Introduction

The problem of left censoring was discussed by Professor Amemiya in his book Advanced Econometrics in 1985 (see Amemiya 1985, pp. 447–448). Consider the estimation of unemployment duration when the statistician samples individuals from the pool of unemployed persons and later interviews them and asks how long they have been unemployed. This incurs the problem of left censoring; ignoring it leads to overestimating the mean duration, since longer spells tend to be observed more frequently than shorter spells. This effect is called selectivity bias and needs a special treatment.

Some other authors have noticed and studied the problem (see, e.g., Lancaster 1979, Chesher and Lancaster 1983, Ridder 1984, and Heckman and Singer 1984). To the best of our knowledge, however, Amemiya (1991) is the first paper to give a complete, unified treatment of this problem, including asymptotic properties. According to it, three types of left censoring arise. We assume that spells in progress at the time of the first observation are either completely or partially observed. Suppose such a spell started at x, continued to 0 (the time of the first observation), and ended at s. The statistician may observe both s and x (by asking how long the spell lasted), only s, or only x. Amemiya (1991) calls those cases types 1, 2, and 3, respectively.

In each possible case we have to consider how the selectivity bias is eliminated.