Advanced Econometrics Takeshi Amemiya
Tobit Maximum Likelihood Estimator
The Tobit MLE maximizes the likelihood function (10.2.5). Under the assumptions given after (10.2.4), Amemiya (1973c) proved its consistency and asymptotic normality. If we define 0 = (/?', a2)', the asymptotic variance-covariance matrix of the Tobit MLE 0 is given by
-i-i
;-i t-i
X W І c‘
1-1 i-1 J where
а і: = - а 2{хаф, - [ф?/(1 ~ Ф,)] - Ф,},
b, = (/2)а-(х'а)2фі + ф,- Ш/(1 - Ф,)]}, and
q = -(1/4)ст-4{(х'а)3ф, + (х'а)фі - [{х'а)фУ( - Ф,)] - 2Ф,);
and фі and Ф, stand for ф(х,-0£) and Ф(х<а), respectively.
The Tobit MLE must be computed iteratively. Olsen (1978) proved the global concavity of log L in the Tobit model in terms of the transformed parameters a = fit a and h = er’.a result that implies that a standard iterative method such as Newton-Raphson or the method of scoring always converges to the global maximum of log £.® The log L in terms of the new parameters can be written as
log L — 2 log [1 — Ф(х-а)] + Л| log h о
“ r X ~ x<a)2>
Z 1
from which Olsen obtained
d2 log L d log L дада' dadh
 |
dMog L dMogL dhda' dh2
Because x'a — [1 — Ф(х|а)]-Іф(х[а) < 0, the right-hand side of (10.4.38) is the sum of two negative-definite matrices and hence is negative definite.
Even though convergence is assured by global concavity, it is a good idea to start an iteration with a good estimator because it will improve the speed of convergence. Tobin (1958) used a simple estimator based on a linear approximation of the reciprocal of Mills’ ratio to start his iteration for obtaining the MLE. Although Amemiya (1973c) showed that Tobin’s initial estimator is inconsistent, empirical researchers have found it to be a good starting value for iteration.
Amemiya (1973) proposed the following simple consistent estimator. We have
Е(у}Уі > 0) = (x'tfi)2 + ах'іРЦх'а) + a2. (10.4.39)
Combining (10.4.6) and (10.4.39) yields
E(y2iy, > 0) = х'рЕ(УіУі > 0) + a2, (10.4.40)
which can be alternatively written as
у} = Уіх'іР+ о2 + Сі, for і such that yt > 0, (10.4.41)
where £(С, ІУ/ > 0) = 0. Then consistent estimates of and a2 are obtained by applying an instrumental variables method to (10.4.41) using (ypicj, 1) as the instrumental variables, where pt is the predictor of yt obtained by regressing
positive у і on x, and, perhaps, powers of xf. The asymptotic distribution of the estimator has been given by Amemiya (1973c). A simulation study by Wales and Woodland (1980) indicated that this estimator is rather inefficient.
