Is parental involvement lower at larger schools?

Economics of Education Review 29 (2010) 959–970

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Is parental involvement lower at larger schools? Patrick Walsh Department of Economics, St. Michael’s College, Colchester, VT 05439, USA

a r t i c l e

i n f o

Article history: Received 15 January 2009 Received in revised form 23 November 2009 Accepted 6 April 2010 JEL classification: H41 I20 Keywords: Economies of scale Educational economics

a b s t r a c t Parents who volunteer, or who lobby for improvements in school quality, are generally seen as providing a school-wide public good. If so, straightforward public-good theory predicts that free-riding will reduce average involvement at larger schools. This study uses longitudinal data to follow families over time, as their children move from middle schools to high schools, thus netting out unobservable differences among families. Increases in school size result in significant reductions in parental involvement, although the magnitude of the effect is small. If parents experience a doubling in school size, they are 2 percentage points less likely to increase their contacts with the school, and 5 percentage points less likely to increase their volunteering. A continuous-treatment propensity-score method tests whether the results are driven by selection into treatment. The parental contact results are robust to this test, while the volunteer results are not. Also, there is some evidence that parents see their involvement as a substitute, rather than a complement, for perceived school quality. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Parental involvement is seen as a key input to education. Recent studies by sociologists and education scholars (Ou, Mersky, Reynolds, & Kohler, 2007; Pong, Hao, & Gardner, 2005) associate parental involvement with higher student GPA, higher attainment, and better post-school outcomes. Economists Houtenville and Conway (2008) find that parental involvement (both at home and at the school) strongly raises student achievement, even controlling for family background. Economists understand that parental involvement is not exogenous, but rather is the object of optimizing behavior. If parental involvement has genuine educational benefits, and parents actively choose their level of involvement, it is natural to ask what factors increase or decrease that involvement. This paper considers the public-good aspects of parental involvement, where one parent’s efforts may provide spillover benefits to another family at the same school.

E-mail address: [email protected]. 0272-7757/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.econedurev.2010.04.003

Therefore, a distinction must be made between homebased parental involvement (such as reading to young children, help with homework, or stressing the importance of grades) and school-based parental involvement (lobbying school administrators for different policies, better teachers or facilities, or volunteering at the school). Aside from peereffect spillovers, home-based parental involvement has only private returns to the family. In the case school-based parental involvement, further distinctions must be made between activities such as complaints about grades, meeting about student behavior, or discussing college plans, and activities that improve academic quality or provide resources. School-based involvement of the latter type can potentially be a public good. Confirming this, Walsh (2008) uses an instrument for parental involvement, and finds that high participation in the PTO leads principals to monitor teachers more closely. McMillan (2000) also instruments for parental involvement, and finds that involvement in parent–teacher organizations (PTO) raises school-wide test scores. A straightforward prediction of public economics is that, as the size of the community increases, free-riding also increases, causing average voluntary public-good provision

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P. Walsh / Economics of Education Review 29 (2010) 959–970

to fall (Bergstrom, Blume, & Varian, 1986). In the context of schools, we should therefore expect average parental involvement to be lower at larger schools, holding all other factors constant. Despite this prediction, much of the research into school size and parental involvement has proceeded in parallel, without much intersection. Andrews, Duncombe, and Yinger (2002) summarize early literature on school size. More recent contributions include Lee and Smith (1997), Foreman-Peck and Foreman-Peck (2006), and Kuziemko (2006). These studies generally find modest performance advantages for small schools. Scott-Stein and Thorkildsen (1999) review studies on parental involvement. Studies uniting parental involvement and school size include Bonesronning (2004), which looks at the impact of class size (rather than school size) on parental involvement. Brunner and Sonstelie (2003) look at the effects of school size on voluntary financial contributions by families to public schools. Both papers find evidence for free-riding in larger classes/schools. Datar and Mason (2008) find mixed parental responses to smaller classes. This paper investigates whether free-riding causes less school-based parental involvement at larger schools. It bridges the gap between studies on school size and studies on parental involvement by applying the insights of private public-good provision to schools. The National Educational Longitudinal Survey (NELS:88) provides data on parental involvement, school size, and a host of family and school characteristics. This data set is sufficiently rich to differentiate parentinitiated contacts from school-initiated contacts, and to differentiate parental contacts about the quality of the academic program from parental contacts regarding grades, behavior, or college admissions. A simple cross-sectional comparison of school size and parental involvement would be biased, since the type of parents who select smaller schools are also more likely to get involved. Instead, this paper utilizes the longitudinal aspect of the dataset to follow the same families as they move from relatively smaller middle schools (8th grade) to relatively larger high schools (12th grade). This approach nets out any unobservable family characteristics that are fixed over time, and any trends that are common to all families. Identification therefore comes from the differences across families in the jumps in school size between 8th and 12th grade. Other factors that affect involvement, such as changes in school quality, family income, family composition, and student test scores, are controlled. Larger jumps in school size are associated with significantly less parental involvement, although the magnitude of the effect is small. A doubling in grade enrollment, for example, is associated with a 2 percentage point decline in the probability that parents will begin contacting the school, or raise their frequency of contact. Likewise, a doubling of grade enrollment is associated with a 4–5 percentage point decline in the probability that parents will begin volunteering, or raise their frequency of volunteering. A doubling of grade enrollment yields a drop in involvement as large as that brought about by a parent starting to work outside the home. This result is robust across different definitions of involvement change.

The jump in enrollment between 8th and 12th grades may not be randomly assigned. If different types of parents select school districts with different “jumps”, even the longitudinal approach may be biased. Therefore, a continuous-treatment propensity-score method (the “generalized propensity score”, or GPS) is used to test these results by balancing large-jump and small-jump families based on their observables. The GPS results support the finding that larger schools reduce parental contact, suggesting that parents change their contacting behavior as theory would predict. However, the effect of school size on volunteering is insignificant in the GPS estimation. This implies that changes in volunteering behavior arise from the fact that different “volunteering types” of parents experience different changes in school size. Finally, there is evidence that parents see their involvement as a substitute, rather than a complement, for perceived school quality. Increases in teacher salary (an indicator of school funding levels) result in reduced parental volunteering. Likewise, parents tend to increase their volunteering when they move to a school with a higher share of students receiving free lunches. 2. Free-riding among parents An original model is not required to explore the familiar free-riding problem. Instead, a simple adaptation of this dynamic to the parent/school setting will be provided. All families at the same school commonly experience the same level of school quality, Q. School quality is therefore treated as a public good. Parents receive positive utility from the school quality Q, and negative utility from expending lobbying effort Ei according to utility function Ui (Q, Ei ). In order to raise Q, a parent must expend lobbying effort (Ei ). Schools are treated as black boxes which respond mechanically to lobbying. The lobbying effort of all other parents (z ≡ E ) also raises Q, introducing the possibility of j= / i j free-riding. 2.1. Individual parents’ problem (assume z exogenous) The parent’s problem is given by: max Ei

s.t.

Ui (Q, Ei ) 

Q = (Q0 + (Ei + z) )

(1) where Q0 and z > 0, and 0 <  < 1

The constraint can be understood as follows.1 Any lobbying effort provided by family i (Ei ) or by all other parents (z) increases aggregate quality. This aggregate quality is simultaneously experienced by all families as a public good. Finding FOC and solving for Ei gives:



Ei∗ = −

∂Ui /∂Q ∂Ui /∂Ei

1/(1−)

−z

(2)

Recall that (∂Ui /∂Ei ) < 0, rendering the entire expression positive. Eq. (2) has all the expected properties. Effort

1

This functional form is chosen to retain Ei as an explicit term.

P. Walsh / Economics of Education Review 29 (2010) 959–970

Ei∗ is: (i) decreasing in z; (ii) decreasing in n; (iii) increasing in ; (iv) increasing in marginal utility of Q; (v) decreasing in marginal dis-utility of Ei ; (vi) decreasing in Q0 (higher Q0 reduces marginal utility of Q).

the marginal dis-utility of effort. Accordingly, the following structural equation can describe parental involvement: Esti = ı0 + ı1 (ln(Nst )) + ı2 (Qst ) + ı3 (C Xti ) + ı4 (F Xi ) + ı5 (Ui ) + t + ist

2.2. General equilibrium among identical agents Of course, z is not exogenous – parents will best-respond to one another’s lobbying efforts. z may be endogenized by assuming identical agents. With common utility functions across all agents, the optimal level of effort will be the same for all agents. This gives z = E = (N − 1)E, where N is j= / i j the number of families at the school. Substituting this into Eq. (2) gives:



Ei∗

1 ∂Ui /∂Q = − N ∂Ui /∂Ei

1/(1−) (3)

Recall that (∂Ui /∂Ei ) < 0, rendering the entire expression positive. Eq. (3) provides a more general solution for individually optimal effort, where parents best-respond to one another. 2.3. Socially optimal effort among identical agents In the identical-agent case, the socially optimal level of effort is easily solved. The familiar Lindahl–Samuelson condition, where MRTEi ,Qi =



∂Q − ∂Ei

MRSEi ,Qi , is given by:

i

 =N

∂Ui /∂Ei ∂Ui /∂Q

(4)

Substituting in for ∂Q/∂Ei , replacing (Ei + z) with NEi , and solving for E gives:



opt Ei

1 ∂Ui /∂Q = −N N ∂Ui /∂Ei

1/(1−)

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(6)

where Esti is parental involvement effort, Nst is the size of the school, Qst is the quality of the school, C Xti is a vector of changing observable family characteristics, F Xi is a vector of fixed observable family characteristics, Ui is a vector of fixed unobservable family characteristics, and t is a time trend that is common across families. Since Ei varies by the inverse of N in Eq. (3), ln(N) is used in linear Eqs. (6)–(8). If parents were randomly assigned to schools of different size, we could test the hypothesis that parental involvement is decreasing in school size by simply estimating: Esti = ˛0 + ˛1 (ln(Nst )) + ˛2 (Qst )+˛3 (C Xti ) + ˛4 (F Xi ) + ist (7) In reality, parents have some influence (largely through residential choice and the availability of private schools) over which school their child attends. If the choice of school is correlated with unobserved family characteristics (as seems likely), the estimate of ı1 will be biased. In particular, if parents who place a higher value on education are (i) more involved and (ii) choose smaller schools, ˛1 in Eq. (7) would be biased downwards (upwards in absolute value) relative to ı1 because Ui has been omitted. However, by following the same family over time, we can net out the unobservable but fixed factors that influence involvement effort: Etsi = ˇ0 + ˇ1 (ln(Nts )) + ˇ2 (Qts ) + ˇ3 (C Xti ) + ˇ4 (F Xi ) + ˇ5 (Ui ) + t + ist

(5)

Again, recall that (∂Ui /∂Ei ) < 0, rendering the entire expression positive. Note that 0 <  < 1 gives (1/(1 − )) > 1. This means that optimal effort is rising in N.

−[E(t−1)si = ˇ0 + ˇ1 (ln(N(t−1)s )) + ˇ2 (Q(t−1)s ) + ˇ3 (C X(t−1)i ) + ˇ4 (F Xi ) + ˇ5 (Ui ) + (t−1) + is(t−1) ]

Eti − E(t−1)si = ˇ1 [ln(Nts ) − ln(N(t−1)s )] + ˇ2 (Qts − Q(t−1)s ) 2.4. Testing the model’s predictions The model presents us with very clear predictions. As N opt grows, Ei∗ falls due to free-riding, while Ei rises. This paper will primarily test the sign of ∂Ei /∂N. If (∂Ei /∂N) < 0, we can conclude that larger schools reduce parental involvement through increased free-riding. 3. Estimating the involvement response to school size 3.1. Empirical strategy Eq. (3) shows that parental effort is affected by N, the marginal utility of Q, and the marginal dis-utility of effort. Exogenous school quality Q0 impacts the marginal utility of Q, while family circumstances such as language, work schedules, family composition, and opportunity cost affect

+ ˇ3 (C Xti − C X(t−1)i )+(t −(t−1) )+εist

(8)

Free-riding would appear as involvement drops that are larger for families that experience larger jumps in school size. In the context of Eq. (8), the free-riding question is whether ˇ1 < 0. This differences-in-differences approach nets out two kinds of confounders. First, it eliminates all fixed, unobservable family characteristics (Ui ) that may drive both involvement and selection of school size. For example, high-ability families that place a high value on education may select small schools, and also exhibit high involvement. Effects such as these are netted out by the longitudinal approach. Second, any time trends in involvement that are common across all families (t − (t−1) ) are treated as a constant. For example, most families may lower their involvement, or change the nature of their involvement, as their child moves from 8th grade to 12th grade. These

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P. Walsh / Economics of Education Review 29 (2010) 959–970

Table 1 Urbanicity and mean enrollment. Full Sample

Urban

Suburban

Rural

Middle School Total Enrollment (S.D.) High School Total Enrollment (S.D.) 8th Grade Enrollment (S.D.) 12th Grade Enrollment (S.D.) ln(Grade Enrollment) (S.D.)

666 (362) 1123 (721) 198 (152) 248 (178) 0.257 (0.607)

774 (374) 1349 (800) 214 (166) 280 (195) 0.351 (0.600)

692 (382) 1229 (739) 217 (163) 286 (186) 0.339 (0.674)

539 (275) 785 (463) 158 (109) 169 (115) 0.065 (0.451)

N

5089

1312

2195

1574

common trends are simply wrapped into the constant of Eq. (8). This empirical approach requires controls for factors that affect involvement which could change over time. Since changes in school quality (Qts − Q(t−1)s ) can lead to a changes in parental involvement, the changes in several school characteristics relating to student body composition, funding, and student/teacher ratios were included in the model. Likewise, changes in family structure or student performance (C Xti − C X(t−1)i ) can also drive changes in parental involvement. Therefore, changes in number of siblings in the home, in parental marital status, in student test scores, in family income, and in parental work status are included. Unchanging family characteristics, like language or parental education, would be netted out of the difference-in-difference estimation and are therefore excluded. Even if the longitudinal approach nets out a constant Ui , a change in the availability of choices could change the effect that a constant Ui has on school selection. For example, a geographic area may offer a choice of private middle schools, but have only one (public) high school, or vice versa. Families that switched between public and private sectors due to changes in availability may have unusual changes in school size, and unusual changes in involvement. This would also bias ˇ1 . Therefore, the sample is restricted to families who were in traditional (non-magnet, non-choice) public schools in both the base year and follow-up; or if they were in private schools in both years. As will be described in Section 3.3, the LHS variable will have three categories: parents may begin or raise involvement; no change; or stop or lower involvement. These categories are given the values of {1, 0, −1}, respectively. Accordingly, Eq. (8) (the baseline specification) is estimated with an ordered logit. The advantage of this estimator (relative to OLS run on a three-category LHS variable) is that the ordered logit does not impose the assumption that beginning or raising involvement (1) is quantitatively symmetric to stopping or lowering involvement (−1).

ited less variation across urbanicity. Bias would be introduced if the trend in urban families’ involvement were significantly different from that of other families. Table 1 shows data on school size across urbanicity type. The first pattern to note is that within-urban/suburban/rural variation is so high as to swamp the difference between urban and suburban schools. As will be explained in Section 3.3, grade enrollment (rather than total school enrollment) is used as the measure of school size. On this measure, urban schools are not significantly different from suburban schools at both the 8th and 12th grade levels. Higher urban dropout rates may explain small urban 12th grades coupled with somewhat larger total high school enrollments in urban areas. More importantly, the change in log enrollment is not significantly different between urban and suburban. In contrast, rural schools seem to be the outlier, with significantly smaller grade enrollments and enrollment changes than urban/suburban. This does not affect the results: although the baseline estimates do not control for urbanicity, specifications with these controls produced virtually identical (insignificantly different) results. This is not surprising, given how much variation comes within urbanicity type. A second objection to this approach could be that, holding population density constant, a larger school mechanically implies a farther average distance to the school. If distance is an obstacle to involvement, differences in the jump in distance could explain differences in changes in involvement. However, recent research suggests that distance to schools is not well correlated with parental involvement (Sipple & Brent, 2008). A third objection to this approach could be that while the longitudinal approach controls for family fixed effects, family-specific trends in involvement over time could be correlated with unobserved family characteristics. If these unobserved characteristics are also correlated with the jump in school size, bias will be introduced. In the context of the empirical model, this suggests that the structural equation is:

3.2. Controlling for selection into treatment

Esti = ı0 + ı1 (ln(Nst )) + ı2 (Qst ) + ı3 (C Xti ) + ı4 (F Xi )

One objection to this approach could be that the difference in enrollment [ln(Nts ) − ln(N(t−1)s )] is confounded with urbanicity. This would result if, for example, urban high schools were very large while middle schools exhib-

+ ı5 (C Uti ) + ı6 (F Ui ) + t + ist

(9)

where C Uti is a vector of changing unobservable family characteristics and F Ui is a vector of fixed unobservable family

P. Walsh / Economics of Education Review 29 (2010) 959–970

characteristics. The longitudinal approach would yield: Etsi = ˇ0 + ˇ1 (ln(Nts )) + ˇ2 (Qts ) + ˇ3 (C Xti ) + ˇ4 (F Xi ) + ˇ5 (C Uti ) + ˇ5 (F Ui ) + t + ist −[E(t−1)si = ˇ0 + ˇ1 (ln(N(t−1)s )) + ˇ2 (Q(t−1)s ) + ˇ3 (C X(t−1)i ) + ˇ4 (F Xi ) + ˇ5 (C U(t−1)i ) + ˇ5 (F Ui ) + (t−1) + is(t−1) ]

Eti − E(t−1)si = ˇ1 [ln(Nts ) − ln(N(t−1)s )] + ˇ2 (Qts − Q(t−1)s )

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were 8th graders in 1988. In that base year, schools were randomly sampled within each geographical strata, and an average of 22 students were sampled at each school. Parents provide family background data, including contact with the school, family size and composition, income, their own education, and attainment expectations. Principals provide information on school-level policies, programs, and student composition. Follow-up surveys were conducted in the 10th and 12th grades. However, only the 12th-grade follow-up surveyed parents. Therefore, the longitudinal identification will use the base year (8th grade) and 2nd follow-up (12th grade) surveys.

+ ˇ3 (C Xti − C X(t−1)i ) + ˇ5 (C Uti − C U(t−1)i ) + (t − (t−1) ) (C Uti − C U(t−1)i ) can be thought of as an omitted variable that will bias the estimate if it is correlated with [ln(Nts ) − ln(N(t−1)s )]. Plausible scenarios can produce both positive and negative biases. However, if the trend in unobservables (C Uti − C U(t−1)i ) is correlated with observable family type, it is possible to use those same observables to control for selection into treatment using a propensity-score matching method. For example, suppose that low-income parents become disproportionately disaffected with schools as their children age, and that these families experience larger-than-average jumps in school size. We cannot observe “disaffection”, but we can observe income (and other family characteristics) that drives selection into school type. As long as the unobservable trend in “disaffection” is well correlated with observables such as income, matching on these observables will address this bias. The treatment in this case, [ln(Nts ) − ln(N(t−1)s )], is continuous. Therefore, a continuous-treatment version of the propensity score estimator called the “general propensity score” or GPS (Hirano & Imbens, 2004) is used to verify the results of the difference-in-difference ordered logit. Intuitively, the GPS matches individuals with very similar predicted treatments (similar propensity scores) but different actual treatments. The treatment effect is then identified off these pairings. In this application, the GPS involves the following steps: (i) regress ln(Nts ) − ln(N(t−1)s ) on a list of family characteristics, (ii) use the results of (i) to compute a propensity score, (iii) verify that the propensity score balances the distributions of family characteristics across different treatment groups, and (iv) estimate the treatment effect of changes in enrollment on parental involvement by regressing changes in involvement on a quadratic function of ln(Nts ) − ln(N(t−1)s ) and the propensity score, again using an ordered logit. A STATA do-file provided by Michela Bia was used to implement this process (Bia & Mattei, 2008). 3.3. Data The data used for this study come from the National Educational Longitudinal Survey (NELS88). This dataset consists of individual-level observations of students who

Table 2 NELS:88 variables pertaining to parental involvement. Since school started in the fall, have you or your spouse/partner BEEN CONTACTED BY THE SCHOOL regarding Base year (8th grade) Parent survey

2nd-Follow-up (12th grade) Parent survey (no parent survey done in 1st follow-up)

Grades Quality of academic program

Grades Quality of academic program Post-HS plans Course selection for post-HS Attendance

Course selection for HS HS program Behavior Fundraising Obtain info for records Volunteering

Behavior

Volunteering How to help with skills/HW

Since school started in the fall, have you or your spouse CONTACTED THE SCHOOL regarding Base year (8th grade) Parent Survey

2nd-Follow-Up (12th grade) Parent survey (no parent survey done in 1st follow-up)

Grades Quality of academic program

Grades Quality of academic program Post-HS plans Course selection for post-HS Attendance Behavior

Behavior Fundraising Send info for records Volunteering

Volunteering

Do you or your spouse/partner do any of the following Belong to PTO Attend PTO Take part in PTO Act as volunteer at school Note: This table displays the possible parental involvement variables available in the NELS:88 dataset. The question of public-good provision requires a measure of parental involvement that is (i) school-based (as opposed to help with homework), (ii) parent-initiated, (iii) focused on volunteering or academic quality (as opposed to behavior, grade complaints, or college counseling). Further, the longitudinal estimation strategy requires (iv) the same question is asked in both the base year and follow-up surveys. Parents contacting the school regarding the quality of the student’s academic program, and parents contacting the school to volunteer (highlighted in grey above) are the only parental involvement variables that meet these criteria. Parental involvement in PTO would be an ideal measure, but this question was not asked in the follow-up, so it does not meet criteria (iv).

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P. Walsh / Economics of Education Review 29 (2010) 959–970

Table 3 Summary statistics. 8th Grade

Any contact Re: Academic Program {0, 1} Any Volunteering {0, 1} School Enrollment Grade Enrollment Student Standardized Reading Score Student Standardized Math Score Parents Work Full Time {0, 1} School Share Free Lunch School Share Hispanic School Share Black School Starting Teacher Salary School Student/Teacher Ratio

12th Grade

Mean

S.D.

Mean

S.D.

0.35 0.24 656 197 53.1 53.5 0.585 0.197 0.076 0.098 $17,558 16.91

0.48 0.43 355 152 10.1 10.1 0.493 0.206 0.183 0.181 $2,563 4.88

0.45 0.46 1123 246 52.6 53.2 0.608 0.181 0.079 0.096 $21,462 16.94

0.50 0.50 855 177 9.6 9.6 0.488 0.196 0.176 0.172 $3,580 12.01

3.4. Definition of variables 3.4.1. Parental involvement First, since public-good effects are only expected to operate for school-based parental involvement, the homebased variables are disregarded. Table 2 lists the variables in NELS that relate to school-based parental involvement. Second, since school-initiated contacts do not capture the idea of voluntary public goods provision, these variables are disregarded. Third, since parent-initiated contacts regarding student behavior, student grades, or college admissions are also unlikely to generate public-good effects, these variables are also disregarded. Fourth, in order to implement the longitudinal identification strategy, the parental involvement variables must be asked in both the parental surveys. Parents are surveyed in the base year, when the student is in 8th grade, and again in the 2nd followup, when the student is in the 12th grade. This step excludes variables regarding PTO involvement, which is not included in the follow-up. By process of elimination, this leaves (1) parent-initiated contacts to discuss the student’s academic program, and (2) parent-initiated contacts to offer volunteer services as the two measures of public-good-providing parental involvement. Eq. (8) is estimated separately for both “contact” and “volunteer” activities. For each of these two measures of involvement, two types of “difference” are calculated: changes in the extensive margin, and changes in the intensive margin. The extensive variable takes a value of −1 if a parent contacted or volunteered at all in the base year but not in the follow-up; 0 if there was no change; and 1 if a parent contacted or volunteered at all in the follow-up but not in the base year. The intensive variable takes a variable of −1 if contact or volunteering fell (possibly to zero) between the base year and the follow-up; 0 if there was no change; and 1 if contact or volunteering rose (possibly from zero) between the base year and the follow-up. A disadvantage of this construction is that it assumes that falling from “more than 4 times” to “3–4 times” is the same as falling from “more than 4 times” to “1–2 times”. The advantage of this construction is that somewhat more information is included by considering people who changed their involvement level but did not stop altogether.

3.4.2. School size The premise of this paper is that public-good effects are likely to operate on the school level, as opposed to the classroom level or district level. Two possible measures could be used to capture school size: total school enrollment, and enrollment at the student’s grade level. Total enrollment would seem to be the best indicator of a school’s size. However, much of its variation is explained by the number of grades in a school, rather than the size of each grade. Consider a K-8 school with 50 students in each grade, and a grade 6–8 school with 150 students in each grade. Both would have a total enrollment of 450, but the K-8 school would be less susceptible to free-riding for two reasons. First, the K-8 school is more likely to have multiple students from the same family than the grade 6–8 school, resulting in fewer families in the K-8 school. Since the number of families more closely reflects “N” than total enrollment, this would lead to less free-riding at the K-8 school. Second, it is likely that school policies (targets of parental contacts) and school activities (targets of volunteering) vary by grade. Intuitively, there may be less free-riding in the K-8 school because involvement by parents of kindergartners does not

Table 4 Changes in involvement between 8th and 12th grade. Measure of involvement Contact Re: Academic Program

Volunteer

Change in activity None to some No change Some to none

1,159 2,854 645

1,409 2,852 386

Increased No change Decreased

1,387 2,536 735

1,681 2,473 493

Shares None to some No change Some to none

0.249 0.613 0.138

0.303 0.614 0.083

Total

1.000

1.000

Increased No change Decreased

0.298 0.544 0.158

0.362 0.532 0.106

Total

1.000

1.000

P. Walsh / Economics of Education Review 29 (2010) 959–970

Fig. 1. Average difference in log grade enrollment by extensive change in parental involvement. Note: This figure should be interpreted as showing (for example) that among parents who stopped (between 8th and 12th grades) contacting the school regarding their student’s academic program, the average change in ln(Grade Enrollment) was 0.34; while among parents who started contacting the school the average change was 0.21. Parental contact is differentiated from school-initiated contacts, and from contacts regarding behavior or grades.

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Fig. 2. Average difference in log grade enrollment by intensive change in parental involvement. Note: This figure should be interpreted as showing (for example) that among parents who decreased (between 8th and 12th grades) their contact with the school regarding their student’s academic program, the average change in ln(Grade Enrollment) was 0.33; while among parents who increased their contact with the school the average change was 0.22. Parental contact is differentiated from school-initiated contacts, and from contacts regarding behavior or grades.

crowd out involvement by parents of 8th graders. Focusing on total enrollment would make the two schools appear equal in their effects on parental involvement. Therefore, the grade enrollment provides a cleaner measure of a school’s scale than the total enrollment. Accordingly, the sizes of the 8th grade and 12th grade are used as the measures of school size for their respective schools. Nevertheless, regressions using changes in total enrollment produced similar results to those using grade enrollment, although (as predicted by the two factors men-

tioned above) the standard errors were larger, and the coefficients lower. 3.4.3. Family characteristics Family characteristics must be included if they are not fixed over time. Therefore changes in family size, parental work status, family income, parental marital status, and student standardized test scores are included in Eq. (8).

Table 5 Effect of ln(Grade Enrollment) on parental involvement. Extensive margin Marginal effects from ordered logit reported Measure of involvement

Effect of  Enrollment on Prob of:  ln(Grade Enrollment)  Student’s Math Score  Student’s Reading Score  School % Free Lunch  School % Hispanic  School % Black  School Teacher Salary (thousands)  Student/Teacher Ratio  Family Income  Parent Work Status  Family Size Dummies for changes in marital status N Wald Mean of LHS (Unconditional Prob) Effect on Prob of Doubling Enrollment

Contact Re: Academic Program

Volunteer

0 to 1

0 to 1

1 to 0

−0.0331 −0.0006 −0.0008 0.0001 −0.0012 −0.0010 −0.0015 0.0005 0.0054* 0.0154 −0.0048

***

***

0.0209 0.0004 0.0005 −0.0001 0.0008 0.0006 0.0009 −0.0003 −0.0034* −0.0097 0.0030

1 to 0

−0.0531 −0.0010 −0.0005 0.0007 0.0014 −0.0015* −0.0067*** −0.0003 −0.0001 −0.0244* 0.0037

***

Yes

Yes

4840 64.79 0.246 −0.02284***

0.0187*** 0.0004 0.0002 −0.0002 −0.0005 0.0005* 0.0024*** 0.0001 0.0000 0.0086* −0.0013 4829 72.54

0.135 0.01441***

0.300 −0.03662***

0.080 0.01293***

Standard errors clustered on base-year schools. Note: Unit of observation is a family, as the student progresses from 8th to 12th grade. Reports the marginal effects for an ordered logit regressing a 3-category measure of involvement change between 8th and 12th grade (start, no change, stop) on the change in ln(Grade Enrollment) and the changes in covariates. Standard errors are not reported but significance is indicated with stars. Since a doubling represents 0.69 log points, the coefficients on change in log enrollment are multiplied by 0.69 to provide the effect of a doubling of enrollment (for greater ease of interpretation). Figures in Bold are the RHS variable of interest, and the same variable re-scaled to show the effect of doubling enrollment. * Significant at the 10% level. ** Significant at the 5% level. *** Significant at the 1% level.

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3.4.4. School quality Parents may change their involvement level in response to a change in perceived school quality. Therefore, changes in the following measures (reported by the principals) are included: percent of students receiving free/reduced price lunches; percent of students who are black; percent of students who are Hispanic; teacher salary; and student/teacher ratio. Variables such as teacher experience and class size are reported by teachers rather than principals, and missing data in the teacher survey means the use of these variables would impose a nearly 40% drop in the number of observations. Other variables, relating to school safety or academic performance, may be endogenous to parental involvement. For these reasons, the teacher-level variables and safety/performance variables are excluded. The relevant variables are summarized in Table 3. Table 4 shows the distribution of changes in parental contact and volunteering between 8th and 12th grade. Figs. 1 and 2 break down the average 12th grade − 8th grade change in log enrollment by the involvement behavior of the parents. Fig. 1 provides initial support to the hypothesis by showing that stopping involvement is associated with larger jumps in school size, while starting involvement is associated with smaller jumps in size. Likewise, Fig. 2 shows that that decreases in involvement are associated with larger jumps in school size, while increases in involvement are associated with smaller jumps in size. Between the 8th and 12th grades, the average family experiences an increase in enrollment and raises their level

of involvement (Table 4). This does not, however, contradict the paper’s main hypothesis, as the “holding all else constant” assumption cannot be made in a simple comparison of tables. In the context of generally rising levels of involvement, the differences-in-differences approach is simply asking, “was the rise in involvement weaker – in probability or magnitude – among families who experienced larger jumps in enrollment?” 4. The effect of school size on parental involvement 4.1. Main results Since both the “extensive” and “intensive” versions of the LHS variables have three categories (increase; no change; decrease), Eq. (8) will be estimated with an ordered logit. The marginal effects from these ordered logits for the “increase” and “decrease” values of the LHS are reported in Tables 5 and 6. Table 5 reports the marginal effects for the extensive margin (parent starts or stops involvement entirely), for both the “Contact Re: Academic Program” and the “Volunteer” definitions of involvement. Table 6 repeats Table 5, but uses the intensive margin (parent increases or decreases involvement). Marginals for school size should be interpreted as the “change in the (0,1) probability of observing this behavior change as a result of a 1 log-point rise in enrollment”. Since 1 log point is a factor of about 2.72, the marginal effect of a doubling of enrollment (a factor of 2, or rise of 0.69 log points) is also provided

Table 6 Effect of ln(Grade Enrollment) on parental involvement. Intensive margin Marginal effects from ordered logit reported Measure of involvement

Effect of  Enrollment on Prob of:  ln(Grade Enrollment)  Student’s Math Score  Student’s Reading Score  School % Free Lunch  School % Hispanic  School % Black  School Teacher Salary (thousands)  Student/Teacher Ratio  Family Income  Parent Work Status  Family Size Dummies for Changes in Marital Status N Wald Mean of LHS (Unconditional Prob) Effect on Prob of Doubling Enrollment

Contact Re: Academic Program

Volunteer

Increase

Increase

Decrease

−0.0343 −0.0003 −0.0012 −0.0001 −0.0013 −0.0013* −0.0018 0.0005 0.0061* 0.0129 −0.0026

***

***

0.0216 0.0002 0.0008 0.0001 0.0008 0.0008* 0.0011 −0.0003 −0.0039* −0.0081 0.0016

Decrease

−0.0650 −0.0006 −0.0007 0.0010** 0.0012 −0.0017* −0.0095*** −0.0002 −0.0029 −0.0164 0.0020

***

Yes

Yes

4840 60.35 0.296 −0.02366***

0.0261*** 0.0002 0.0003 −0.0004** −0.0005 0.0007* 0.0038*** 0.0001 0.0012 0.0066 −0.0008

4829 75.28 0.155 0.01491***

0.357 −0.04488***

0.103 0.018026***

Standard errors clustered on base-year schools. Note: Unit of observation is a family, as the student progresses from 8th to 12th grade. Reports the marginal effects for an ordered logit regressing a 3-category measure of involvement change between 8th and 12th grade (increase, no change, decrease) on the change in ln(Grade Enrollment) and the changes in covariates. Standard errors are not reported but significance is indicated with stars. Since a doubling represents 0.69 log points, the coefficients on change in log enrollment are multiplied by 0.69 to provide the effect of a doubling of enrollment (for greater ease of interpretation). Figures in Bold are the RHS variable of interest, and the same variable re-scaled to show the effect of doubling enrollment. * Significant at the 10% level. ** Significant at the 5% level. *** Significant at the 1% level.

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at the bottom of Tables 1 and 5. Extensive dummies for changes in parents’ marital status (for each of the years between the base year and follow-up, one dummy each for divorce, separation, death, marriage, and cohabitation) are included in regressions but are not reported in the tables. The larger the jump in school size, the less likely parents are to raise their involvement level, and the more likely parents are to reduce their involvement level. Table 5 shows that a doubling in grade enrollment, for example, is associated with a 2% decline in the probability that parents will begin contacting the school, and a 3.6% decline in the probability that parents will begin volunteering. Unconditionally, 25% of parents begin contacting, and 30% begin volunteering. Likewise, Table 6 shows that a doubling of grade enrollment is associated with 2% decline in the probability that parents will increase their contacts with the school, and a 4.5% decline in the probability that parents will increase their volunteering. Unconditionally, 30% of parents increase their contacts, and 35% increase their volunteering. The drop in involvement from a doubling of enrollment is on the same order or magnitude, and often somewhat larger, than the drop in involvement from a parent starting to work outside the home. Both the direction and magnitude of these effects are consistent with public-good theory: a larger community will see less voluntary provision. This result is robust across different definitions of involvement (“Contact Re: Academic Program” vs. “Volunteer”), and across definitions of change in involvement (extensive vs. intensive margins). By creating larger communities, larger schools induce more free-riding among parents. 4.2. Correction for selection into treatment As noted in Section 3.2, the difference-in-difference approach may still produce biased estimates due to the non-random assignment of jumps in school size between 8th and 12th grades. Assuming that the selection intro treatment is based on observables, the baseline results can be tested with a continuous-treatment propensity-score matching procedure, called the generalized propensity score (Hirano & Imbens, 2004). GPS matches individuals who have very similar predicted treatments (similar propensity scores) but different actual treatments, and identifies the treatment effect from these pairings. The first step of the GPS process is to estimate the treatment (ln(Grade Enrollment)) as a function of observable family characteristics. The results of this estimation is given in Table 7. Less parsimonious specifications yielded very similar results. The propensity score is essentially the change in enrollment predicted by this model. The second step tests the propensity score’s ability to balance the distributions of observables between more-treated and less-treated individuals. This is done by cutting the sample into two parts: those with ln(Grade Enrollment) below 0.4, and those with (ln(Grade Enroll-

967

Table 7 Estimation of generalized propensity score (GPS). LHS variable: ln(Grade Enrollment) Parents Expect Student to Attain > BA Parents Married, base year Computer in Home, base year Newspaper Subscription in Home, base year Student is Black Student is Hispanic Base Year Family Income <$10,000 $25,000 < Base Year Family Income < $50,000 $50,000 < Base Year Family Income < $75,000 Base Year Family Income >$75,000 Parent Education: No HS Diploma Parent Education: Some College Parent Education: BA or higher Number of siblings, base year Family Lives in Rural Area, base year Student Standardized Reading Test Score, base year N Wald Mean of LHS

0.046** (0.021) −0.022 (0.027) 0.051*** (0.019) 0.059*** (0.022) 0.042 (0.034) 0.063* (0.033) −0.045 (0.025) −0.008 (0.026) 0.001 (0.040) −0.049 (0.030) −0.022 (0.026) −0.048 (0.031) −0.022 (0.023) 0.009 (0.008) −0.257*** (0.020) 0.003*** (0.001) 4,502 271.3 0.253

Note: This is the first step in a continuous-treatment propensity-score matching procedure that controls for selection into treatment. A family’s change in log enrollment between 8th grade and 12th grade is regressed on a number of family-level characteristics to generate the Generalized Propensity Score, a prediction of the enrollment change. * Significant at the 10% level. ** Significant at the 5% level. *** Significant at the 1% level.

ment)) above 0.4.2 Table 8 shows the t-statistics for difference-in-means tests across these two groups for family characteristics, first unadjusted and then adjusted by the propensity score. In general, these differences are reported for each grouping of the treatment, but since there are only two such groups here, there is only one “difference” for each variable. The GPS dramatically improves the balance of family characteristics between those with small enrollment jumps and those with large jumps. Before the GPS adjustment, 14 out of 16 variables are different between these groups at a 1% level; after adjustment only one is different at that level. Additional tests of the balancing power of the propensity score are discussed in Rubin (2001): (1) the means

2 ln(Grade Enrollment) has a range of −3.25 to 3.86, with a mean of 0.257. A breakpoint of 0.4 separates the “average” jump from a “large” jump. Different breakpoints produce similar results.

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P. Walsh / Economics of Education Review 29 (2010) 959–970 Table 8 Test of GPS balancing property. Reported: t-statistics for difference-in-means tests between “low” and “high” treatment groups

Parents Expect Student to Attain > BA Parents Married, Base Year Computer in Home, base year Newspaper Subscription in Home, base year Student is Black Student is Hispanic Base Year Family Income <$10,000 $25,000 < Base Year Family Income < $50,000 $50,000 < Base Year Family Income < $75,000 Base Year Family Income >$75,000 Parent Education: No HS Diploma Parent Education: Some College Parent Education: BA or higher Number of siblings, base year Family Lives in Rural Area, base year Student Standardized Reading Test Score, base year

Unadjusted

Adjusted

−4.024*** −2.065*** −5.765*** −3.218*** 2.114*** −2.145*** 5.136*** −2.493*** 3.489*** −5.041*** −5.387*** 3.645*** 0.181 −0.994 16.041*** −5.225***

−0.120 −1.199 −0.648 1.227 1.275 −1.647 0.050 0.637 1.215 −1.359 −0.987 0.448 −0.011 −0.288 2.629*** −0.542

Note: This is the second step in a continuous-treatment propensity-score matching procedure that controls for selection into treatment. Observations are split into two Groups: Group 1 experiences a <0.4 change in log enrollment between 8th and 12th grade, while Group 2 experiences a >0.4 change in log enrollment between 8th and 12th grade. The “Unadjusted” column reports the t-statistics on difference-of-means tests between the two groups, for each of the RHS variables in the GPS 1st stage. The “Adjusted” column reports t-statistics on difference-of-means tests between the two groups after their distributions are re-weighted by the propensity score. This re-weighting is quite complex and the reader is referred to Hirano and Imbens (2004) for details. Briefly, the initially large differences in covariates between people of different treatment groups should be reduced (ideally to insignificance) by taking the GPS into account. This means that the distributions of covariates have been balanced across different levels of treatment, negating any bias from selection into treatment. * Significant at the 10% level. ** Significant at the 5% level. *** Significant at the 1% level.

of the propensity score for the treated and control groups must not differ by more than 1/2 standard deviation, (2) the variances of the propensity scores must be similar across these two groups, and (3) the relationship between the 1st stage covariates and the propensity score must the similar across these two groups. The propensity score used here meets these criteria. Finally, the treatment effect (called a dose–response in the continuous-treatment setting) is estimated by regressing the change in involvement on a quadratic function of the treatment and the propensity score. Since the LHS variable is still {−1, 0, 1}, this is again done with an ordered logit. These results are given in Table 9, where Panel A shows results for the extensive margin of involvement changes, and Panel B shows results for the intensive margin. For comparison, the raw coefficient3 on ln(Grade Enrollment) from the baseline ordered logit is also provided. The GPS process, and the 2nd step in particular, is quite complex, and readers are referred to Hirano and Imbens (2004) for more detail. For both the extensive and intensive margins, the GPS procedure confirms that increases in school size significantly reduce parental contact regarding the student’s academic program. In fact, the GPS results are somewhat

3 The baseline results are reported as marginal effects in Tables 5 and 6 to facilitate the economic interpretation (a change in a probability given a change in ln(Grade Enrollment)), while the same results are reported as raw logit coefficients in Table 9 to facilitate the statistical interpretation (a comparison with the GPS dose–response coefficient).

stronger than the baseline, suggesting that selection bias actually weakens the size-contact relationship in the baseline specification. An example of such a bias would be if parents with high attainment expectations and highscoring students (to pick two significant predictors of ln(Grade Enrollment) from Table 6) exogenously increase their contact as their student enters high school. This would spuriously associate larger enrollment jumps with increases in involvement, the opposite of the predicted effect. On the other hand, for both the extensive and intensive margins, the GPS procedure fails to confirm that increases in school size significantly reduce parental volunteering. This suggests that selection bias exaggerates the size-volunteering relationship in the baseline specification. Again, an example of such a bias would be if parents with high attainment expectations and high-scoring students exogenously decrease their volunteering as their student enters high school. This would spuriously associate larger enrollment jumps with decreases in involvement, exaggerating the predicted effect. 4.3. Effect of changes in school quality on parental involvement In addition to the effects of school size, the regressions suggest that parents view involvement as a substitute for, rather than a complement to, school quality. On both the extensive and intensive margins, an increase in teacher salary (roughly indicating a better-funded school) significantly reduces parental volunteering. A similar pattern

P. Walsh / Economics of Education Review 29 (2010) 959–970

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Table 9 Dose–response functions estimated using GPS.

Panel A: Extensive margin ln(Grade Enrollment)

Contact Re: Academic Program

Volunteer

Original coefficient from ordered logita

GPS dose–response

Original coefficient from ordered logita

GPS dose–response

−0.178*** (0.052)

−0.282** (0.134) 0.006 (0.075) −1.387 (1.127) 1.823 (1.184) 0.351 (0.298)

−0.253*** (0.060)

0.082 (0.164) −0.164* (0.092) 1.913 (1.175) −1.940 (1.224) −0.189 (0.334)

−0.164*** (0.056)

−0.266** (0.130) −0.012 (0.076) −1.101 (1.110) 1.295 (1.158) 0.446 (0.288)

−0.283*** (0.065)

−0.075 (0.156) −0.143 (0.089) 1.210 (1.124) −1.054 (1.172) 0.135 (0.320)

[ln(Grade Enrollment)]2 Propensity Score [Propensity Score]2 ln(Grade Enrollment) × Propensity Score Panel B: Intensive margin ln(Grade Enrollment) [ln(Grade Enrollment)]2 Propensity Score [Propensity Score]2 ln(Grade Enrollment) × Propensity Score

Note: This is the third step in a continuous-treatment propensity-score matching procedure that controls for selection into treatment. The same 3-category measure of involvement change between 8th and 12th grade is regressed on a quadratic function of the change in ln(Grade Enrollment) and the Generalized Propensity Score, again using an ordered logit. The dose–response effect (the equivalent of a treatment effect in the continuous-treatment setting) is given by the coefficient on change in ln(Grade Enrollment). Statistically, this effect is comparable to the raw coefficient from the baseline regressions, rather than the marginal effects reported in Tables 5 and 6. Therefore, this baseline coefficient is reported for comparison. Figures in Bold are the RHS variable of interest. a Different from results in Tables 5 and 6, which are marginal effects. * Significant at the 10% level. ** Significant at the 5% level. *** Significant at the 1% level.

is observed for parental contact, although this effect is not significant. Secondly, there is significant evidence that volunteering rises in response to an increased share of students who get free lunches (a generally reliable indicator of school poverty). Taken together, these results suggest that parents raise their involvement level in response to perceived problems, but see less reason to get involved when perceived quality is high. Again, this is consistent with the theory, which predicts that Ei should be falling in Q0 because of lower marginal utility of Q. 5. Conclusion Involved parents are typically thought to be providing public goods. If so, a straightforward implication of publicgood theory is that parental involvement will be lower at larger schools. A cross-sectional approach to this question is inappropriate, since parents are likely to sort into large or small schools based on unobservable factors that also drive involvement. A differences-in-differences approach, however, nets out fixed family effects and identifies the effect of changes in school size as students move from 8th to 12th grade. Larger jumps in school size are associated with significantly less parental involvement, although the magnitude is small. A doubling in grade enrollment, for example, is associated with a 2% decline in the probability that parents will begin contacting the school or

raise their frequency of contact (when, unconditionally, 30% of parents raise their frequency of contact, 25% of them from a baseline of zero contact). A continuous-treatment propensity-score matching technique shows that these results are robust to selection into treatment. A doubling of grade enrollment is associated with a 3.6% decline in the probability that parents will begin volunteering and a 4.5% decline in the probability of raising their frequency of volunteering, but these results are not robust to selection intro treatment. This pattern suggests that larger schools do reduce parental contacts because of free-riding, while parents who are inherently likely to raise their volunteering are also likely to pick small 8th grade schools and experience a large enrollment jump. The drop in involvement from a doubling of enrollment is on the same order or magnitude, and often somewhat larger, than the drop in involvement from a parent starting to work outside the home. Finally, there is evidence that parents see their involvement as a substitute, rather than a complement, for perceived school quality. Acknowledgements The author wishes to thank John Carvellas, Brian Kovak, Brian Cadena, Jim Sallee, Molly Sherlock, and two anonymous referees for very helpful comments. Many thanks to Michela Bia for providing the STATA do-file used to imple-

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