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How do you fit a model when the dependent variable is a proportion?

Title   Logit transformation
Author Allen McDowell, StataCorp
Nicholas J. Cox, Durham University, UK
Date August 2001; updated August 2004; minor revisions March 2014

A traditional solution to this problem is to perform a logit transformation on the data. Suppose that your dependent variable is called y and your independent variables are called X. Then, one assumes that the model that describes y is

        y = invlogit(XB)

If one then performs the logit transformation, the result is

        ln( y / (1 - y) ) = XB

We have now mapped the original variable, which was bounded by 0 and 1, to the real line. One can now fit this model using OLS or WLS, for example by using regress. Of course, one cannot perform the transformation on observations where the dependent variable is zero or one; the result will be a missing value, and that observation would subsequently be dropped from the estimation sample.

A better alternative is to estimate using glm with family(binomial), link(logit), and robust; this is the method proposed by Papke and Wooldridge (1996). At the time this article was published, Stata’s glm command could not fit such models, and this fact is noted in the article. glm has since been enhanced specifically to deal with fractional response data.

In either case, there may well be a substantive issue of interpretation. Let us focus on interpreting zeros: the same kind of issue may well arise for ones. Suppose the y variable is proportion of days workers spend off sick. There are two extreme possibilities. The first extreme is that all observed zeros are in effect sampling zeros: each worker has some nonzero probability of being off sick, and it is merely that some workers were not, in fact, off sick in our sample period. Here, we would often want to include the observed zeros in our analysis and the glm route is attractive. The second extreme is that some or possibly all observed zeros must be considered as structural zeros: these workers will not ever report sick, because of robust health and exemplary dedication. These are extremes, and intermediate cases are also common. In practice, it is often helpful to look at the frequency distribution: a marked spike at zero or one may well raise doubt about a single model fitted to all data.

A second example might be data on trading links between countries. Suppose the y variable is proportion of imports from a certain country. Here a zero might be structural if two countries never trade, say on political or cultural grounds. A model that fits over both the zeros and the nonzeros might not be advisable, so that a different kind of model should be considered.

For an excellent broader discussion, see Baum (2008).

Reference

Baum, C.F. 2008.
Modeling proportions. Stata Journal 8: 299–303.
Papke, L. E. and J. Wooldridge. 1996.
Econometric methods for fractional response variables with an application to 401(k) plan participation rates. Journal of Applied Econometrics 11: 619–632.
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