Causal inference using regression on the treatment variable
CHAPTER 9Causal inference using regression on thetreatment Causal inference and predictive comparisonsSo far, we have been interpreting regressionspredictively: given the values of severalinputs, the fitted model allows us to predicty, considering thendata points as asimple random sample from a hypothetical infinite superpopulation or probabilitydistribution. Then we can make comparisons across differentcombinations of valuesfor these chapter and the next considercausal inference , which concerns whatwouldhappento an outcomeyas a result of a hypothesized treatment or a regression framework, the treatment can be written as a variableT:1Ti={1 if unitireceives the treatment 0 if unitireceives the control, or, for a continuous treatment ,Ti= level of the treatment assigned to the usual regression context, predictive inference relates to comparisonsbetweenunits, whereas Causal inference addresses comparisons of different treatments ifapplied to thesameunits.}
happened to the individual if assigned to control. Conversely, for control units, y0 i is observed and y1 i is counterfactual. In either case, a simple treatment effect for unit i can be defined as treatment effect for unit i = y1 i − y 0 i Figure 9.3 displays hypothetical data for an experiment with 100 units (and thus
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