MLE

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Data: ${w_i: i = 1, \cdots , n } \enspace iid$, $w_i \sim f(w,\theta) , \theta \in \Theta \in \mathbb{R}_k$

Because of independence:

Likelihood Function

MLE Estimator

Information Matrix

Asymptotic Convergence

Binary Choice

  • Probit: $F(u) = \Phi(n)$ - Standard normal $N(0,1)$ CDF
  • Logit:
    • $F(u) = \Lambda(u)$ = $\frac{1}{1+e^{-u}}$
    • $f(u) = \Lambda’(u) = (1-e^{-u})^{-2} e^{-u}$

General case: maximize the following Log Likelihood

Solutions for Logit :

Point Estimate

Variance

Marginal Effects