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Unformatted text preview: Extensions to Tobit and Heckit regression models 1 Multiple censoring points A useful extension to the traditional Tobit model y & i = & + & 1 x i + u i y i = & y & i if y & i > if y & i & is when data are subject to two forms of censoring. For example, y & i could be fully observed only if it is between a lower and an upper threshold, and being set equal to the thresholds otherwise. In this case the model is written: y & i = & + & 1 x i + u i y i = 8 < : l i y & i L i if y & i & l i if l i < y & i < L i if y & i L i where l i and L i are known constants. Figure 1 shows the relationship between y i and y & i in this particular case (the red line). For values of y & i between the two thresholds, y i = y & i . For values above L i or below l i , the data are censored at the thresholds. One example is when data from a survey of incomes are censored at some upper or lower thresholds for con&dentiality reasons. Another (economic) example is in a labor supply model with two possible corner solutions at 0 ( l ) leisure or 24 hours ( L ) leisure per day. Estimation is again by ML. Assume u i N ; 2 for all i . De&ne two dummies d 1 i = 1 f y & i & l i g and d 2 i = 1 f y & i L i g . Clearly, (1 d 1 i d 2 i ) = 1 f l i & y & i & L i g . The contribution of an individual to the maximum likelihood is l i = & l i ( & + & 1 x i ) d 1 i 1 & L i ( & + & 1 x i ) d 2 i 1 y i ( & + & 1 x i ) 1 d 1 i d 2 i and so the likelihood function is L = n Y i =1 l i There are two alternative estimation methods that are based on a two step strategy. 1 y y* l L y y* l L 1.1 Method 1: Use only "complete" observations In this case, one consider the conditional mean E ( y i j x i ;l i < y & i < L i ) = & + & 1 x i + E ( u i j x i ;l i < y & i < L i ) = & + & 1 x i + E ( u i j l i & ( & + & 1 x i ) < u i < L i & ( & + & 1 x i )) If E ( u i j l i & ( & + & 1 x i ) < u i < L i & ( & + & 1 x i )) 6 = 0 , OLS on the sample with "complete" observations will deliver biased and inconsistent estimates of the parameters of interest. This is because there is sample selectivity: in this case, only individuals with no too extreme realizations of u i are in the sample. We can eliminate the sample selectivity bias using a control function strategy. The problem is one of omitted variable bias: the variable E ( u i j l i & ( & + & 1 x i ) < u i < L i & ( & + & 1 x i )) is omitted from the regression. We can solve the problem by including this vari able in the regression. For this purpose, we need to know what this expectation is. With a normality assumption on u , we can use the formulae of the normal distribution to construct an estimate of this expectation. In fact 2 E ( u i j l i & ( & + & 1 x i ) < u i < L i & ( & + & 1 x i )) = Z L i & ( & + & 1 x i ) l i & ( & + & 1 x i ) u i f ( u i j l i & ( & + & 1 x i ) < u i < L i & ( & + & 1 x i )) du i = R L i & ( & + & 1 x i ) l i & ( & + & 1 x i ) u i f ( u i ) du i...
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This note was uploaded on 07/28/2011 for the course ECON 102C taught by Professor Pistaferri,l during the Spring '11 term at Stanford.
 Spring '11
 Pistaferri,L
 Econometrics

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