RCT baseline controls that do not match outcome measures Does anyone have any input on whether it is ok to use baseline measures as control variables in a randomized controlled trial (RCT) if they are not exactly the same as the outcome measures?  I.e., if the questions were worded slightly differently between the baseline survey and the outcome survey, can I still include the baseline survey measure as a control?
For instance, if the baseline survey asks how frequently the individual pays bills on time, and the outcome survey asks how many times in the past two months the individual paid a bill late, can I use the frequency of ontime bills as a control when the dependent variable is the number of late bills in the last two months? Any corrections I should make?
 A: Controlling for covariates can often improve the precision of the experimental treatment effect estimate. Initially, I thought I could cook up an example where that is not the case due to smearing, but that does not seem to be the case as long as the treatment dummy is truly random.
Here's a simple simulation that illustrates this. Let us suppose that propensity to be late on bills consists of a time-invariant fixed effect $u$ plus noise $\varepsilon_{it}$. 
We can also add some autocorrelated disturbances. Imagine that someone had an unexpected car repair or won the lottery. The first makes him more likely to be late on some bills and the second less likely. The effect is likely to persist over multiple periods, and be part of the error today and yesterday's error. This seems a very likely scenario, especially since late bills tend to carry over with increasing penalties.
The estimate with the baseline is more precisely estimated around the true value of 1, compared to a model with just a treatment dummy:

Stata code:
clear
set obs 1000

gen b1 =.
gen b2 =.

forvalues i = 1/1000 {
    gen treat = cond(mod(_n,4),0,1) // 1/4 treated at random

    gen u = rnormal(1,1)

    gen e1    = rnormal(0,1) + 0.9*rnormal(0,1)
    gen e2    = rnormal(0,1) + 0.9*e1

    gen y1 = 2 + u + e1
    gen y2 = 2 - 1*treat + u +  e2

    qui reg y2 i.treat
    qui replace b1 = _b[1.treat] in `i'
    qui reg y2 i.treat c.y1
    qui replace b2 = _b[1.treat] in `i'

    drop e* treat y* u
}

tw (kdens b1) (kdens b2), xline(-1) legend(label(1 "Without Baseline") label(2 "With Baseline"))

