## Appendix 15a

15A.1 MAXIMUM LIKELIHOOD ESTIMATION OF THE LOGIT AND PROBIT MODELS FOR INDIVIDUAL (UNGROUPED) DATA*

As in the text, assume that we are interested in estimating the probability that an individual owns a house, given the individual's income X. We assume that this probability can be expressed by the logistic function (15.5.2), which is reproduced below for convenience.

We do not actually observe Pi, but only observe the outcome Y = 1, if an individual owns a house, and Y = 0, if the individual does not own a house. Since each Y, is a Bernoulli random variable, we can write

The following discussion leans heavily on John Neter, Michael H. Kutner, Christopher J. Nachsteim, and William Wasserman, Applied Linear Statistical Models, 4th ed., Irwin, 1996, pp. 573-574.

634 PART THREE: TOPICS IN ECONOMETRICS

Suppose we have a random sample of n observations. Letting fi(Y) denote the probability that Yi = 1 or 0, the joint probability of observing the n Y values, i.e., f (Y1, Y2,..., Yn) is given as:

n f (Yi, Y2,..., Yn) = n fi(Yi) = n PY(1 - Pi)1-Yi (4)