Advanced Econometrics Takeshi Amemiya
Type 2 Tobit Model: P(y1 < 0) • P(y, > 0, y2)
10.7.1 Definition and Estimation
The Type 2 Tobit model is defined as follows:
у*и = хА + Щі (Ю.7.1)
У 21 = *2ifi2 + U2i
У2і = У*і if ^>0
= 0 if /= 1, 2,. . . , n,
where {uu, u2i} are i. i.d. drawings from a bivariate normal distribution with zero mean, variances a and a, and covariance a12. It is assumed that only the sign of yf,- is observed and that y* is observed only when y* > 0. It is assumed that x1( are observed for all і but that x2i need not be observed for і such that yf, S 0. We may also define, as in (10.4.3),
w„-l if yf(> 0 (10.7.2)
= 0 if yfi S 0.
Then {wu, y2i] constitute the observed sample of the model. It should be noted that, unlike the Type 1 Tobit, y2i may take negative values.11 As in (10.2.4), y2i = 0 merely signifies the event yf, S 0.
The likelihood function of the model is given by
L = П w, ^ 0) Y[Ay2iyt, > 0)P(yf, > 0), (10.7.3)
0 1
where П0 and IIj stand for the product over those і for which y2i = 0 and y2/ Ф 0, respectively, and /( • |yf, > 0) stands for the conditional density of yf, given yf, > 0. Note the similarity between (10.4.1) and (10.7.3). As in Type 1
Tobit, we can obtain a consistent estimate of /ст( by maximizing the probit part of (10.7.3).
Probit L = ПP(y*u ^ 0) П P(y*u > 0). (10.7.4)
о і
Also, (10.7.4) is a part of the likelihood function for every one of the five types of models; therefore a consistent estimate of can be obtained by the probit MLE in each of these types of models.
We can rewrite (10.7.3) as
L = nw, si0)n ГлуЪ. Уи)^ 00.7.5)
0 1 Jo
where/( • , • ) denotes the joint density ofy* andy*,. We can write the joint density as the product of a conditional density and a marginal density, that is Уіі) =КУиУн)ЯУіі and can determine a specific form for/(yft|y2l) from the well-known fact that the conditional distribution of y* given Угі — Угі is normal with mean 'UP1 + ol2o22(y2i ~ x2l/l2) and variance a — с и o'!2- Thus we can further rewrite (10.7.5) as
І = П[1-Ф»Г1)] (Ю.7.6)
0
ХП Ф{[х»Ді(7Г1 + а2а1^агУи - *2іРг)
1
Х[1 - erf2СГ72СТ J2]- 1/2}crj ‘Ф [crj ‘(^2/ - *2iA)]-
Note that L depends on only through fJxay1 and ct12o7 1; therefore, if there is
no constraint on the parameters, we can put ax = 1 without any loss of generality. Then the remaining parameters can be identified. If, however, there is at least one common element in and fi2, °i can also be identified.
We shall show how Heckman’s two-step estimator can be used in this model. To obtain an equation comparable to (10.4.11), we need to evaluate E(y*iy*i > 0). For this purpose we use
y*2i = *2ip2 + WT2(yti - *i, A) + Си. (10.7.7)
where C2і is normally distributed independently of y* with zero mean and variance a — a 2a^2. Using (10.7.7), we can express £(y2i|>’* > 0) as a simple linear function of E{y'f,> 0), which was already obtained in Section
10.4. Using (10.7.7), we can also derive V(y2iy* > 0) easily. Thus we obtain
Угі = *2 A + о-12<7Г1А(х,1,а1) + e2f,
for і such that y2i Ф 0,
where a, = Ee2i = 0, and
Ve2l = a- (т212<7Т2[^иаіМ^и<Хі) + Я(х'1Іа1)2]. (10.7.9)
As in the case of the Type 1 Tobit, Heckman’s two-step estimator is the LS estimator applied to (10.7.8) after replacing at, with the probit MLE. The asymptotic distribution of the estimator can be obtained in a manner similar to that in Section 10.4.3 by defining rj2i in the same way as before. It was first derived by Heckman (1979).
The standard Tobit (Type 1) is a special case of Type 2 in which yf, = y*. Therefore (10.7.8) and (10.7.9) will be reduced to (10.4.11) and (10.4.12) by putting x'upx = x2ifl2 and a = a = an.
A generalization of the two-step method applied to (10.4.23) can easily be defined for this model but will not be discussed.
Note that the consistency of Heckman’s estimator does not require the joint normality of yf and yf provided that yf is normal and that Eq. (10.7.7) holds with C2 independently distributed of yf but not necessarily normal (Olsen, 1980). For then (10.7.8) would still be valid. As pointed out by Lee (1982c), the asymptotic variance-covariance matrix of Heckman’s estimator can be consistently estimated under these less restrictive assumptions by using White’s estimator analogous to the one mentioned after Eq. (10.4.22). Note that White’s estimator does not require (10.7.9) to be valid.
