Advanced Econometrics Takeshi Amemiya

Heteroscedasticity

Hurd (1979) evaluated the probability limit of the truncated T obit MLE when a certain type of heteroscedasticity is present in two simple truncated Tobit models: (1) the i. i.d. case (that is, the case of the regressor consisting only of a constant term) and (2) the case of a constant term plus one independent variable. Recall that the truncated Tobit model is the one in which no infor­mation is available for those observations for which yf < 0 and therefore the MLE maximizes (10.2.6) rather than (10.2.5).

In the i. i.d. case Hurd created heteroscedasticity by generating rn observa­tions from N(n, (72i) and (1 — r)n observations from N(p, a). In each case he recorded only positive observations. Let >>(,/= 1, 2,. . . , и,, be the recorded observations. (Note nx = rt). We can show that the truncated Tobit MLE of ц and a2, denoted p and a2, are defined by equating the first two population moments of yt to their respective sample moments:

/2 + <tA(/2/<j) = nr1 2^. (10.5.1)

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and

p.2 + dfik(p/a) + a2 = «71 ^y2. (10.5.2)

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Taking the probability limit of both sides of (10.5.1) and (10.5.2) and express­ing plim n712y, and plim njl1,y2 as certain functions of the parameters p, a2, 0J, and r, we can define plim p and plim a2 implicitly as functions of these parameters. Hurd evaluated the probability limits for various values of p and <7j after having fixed r = 0.5 and a2 = 1. Hurd found large asymptotic biases in certain cases.

In the case of one independent variable, Hurd generated observations from N(a + 0xt, a2) after having generated x, and log|<r,| from bivariate N(0, 0, V, V, p). For given values of a, /?, V{, V2, and p, Hurd found the values of a, 0, and a2 that maximize E log L, where L is as given in (10.2.6). Those values are the probability limits of the MLE of a, 0, and a2 under Hurd’s model if the expectation of log L is taken using the same model. Again, Hurd found extremely large asymptotic biases in certain cases.

Arabmazar and Schmidt (1981) showed that the asymptotic biases of the censored Tobit MLE in the i. i.d. case are not as large as those obtained by Hurd.