derivative elements, and become purely himself. And most people never get there. In his own pure individuality a man surpasses his father and mother, and is utterly unknown to them.”
    —D.H. (David Herbert)

    “What likelihood is there of corrupting a man who has no ambition?”
    —Samuel Richardson (1689–1761)

    “No man ever stood lower in my estimation for having a patch in his clothes; yet I am sure that there is greater anxiety, commonly, to have fashionable, or at least clean and unpatched clothes, than to have a sound conscience.”
    —Henry David Thoreau (1817–1862)