Advanced Econometrics Takeshi Amemiya
Least Squares Estimator
From Figure 10.1 it is clear that the least squares regression of expenditure on income using all the observations including zero expenditures yields biased estimates. Although it is not so clear from the figure, the least squares regression using only the positive expenditures also yields biased estimates. These facts can be mathematically demonstrated as follows.
First, consider the regression using only positive observations of yt. We obtain from (10.2.3) and (10.2.4)
Е(Уіу, > 0) = x't0 + E(UjUj > - xtfi). (10.4.5)
The last term of the right-hand side of (10.4.5) is generally nonzero (even without assuming щ is normal). This implies the biasedness of the LS estimator using positive observation on yt under more general models than the standard Tobit model. When we assume normality of u, as in the Tobit model,
(10.4.5) can be shown by straightforward integration to be
Е(УіУі > 0) = x'ifi + аЛ(х',Р/<т), (10.4.6)
where X(z) = ф(г)/Ф(г).2 As shown later, this equation plays a key role in the derivation of Heckman’s two-step, NLLS, and NLWLS estimators.
Equation (10.4.6) clearly indicates that the LS estimator of 0 is biased and inconsistent, but the direction and magnitude of the bias or inconsistency cannot be shown without further assumptions. Goldberger (1981) evaluated the asymptotic bias (the probability limit minus the true value) assuming that the elements of x, (except the first element, which is assumed to be a constant) are normally distributed. More specifically, Goldberger rewrote (10.2.3) as
yf-Д + зВД + и, (10.4.7)
and assumed x,- ~ JV(0, X), distributed independently of u. t. (Here, the assumption of zero mean involves no loss of generality because a nonzero mean can be absorbed into 0O.) Under this assumption he obtained
Plim A = i-fy A > (10-4-8)
where у = OylX(P0/oy)[fi0 + (ТуЛ(Ро/(ту)] and p2 = Oy20 {XA, where a2 = a2 + PXPi. It can be shown that 0 < у < 1 and 0 < p2 < 1; therefore (10.4.8) shows that Д shrinks jff, toward 0. It is remarkable that the degree of shrinkage is uniform in all the elements of 0y. However, the result may not hold if x, is not normal; Goldberger gave a nonnormal example where Д = (1, 1)'and plim Д = (1.111,0.887)'.
Consider the regression using all the observations of yh both positive and 0. Tо see that the least squares estimator is also biased in this case, we should look at the unconditional mean of yt,
Eyt = Ф(х;у?/ст)х;у? + стф(х'у?/сг). (10.4.9)
Writing (10.2.3) again as (10.4.7) and using the same assumptions as Gold - berger, Greene (1981) showed
р1ітД = Ф(/?0 /0y)fa, (10.4.10)
where fii is the LS estimator of fix in the regression of y, on x, using all the observations. This result is more useful than (10.4.8) because it implies that (и/«,)Д is a consistent estimator of fix, where л, is the number of positive observations of yt. A simple consistent estimator of fi0 can be similarly obtained. Greene (1983) gave the asymptotic variances of these estimators. Unfortunately, however, we cannot confidently use this estimator without knowing its properties when the true distribution of x( is not normal.3
