The Tobit model. Herman J. Bierens. September 17, Y j = max ³ Y ; (1) j ;0

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1 The Tobit model Herman J. Bierens September 17, The model The Tobit 1 model assumes that the observed dependent variablesy j for observationsj 1;::;n satisfy Y j max ³ Y j ;0 ; (1) where they j s are latent variables generated by the classical linear regression model Y j 0X j +U j ; (2) withx j a vector of regressors, possibly including 1 for the intercept, and the corresponding vector of parameters. The model errorsu j are assumed to be independentn(0;¾ 2 ) distributed, conditional on thex j s. Denoting by f(z) exp( z 2 2) p 2¼ the density of the N(0; 1) distribution, with corresponding cumulative distribution function (c.d.f.) F(z) Z z 1 f(v)dv; 1 The model is called Tobit because it was rst proposed by Tobi n (1958), and involves aspects of Probit analysis. See: Tobin, J. (1958), Estimation of Relationships for Limited Dependent Variables, Econometrica 26,

2 the conditional c.d.f. ofy j giveny j >0 andx j is H(yjY j > 0;X j ; ;¾) P(y j yjy j > 0;X j ) P(0<Y j yjx j) P( 0X j <U j y 0X j jx j ) P(Y j > 0 jx j ) P(U j > 0X j jx j ) F ((y 0X j )¾) F ( 0X j ¾) ; F ( 0X j ¾) and the corresponding conditional density is h(yjy j > 0;X j ; ;¾) dh(y jy j>0;x j ; ;¾) dy f ((y 0X j )¾) ;y>0. ¾F ( 0X j ¾) Thus, the conditional distribution ofy j giveny j >0 andx j is continuous. De ne the dummy variabled j by Then D j 1 ify j > 0; D j 0 ify j 0: P [D j 1 jx j ] F ( 0X j ¾); P [D j 0 jx j ] 1 F ( 0X j ¾); and Y j D j Y j : 2 Truncation bias Now the conditional expectation ofy j give j andd j 1 is E[Y j jx j ;D j 1] Z 1 0 yh(y jy j > 0;X j ; ;¾)dy Z 1 1 yf ((y 0X ¾F ( 0X j ¾) j )¾)dy 0 Z 1 1 F ( 0X ( 0X j +¾z)f (z)dz j ¾) 0X j ¾ 2

3 ( 0X j ) R 1 0Xj¾ f (z)dz +¾R 1 0Xj¾ zf (z)dz F ( 0X j ( 0X ¾) j )F( 0X j ) ¾ R 1 0X j ¾ f0 (z)dz F ( 0X j ¾) 0X j +¾ f ( 0X j ¾) F ( 0X j ¾) : Therefore, if you regress only the positivey j s on the correspondingx j s then, due to the latter term, the OLS parameters estimate of will be biased and inconsistent. Moreover, note that E[Y j jx j ] ( 0X j )F ( 0X j ¾) +¾f ( 0X j ¾): (3) Therefore, if you treat the zero values ofy j as regular dependent variable values in a linear regression model the OLS parameters estimate of will be biased and inconsistent as well. 3 The log-likelihood The conditional c.d.f. ofy j give j is now G(yjX j ; ;¾) P [Y j y jx j ] P [Y j yjx j ;D j 1]P [D j 1jX j ] +P [Y j y jx j ;D j 0]P [D j 0jX j ] I(y>0)H(y jy j >0;X j ; ;¾)F ( 0X j ¾) +I(y 0)(1 F ( 0X j ¾)); wherei(:) is the indicator function: I(true) 1,I(false) 0: Hence, the corresponding conditional density is g(yjx j ; ;¾) I(y>0)h(y jy j >0;X j ; ;¾)F ( 0X j ¾) +I(y 0)(1 F ( 0X j ¾)) Therefore, the log-likelihood function of the Tobit model is L( ;¾) ln[g(y j jx j ; ;¾)] 3

4 + + D j ln[h(y j jy j > 0;X j ; ;¾)] D j lnf ( 0X j ¾) + D j µ 1 2 (Y j 0X j ) 2 ¾ 2 ln(¾) (1 D j )ln(1 F ( 0X j ¾)) (1 D j )ln(1 F ( 0X j ¾)) D j ln ³p 2¼ However, before maximizing this likelihood function it is convenient to reparametrize it by replacing by¾ and¾by 1µ; so that L ( ;µ) L(¾ ;1¾) D j ³ (µyj 0 X j ) 2 +ln ³ µ 2 (1 D j )ln(1 F ( 0 X j )) D j ln ³p 2¼ Moreover, if can be shown 2 (but this is nontrivial) that the Hessian 2 L ( 0 ;µ) 0 ;µ) is negative de nite for all values of andµ>0; so that the log-likelihood L ( ;µ) has no local maxima. This is the main reason for the reparametrization involved. 4 The pseudor 2 EasyReg computes the (pseudo)r 2 of the model as follows. Given the maximum likelihood estimators b and b¾; the residuals are computed as bu j Y j ³b 0X j F ³b 0X j b¾ b¾f ³b 0X j b¾ 2 Reference: Olsen, R. (1978), A Note on the Uniqueness of the Maximum Likelihood Estimator in the Tobit Model, Econometrica 46,

5 [see (3)], and then ther 2 is computed in the same way as for OLS: R 2 1 P n b U 2 j P n ³ Yj Y 2 ; wherey is the sample mean of they j s. 5 When does EasyReg refuse to conduct Tobit analysis? There are three cases for which EasyReg does not allow you to conduct Tobit analysis: 1. If the dependent variablesy j take negative values. 2. If the dependent variablesy j take only positive values. 3. If the dependent variablesy j are non-negative, with some of they j s zero, but all they j s integer valued. In the rst two cases the reason for refusing to conduct Tobit analysis is obvious, but I have gotten quite a few queries from EasyReg users why in the last case EasyReg refuses to continue. In the third case they j s satisfy P [Y j 2 f0;1;2;::::::g] 1; and therefore the conditional distribution ofy j given Y j > 0 and X j is discrete, i.e., the conditional c.d.f. H(yjY j > 0;X j ; ;¾) in this case is a step function, with jumps at some integer values ofy: This violates the basic assumption of the Tobit model, i.e., (1), (2) and the normality of the U j s, and therefore EasyReg will not allow you to conduct Tobit analysis. However, if you insist on conducting Tobit analysis with this dependent variable there is a trick to fool EasyReg: Multiply they j s by a factor10 m ; wherem> 0 is such that at least one of the new variables10 m Y j has one or more decimal digits. 3 For example, suppose that the originaly j s are dollar amounts, roundedo to multiples of1000 dollar, and suppose that there exits at least oney j value with a non-zero digit next to the last three zeros, say 3 This transformation can be done in EasyReg: Click Menu! Input! Transform variables! Linear combination of variables, double click the variable involved, click Selection OK, enter the coe cient involved, and click OK. 5

6 Y j : Then 10 4 Y j 0: :3; hence at least one of the new variables 10 4 Y j has a decimal digit. Since EasyReg validates your data by scanning for decimal digits in the values of the dependent variable, it will allow you to conduct Tobit analysis for the rescaled dependent variable involved. 6