Should we regress x or use logistic regression on x>5000

Question

Suppose we need to take an action on a population with income (x) more than $5,000. Income is not observed directly.

Should we use logistic regression to estimate x, or should we use logistic regression to estimate the probability of x>5000 directly? (What are the drawbacks/advantage of the methods?)

Edit: Yes - by logistic I meant logistic regression. The other variables I have are Financial history, Demograpgic and Credit Bureau variables. For example, the balance, utilization, gender, # cars, house owned or not, external card balance, external # bad debts, etc.

Thanks

Answer 1

If your only interest is whether their income is over $\$$5,000 or not, and it doesn't make a difference how far from that threshold their income actually is, then I would recommend using a classification technique (not necessarily logistic regression, try a range of methods and use whatever gives best out-of-sample performance) rather than regression. Vladimir Vapnik (inventor of the support vector machine) says you should always aim to solve the problem at hand directly, rather than use a more general method and post-process the result. That is a reasonable argument; if you are not interested how far above $5000 someones income is, why expend resources modelling the regression function a long way above that threshold? So if you have a classification problem, use a classifier, rather than threshold a regression.

Note however it is likely that false-positive and false-negative costs may be different though and factor that into your classifier design.

HTH

Answer 2

What are you trying to predict? Is your outcome just an indicator of whether income is above 5000 dollars or not? If so, that is the best that you can predict; that is, you can't predict anyone's income, only whether he has high (above 5000) income or not.

If this is your outcome and what you'd like to predict, the question is why would we favor using logit versus a linear probability model (standard OLS) for a binary outcome. Logit forces your prediction to be between 0 and 1, while LPM can predict any value for the probability of having high income (which is also the expected value of an indicator for having high income). This distinction is especially important if you have multiple predictors in your model or are trying to extrapolate to values of your predictors outside those of your sample. The downside to logit is the coefficients are difficult to interpret directly; instead, you typically report and consider marginal effects. Marginal effects depend upon the levels of your covariates, which can be confusing. Logit is also less robust to deviations from the normality and identical distribution assumptions about the error terms as I recall.