Advanced Econometrics Takeshi Amemiya

Tests for Normality

The fact that the Tobit MLE is generally inconsistent when the true distribu­tion is nonnormal makes it important for a researcher to test whether the data are generated by a normal distribution. Nelson (1981) devised tests for nor­mality in the i. i.d. censored sample model and the Tobit model. His tests are applications of the specification test of Hausman (1978) (see Section 4.5.1).

Nelson’s i. i.d. censored model is defined by

Уі = уТ if yf>0 = 0 if yf = 0, /'= 1,2,. . . ,n,

where yf — Щц, a2) under the null hypothesis. Nelson considered the esti­mation of P(yf > 0). Its MLE is Ф(р/а), where fi and a are the MLE of the respective parameters. A consistent estimator is provided by щ/п, where, as before, n{ is the number of positive observations of yt. Clearly, njn is a consistent estimator of P(yf > 0) under any distribution, provided that it is i. i.d. The difference between the MLE and the consistent estimator is used as a test statistic in Hausman’s test. Nelson derived the asymptotic variances of the two estimators under normality.

If we interpret what is being estimated by the two estimators as lim„_„ и" ‘2-L [P(yf > 0), Nelson’s test can be interpreted as a test of the null hypothesis against a more general misspecification than just nonnormality. In fact, Nelson conducted a simulation study to evaluate the power of the test against a heteroscedastic alternative. The performance of the test was satisfac­tory but not especially encouraging.

In the Tobit model Nelson considered the estimation of n~lEX'y = nr1,x([Ф(х£ск)х[fi + оф(х'(а)]. Its MLE is given by the right - hand side of this equation evaluated at the Tobit MLE, and its consistent estimator is provided by /Г1 X'y. Hausman’s test based on these two estima­tors will work because this consistent estimator is consistent under general distributional assumptions on y. Nelson derived the asymptotic variance-co­variance matrices of the two estimators.

Nelson was ingenious in that he considered certain functions of the original parameters for which estimators that are consistent under very general as­sumptions can easily be obtained. However, it would be better if a general consistent estimator for the original parameters themselves could be found. An example is Powell’s least absolute deviations estimator, to be discussed in the next subsection.

Bera, Jarque, and Lee (1982) proposed using Rao’s score test in testing for normality in the standard Tobit model where the error term follows the two-parameter Pearson family of distributions, which contains normal as a special case.