A Variational Framework for Estimating Continuous Treatment Effects with Measurement Error

The term 'treatment effect' refers to the causal effect of a binary (0-1) variable on an outcome variable of scientific or policy interest. Economics examples include the effects of government programmes and policies, such as those that subsidize training for disadvantaged workers, and the effects of individual choices like college attendance. The principal econometric problem in the estimation of treatment effects is selection bias, which arises from the fact that treated individuals differ from the non-treated for reasons other than treatment status per se. Treatment effects can be estimated using social experiments, regression models, matching estimators, and instrumental variables. A 'treatment effect' is the average causal effect of a binary (0-1) variable on an outcome variable of scientific or policy interest. The term 'treatment effect' originates in a medical literature concerned with the causal effects of binary, yes-or-no 'treatments', such as an experimental drug or a new surgical procedure. But the term is now used much more generally. The causal effect of a subsidized training programme is probably the mostly widely analysed treatment effect in economics (see, for example, Ashenfelter, 1978, for one of the first examples, or Heckman and Robb, 1985 for an early survey). Given a data-set describing the labour market circumstances of trainees and a non-trainee comparison group, we can compare the earnings of those who did participate in the programme and those who did not. Any empirical study of treatment effects would typically start with such simple comparisons. We might also use regression methods or matching to control for demographic or background characteristics. In practice, simple comparisons or even regression-adjusted comparisons may provide misleading estimates of causal effects. For example, participants in subsidized training programmes are often observed to earn less than ostensibly comparable controls, even after adjusting for observed differences (see, for example, Ashenfelter and Card, 1985) . This may reflect some sort of omitted variables bias, that is, a bias arising from unobserved and uncontrolled differences in earnings potential between the two groups being compared. In