Mostly Harmless Econometrics: An Empiricist’s Companion

# Random Assignment Solves the Selection Problem

Random assignment of Dj solves the selection problem because random assignment makes Dj independent of potential outcomes. To see this, note that

E[yj |dj = 1] - E[Yj|Dj =0] = E[yli IDj = 1] - E[Yoj|Dj =0]

= E[yij |Dj = 1] - E[Yoj|Dj = 1],

where the independence of Yoj and Dj allows us to swap E[Yoj|Dj = 1] for E[Yoj|Dj = 0] in the second line. In fact, given random assignment, this simplifies further to

E [Yij|Dj = 1] - E [Yo j | D j = 1] = E [y ij - Yoj|Dj = 1]

= E [yij - Yoj] .

The effect of randomly-assigned hospitalization on the hospitalized is the same as the effect of hospitalization on a randomly chosen patient. The main thing, however, is that random assignment of Dj eliminates selection bias. This does not mean that randomized trials are problem-free, but in principle they solve the most important problem that arises in empirical research.

How relevant is our hospitalization allegory? Experiments often reveal things that are not what they seem on the basis of naive comparisons alone. A recent example from medicine is the evaluation of hormone replacement therapy (HRT). This is a medical intervention that was recommended for middle-aged women to reduce menopausal symptoms. Evidence from the Nurses Health Study, a large and influential non­experimental survey of nurses, showed better health among the HRT users. In contrast, the results of a recently completed randomized trial shows few benefits of HRT. What’s worse, the randomized trial revealed serious side effects that were not apparent in the non-experimental data (see, e. g., Women’s Health Initiative [WHI], Hsia, et al., 2006).

An iconic example from our own field of labor economics is the evaluation of government-subsidized training programs. These are programs that provide a combination of classroom instruction and on - the-job training for groups of disadvantaged workers such as the long-term unemployed, drug addicts, and ex-offenders. The idea is to increase employment and earnings. Paradoxically, studies based on non­experimental comparisons of participants and non-participants often show that after training, the trainees earn less than plausible comparison groups (see, e. g., Ashenfelter, 1978; Ashenfelter and Card, 1985; Lalonde 1995). Here too, selection bias is a natural concern since subsidized training programs are meant to serve men and women with low earnings potential. Not surprisingly, therefore, simple comparisons of program participants with non-participants often show lower earnings for the participants. In contrast, evidence from randomized evaluations of training programs generate mostly positive effects (see, e. g., Lalonde, 1986; Orr, et al, 1996).

Randomized trials are not yet as common in social science as in medicine but they are becoming more prevalent. One area where the importance of random assignment is growing rapidly is education research (Angrist, 2004). The 2002 Education Sciences Reform Act passed by the U. S. Congress mandates the use of rigorous experimental or quasi-experimental research designs for all federally-funded education studies. We can therefore expect to see many more randomized trials in education research in the years to come. A pioneering randomized study from the field of education is the Tennessee STAR experiment designed to estimate the effects of smaller classes in primary school.

Labor economists and others have a long tradition of trying to establish causal links between features of the classroom environment and children’s learning, an area of investigation that we call “education pro­duction.” This terminology reflects the fact that we think of features of the school environment as inputs that cost money, while the output that schools produce is student learning. A key question in research on education production is which inputs produce the most learning given their costs. One of the most expensive inputs is class size - since smaller classes can only be had by hiring more teachers. It is therefore important to know whether the expense of smaller classes has a payoff in terms of higher student achievement. The STAR experiment was meant to answer this question.

Many studies of education production using non-experimental data suggest there is little or no link be­tween class size and student learning. So perhaps school systems can save money by hiring fewer teachers with no consequent reduction in achievement. The observed relation between class size and student achieve­ment should not be taken at face value, however, since weaker students are often deliberately grouped into smaller classes. A randomized trial overcomes this problem by ensuring that we are comparing apples to apples, i. e., that the students assigned to classes of different sizes are otherwise comparable. Results from the Tennessee STAR experiment point to a strong and lasting payoff to smaller classes (see Finn and Achilles, 1990, for the original study, and Krueger, 1999, for an econometric analysis of the STAR data).

The STAR experiment was unusually ambitious and influential, and therefore worth describing in some detail. It cost about \$12 million and was implemented for a cohort of kindergartners in 1985/86. The study ran for four years, i. e. until the original cohort of kindergartners was in third grade, and involved about 11,600 children. The average class size in regular Tennessee classes in 1985/86 was about 22.3. The experiment assigned students to one of three treatments: small classes with 13-17 children, regular classes with 22-25 children and a part-time teacher’s aide, or regular classes with a full time teacher’s aide. Schools with at least three classes in each grade could choose to participate in the experiment.

The first question to ask about a randomized experiment is whether the randomization successfully balanced subject’s characteristics across the different treatment groups. To assess this, it’s common to compare pre-treatment outcomes or other covariates across groups. Unfortunately, the STAR data fail to include any pre-treatment test scores, though it is possible to look at characteristics of children such as race and age. Table 2.2.1, reproduced from Krueger (1999), compares the means of these variables. The student

Table 2.2.1: Comparison of treatment and control characteristics in the Tennessee STAR experiment

 Students who entered STAR in kindergarten Variable Small Regular Regular/Aide Joint P-value 1. Free lunch .47 .48 .50 .09 2. White/Asian .68 .67 .66 .26 3. Age in 1985 5.44 5.43 5.42 .32 4. Attrition rate .49 .52 .53 .02 5. Class size in kindergarten 15.10 22.40 22.80 .00 6. Percentile score in kindergarten 54.70 48.90 50.00 .00

Notes: Adapted from Krueger (1999), Table 1. The table shows means of variables by treatment status. The P-value in the last column is for the F-test of equality of variable means across all three groups. All variables except attrition are for the first year a student is observed, The free lunch variable is the fraction receiving a free lunch. The percentile score is the average percentile score on three Stanford Achievement Tests. The attrition rate is the proportion lost to follow up before completing third grade.

characteristics in the table are a free lunch variable, student race, and student age. Free lunch status is a good measure of family income, since only poor children qualify for a free school lunch. Differences in these characteristics across the three class types are small and none are significantly different from zero. This suggests the random assignment worked as intended.

Table 2.2.1 also presents information on average class size, the attrition rate, and test scores, measured here on a percentile scale. The attrition rate was lower in small kindergarten classrooms. This is potential a problem, at least in principle.[5] Class sizes are significantly lower in the assigned-to-be-small class rooms, which means that the experiment succeeded in creating the desired variation. If many of the parents of children assigned to regular classes had effectively lobbied teachers and principals to get their children assigned to small classes, the gap in class size across groups would be much smaller.

Because randomization eliminates selection bias, the difference in outcomes across treatment groups captures the average causal effect of class size (relative to regular classes with a part-time aide). In practice, the difference in means between treatment and control groups can be obtained from a regression of test scores on dummies for each treatment group, a point we expand on below. The estimated treatment-control differences for kindergartners, reported in Table 2.2.2 (derived from Krueger, 1999, Table 5), show a small - class effect of about 5 to 6 percentile points. The effect size is about.2a, where a is the standard deviation of the percentile score in kindergarten. The small-class effect is significantly different from zero, while the

Table 2.2.2: Experimental estimates of the effect of class-size assignment on test scores

 Explanatory variable (1) (2) (3) (4) Small class 4.82 5.37 5.36 5.37 (2.19) (1.26) (1.21) (1.19) Regular/aide class .12 .29 .53 .31 (2.23) (1.13) (1.09) (1.07) White/Asian (1 = yes) 8.35 (1.35) 8.44 (1.36) Girl (1 = yes) 4.48 (.63) 4.39 (.63) Free lunch (1 = yes) -13.15 (.77) -13.07 (.77) White teacher -.57 (2.10) Teacher experience .26 (.10) Master’s degree -0.51 (1.06) School fixed effects No Yes Yes Yes R2 .01 .25 .31 .31

Note: Adapted from Krueger (1999), Table 5. The dependent variable is the Stanford Achievement Test percentile score. Robust standard errors that allow for correlated residuals within classes are shown in parentheses. The sample size is 5681.

regular/aide effect is small and insignificant.

The STAR study, an exemplary randomized trial in the annals of social science, also highlights the logistical difficulty, long duration, and potentially high cost of randomized trials. In many cases, such trials are impractical.[6] In other cases, we would like an answer sooner rather than later. Much of the research we do, therefore, attempts to exploit cheaper and more readily available sources of variation. We hope to find natural or quasi-experiments that mimic a randomized trial by changing the variable of interest while other factors are kept balanced. Can we always find a convincing natural experiment? Of course not. Nevertheless, we take the position that a notional randomized trial is our benchmark. Not all researchers share this view, but many do. We heard it first from our teacher and thesis advisor, Orley Ashenfelter, a pioneering proponent of experiments and quasi-experimental research designs in social science. Here is Ashenfelter (1991) assessing the credibility of the observational studies linking schooling and income:

How convincing is the evidence linking education and income? Here is my answer: Pretty con­vincing. If I had to bet on what an ideal experiment would indicate, I bet that it would show that better educated workers earn more.

The quasi-experimental study of class size by Angrist and Lavy (1999) illustrates the manner in which non-experimental data can be analyzed in an experimental spirit. The Angrist and Lavy study relies on the fact that in Israel, class size is capped at 40. Therefore, a child in a fifth grade cohort of 40 students ends up in a class of 40 while a child in fifth grade cohort of 41 students ends up in a class only half as large because the cohort is split. Since students in cohorts of size 40 and 41 are likely to be similar on other dimensions such as ability and family background, we can think of the difference between 40 and 41 students enrolled as being “as good as randomly assigned.”

The Angrist-Lavy study compares students in grades with enrollments above and below the class-size cutoffs to construct well-controlled estimates of the effects of a sharp change in class size without the benefit of a real experiment. As in Tennessee STAR, the Angrist and Lavy (1999) results point to a strong link between class size and achievement. This is in marked contrast with naive analyses, also reported by Angrist and Lavy, based on simple comparisons between those enrolled in larger and smaller classes. These comparisons show students in smaller classes doing worse on standardized tests. The hospital allegory of selection bias would therefore seem to apply to the class-size question as well.[7]

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