Controlling for covariates

Abstract

Abstract In finding a treatment (d) effect on a response variable (y) with observational data, the control group with d = 0 (or C group) may be different from the treatment group with d = 1 (or T group) in observed variables x. This can cause an ‘overt bias’, because the difference between the two groups’ responses may be due to the difference in x, and not due to the difference in d. Also, if the T group differs from the C group in unobserved variables e, then this can cause a ‘hidden (or covert) bias’. This and the following chapter deal with controlling for x to avoid overt bias. The basic way to control for x is ‘matching’ (examined in the next chapter). All the other issues for overt bias is discussed here; no hidden bias will be assumed throughout. Controlling for x involves comparing treated and control subjects sharing the same value of x. This raises a number of issues: (i) which variables to control for, (ii) what is identified with x controlled for, (iii) what if the C and T groups do not overlap in x. In general, (i) pre-treatment variables should be controlled for; (ii) with x controlled for, the conditional effect E(y1 - y0¦x) is identified where yj is the potential response when d = j; (iii) ‘regression discontinuity design’ may be useful if the values of x do not overlap. Instead of getting E(y1 - y0) via E(y1 - 0¦x), there is a ‘weighting’ approach to find E(y1 - y0).