A NUMBER OF ARTICLES have been written in recent as well as more distant times on the subject of limited dependent variables. This broad classification includes two particular special cases. First, the dependent variable has finite probability mass concentrated at some limit point-the so called Tobit model. This case is referred to as the censored model in the statistical literature. The second case is when the dependent variable simply has a limited range and follows a continuous density on this support-the truncated case. In this note we show that a fairly powerful technique for truncated normal distributions, first suggested over 70 years ago by Karl Pearson and Alice Lee [9], can be used in the place of more sophisticated iterative maximum likelihood or instrumental variables techniques. Approximations based upon least squares regressions can be generated which require only a calculator and the table produced here. The censored or limited variable case can also be treated in much the same way. Although the estimator is inconsistent, a consistent estimator based upon this general approach could probably be constructed. An example is given comparing this approximation with the results produced by least squares, maximum likelihood, and Amemiya's consistent estimator. The estimator suggested here compares very favorably with Amemiya's. In a more literary vein, Pearson and Lee should be given substantial credit for their work without detracting in any way from those who independently advanced the area without the benefit of reference to the rather obscure article by Pearson and Lee.2

WHEN ESTIMATING REGRESSION MODELS it is very nearly always assumed that the sample is random. The recent literature has begun to deal with the problems which arise when estimating a regression model with samples which may not be random. The most general case in which one only has access to a single nonrandom sample has not been addressed since it is a very imposing problem. The case which has been addressed starts with a random sample but considers the problem of missing values for the dependent variable of a regression. If the determination of which values are to be observed is related to the unobservable error term in the regression, then methods such as ordinary least squares are in general inappropriate. By constructing a joint model which represents both the regression model to be estimated and the process determining when the dependent variable is to be observed, some progress can be made towards taking into account nonrandomness for the observed values of the dependent variable. The actual techniques employed fall into two rough groups, full information maximum likelihood models, and limited information methods which are more easily estimated. In the full information category are two methods. One model combines the probit and the normal regression models, and the other combines the Tobit or limited dependent variable model with the normal regression model. The form of the probit regression model is

In this paper [the authors] develop and implement an econometric methodology estimating a family-specific exogenous component of life-expectancy in order to determine the responsiveness of fertility to exogenous changes in child mortality. [They] use a generalized waiting time regression model applied to length of life which is viewed as the output of a production process. [They] allow for family-specific heterogeneity in duration of life and for time-varying explanatory variables. The heterogeneity component retrieved from the production function estimation is used to estimate the impact of exogenous child mortality on a measure of fertility. The data concern 1938 children from 311 families included in the 1976 Malaysian Family Life Survey. This paper was previously published in Econometrica (Chicago Ill.) Vol. 51 No. 3 May 1983 pp. 731-49. (EXCERPT)

Randall J. Olsen, D. Alton Smith, George Farkas, Structural and Reduced-Form Models of Choice among Alternatives in Continuous Time: Youth Employment under a Guaranteed Jobs Program, Econometrica, Vol. 54, No. 2 (Mar., 1986), pp. 375-394