204 S.G. Rivkin Economics of Education Review 20 2001 201–209
3. Empirical model
The acquisition of knowledge and skills is a cumulat- ive process that takes place over many years. Eq. 1
describes the relationship between achievement A, fam- ily background X and peer group quality P for student
i who attends school j in year T. The effects of both family background and peer group quality are allowed
to accumulate over time, but there is no serial correlation in the error term
e. A
ijT
5
O
T 21
t 51
f
t
X
ij
, P
jt
1e
ijT
1 Estimates of peer effects for a single time period
almost certainly confound the influences of current peers with those of peer groups and families from prior years.
In order to isolate the effects of current peers, a pretest score is included as a control. Eq. 2 presents a linear,
value added specification in which the peer group effect is presumed to depend in part upon family characteristics
relative to those of peers.
A
ijT
5W
ijT −
1
a1X
ij
x1P
jT −
1
d1P
jT −
1
2X
ij
D1e
ijT
2 Despite the inclusion of a measure of academic
achievement in the sophomore year of high school, the estimation of Eq. 2 may still generate biased peer effect
coefficients if relevant family background or school vari- ables are omitted. As previously discussed, one treatment
for this problem has been the use of more aggregate information as instruments for the school or neighbor-
hood data. If valid instruments are identified, i.e. instru- ments that are uncorrelated with
e
ijT
and highly corre- lated with P
jT 21
, consistent peer effect estimates can be obtained.
The key issue is whether the use of aggregate infor- mation as instruments reduces endogeneity bias. This
question is considered with the following simple model in which exogenous explanatory variables are omitted
without loss of generality.
11
Note that in a linear specifi- cation, the coefficient that captures the common peer
effect for all students, d, cannot be separately identified
from the coefficient that captures peer influences that depend upon student background relative to school-
mates, D
11
It is straightforward to show that IV estimation using the aggregate information as an instrument and OLS estimation that
substitutes the aggregate information in place of the school level measure of peer group quality produce estimates that have the
same expected value, because within county deviations in peer group quality are orthogonal to county average peer group qual-
ity by construction.
A
ijT
5P
cT −
1
1P
dT −
1
b1u
ijT
3 In Eq. 3,
b is the combined peer effect, P
c
is the aver- age peer group quality in county c and P
d
is the deviation of school peer group quality from the county average.
The decomposition of the variation in peer group quality into orthogonal within county and between county
components makes explicit the fact that b
ols
is determ- ined by both sources of variation, and that the bias is
proportional to the sum of the covariation between the county average peer group quality and the error and the
covariation between within county deviations in peer group quality and the error:
p lim bˆ
ols
1s
u,P
c
1s
u,p
d
s
2 P
c
1s
2 P
d
4 By comparison,
b
IV
is identified solely by between county variation:
p lim bˆ
IV
5b1s
u,p
c
s
2 P
c
5 The instrumental variable estimate is consistent as
long as s
Pc,u
equals 0. However, if s
Pc,u
does not equal 0, the use of aggregate data as instruments may move
the estimate away from its true value. This will occur if aggregation reduces the denominator of the second term
of Eq. 4 and Eq. 5 proportionately more than the numerator, i.e. if the between county variation is rela-
tively more contaminated by endogeneity bias than the within county variation.
Unfortunately, it is not possible to observe the covari- ations between the within and between county compo-
nents of the peer group characteristics and the error. In the case where there are more instruments than endogen-
ous explanatory variables, tests of over identifying restrictions such as the Sargan test can be used to test
the hypothesis that the instruments are uncorrelated with the structural error.
12
However, this test lacks power against some alternative hypotheses,
13
and it is difficult to interpret the results. Because the instruments are valid
only in the case where they have zero explanatory power, the inability to reject the null of zero explanatory power
at the 95, 90 or even 50 percent significance levels is
12
The Sargan test statistic equals Sargan
5T2kR
2
|c
2
r where R
2
= the value of R
2
from a regression of the IV residuals from the second stage on the exogenous explanatory variables
and the instruments; T =
number of observations; k =
number of parameters in the outcome equation; and r
= number of over-
identifying restrictions instruments minus endogenous explana- tory variables.
13
For example, the test cannot distinguish between a set of instruments that is truly unrelated to the structural error and a
set in which each instrument has the same relationship with the structural error.
205 S.G. Rivkin Economics of Education Review 20 2001 201–209
certainly not evidence that the true correlation between the instruments and the error is zero. Moreover, the
addition of instruments orthogonal to both the endogen- ous explanatory variable and the error increases the prob-
ability that the null of zero is not rejected without reduc- ing the correlation between the instrument and the error,
raising doubts about the value of information gained from this test. Despite these problems, the Sargan Test
statistic will be reported where appropriate.
In cases where tests of over identifying restrictions cannot be used, such as just identified specifications or
models with binary dependent variables, there is no asymptotically consistent test of the relationship between
the structural error and the instruments. One potential way to obtain information on the correlation between the
instrument and the error is to examine the explanatory power of the instruments in a regression of the outcome
variable on the endogenous explanatory variable, the instruments, and the included exogenous variables. This
is the evidence offered by Evans, Oates, and Schwab in support of the validity of their instruments. However, it
is straightforward to show that in both the just identified and over-identified cases, the explanatory power of the
instruments in such a regression offer no useful infor- mation concerning the relationship between the instru-
ments and the structural error Rivkin Woglom, 1999.
Though direct tests of instrument validity are either weak or uninformative, the OLS and IV estimates them-
selves can provide information on the desirability of using aggregate information as instruments. Theory in
favor of aggregation makes explicit the argument that the within county or metropolitan area deviation in peer
group quality is contaminated by unobserved family characteristics related to the choice of neighborhoods
and schools, and that such contamination introduces an upward bias into estimates of peer group effects. There-
fore precisely estimated IV coefficients that are smaller than single equation estimates are consistent with the
view that aggregation reduces endogeneity bias, while precisely estimated IV coefficients that are larger than
the single equation estimates contradict that view. Of course imprecise IV estimates provide little useful infor-
mation.
4. Results
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