A vast literature in social science is concerned with peer effects. Loosely speaking, this means the causal effect of group characteristics on individual outcomes. Sometimes regression is used in an attempt to uncover these effects. In practice, the use of regression models to estimate peer effects is fraught with peril. Although this is not really an IV issue per se, the language and algebra of 2SLS helps us understand why peer effects are hard to identify.
Broadly speaking, there are two types of peer effects. The first concerns the effect of group characteristics such as the average schooling in a state or city on individually-measured outcome variable. This peer effect links the average of one variable to individual outcomes as described by another variable. For example, Acemoglu and Angrist (2000) ask whether a given individual’s earnings are affected by the average schooling in his or her state of residence. The theory of human capital externalities suggests that living in a state with a more educated workforce may make everyone in the state more productive, not just those who are more educated. This kind of spillover is said to be a social return to schooling: human capital that benefits everyone, whether or not they are more educated.
A causal model which allows for such externalities can be written
where Yijt is the log weekly wage of individual i in state j in year t, Ujt is a state-year error component, and p is an individual error term. The controls Sj and At are state-of-residence and year effects. The coefficient p is the returns to schooling for an individual, while the coefficient 7 is meant to capture the effect of average schooling, Sjt, in state j and year t.
In addition to the usual concerns about Sj, the most important identification problem raised by equation (4.6.6) is omitted variables bias from correlation between average schooling and other state-year effects embodied in the error component ujt. For example, public university systems may expand during cyclical upturns, generating a common trend in state average schooling levels and state average earnings. Ace – moglu and Angrist (2000) attempt to solve this problem using instrumental variables derived from historical compulsory attendance laws that are correlated with Sjt but uncorrelated with contemporary ujt and pj.
While omitted state-year effects are the primary concern motivating Acemoglu and Angrist’s (2000) instrumental variables estimation, the fact that one regressor, Sjt, is the average of another regressor, Sj, also complicates the interpretation of OLS estimates of equation (4.6.6). To see this, consider a simpler version of (4.6.6) with a cross-section dimension only. This can be written
where Yjj is he log weekly wage of individual i in state j and Sj is average schooling in the state. Now, let Po denote the coefficient from a bivariate regression of Yjj on Sj only and let p1 denote the coefficient from a bivariate regression of Yjj on Sj only. From the discussion of grouping and 2SLS earlier in this chapter, it’s clear that p1 is the 2SLS estimate of the coefficient on Sj in a bivariate regression of Yjj on Sj using a full set of state dummies as instruments. The Appendix uses this fact to show that the parameters in equation (4.6.7) can be written in terms of Po and p1 as
^0 = Pi + Ф(Ро – Pi) (4.6.8)
^1 = ф(р1 _ po)
where ф = 17R2 > 1; and R2 is the first-stage R-squared.
The upshot of (4.6.8) is that if, for any reason, OLS estimates of the bivariate regression of wages on individual schooling differ from 2SLS estimates using state-dummy instruments, the coeff cient on average schooling in (4.6.7) will be nonzero. For example, if instrumenting with state dummies corrects for attenuation bias due to measurement error in Sj, we have p1 > po and the spurious appearance of positive external returns. In contrast, if instrumenting with state dummies eliminates the bias from positive correlation between Sj and unobserved earnings potential, we have p1 < po, and the appearance of negative social returns. In practice, therefore, it is very difficult to substantiate social effects by OLS estimation of an equation like 4.6.6, though more sophisticated strategies where both the individual and group averages are treated as endogenous may work.
A second and even more difficult peer effect to uncover is the effect of the group average of a variable on the individual level of this same variable. This is not really an IV problem; it takes us back to basic regression issues. To see this point, suppose that Sj is the high-school graduation rate in school j, and we would like to know whether students are more likely to graduate from high school when everyone around them is more likely to graduate from high school. To uncover the peer effect in high school graduation rates, we might work with a regression model like:
Sij = M + ^2 Sj + £ij; (4.6.9)
where Sij is individual i’s high school graduation status and Sj is the average high school graduation rate in school j, which i attends.
At first blush, equation (4.6.9) seems like a sensible formulation of a well-defined causal question, but in fact it is nonsense. The regression of Sij on Sj always has a coefficient of 1, a conclusion that can be drawn immediately once you recognize Sj as the first-stage fitted value from a regression of Sij on a full set of school dummies. Thus, an equation like (4.6.9) cannot possibly be informative about causal effects.
A modestly improved version of the bad peer regression changes (4.6.9) to
where S(i)j is the mean of Sij in school j, excluding student i. This is a step in the right direction – by definition, i is not in the group used to construct S(i)j – but still problematic because Sij and S(i)j are both affected by school-level random shocks. The presence of random effects in the error term raises important issues for statistical inference, issues discussed at length in Chapter 8. But in an equation like (4.6.10), group-level random shocks are more that a problem for standard errors: any shock common to the group (school) creates spurious peer effects. For example, particularly effective school principals may raise graduation rates for everyone in the schools at which they work. This looks like a peer effect since it induces correlation between Sij and S(i)j even if there is no causal link between peer means and individual student achievement. We therefore prefer not see regressions like (4.6.10) either.
The best shot at a causal investigation of peer effects focuses on variation in ex ante peer characteristics, that is, some measure of peer quality which predates the outcome variable and is therefore unaffected by common shocks. A recent example is Ammermueller and Pischke (2006), who study the link between classmates’ family background, as measured by the number of books in their homes, and student achievement in European primary schools. The Ammermueller and Pischke regressions are versions of
sij = M* + ^4B(i)j + Cij;
where B(i)j is the average number of books in the home of student i’s peers. This looks like (4.6.10), but with an important difference. The variable B(ij is a feature of the home environment that predates test scores and is therefore unaffected by school-level random shocks.
Angrist and Lang (2004) provide another example of an attempt to link student achievement with the ex ante characteristics of peers. The Angrist and Lang study looks at the impact of bused-in low-achieving newcomers on high-achieving residents’test scores. The regression of interest in this case is a version of
s ij = M + ^3 mj + , (4.6.11)
where mj is the number of bused-in low-achievers in school j and Sij is resident-student i’s test score. Spurious correlation due to common shocks is not a concern in this context for two reasons. First, mj is a feature of the school population determined by students outside the sample used to estimate (4.6.11). Second, the number of low-achievers is an ex ante variable biased on prior information about where the students come from and not the outcome variable, Sij. School-level random effects remain an important issue for inference, however, since mj is a group-level variable.