Censored Data Models

The censored normal regression model considered by Tobin (1958), also commonly known as the ‘tobit’ model, is the following: <math display='block'> <mrow> <msubsup> <mi>y</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>=</mo><mi>&#x03B2;</mi><msub> <mi>x</mi> <mi>i</mi> </msub> <mo>+</mo><msub> <mi>u</mi> <mi>i</mi> </msub> <mo>.</mo><mtext>&#x2003;</mtext><msub> <mi>u</mi> <mi>i</mi> </msub> <mo>&#x223C;</mo><mi>I</mi><mi>N</mi><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><msup> <mi>&#x03C3;</mi> <mn>2</mn> </msup> <mo stretchy='false'>)</mo> </mrow> </math>$$y_i^* = \beta {x_i} + {u_i}.\quad {u_i} \sim IN(0,{\sigma ^2})$$ The observed y _i are related to y _i * according to the relationship (1)<math display='block'> <mtable columnalign='left'> <mtr> <mtd> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>=</mo><msubsup> <mi>y</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mtext>&#x2003;</mtext><mi>i</mi><mi>f</mi><mtext>&#x2009;</mtext><msubsup> <mi>y</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>&#x003E;</mo><msub> <mi>y</mi> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo><msub> <mi>y</mi> <mn>0</mn> </msub> <mtext>&#x2003;</mtext><mtext>&#x2009;</mtext><mtext>&#x2003;</mtext><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi> </mtd> </mtr> </mtable> </math>$$\eqalign{ & {y_i} = y_i^*\quad if\,y_i^* > {y_0} \cr & = {y_0}\quad \;\quad otherwise \cr} $$ where y₀ is a prespecified constant (usually zero). The y _i * could take values <y₀. The only thing is that they are not observed. Thus, y _i is set equal to y₀ because of non-observability. The values x _i are observed for all the observations. If both y _i and x _i are unobserved for y _i ⩽ y₀ then we have what is known as a truncated regression model.