Models for Binary Outcomes and Proportions

This is probably the simplest form of a limited dependent variable (LDV) setup. In the data available for analysis an outcome or attribute is present or it isn’t, and this dichotomy is exhaustive. This typically occurs because of fairly natural limits to data collection and our observational powers. For example, someone either votes or doesn’t. Someone is employed or isn’t. It isn’t easy to collect data on, say, the enthusiasm with which someone voted, though it might be worthwhile to know that if we were interested in making a prediction about the probability of a similar person voting in future elections. Someone may be ‘‘on the cusp’ ’ of unemployment, but our data generationmechanism is unable to provide that type of information, because like the voting example, the concept is rather vague, difficult and probably very expensive to operationalize. Someone owns a house or doesn’t. A pair of countries (a dyad) is at war, or isn’t. And so on. For the purposes of doing quantitative analysis (and without loss of gen-erality) we conventionally code the presence of the outcome or attribute of interest with as yi = 1, and its absence as yi = 0, in a sample of n observations, indexed by i = 1,..., n. Wemake no additional (or fewer) assumptions about independent variables than for LS regression. Proportions data come from aggregating these unit-level binary outcomes or attributes into groups; the first applications of probit analysis come from bioassay, modeling the proportion of moths dying in response to a varying levels of exposure to a toxin (Bliss 1935). It will be useful to consider binary responses and proportions data simultaneously; arguments as to the inappropriateness of LS regression and alternative modeling and estimation strategies are similar in both contexts