Attempts to estimate peer effects on educational achievement directly have been relatively limited. Hanushek (1972, 1992) finds no peer achievement effects, while Henderson et al. (1976), Summers and Wolfe (1977) and Zimmer and Toma (2000) report positive influences of higher achieving peers, at least for some students. Consideration of ability tracking in schools likewise has yielded mixed results, even though policy has presumed that tracking is generally bad for achievement (e.g. see Oakes, 1992; Argys et al., 1996). The evidence on achievement effects of racial composition has been much more voluminous, although the results are no easier to summarize or interpret (cf. Armor, 1995).
2.2 The Importance of Measurement in Peer Effect Estimation
In reality, researchers virtually never possess the entire history of the relevant inputs. Consequently specifications based on equation (1) are rarely if ever directly estimated. The most common alternative, lacking historical information, bases estimation solely on measures of the current values of X, S, P̄(−i) and Ā(−i). But, estimation of specifications of this form offers little hope of providing consistent estimates of the peer parameters (λ and γ). The main issue—one that is not specific to peer effect estimation—is that current characteristics will generally be correlated with unobserved past determinants of achievement, introducing the standard problem of omitted variables bias.
In peer estimation, ignoring history has a stronger impact. Because members of peer groups tend to have similar experiences over time through systematic neighbourhood and school choice, many omitted historical factors will be common to the peer group. Perhaps even more relevant, many left out or poorly measured contemporaneous inputs will also tend to be common to the group. Common past and current omitted factors that affect both individual i's achievement and peer achievement will induce a correlation of contemporaneous peer factors and the individual error term (e), making peer effects appear important even when they have no true impact, i.e. even when λ and γ are identically zero.6
Such problems have been widely discussed within the general achievement literature. For example, in discussions of the interpretations of peer influences contained in the 1966 Coleman Report (Coleman et al., 1966), the possible interactions of model misspecification and peer group measurement entered into early critiques (Hanushek and Kain, 1972; Smith, 1972). In other words, the nature and measurement of peer group factors implies that common model misspecification is particularly damaging to inferences about the importance of peers.
An approach to the general problem of estimating achievement relationships, which we follow below, begins by taking the first difference of equation (1). The value added specification reduces the data requirements to the inputs relevant for grade G, since all of the historical influences on the current achievement level drop out, as in equation (2):
where is the achievement gain (difference between current grade and previous grade test scores) for student i in grade G in school s in cohort c.7 Student achievement growth is related to the contemporaneous inputs (which are the flows of these factors over the observed time period), and the generic problems of omitted historical variables are circumvented.
Even with the value added form, consideration of peer influences complicates the estimation of achievement models. The problems of poorly measured individual school and background factors (either because of omitted or error-prone measures) have the same extra impact in the value added models because of the ‘strong proxy’ nature of peer measures. One important and relevant example is systematic but unmeasured elements of teacher quality. As a simple illustration, assume that the error term in equation (2) omits an aspect of teacher quality (κG) that, while uncorrelated with XG and the measured school aspects of SG, is common to the achievement of peers. Even in the case where current peer achievement is irrelevant (i.e. γ = 0), the estimation of the effects of Ā(−i) will yield an estimator of peer effects, , with upward bias that systematically makes peers look more important. The magnitude of the bias is directly related to the importance of the omitted factor in determining achievement.8
In part to circumvent such problems of mismeasured current inputs commonly affecting all peers, a number of studies have simply dropped Ā(−i) from the specification and included its lagged value in its place. Unfortunately, by itself this introduces a series of statistical and interpretative problems (depending on the precise nature of the underlying behaviour). For example, the lagged average achievement score is likely to remain correlated with the error term because of the serial correlation in unobserved teacher, school and individual factors. Thus, the simultaneous equations and omitted variables biases in estimating peer effects, while altered in form, are not eliminated. Additionally, the substitution of lagged achievement introduces another type of bias that we discuss in detail below.
Our primary strategy for dealing with these general issues begins by extracting fixed components of individuals and schools to deal not only with the most significant omitted variables problems but also the key elements of neighbourhood and school selectivity. Importantly, the substitution of lagged values of peer achievement in place of the current value within such a fixed effects framework is not subject to the problems of simultaneous equations and omitted variables biases, because the fixed effects eliminate the systematic family and school influences that are correlated over time.
From the starting point of equation (2), equation (3) decomposes the error, υ, into a series of components that highlights those factors most likely to contaminate the peer estimates:
The first three terms capture time invariant individual (ωi), school (ωs) and school-by-grade effects on achievement (ωGs); the fourth factor captures cohort-by-year differences in the testing regime; the fifth component captures school-by-grade effects that vary from cohort to cohort, most notably the quality of teaching; and the final factor (ε) is a random error capturing individual shocks that vary over time.
Our approach makes use of matched panel data to remove explicitly the first four components: fixed individual, school, school-by-grade and cohort-by-grade effects. Notice how these fixed effects account for the primary systematic but unobserved differences in students and schools. The student fixed effects (in the gains formulation of equation (2)) account for all student and family factors that do not vary over the period of achievement observation and that affect the rate of learning—including ability differences, family child rearing practices, general material inputs, consistent motivational influences, and parental attitudes towards schools and peers. This approach thus directly deals with many of the most difficult issues of potential bias in the peer estimates arising from omitted and mismeasured individual and family factors.
Next consider any fixed differences in schools that are not perfectly correlated with the student fixed effects or included covariates (S and X). While these are typically correlated with peer group composition through school and neighbourhood choice, they are accounted for by school fixed effects. Finally, even systematic within-school changes in achievement gains across grades can be accounted for through the use of school-by-grade fixed effects.
The importance of the multiple cohorts should not be underestimated. For example, consider the possibility that achievement for students in some schools tends to decline as the students age due to factors other than peer achievement (e.g., adolescence may be more disruptive for economically disadvantaged students). If only fixed individual and school effects were removed (as is possible with panel data for a single cohort) in the estimation, the resulting positive peer effect estimate would suggest that students were responding to peers when in fact other factors had introduced a spurious relationship between the achievement gains of all students in a school. On the other hand, if fixed student and school-by-grade effects are removed—as is possible with data for multiple cohorts—such systematic changes in specific schools cannot drive the results.
The estimation of peer effects along with the fixed individual and school-by-grade effects intuitively relies on perturbations in the pattern of peers across grades and cohorts, i.e. the estimates are identified by small within-school and grade differences in peer group characteristics between cohorts. Such differences emanate from two sources: mobility into or out of the school and, less importantly, changes in student circumstances (e.g. income or achievement). The large annual mobility of students, averaging greater than 20% per year in the Texas public schools, accounts for much of the differences across grades and cohorts.9
One significant concern of course is the possibility that the observed changes in peers simply act as proxies for other changes in family or school inputs. Whether our estimation strategy can generate consistent estimates hinges upon whether the two time-varying components of the error terms (θ and ε in equation (3)) are orthogonal to the included peer variables. Three possibilities seem most important. First, the small observed changes in peer circumstances may be related to changes in family conditions that could bias the estimates. An increase or decrease in peer average income or achievement may result from similar changes in own family income that precipitate a school transfer and exert a direct effect on outcomes. Alternatively, shifts in local labour market conditions may cause changes in both own family and peer group average income, making it difficult to disentangle the influences of peers and family. Second, changes in school characteristics may affect both own achievement and that of peers. For example, the funding and availability of compensatory education programmes is linked to school average income, possibly building in a correlation between peer average income and programmatic effects. Or teacher differences in a specific grade may vary with peer characteristics. Finally, school selection by other families may be driven by attributes of the school, and the effects of such attributes may be confused with peer effects. Consider a school that is becoming dysfunctional, say because of an ineffective principal, and finds that all of its upper income families flee over time. In such a case, achievement of the remaining students could fall along with the incomes of peers, erroneously suggesting that peer income affects achievement even when there is no such relationship.
The severity of these potential problems depends in part on the ability to control for changes in families and schools. In this analysis (described below) we include time-varying measures of family income, school characteristics, compensatory programme status and overall school transfer behaviour. Perhaps more important given the controls for student and school-by-grade fixed effects, the potential severity also depends upon the speed with which families relocate in response to school conditions. The concern is simply that families adjust to changes in school quality (including peer composition) and thereby might induce bias through equilibrium selection behaviour. But, because residential moving is a costly process that undoubtedly includes some slow adjustment, movement due to parental selectivity of schools is almost certainly much slower than the movement of peer characteristics found in exogenous year-to-year variations. The assumption that families also react slowly (i.e. not in the current year) to specific variations in teacher quality seems natural, implying that there is no reason to believe that the choice behaviour of parents to current changes in teachers leads to any presumption about correlation of peer factors and annual variations in teacher quality. It seems plausible that differences captured by school or school-by-grade quality provide the prime motivation for any family selection of schools. Any remaining variations in annual teacher quality, even if large, must be orthogonal to the school-by-grade estimates of quality. Particularly because the average family has more than one child, mobility reactions to current shocks to teacher quality are likely to be minimal.
In sum, the choice of neighbourhood and school will tend to bias upwards the estimated effect of peer achievement unless an exogenous source of variation in peer achievement can be identified. Our estimation strategy, which relies on small changes over time and grades in peer characteristics, will provide consistent estimates of the underlying peer parameters unless systematic changes in the contemporaneous innovations to achievement (θ and ε) are correlated with the predetermined peer effects. Given the available measures of year-to-year changes in family and school characteristics and the structure of the data allowing for the removal of student and school-by-grade fixed effects, such correlations are likely to be of a very low order of importance.
2.3 The Reflection Problem
Before completing this discussion, an additional issue of peer influences must be introduced. The most vexing estimation problem, formulated in detail by Manski (1993), is the possible simultaneous determination of achievement for all classmates, with high achievement by one student directly improving the achievement of classmates and vice versa. This possibility, which has also received the most theoretical attention, is captured in equation (2) by the inclusion of Ā(−i)G, the average achievement of peers. If the achievement of each peer is also governed by equation (2), we would have Ā(−i)G directly related to AiGs through individual i's influence on the others in the class. As Moffitt (2001) shows, this situation can be thought of as a standard simultaneous equations problem, where the induced correlation of Ā(−i)G and υiG leads to inconsistent estimation of the peer effect parameter. The reflection problem (in the terminology of Manski, 1993) presents a conundrum, because it is extremely difficult to identify the separate structural peer parameters (λ and γ) of equation (2) through standard exclusion restrictions.10 Without imposing functional form restrictions, one needs to find aggregate peer factors that do not have an individual analogue in the achievement relationship, something that is difficult given the underlying conceptual basis that portrays peers as essentially extended families. Nor does randomization help, because current behaviour of the individual and peers will still be important.
The estimation and interpretation issue in this framework is whether the contemporaneous behaviour of peers is important or whether any peer relationship is essentially captured by the underlying characteristics including prior achievement. Understanding the dimensions of this issue requires more detailed consideration of the peer components in equation (2). At the outset, it is important to note that the standard terminology in the reflection problem—distinguishing between behavioural and contextual factors—can be confusing in the case of peers and achievement. Similar to measures of family background, the predetermined measures of peers, such as aggregate parental education levels or racial composition of classmates, are in part proxies for attitudes, behavioural patterns and learning related activities that systematically enter into the behaviour and learning of each student.
Of course the endogenous peer component represented here by current aggregate achievement is distinguished from the other factors mainly because of the reciprocal nature of the determination of peer achievement. It is ‘behavioural’ in the sense that each student's actions directly affect the rest of the class. It is this issue of simultaneity that severely complicates the estimation of γ, not the fact that exogenous characteristics are unrelated to peer group behaviour.
Our estimation concentrates on models that employ lagged peer achievement instead of the contemporaneous value. Interpretation of estimation built on lagged peer achievement depends on the relationship between lagged and current behaviour.11 If lagged achievement captures all of the relevant variation in current peer behaviour (i.e. there are no year-to-year shocks in current behaviour), there is no bias. Of course in the fixed effects framework there would be no need to substitute for current peer achievement if this were the case. More realistically, lagged peer achievement is likely to be an imperfect proxy for the current value. If the difference between current and lagged measures of peer achievement is random (e.g. the probably of a family shock in grade G such as divorce is randomly distributed), the estimated effect of peer achievement will generally be biased towards zero in a normal proxy variable effect. Even if the current innovation to peer behaviour is correlated with lagged peer achievement, under most conceivable circumstances the estimated effect will still be downward biased. Therefore the estimated effect of lagged achievement should provide a lower bound estimate of γ, and we find little reason to believe, at least based on past estimation and descriptions of classroom behaviour and interactions, that the changes in individual behaviour in a particular grade are especially important when compared to the underlying systematic differences captured by lagged achievement.