Probit and Logit Model

A Tobit regression is very similar to a probit regression (hence the name, originally Tobin's probit).

In the Probit Model,  we hypothesize y∗$y^{<span\; class="mjxassistivemathml"><span\; style="border:\; 1pt\; none\; windowtext;\; font-family:\; "Cambria\; Math",\; serif;\; font-size:\; 11.5pt;\; padding:\; 0in;">\ast </span></span><span\; class="apple-converted-space"><span\; style="font-family:\; Georgia,\; serif;\; font-size:\; 11.5pt;"> </span></span><span\; style="font-family:\; Georgia,\; serif;\; font-size:\; 11.5pt;">being\; a\; Gaussian\; variable\; with\; mean<o:p></o:p></span>}$

E(y∗)=x′i

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β

β and a variance conventionally fixed to 1 as an identification constraint. All we observe in the probit model is the signs and make use of the standard Gaussian function to define probabilities for the Bernoulli likelihood. 

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In a   Tobit Model, we observe actual values of y∗$y^{<span\; class="mjxassistivemathml"><span\; style="border:\; none\; windowtext\; 1.0pt;\; color:\; \#333333;\; font-family:\; "Cambria\; Math","serif";\; font-size:\; 11.5pt;\; mso-bidi-font-family:\; "Cambria\; Math";\; mso-border-alt:\; none\; windowtext\; 0in;\; padding:\; 0in;">\ast </span></span><span\; class="apple-converted-space"><span\; style="color:\; \#333333;\; font-family:\; "Georgia","serif";\; font-size:\; 11.5pt;"> </span></span><span\; style="color:\; \#333333;\; font-family:\; "Georgia","serif";\; font-size:\; 11.5pt;">above\; some\; threshold\; (conventionally\; 0)\; but\; only\; observe\; that\; a\; case\; is\; below\; the\; threshold,\; without\; seeing\; its\; magnitude.\; The\; likelihood\; is\; a\; censored\; Gaussian,\; which\; has\; a\; Bernoulli\; part\; for\; the\; censoring\; and\; a\; modified\; Gaussian\; for\; the\; rest.\; These\; models\; have\; very\; similar\; math,\; though\; the\; probit\; model\; is\; less\; sensitive\; to\; the\; distributional\; specification\; than\; the\; tobit\; model,\; where\; you\; observe\; part\; of\; the\; actual\; variable.<o:p></o:p></span>}$

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