Advanced Econometrics Takeshi Amemiya
Serial Correlation
Robinson (1982a) proved the strong consistency and the asymptotic normality of the Tobit MLE under very general assumptions about ut (normality is presupposed) and obtained its asymptotic variance-covariance matrix, which is complicated and therefore not reproduced here. His assumptions are slightly stronger than the stationarity assumption but are weaker than the assumption that щ possesses a continuous spectral density (see Section 5.1.3). His results are especially useful because the full MLE that takes account of even a simple type of serial correlation seems computationally intractable. The autocorrelations of u, need not be estimated to compute the Tobit MLE but must be estimated to estimate its asymptotic variance-covariance matrix. The consistent estimator proposed by Robinson (1982b) may be used for that purpose.
10.5.1 Nonnormality
Goldberger (1983) considered an i. i.d. truncated sample model in which data are generated by a certain nonnormal distribution with mean fi and variance 1 and are recorded only when the value is smaller than a constant c. Let у represent the recorded random variable and let у be the sample mean. The researcher is to estimate ц by the MLE, assuming that the data are generated by Щц, 1). As in Hurd’s i. i.d. model, the MLE p. is defined by equating the population mean of у to its sample mean:
р-Цс-р) = у.
T aking the probability limit of both sides of (10.5.3) under the true model and putting plim fi = n* yield
/і* - Цс - fi*) =/л - h(c - fi),
where h{c — ц) = E(fi — yy< c), the expectation being taken using the true model. Defining т=ц* — ц and в = c — fi, we can rewrite (10.5.4) as
m = A(0 — m) — h(6).
Goldberger calculated mas a function of в when the data are generated by Student’s t with various degrees of freedom, Laplace, and logistic distributions. The asymptotic bias was found to be especially great when the true distribution was Laplace. Goldberger also extended the analysis to the regression model with a constant term and one discrete independent variable. Arabmazar and Schmidt (1982) extended Goldberger’s analysis to the case of an unknown variance and found that the asymptotic bias was further accentuated.
