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524 Chapter 13 Experiments and Quasi-Experiments

Econometric Methods for Analyzing Experimental Data

Data from a randomized controlled experiment can be analyzed by comparing dif-
ferences in means or by a regression that includes the treatment indicator and
additional control variables. This latter specification, the differences estimator with
additional regressors, can also be used in more complicated randomization schemes,
in which the randomization probabilities depend on observable covariates.

The differences estimator. The differences estimator is the difference in the sam-
ple averages for the treatment and control groups (Section 3.5), which can be com-
puted by regressing the outcome variable Y on a binary treatment indicator X:

Yi = b0 + b1Xi + ui, i = 1, c, n. (13.1)

As discussed in Section 4.4, if X is randomly assigned, then E(ui ͉ Xi) = 0 and the
OLS estimator of b1 in Equation (13.1) is an unbiased and consistent estimator of
the causal effect.

The differences estimator with additional regressors. The efficiency of the differ-
ence estimator often can be improved by including some control variables W in the
regression; doing so leads to the differences estimator with additional regressors:

Yi = b0 + b1Xi + b2W1i + g + b1 + rWri + ui, i = 1, c, n. (13.2)

If W helps to explain the variation in Y, then including W reduces the standard
error of the regression and, typically, the standard error of bn1. As discussed in
Section 7.5 and Appendix 7.2, for the estimator bn1 of the causal effect b1 in Equa-
tion (13.2) to be unbiased, the control variables W must be such that ui satisfies
conditional mean independence, that is, E(ui ͉ Xi,Wi) = E(ui ͉ Wi). This condition
is satisfied if Wi are pretreatment individual characteristics, such as gender: If Wi
is a pretreatment characteristic and Xi is randomly assigned, then Xi is indepen-
dent of ui and Wi which implies that E(ui ͉ Xi,Wi) = E(ui ͉ Wi). The W regressors
in Equation (13.2) should not include experimental outcomes (Xi is not randomly
assigned, given an experimental outcome). As always with control variables under
conditional mean independence, the coefficient on the control variable does not
have a causal interpretation.

Estimating causal effects that depend on observables. As discussed in Chapter 8,
variation in causal effects that depends on observables can be estimated by including
suitable nonlinear functions of, or interactions with, Xi. For example, if W1i is a