When Can School Inputs Improve Test Scores?
Jishnu Das (Development Research Group, World Bank)
Stefan Dercon (Oxford University)
James Habyarimana (Harvard University)
Pramila Krishnan (Cambridge University)
February 5, 2004
Abstract
The relationship between school inputs and educational outcomes is
critical for educational policy. We recognize that households will respond
optimally to changes in school inputs and study how such responses affect
the link between school inputs and cognitive achievement. To incorpo-
rate the forward-looking behavior of households, we present a household
optimization model relating household resources and cognitive achieve-
ment to school inputs. In this framework if household and school in-
puts are technical substitutes in the production function for cognitive
achievement, the impact of unanticipated inputs is larger than that of
anticipated inputs. We test the predictions of the model for non-salary
cash grants to schools using a unique data set from Zambia. We find that
household educational expenditures and school cash grants are substitutes
with a coefficient of elasticity between -0.35 and -0.52. Consistent with
the optimization model, anticipated funds have no impact on cognitive
achievement, but unanticipated funds lead to significant improvements in
learning. This methodology has important implications for educational
research and policy.
Corresponding Author: Jishnu Das (jdas1@worldbank.org). We thank Hanan Jacoby and
workshop participants at the World Bank and NEUDC (Yale) for comments. Funding for the
survey was provided by the Department for International Development (UK).The findings,
interpretations, and conclusions expressed in this paper are those of the authors and do not
necessarily represent the views of the World Bank, its Executive Directors, or the governments
they represent. Working papers describe research in progress by the authors and are published
to elicit comments and to further debate.
1
W C S I I T -S ? 2
1. Introduction
In the last two decades the relationship between schooling inputs and educational outcomes
has received considerable attention in academic and policy forums. Although it is recognized that
households play a critical role in the determination of such outcomes, the literature has bifurcated
in two distinct strands.1 One strand, concerned with the impact of school inputs on cognitive
achievement, has focused on estimating educational production functions where cognitive achieve-
ment is determined as a function of schooling inputs. The second strand is concerned with the effect
of household characteristics on cognitive achievement independent of school inputs. However, the
notion that household responses themselves affect the relationship between cognitive achievement
and school inputs has received little attention in the literature.
In a recent paper Todd and Wolpin (2003) point out that in the presence of household responses,
estimates based on the production function approach will capture a "policy-effect" that incorpo-
rates both the marginal impact of school inputs on outcomes as well as household responses to
such inputs. This raises an important question that defines the central problem addressed here:
How are we to understand the relationship between school inputs and cognitive achievement in an
environment where households respond to the provision of such inputs?
To examine this question we first derive a household optimization model of cognitive achieve-
ment. The model incorporates the basic assumption that households respond optimally to the
provision of inputs at the school level. However, we also need to incorporate the notion that house-
holds may be forward-looking so that responses occur not only in the period that school inputs are
provided, but the moment that new information becomes available. Our explicit consideration of
the dynamics has important implicationssince household adjustments occur with any new infor-
mation, production function parameters can be identified only through the impact of unanticipated
inputs on cognitive achievement.
In this framework, we show that the impact of school inputs depends on (a) whether they are
anticipated or not and (b) the extent of substitutability between household and school inputs in
the production function for cognitive achievement. If household and school inputs are technical
substitutes, an anticipated increase in inputs in the next period increases household contributions
in the current period and decreases them in the next, whereas if they are technical complements, the
impact of anticipated increases in school inputs on current contributions depends on the strength
of the households' preferences for a smooth consumption path. Unanticipated increases in school
1The path-breaking Coleman Report (1966) for instance stresses the importance of household characteristics for
child achievement.
W C S I I T -S ? 3
inputs in the next period preclude household responses in either the current period or the next.
These differences lead to a testable prediction: If household and school inputs are (technical) sub-
stitutes, unanticipated inputs will have a larger impact on cognitive achievement than anticipated
inputs; if they are complements, the relative effects depend on household preferences.
We test this prediction using data for Zambia in 2002-2003. The educational environment in
the country is particularly well suited for our empirical exercise. The system is largely based on
public schools (less than 2 percent of all schools are privately run) and the country has a history of
high enrollment rates and school participation, suggesting that household involvement in children's
education is high.2 In 2000 the government legislated a fixed cash grant to every school. These
grants were large. Among rural schools, they represented 66 percent of household level educational
expenditures for the lowest wealth deciles and 19 percent for the top wealth decile. Moreover the
simplicity of the allocation rule ensured that the grants reached their intended recipients (see Das,
Dercon, Habyarimana, and Krishnan 2003), suggesting that in the year of the survey the fixed cash
grants would be anticipated by households making their educational investment decisions for the
year.
In addition schools could also receive cash from other sources, but these alternative sources were
highly unreliable and unpredictable. In the year of the survey, less than 25 percent of all schools
received such grants and conditional on receipt, there was tremendous variation with some schools
receiving 30 times as much as others. Apart from cash, few resources were distributed to schools
during the year of the survey. Finally, following an agenda of "free" education, all institutionalized
parental contributions to schools were banned in April 2001 (typically these took the form of Parent
Teacher Association fees). Taken together this implied that educational expenditures for children
could be met either through cash grants to schools or through direct parental contribution at the
level of the household. These two factors present exactly the framework required to test our model
with high parental contributions towards educational inputs on the one hand and two different
streams of cash disbursements to schools, one steady and the other unpredictable, on the other.
To exploit the characteristics of the educational environment, we collected a unique data set
for a representative sample of schools in four provinces of the country (covering 58 percent of the
population). The survey includes data on school inputs as well as two test scores on the same
sample of students one year apart. To supplement this data, we also collected information for
households matched to a sub-sample of schools identified as "remote" using GIS mapping tools.
This allows us to directly relate household and school inputs in an environment where issues of
2Net enrollments are upwards of 80 percent for both boys and girls (Figures 2a and 2b).
W C S I I T -S ? 4
school choice are eliminated. We are then interested in the effect of anticipated and unanticipated
cash grants to schools on household educational expenditures and cognitive achievement.
We find strong support for the household approach to cognitive achievement. Using the matched
school-household data, our results suggest that household educational expenditures and cash grants
to schools are substitutes. The elasticity of substitution between the two is high and significant,
with estimates ranging from --0.35 to -0.52 depending on the specification used. In line with the
predictions of our model we then find evidence that unanticipated grants have a significant and
substantial impact on the growth of cognitive achievement while the effect of anticipated grants is
small and insignificant. These results hold for the subjects of Mathematics and English (although
the difference is more pronounced for the latter), and are robust to potential mis-specification
arising from omitted variables in the regression.
The significance of our results goes beyond the particular policy environment considered here.
A failure to reject the null hypothesis in studies that use the production function approach could
arise either because the effect of school inputs on cognitive achievement through the production
function is zero or because households substitute their own resources for such inputs. While in our
case the substitution may take the form of textbooks or writing materials, in a more general setting
it may include parental time, private tuition and other inputs. Our results show that the policy
effect of school inputs are different from the production function parameters. This has important
consequences both for estimation techniques and for educational policy; a detailed discussion comes
later.
This work is new and innovative for a number of reasons. First, the methodology adopted
here extends the work of Becker and Tomes (1986) to the determination of cognitive achievement
and thus allows us to incorporate household responses and school inputs in a single conceptual
framework. Second, the unique data collected on matched schools and households permits the
direct estimation of household responses to school inputs; while this is clearly an important issue
for policy, ours is the one of the first papers to provide an estimate of this relationship in the
context of education. Third, the combination of funding patterns in the country and panel data on
cognitive achievement provides an excellent opportunity to separate policy effects and production
function parameters of schooling inputs; doing so yields new insights on the process through which
school inputs may affect educational outcomes. Thus, the combination of the methodology and the
unique data collected allows us to provide a firm microeconomic foundation for the relationship
between school inputs and cognitive achievement in the context of a household optimization model.
The remainder of the paper is structured as follows. We review the literature in Section 2. In
section 3 we present the model and we derive the empirical specification in section 4. Section 5
W C S I I T -S ? 5
describes the data and sampling technique and Section 6 presents the results from the matched
school-household data. Section 7 presents the results of anticipated and unanticipated cash grants
on the growth of cognitive achievement while Section 8 discusses the policy relevance of our work
and possibilities for future research.
2. Review of the Literature
This work relates to three strands of the literature. The first strand examines the relationship
between schooling inputs and cognitive achievement in the context of production functions. The
second strand examines the impact of household characteristics on educational outcomes. Finally,
a third strand examines the impact of public subsidies on private outcomes, mostly in the context
of labor supply and private transfers. We describe each of these briefly.
2.1. Production Function
The literature on educational production functions (for a review, see Hanushek 1997) attempts
to estimate the effect of school inputs on cognitive achievement. The estimation concern that
most studies have dealt with is the presence of unobserved heterogeneity, which could contaminate
estimates if correlated with the provision of inputs. In response to the omitted variable problem that
such heterogeneity creates, studies have tried to exploit "natural-experiments" (Angrist and Lavy
1999; Case and Deaton 1999; Urquiola 2003), "value-added" specifications (Hanushek 1971) or more
recently, randomized treatment-control designs (Banerjee, Cole, Duflo, and Linden 2003; Glewwe
2002 provides a review) to argue for causality. Below, we show that our approach allows for greater
flexibility in the treatment of unobserved heterogeneityheuristically, since such heterogeneity is
already known at time period t - 1, it has no effect on growth rates between t - 1 and t.3
Further, the methodology adopted here provides a context for results obtained in the educational
production function literature in the presence of household responses. Specifically, the policy effect
that is captured in these studies tells us little about why certain inputs are successful (or not) in
improving cognitive achievement. Clearly this information is important for policyif the provision
of textbooks does not improve test scores, is it because textbooks are insignificant in the production
function or is it due to optimal compensating responses at either the level of the household or the
school? The household optimization framework makes clear that unanticipated inputs provide the
key to understanding these differences.4
3This is not without restrictions; for a more detailed comment see Footnote 10.
4Hoxby (2000) provides another example of using "surprises" in the context of estimating the effect of peers on
cognitive acheivement.
W C S I I T -S ? 6
2.2. Household Characteristics
At the level of the household, studies have examined the relationship between educational out-
comes such as enrollment and drop-outs and household characteristics in the context of a household
optimization model (Glewwe 2002 and Jacoby and Skoufias 1997).5 Fewer studies have examined
the role of household characteristics on cognitive achievement; exceptions include Brown (2003),
Case and Deaton (1999) and Glewwe and Jacoby (1994) who look at the relationship between
parental education and child learning, and Alderman and others (1997) who examine the effect of
household income on test scores.
Closer to the question on how optimal responses may alter the relationship between inputs and
outcomes is Lazear's (2001) study on class size and achievement. In a theoretical examination of
school responses, Lazear (2001) argues that learning in classrooms depends both on the size of the
class as well as the number of "disruptive" children allocated to every class. Optimizing behavior
on the part of the school then implies that less "disruptive" students are allocated to larger classes
and this creates a spurious positive relationship between class size and cognitive achievement. This
paper extends the notion of optimal responses to the household, arguing that similar processes will
attenuate the relationship between school inputs and achievement.
2.3. Effect of Government Subsidies
Our findings on household responses do however have an established precedent in the literature
on private responses to public transfer programs. The research on this front has typically examined
labor supply responses (Moffitt 1992; Ravallion and Datt 1995) and private transfers (Cox and
Jimenez 1995 and Rosenzweig and Wolpin 1994) to find that the effect of government subsidies is
generally attenuated through the presence of household responses. The Euler framework developed
here has also been used to assess the extent of household responses to school feeding programs.
Jacoby (2002) for instance, tests for a "fly-paper" effect in the Philippines by examining the differ-
ence in household calorific intake for children on school and non-school days. There is, however, a
gap in the literature on household responses to school inputs, partly due to tricky sampling issues
(more on this below) and partly due to the predominance of the production function approach in
the literature relating schooling inputs to outcomes. By providing estimates on the size of these
responses, we thus suggest areas for future research.
5We also follow a close parallel literature on consumption and health. For instance, optimal growth paths derived
in our model are similar to those in Foster (1995) who considers the impact of rice prices on child weight in Bangladesh
and Dercon and Krishnan (2000) who relate adult sickness to weight in Ethiopia.
W C S I I T -S ? 7
3. Theoretical Framework
3.1. The Conceptual Experiment
Consider a household that receives a single school input, that can either be anticipated or
unanticipated. The anticipated input is fully incorporated into the utility maximization problem.
For the unanticipated input, households expectations at time t - 1 (when household decisions are
made) are zero, so that they are unable to respond by adjusting their own expenditures. How do
these different types of inputs affect cognitive achievement?
)
X(
oods
Gr
het
O
e
ris
rp
su
=X o
X
Eo Esurprise Cognitive Achievement
(Test Scores)
Figure 1. Household Substitution
Figure 1 depicts this conceptual exercise in the case where household preferences are defined
over cognitive achievement and other goods. For simplicity, we assume that cognitive achievement is
related to schooling inputs through a single-input linear production function. In this framework, any
additional unanticipated input will sustain expenditure on all other goods at X0. Consequently the
change in cognitive achievement will reflect entirely the characteristics of the production function
mapping inputs to attainment, moving the household (for instance) from E0 to Esurprise. Consider
now the impact of inputs of the same magnitude that are fully anticipated by the household at
time t - 1. To the extent that this change is viewed as permanent by the household, the budget
constraint will shift out to the outer (non-dashed) linear at time t -1 itself reflecting the fact that
households will optimally incorporate all future information into their decisions at t - 1. More
generally, it could be that the change is (rationally) expected to last a fixed number of years, in
which case the budget constraint would shift to an intermediate level (shown by the dashed line)
representing the change in permanent income secondary to the anticipated inputs. In either case,
W C S I I T -S ? 8
there will be no difference in educational inputs provided by the household between t - 1 and t.
This difference forms the basis of our statistical test: Cognitive achievement should respond to
unanticipated rather than anticipated inputs.
This special case implicitly assumes that household and school inputs are fully substitutable in
the production function for cognitive achievement and therefore anticipated changes have no impact
between two subsequent time periods. In the formalization of the model below we introduce two new
components. First, we view cognitive achievement as a durable good with household preferences
defined over it's stock. Second, we incorporate the production function for cognitive achievement
as a constraint in our optimization. Below we see that this affects the program directly through the
user-cost of the durable good. In general, anticipated and unanticipated inputs will have differential
effects, but the relative size of these effects will depend on the extent of substitutability between
household and school provision of the input.
We start with two general assumptions on preferences and the production function for cognitive
achievement. The Euler equation derived defines conditions governing the growth of test scores
this development relates closely to the discussion on durable goods and inter-temporal household
optimization discussed, for instance, in Deaton and Muellbauer (1980), Jacoby and Skoufias (1997)
and Foster (1995). Based on this solution we discuss the differential impact of anticipated and
unanticipated school inputs on test-score growth. Finally, we consider how credit constraints can
affect our results.
3.2. Model
A household (with a single child attending school) derives (instantaneous) utility from the
cognitive achievement of the child TS and the consumption of other goods X. The household
maximizes an inter-temporal utility function U(.), additive over time and states of the world with
discount rate (< 1) subject to an inter-temporal budget constraint (IBC) relating assets in the
current period to assets in the previous period, current expenditure and current income. Finally,
cognitive achievement is determined by a production function relating current achievement (TSt) to
past achievement (TSt-1), household educational inputs (zt), school inputs (wt), non time-varying
child characteristics (µ) and non time-varying school characteristics (). We impose the following
structure on preferences and the production function for cognitive achievement:
[A1] Household utility is additively separable, increasing and concave in cognitive achievement
and other goods.
[A2] The production function for cognitive achievement is given by TSt = F(TSt-1,wt,zt,µ,)
where F(.) is concave in its arguments.
W C S I I T -S ? 9
Under [A1] and [A2] the household problem is
T
Max(Xt,zt) U = E t-[u(TSt) + v(Xt)] s.t. (1)
t=
At+1 = (1 + r).(At + yt - PtXt - zt) (2)
TSt = F(TSt-1,wt,zt,µ,) (3)
AT+1 = 0 (4)
Here u and v are concave in each of its arguments. The inter-temporal budget constraint (2) links
asset levels At+1 at t+1 with initial assets At, private spending on educational inputs zt, income yt
and the consumption of other goods, Xt. The price of educational inputs is the numeraire, the price
of other consumption goods is Pt and r is the interest rate. The production function constraint (3)
dictates how inputs are converted to educational outcomes and the boundary condition (4) requires
that at t = T, the household must have zero assets so that all loans are paid back and there is no
bequest motive.6
In this formulation credit markets are perfect so that there are no bounds on At+1 apart from
(4); the perfect credit market assumption is relaxed in our discussion on the impact of liquidity
constraints below. Moreover, households choose only the levels of Xt and zt so that school inputs
(wt) are beyond its control. At the time the household has to make its decision, it knows the
underlying stochastic process governing wt but not the actual level. In other words, we assume
that school inputs are a source of uncertainty in the modelfor simplicity the only source. This
assumption is retained throughout the theoretical discussion, but is later relaxed in the empirical
test, where we allow for unobserved time-varying characteristics of the household that may influence
school inputs.
Maximization of (1) subject to (2) and (3) provides a decision rule related to TSt, character-
izing the demand for cognitive achievement. To arrive at this decision rule, we define a price for
cognitive achievement as the "user-cost" of increasing the stock in one period by one unit, i.e.,
6There are two observations regarding the form of the utility function. First, an alternative assumption, that the
benefits from the child's cognitive achievement are only felt in the future, would not change the model fundamentally.
If these benefits are only related to the flow of earnings in the future from the child's cognitive achievement, then
the education decision becomes similar to a pure investment decision. As long as the benefits from education are
concave in its arguments, the results would be similar. Note that this of course, does not imply that the steady state
value of human capital will be the same in either case, but only that along the growth path first-order conditions
remain unchanged (see Banerjee 2003 for a detailed discussion of steady states under different assumptions regarding
the form of the utility function). Second, the utility function uses a stock as one of its arguments: We assume that
households care about the level of educational achievement. The results below are unaffected if one assumes that
households care about the (instantaneous) flow from educational outcomes, provided that this flow is linear in the
stock.
W C S I I T -S ? 10
the relevant (shadow) price in each period for the household. Once such a price is defined, the
program is transformed into a standard consumer optimization problem (see for instance, Deaton
and Muelbauer 1980).
To define this price, note that if cognitive achievement could be bought and sold at the price,
v, households would pay v in the first period to buy one unit. In the next period, they could then
sell (1-) units (if depreciation is at the rate 1-) and receive the current value (1-)v . Thus, the
1+r
cost of holding one unit of test scores for one period is v - (1-)v and this defines the user cost. In
1+r
the context of a production function, households "buy" test scores in period t is by increasing zt.
Since we are interested in the cost of boosting achievement in one period only we assume that in
the next period they can reduce zt+1 to ensure that the overall stock of test scores at t +1 remains
unchanged. The user cost, evaluated at period t, is then (see Appendix 1 for the derivation):
1 FTSt(.)
t = (5)
Fzt(.) - (1 + r)Fzt (.)
+1
Similar to the expression (v - (1-)v ) derived above, the first term measures the cost of taking
1+r
resources at t and transforming it into one extra unit of cognitive achievement. When implemented
through a production function, the price is no longer constantif the production function is concave,
the higher the initial levels of cognitive achievement, the greater the cost of buying an extra unit as
reflected in the marginal value, Fzt(.). Of the additional unit bought in period t, the amount left to
sell in period t + 1 is FTSt(.) and the second term thus measures the present value of how much of
this one unit will be left in the next period expressed in monetary terms. Once this is defined, the
standard first-order Euler condition related to the optimal path education of educational outcomes
between period t - 1 and t can be derived as:
U
Et-1 t-1 TSt = 1 (6)
t U
TSt -1
Intuitively this expression (ignoring uncertainty for the moment) suggests that if the user-cost of
test scores increases in one period t relative to t-1, along the optimal path this would increase the
marginal utility at t, so that TSt will be lower. This is a standard Euler equation stating that along
the optimal path, cognitive achievement will be smooth, so that the marginal utilities of educational
outcomes will be equal in expectations, appropriately discounted and priced. Finally, the concavity
of the production function will limit the willingness of households to boost education fast since the
cost is increasing in household inputs. Thus, under reasonable restrictions, the optimal path will be
characterized by a gradual increase in educational achievement over time (for an explicit derivation
of the Euler equation with durables, see e.g., Deaton and Muellbauer, 1980 and Foster, 1995).
To proceed with the empirical specification we impose the following conditions on preferences
and the production function:
W C S I I T -S ? 11
[A1] Household utility is additively separable and of the CRRA form.
[A2] TSt = (1 - )TSt-1 + F(wt,zt,µ,) where the Hessian of F(.) is negative semi-definite.
Under [A1] marginal utility is defined as TSt-, with the coefficient of relative risk aversion.
Then, (6) can be rewritten as:
TSt - t-1 = 1 + et (7)
TSt-1 t
where et is an expectation error, uncorrelated with information at t-1. Taking logs and expressed
for child i, we obtain:
TSit 1 1 it 1
ln = ln(1 + eit) (8)
TSit-1 ln - ln(it-1 ) +
or, the growth path is determined by the path of user-costs, and a term capturing expectational
surprises.
3.3. Anticipated and Unanticipated Inputs
A key issue is how increases in school level inputs wt impact on the optimal path of cognitive
achievement. Since school resources are not known with certainty until after households make deci-
sions regarding their own inputs, this impact will depend on whether such increases are anticipated.
Thus, let wt (wt ) be inputs at time t that were anticipated (unanticipated) at t-1. First, consider
a u
increases that are anticipated. In this case, the impact on the path of outcomes will depend on
its impact on the user-cost of educational achievement at t, since there is no direct impact on the
budget constraint (all information included anticipated inputs will have been incorporated into the
budget constraint at time t - 1). In particular, using the implicit function theorem with (5) and
assuming [A2], we have
dt Fztwt
0 if Fztwt 0 (9)
dwta = - Fzt
2
This implies that if household and school inputs are technical substitutes (Fztwt < 0), anticipated
increases in school inputs at t will increase the relative user-cost of boosting at t, resulting in lower
growth of cognitive achievement, ceteris paribus, between t and t - 1, consistent with the optimal
path (6). Alternatively, if school and households inputs are technical complements, increases in
school inputs at t will increase the marginal productivity of household inputs at t, and through the
decline in user-costs lead to higher growth in cognitive achievement along the optimal path between
t and t - 1.
To clarify the dynamics between t - 1 and t further, note that there are two effects we need to
distinguish. The first is due to the change in relative user-costs while the second is governed by
W C S I I T -S ? 12
the households desire to smooth consumption (6). The second effect will always provide incentives
to spend more at t - 1 to take advantage of the additional government spending at t. If household
and school inputs are substitutes, households will optimally recognize that relative user-costs at t
will be higher than at t - 1the implicit price of buying test scores will increase in the future.
Consequently, to retain the optimal growth path of (6) households will choose to increase their own
spending at t - 1. Thus, the growth in cognitive achievement will be lower relative to the case
where no household responses are possible.
In the case of technical complements, the behavioral response is exactly the opposite, since
relative user-costs will be lower at t, households will optimally delay spending. However, in this
case the user-cost and the smoothing effects move in opposite directions so that the overall growth
could still be higher relative to the case where wt = 0. Comparing the two cases of complements
and substitutes, household spending is thus counter-cyclical relative to government spending when
household and school inputs are substitutes. When they are complements, the smoothing and the
user-cost effects move in opposite directions, although in the special CRRA case that we consider
here ( A1) the user-cost effect is higher than the smoothing effect so that the pro-cyclicality of
household inputs is maintained.
For unanticipated increases in school inputs, since households are unable to respond, they are
pushed off their optimal path and the increase in educational achievement in period t is given by
Fwtdwt. What is the size of this effect compared to a similar increase in anticipated inputs? When
inputs are anticipated, using (8) the change in the optimal growth path is given by
(tt-1 lnTS) 1 ln t )
wt anticipated = -( wt
1 1 Fzw
= 0 if Fzw 0 (10)
t Fz 2
For unanticipated increases the change in the growth path is give by ln(TSt + wt unantFw) which is
strictly positive. Thus, the effect of an unanticipated change is higher than that of an anticipated
change in the case of substitutes, and relative sizes cannot be ranked when they are complements
without further restrictions on the form of the utility function. These results are summarized in
Table A for the case of the CRRA.
W C S I I T -S ? 13
Table A
Type Inputs Cross- Spending at Spending at Effect on Relative
Derivative t-1 t Growth Ranking
A Anticipated Substitutes Increases Decreases Lower A<{B,C,D}
B Anticipated Complements Decreases Increases Higher B>A
C Unanticipated Substitutes Unchanged Unchanged Higher C>A ;C=D
D Unanticipated Complements Unchanged Unchanged Higher D>A; C=D
We stress here that increases in outcomes due to unanticipated inputs (in the case of substitutes)
are sub-optimal; household education spending will be higher than that justified by the decline in
the user-cost of boosting educational achievement. Consequently, in the next period, the household
will implement a correction using the "correct" user-cost to restore themselves to the optimal path.
Cognitive achievement being a durable, however, implies that the increase in outcomes will not be
entirely undone, since incentives will exist to sustain the stock at a higher level than before. It is
important to realize therefore that the distinction between anticipated and unanticipated inputs has
greater relevance for identification purposes than it has for policyalthough unanticipated inputs
lead to larger increases in cognitive achievement in the current period, this will be smoothed out
subsequently.
3.4. Credit Constraints
A straightforward way to incorporate credit constraints is to assume that any point in time
assets have to be nonnegative, i.e., credit is impossible unless fully collateralized (At 0). The
definition of the user-cost remains unaffected so that the main impact is that credit constraints may
limit the ability to equate appropriately discounted and priced marginal utilities. More specifically,
let t be the multiplier linked to the non-negativity constraint of carrying over assets between t-1
and t, then the optimal path, linking t - 1 and t, can be defined as:
U t-1 U + t (11)
TSt-1 = Et-1 t TSt
in which a binding credit constraint (t > 0) would result in higher marginal utility at t - 1
(lower educational achievement), than what would have been implied without credit constraints
(t = 0)see also Equation (6). What is the impact of unanticipated and anticipated changes
in wt in this case? First, the impact of unanticipated changes in unaffected: since no behavioral
response is possible, the effect still works through the production function of cognitive achievement.
A difference in this case is that with binding credit constraints the impact may be to bring the
household closer to its unconstrained optimal path by alleviating the credit constraint (since fewer
W C S I I T -S ? 14
private resources would be needed in the unconstrained case, relaxing the budget constraint). The
impact of anticipated changes now works via two effects. First the effect through the change in
user-costs has exactly the same impact as before. Second, there is a further effect through the
budget constraint: since fewer household resources are required than before to achieve the same
level of educational achievement, this would alleviate the budget constraint (i.e., reduce the shadow
cost of the constraint, t).
The size of the overall effect on cognitive achievement would depend on the income effects related
to the alleviation of the budget constraint. If the reduction in credit constraints leaves spending
on private inputs unaffected (so that no more other goods are consumed despite the relaxation of
the budget constraint), then the impact would be indistinguishable from unanticipated changes.
However, in the more plausible case that other goods are normal, in general, the effect of the
anticipated relative to unanticipated changes will retain the same ordering (i.e., anticipated changes
have a lower (higher) effect if school and household inputs are substitutes (complements) caveated,
as usual, by our discussion on the relative ranking of growth effects in the case of complements.
3.4.1. Summary
The key results of this section can thus be summarized as follows:
1. When household and school inputs are substitutes, an increase in anticipated inputs at t will
lead to an increase in household inputs at t-1, a decrease at t and subsequently a lower rate
of growth of cognitive achievement.
2. When they are complements, the opposing directions of the user-cost effect and the house-
hold's desire to smooth implies that the overall effect depends on household preferences.
3. Unanticipated inputs, on the other hand, have no impact on household inputs at t and t - 1
and thus always lead to a higher growth in cognitive achievement between t and t - 1.
4. Finally, these results remain unchanged (though attenuated) by the imposition of credit con-
straints under the mild assumption that other goods consumed by the household are normal.
In testing the predictions from this model using cash grants as the relevant input, we recognize
that schools should optimally allocate these grants across different inputs. One way to interpret
these results is that schools are constrained in what they can do and are hence unable to spend cash
grants optimally. These constraints could arise either due to thin markets (for instance, in the case
W C S I I T -S ? 15
of teachers) or lack of scale economies (for instance, to improve infrastructure).7 The estimated
equations are thus a "reduce-form" in the sense that they represent the effect of grants taking into
account constrained maximization at the school level.
4. Empirical Specification
Our empirical strategy is based on two related tests. We first test whether household educational
expenditures and school cash grants are substitutes. The theory implies that if this is the case,
contemporary household funding zt should decline with an increase in wt. This is a cross-sectional
testas long as the assumptions of the regression framework are maintained we should find that
households matched to schools with higher cash grants spend less on their children's education.
Once we have established that household and school inputs are technical substitutes, we proceed
to examine the hypothesis that (under the assumption of technical substitution) the impact of
anticipated grants is smaller than that of an equivalent unanticipated amount. We detail each of
these in turn.
4.1. Testing Substitutability between Household and Cash Grants to Schools
We estimate a generic demand model in which household spending on school-related inputs is
regressed on wealth (proxying permanent income), school attributes and school grants according
to the following system of equations.
ln zij = + 1Ai + 2 ln wj(match a u
i)+ 3 ln wj(match i)+ 4Xi + DiPi + i + j (12)
zij = zij if zij 0
(13)
0 if zij < 0
wj(match
a (14)
i) = + 1j + j
In Equation (12) zij is the spending by the household on child i enrolled in school j, Ai are assets
owned by household of which i is a member (as a proxy variable for the permanent income of the
household), wj is anticipated grants in school j that matches to child i, wj is unanticipated grants
a u
received by school j, Xi are other characteristics of child i and Pi are province level dummies.
The error term is decomposed into two components where i and j are child and school specific
error terms respectively. Further, zij itself is a censored variable; we observe zij only for strictly
positive values (corresponding to an enrolled child) and for cases where the optimal zij is negative,
we observe censoring at zero. We test 2 < 3 = 0, i.e., households respond negatively to expected
7As an example, in the case of teachers, while most head-teachers complained of shortages, only in two cases were
teachers hired by the school. Both turned out to be significantly worse (in terms of education and training) than
government teachers, leading to considerable dissatisfaction among the community.
W C S I I T -S ? 16
grants at the school level by cutting back their own funding, but are unable to respond to cash
grants that are unanticipated.
A potential estimation problem is that our estimate of 2 is inconsistent if the error in the
selection equation (14) is correlated to that in the demand equation, so that cov(j,j) = 0. In
particular, we will see that in the Zambian case, wit = a Constant , so that omitted variables that
enrollment
increase the probability of sending a child to a specific school as well as the unconditional (on
school choice) spending on educational materials at the household level will lead to the inconsistent
estimation of 2. Such a problem may arise, for instance, if there are rich villages where households
send their children to school but also spend more on education. The coefficient 1 would then
capture the differences in underlying wealth rather than a causal response to rule-based grants. To
some extent, we control for such wealth differences by including three different wealth indicators
in the regression; the household wealth index, the average wealth index for the village and the
average wealth index of students attending the school. Nevertheless, we may be worried that there
are other omitted variables that lead to the inconsistent estimation of response elasticities.
To address this issue, note first that by restricting the sample of villages to only those where there
was no school choice, we reduce the extent of the selection problem considerably. This strategy has
been used previously by Case and Deaton (1999) and Urquiola (2003) in their studies of schooling
inputs and cognitive achievement. However, we are still left with the parental choice of sending
children to school in the first place. To address this issue we base our identification on careful
sampling taking advantage of the very high historical enrolment rates in Zambia, even in remote
rural areas. Thus, we restrict attention only to those villages where the distance costs of travelling
to a school other than the one surveyed are very high (more on this below) so that any potential
benefits of choosing an alternative school are unlikely to outweigh the cost of transportation. Under
this assumption the problem is simplified to a two-dimensional choice between schooling and no-
schooling.
We then test for the weak exogeneity of lnws using the methodology proposed by Blundell and
Smith (1986). The exclusion restriction for this test is satisfied if there is a variable, j, which is
correlated with lnws, but not zij.We use the size of the eligible cohort in the catchment area and
the distance to school on the assumption that these variables are correlated to enrollment, but not
expenditure on the child conditional on enrollment. The Blundell and Smith (1986) test rejects
exogeneity if the coefficient on the residual obtained from Equation (14) is significant in Equation
(12) above. The inability to reject the null hypothesis thus establishes the exogeneity of lnws under
the assumption that the size of the eligible cohort and the distance from the school are exogenous
to household spending on children's schooling.
W C S I I T -S ? 17
From the basic hypothesis, 2 < 3 = 0, we can further exploit the data to test whether
the grants are "truly" anticipated in the sense that households make their own decisions taking
anticipated cash grants into account even before such grants are actually received. If there are
schools where, at the time of the survey, wa = 0 but wreceived = 0, a test can be based on the
a
difference in the estimate of 2 depending on whether we use wa or wreceived as the explanatory
a
variable.8 Specifically if household decisions are based on anticipated rather than received grants,
2(wa) < 2(wreceived) since if wa > 0 household spending will be less than what would be predicted
a
by using wreceived(= 0) as the explanatory variable. Thus, we can estimate a second equation
a
ln(zi) = + 1Ai + 2 ln wj(received) + 3 ln wj + DiPi + i + j
a u (15)
and test 2 < 2. A rejection of the null would lead to greater confidence in the technical-
substitution results since it would not only imply that cash grants to schools crowd-out household
funding (2 < 0), but further that households anticipate such grants and make their expenditure
decisions before the grants are actually disbursed.
4.2. Test-Score Hypothesis
Once we have established that household and school grants are technical substitutes, we can re-
strict ourselves to testing whether the impact of unanticipated grants on gains in cognitive achieve-
ment is higher than that of anticipated grants. A major concern in the parallel literature on
production functions when testing for the relationship between school inputs and cognitive achieve-
ment has been the presence of unobserved child and school level heterogeneity. In the context of
the Euler framework, consistent estimates may be obtained with a fairly flexible form of hetero-
geneity in the production function. To see this return to Equation (8) and the production function
given by [A2]. We showed previously that t = 1
Fzt(.)- (1+r)Fzt
(1-) (.). Now, as long as Fzµi = 0 or
+1
Fzt = µi(g(wt,zt)), so that lnit = lnti, the unobserved heterogeneity embodied in µi is elimi-
nated from the estimating equation. This formulation is then sufficient to ensure that the path of
user-costs is defined only in contemporaneous variables and is unaffected by fixed heterogeneity and
past school achievement.9 Assuming identical risk preferences, an empirical specification consistent
8Satisfying the requirement that wa = 0 at the time of the survey is uncorrelated to the error term of the regression.
Our identification is based on the fact that these were schools that were surveyed earlier in the month combined with
delays in disbursement.
9How restrictive is this particular formulation of the production function? Note first that we can either write the
production function as
TSit = (1 - )TSit -1 + 1wt + 2zt + 3µi + 4j
or as
TSit = (1-)TSit -1 +f(wt, zt,µij) if we make sure that Fzt = µi(g(wt,zt). We can compare this to three popular
specifications used in the literature on the estimation of production functions for cognitive achievement, discussed in
Todd and Wolpin (2003). The first, the contemporaneous specification has no role for either past levels of cognitive
achievement or for the possibility that household inputs zt will be correlated to unobserved school and child level
W C S I I T -S ? 18
with (8) can then be written as:
TSit
ln = o + 1 ln wit + 2 ln wit + 3Xt +
a u
it (16)
TSit-1
in which wit and wit are anticipated and unanticipated input changes, in this paper proxied by flows
a u
of funds, while Xt reflects all other sources of changes in the user cost between t and t - 1. The
core prediction is that the marginal effect of anticipated is lower than unanticipated funds when
household and school inputs are substitutes. This prediction is unaffected by the presence of credit
constraints, even though 1 is likely to be larger in that case.
The first econometric concern relates to the identification of anticipated and unanticipated cash
grants. Below, we will see that anticipated grants are well identifiedsuch grants are based on
a legislated rule and a detailed tracking (implemented in the survey itself) confirms that schools
receive exactly the amount stipulated. Our interpretation of unanticipated grants may be more
problematic since time-series data on cash grants, which could be used to calculate deviations from
the mean, are not available. We assume that grants other than the anticipated amount, which
are determined at the discretion of the District Education Office, are unanticipated. This would
probably overstate the unanticipated componentschools could have been informed previously of
such grants, or there could be differential expectations for different schools. To see how this affects
our estimates, ideally we would like to estimate
TSit
ln = o + 1 ln wit + 2 ln wit + it
a u (17)
TSit-1
but (assuming that observed anticipated grants are zero) our estimating equation is actually
TSit
ln = o + 2 ln(wit + wit) + it
a u (18)
TSit-1
Under the assumption that 2 > 1, 2 < 2, our estimates of the impact of unanticipated
grants constitute a lower bound, where the extent of attenuation depends on the degree to which
variables. The production function that we use here allows for both of these possibilities. The second specification,
which has been widely used in the recent past is the value-added specification
Tit = 1Tit -1 + 2wt + 3zt
Note that in this case, the unobserved heterogeneity is assumed to enter only at time 0 so that Ti = µithe child's
0
mental "endowment" leads to a fixed increase in test scores, instead of an incremental increase in every period. A
more general form is given by the cumulative specification where
Tit = 1Tit -1 + 2wt + 3zt + 4µi
so that child endowment can affect cognitive achievement in every period. These three widely used specifications
have increasing data requirements. In particular, the cumulative specification would require at least three periods of
data to arrive at consistent estimates. Our specification of the production function not only allows for the cumulative
specificaiton, but also allows unobserved heterogeneity to enter in a multiplicative form so that the marginal value
of household inputs can depend on unobserved child and school endowments. Further, the production function also
allows for the possibility that past inputs may affect current cognitive achievement through ways other than lagged
achievementspecifically, as long as we maintain the additive separability of Fzt and past inputs, our user-costs will
remain unaffected. The immense flexibility in the form of the production function for cognitive achievement that the
Euler framework allows for is then a major advantage over attempts to directly estimate such relationships.
W C S I I T -S ? 19
our construction of unanticipated grants may actually contain components that were anticipated.
To see this formally we can write the mis-specification as
True Model : y = b1X1 + b2X2 +
Estimated Model : y = (X1 + X2) +
where we are interested in the relative size of E() compared to b1 and b2 under the assumption
that b1 > b2. Then,
(X1 + X2)y
= Let X1,X2 have 0 mean. Then,
(X1 + X2)2
(X1 + X2)(b1X1 + b2X2 + )
=
(X1 + X2)2
b1V ar(X1) + (b1 + b2)cov(X1,X2) + b2var(X2)
E() = var(X1) + var(X2) + 2cov(X1,X2)
and a sufficient condition for b1 > E() > b2 is that cov(X1,X2) 0. This covariance is necessarily
zero if X1 and X2 are unanticipated and anticipated components, respectively.
The second concern is potential inconsistency in our estimate of 2 if cov(wit,it) = 0, possibly
u
arising from dynamic heterogeneity (time varying school or district-specific effects).10 This may
be the case for instance if there is a change in a school-level variable that leads, on one hand, to
greater unanticipated grants and, on the other hand, to higher learning gains (the introduction of
a highly motivated teacher who both searches for such funds and teaches exceptionally well is an
example). With such omitted variables, our estimate of the impact of unanticipated grants would
be inconsistent.
We correct for this problem through an instrumental variables strategy. From the Euler frame-
work, any variables at t - 1 are available for use as instruments since such information will have
been incorporated into the decision process. In addition we also use variables that were unknown at
t -1 but had an impact on unanticipated grants at t. In particular, based on the detailed tracking
of funds, we use per-pupil grants that the district office received from external (non-governmental)
sources that was allocated without any consultation with the offices.
For our instrumentation strategy to be valid, we require that the instrumental variables are
positively correlated with the amount of unanticipated funds received, but are not correlated with
the gain in cognitive achievement over the year. First, the amount of such grants boosts the
overall funds available at the district level and hence the unanticipated funds passed on to schools.
10Note though that omitted variable bias due to heterogeneity of fixed characteristics, which normally plague
cross-sectional estimates, are accounted for through the Euler framework.
W C S I I T -S ? 20
Simultaneously, since such funds arrive from external sources, it is unlikely that districts were
able to actively influence the amount of cash grants that they would receive. This addresses our
main concern that there might have been changes at the district level such that districts that
received more grants were also more likely to "place" this in a targeted manner. Consequently,
our strategy of using district-level aggregates combined with interactions of lagged stock variables
at the school-level isolates that portion of unanticipated grants that are uncorrelated to the error
term in Equation (16) above.
Finally, our results are reported at the school level. Since this is a straightforward linear
aggregation, there is no reason to expect results to change; simultaneously, we are better able to
handle the clustering of errors at the level of the district. To ensure that the sample remains the
same, we compute school level scores only for those students who were present in both years for
the test.
4.3. Other Econometric Concerns
There are two other specific concerns that arise due to the specific nature of our data. We
briefly detail our strategy in dealing with each of these below.
4.3.1. Treatment of Zero Cash Grants
The first is that a large number of schools receive zero unanticipated grants in the sample. Since
our estimation equation is based on the log transformation, we need to modify this variable in order
for the log to be defined. Moreover we need to address this problem both when cash grants are
explanatory (in estimations of the impact on cognitive achievement) as well as dependant variables
(in the first stage of our instrumentation strategy). We address each case in turn.
We use a modification of the method developed by Johnson and Rausser (1971) to derive the
optimal constant to be added on to zero values. The basic intuition behind this approach is that
the constant should be chosen so that the estimated relationship between cash grants and cognitive
achievement is identical for both schools with zero and non-zero cash grant values (dealing with
potential selection issues through the IV strategy above). In particular, we can treat the sample
of schools as two separate samples consisting of m observations of zero grant values and n - m
W C S I I T -S ? 21
observations of positive grant values. The regressions can then be represented as
ln(TSi) = 1 ln(Xi + k) + i
i = 1,2,....,m
ln(TSi) = 2 ln(Xi) + i
i = m + 1,...,n
and k is estimated under the restriction that 1 = 2. In addition to presenting estimated coefficients
based on the estimated k, we also present robustness tests based on estimated coefficients when the
unanticipated grants are treated as dummy variables (i.e., making a distinction only between those
who received and those who did not). We show that our results are robust to these alternative
specifications.
For the case where cash grants are the dependant variable, we estimate a hurdle model (Wooldridge
2001) where the probability of receiving such grants is estimated separately from the amount re-
ceived conditional on receipt. Under the assumption that the dependant variable, y, is distributed
log-normally conditional on y > 0, the maximum likelihood estimate of the unconditional E(y) can
be shown to be
2
E(y|x) = exp(x + )(x)
2
where x is the predicted value from the OLS regression of lny = x + (restricted to y > 0)
and (x) is the predicted probabilities estimated from a probit. In this instance, a hurdle model
is preferable to the Tobit since the latter requires the probability of receipt as well as the amount
received (conditional on receipt) to be governed by the same process. Contrary to this, the hurdle
allows for these processes to differ, so that in the predicting equations the process of determining
which schools receive positive grants is separate from the determination of how much the schools
receive. Note that since we are interested in the E(lny|x) and not ln(E(y|x)) the above simplifies
to
E(lny|x) = (x)(x)
Finally, the prediction for this hurdle model is then used in the second stage of our instrumen-
tation strategy, with an appropriate standard error correction for the use of generated regressors
(Murphy and Topel 1985).
W C S I I T -S ? 22
4.3.2. Measurement Error: Test Scores
Our second concern relates to the treatment of test scores. Our measurement of the child's
human capital, TSt, is based on tests administered in English and Mathematics. We model scores
in the test as arising from the distribution of the underlying latent variable (TSt) following the
literature on Item Response Theory (Birnbaum 1967). This method has several advantages in that
the properties of the estimated latent variables are easy to interpret and the importance of the
characteristics of the test are made explicit in the estimation. Further, the maximum likelihood
procedure used to estimate the latent variable generates weights that are locally-optimal in the
sense that they minimize the error of classification. The latent variable, TSt, is estimated through
a maximum likelihood procedure using a structural assumption regarding the mapping between
TSt and the probability of a correct response. The standard error of this estimate can then be
computed as 1 where Ij(TSt) is Fisher's information for a particular question, j, and the
jIj(TSt)
sum is over all questions in the test (see Appendix 1).
How does this error of measurement effect our estimates? To the extent that the change in TSt
is the dependant variable in our regressions, this increases the standard error of the regression, but
our estimates remain consistent and unbiased. A correction is required, however, since the error
structure is now characterized by a variance-covariance matrix that violates homoskedasticity of
the disturbance terms. Since the standard error of our estimate is itself a function of TSt, the
variance-covariance matrix consists of terms like 2 +2ui, where is the regression error, and ui
the measurement error for individual i. We account for this by adjusting standard errors for an
arbitrary error-structure due to clustering.11
A more serious problem arises if cash grants are targeted toward poorly performing schools in
the base year. Measurement error in test scores implies that gain-scores are higher for schools that
performed poorly in the base year (Kane and Staiger 2001, 2002). Thus improvements in cognitive
achievement could arise due to mean reversion rather than a causal relationship with such grants.
Although measurement error due to differences in cohorts (a major source of variation in Kane
and Staiger 2002) are eliminated by retesting the same children we do find that in the case of
Mathematics the gradient between gains and initial scores is significantly negative. To address this
issue, we first show that there is no evidence of targeting toward poorly performing schools and we
then instrument for potential endogenous placement using the strategy detailed above.
11Greater efficiency can be obtained by using the standard error of the estimates to implement a (modified) variance
weighted least squares estimation. However, simulations in Das and DeLaat (2003) show that the efficiency gains
from doing so (compared to the robust sandwich estimator) are smallthe major efficiency gains arise from the use
of estimated latent variable rather than the test score itself.
W C S I I T -S ? 23
5. Data
5.1. The Country Context: Education in Zambia
Our data are from Zambia, a landlocked country with a population of 10 million, almost en-
tirely dependant on copper for export revenues. With a decline in copper prices, there has been a
commensurate decrease in income and government resources. As a result, average real per capita
government education expenditure in 1996-98 was only about 73 percent of its 1990-92 level, declin-
ing further to an average of about 60 percent of this level by 1999-2000 (World Bank data based
on Government of Zambia Financial Statements).
This economy-wide decline has also had an impact on educational attainment. Although Zambia
outperforms other African countries with similar per capita income levels, net primary school
enrolment at about 72 percent is at a historically low level, having seen some decline during the
previous decade (Figures 2a and 2b).12 Both the government and households have responded to
this worsening of the education profile. The government for instance initiated a Basic Education
Sub-Sector Investment Program in 2000, which along with administrative changes in the delivery of
educational services and restructuring of the payroll for teachers also led to some direct financing of
schools through cash disbursements. While household responses are clearly harder to interpret, we
will see below that parents tend to be active in their children's education with high contributions,
both in terms of expenditures as well as time. It is precisely this involvement that will be exploited
in our tests below.
5.2. Sampling
The study is based on a survey of 182 schools in four provinces of the country and was collected
by the authors in 2002.13 The choice of schools was based on a probability-proportional-to-size
sampling scheme, where each of 35 districts in the four provinces was surveyed and schools were
randomly chosen within districts with probability weights determined by enrollment in the school.
Thus, every enrolled child in the district had an equal probability of being in a school that partic-
ipated in the survey.
In every school, 20 students were randomly chosen in Grade V in 2001 and an achievement
test was administered in Mathematics, English and the vernacular.14 The same test was admin-
istered again in 2002 to the same students leading to the construction of a two-year panel of test
12These levels are similar to Kenya, higher than Congo or Mozambique, but below those typically attained in other
Southern African countries (see for example, World Bank and UNESCO).
13Lusaka, Northern, Copperbelt, and Eastern provinces were surveyed. These four provinces account for 58 percent
of the total population in Zambia.
14In cases with less than 20 students, all children were tested.
W C S I I T -S ? 24
scores. In addition to the tests and a school questionnaire, questionnaires were administered to the
head-teacher and all teachers who were teaching or had taught the tested children. These children
were also asked to complete a pupil questionnaire in every year with information on basic assets,
demographic information, and educational flows within the household. Further, as part of the ex-
penditure tracking exercise, district and provincial educational offices associated with the surveyed
schools were administered questionnaires detailing financial activity over the year (receipts and
disbursements of cash and materials).
In addition to the school survey, household surveys were administered to 540 households in
35 villages. The choice of villages was designed to eliminate complications arising from school
choice (see Section 4.1): Based on a geographical mapping of all schools, those that satisfied a
"remoteness" criteria (defined as the closest edge of the relevant Thiessen polygon lying at least
2.5 kilometers from the school) were chosen as starting points for villages in the household survey
(so that the school was at least 5 kms from the nearest other school). From these schools, the
closest (or second closest depending on a random number) village was chosen and 15 households
were randomly chosen from households with at least one child of school-going age.
Two different samples are thus used for the empirical section of this paper. The first sample
(the Household Sample) is based on a subset of 35 remote schools, with data on matched school
and household inputs for 540 households. Since 15 households were selected from every village, we
have data on cognitive achievement for only 200 students in this sample and hence can use this data
only to test the resource-substitution hypothesis. The larger sample of 182 schools provides data
on changes in cognitive achievement for 2,600 students with matched data on school, teacher and
head-teacher attributes but not on household expenditures. This sample is used to test the Test-
Score hypothesis. Finally, data from provincial and district levels are used to provide instrumental
variables for the Test-Score Hypothesis. Table B clarifies the use of our data.
Table B
Sample Questionnaires Learning Achievement Used For
Household Household Questionnaire X Substitution Hypothesis
School School Funding, School 3,600 students in 2001
Attributes
Head Teacher and Teacher 2,700 re-tested in 2002 Test Score Hypothesis
Attributes
Tracking District/Province Level X Instrumental Variables
Funding
W C S I I T -S ? 25
5.3. Description: Schools
Reflecting the overall decline in the education sector, schools in our sample are under some
stress (Table 1a). There are over 100 children for every functional classroom, student-teacher ratios
are above the Zambian guideline of 40 and there are a large number of repeaters. Moreover, for
almost every variable rural areas tend to do worse than their urban counterparts and this difference
is further magnified in the case of the schools chosen for the household sample. The difference is
particularly marked for asset holdings where the mean value of the asset index is one standard
deviation lower in rural and 1.2 standard deviations lower in remote villages compared to urban
areas.
Turning to educational inputs, there are three distinct types of inputs that schools may receive
teaching inputs through new teachers or increases in staff remuneration, in-kind receipts in the form
of textbooks or chalk, and cash receipts. The effect of teacher inputs is studied in some detail in
Das, Dercon, Habyarimana, and Krishnan (2003). Further, during the year of the survey, schools
received very little inputs in-kindon average less than 0.05 textbooks, 0.012 desks, 0.001 chairs
and 0.01 boxes of chalk were received per student.15 Consequently, the impact of the third type of
inputs (cash receipts excluding teacher's salaries) on cognitive achievement forms the basis for this
study.
Contrary to the poor record of in-kind receipts, most schools had received some cash and this is
explored further in the "cash grants" rows of Table 1a. There were two kinds of cash receipts that
schools could receive. Rule-based grants were received under legislation that grants $600 ($650
in the case of schools with Grades 8 and 9) to every school irrespective of enrollment. We treat
this as "anticipated" in our analysis. The second kind, discretionary grants were disbursed to
schools at the discretion of the District Education Office as well as external donors. We treat this
as "unanticipated" recognizing that this may overestimate the "true" unanticipated component of
cash grants .
One concern is that legislated allocations may have little to do with received grants and our
treatment of such grants as "anticipated" is thus incorrect.16 A tracking exercise presents some
encouraging results on this front (Table 1a, Cash-Grant Characteristics). In three out of the four
provinces, over 93 percent of the schools surveyed received exactly the amount allocated. In the
15This was largely due to problems in the planning department of the Ministry of Education coupled with problems
in procurement, rather than due to the lack of funds (less than 60 percent of the allotted budget was actually used
during the fiscal year).
16In the case of Uganda for instance, Ablo and Reinikka (2000) showed that less than 30 percent of the allocated
capitation grant was received by schools.
W C S I I T -S ? 26
fourth province, Lusaka, this percentage dropped to 71 percent and this was attributed to delays in
disbursement. Based on receipts in the previous year as well as interviews with head-teachers and
district officials, it appeared that the remaining schools would receive the allocated disbursement
shortly after the survey.17 It is precisely this delay that we exploit in drawing the distinction
between wa and wreceived in Equation (15) above (details of the tracking exercise can be found in
a
Das and others 2003). Note also that the per-pupil rule-based grants are fairly high in absolute
amounts ranging from K 2,400 (urban) to K 8,700 (household sample), which corresponds to the
cost of two textbooks or 36 boxes of chalk.
In contrast to the regularity of rule-based allocations, we find that the probability of receiving
discretionary grants was much lower (24 percent). Conditional on receipts such funds tend to be
either very small or extremely largethe inter-quartile range for log discretionary grants ranges
between 6 and 10 log kwacha per pupil with a coefficient of variation greater than 6. This large
dispersion in discretionary grants is also seen in Table 1a when we compare the logs to the actual
amounts. In the case of actual amounts, discretionary are always larger than rule-based funds,
in logs however, this relative ranking reverses due to the wide dispersion in the former. Finally,
variables such as school or pupil characteristics have almost no predictive power in explaining the
distribution of discretionary fundsless than .05 percent of the variation in such funds can be
explained through differences in student composition, characteristics of teachers (head-teachers) or
the availability of educational resources in the school.
The last point also addresses the potential targeting of discretionary funds (Table 1b). We
find that, at least on the basis of observable outcomes, there is little difference between schools
that received discretionary funds vis-à-vis those that did not. Although schools that received such
funds tend to have students who are marginally wealthier and are located closer to the district
office, these differences are not significant at the 10 percent level. Moreover, there is no difference
in baseline scores between schools that received discretionary funds and those that did notthis
result in particular, addresses the issues of mean reversion discussed earlier.
Thus, on the one hand rule-based allocations are clearly demarcated and defined, and schools
receive the amount stipulated. On the other hand, discretionary funds are more volatileless than
one quarter of all schools receive such funds, and even conditional on receipts, the amount received
varies dramatically. Further, such funds do not seem to have been allocated in a targeted fashion,
at least on the basis of observable school and student characteristics. It is precisely this difference
that forms the basis for our division of cash grants into anticipated and unanticipated components.
17This was checked and confirmed in the case of some schools in Lusaka province, two months after the survey.
W C S I I T -S ? 27
5.4. Description: Households
To complete our description, it is also instructive to examine household funding of school inputs
in comparison to the funding received from the government. In particular, if household educa-
tional expenditures are small compared to school grants, anticipated grants may play a role in the
alleviation of credit constraints at the household level (although our empirical test would remain
valid). Figure 3 explores the importance of different funding sources for educational expenditure,
disaggregated by schools that received high/low anticipated cash grants (with the cutoff at the
median).
We find that in both types of schools, household expenditure accounts for a large share of total
(public and private) spending on education, ranging from 25 percent to 33 percent across schools
that received high/low cash grants. The other significant expenditure share is accounted for by
teachers salaries (roughly 50 percent in both cases); ignoring this component implies that household
expenditure accounts for between 50 percent and 60 percent of total available expenditures. Since
the household data is based on a sample of remote schools that tend to be poorer, this percentage
represents a lower bound on the actual share of household expenditure in total funding. Clearly
then, even in remote and poor areas, households represent an extremely important component of
educational funding and it is likely that they have sufficient leeway to adjust for changes in expected
grants at the school level.
In anticipation of our results, Table 2 then looks at key household and school variables for
schools with high/low enrollments (and hence low/high anticipated grants). The first row (matching
success) shows the percentage of children in the primary age group who were successfully matched
to the surveyed school, verifying our assumption of no school choice through the choice of the
sample. For both high/low grant schools, matches are above 95 percent, but there is a (significant)
3 percent difference suggesting that endogenous enrollment may still be an issue.
The next two rows describe school cash grants and household expenditures. There are large
differences in the means of two groups, although interestingly total funding is roughly equivalent,
at K24,000 in low and K22,000 in high grant schools. The other rows in the table correspond to
observable components of schools and households. For a number of important variables (household's
asset indices, mother's/father's education and village enrollment) there is no significant difference
between the two categories of cash grants. Moreover, in cases where differences are significant
(percentage with mother/father at home) the direction is the opposite of what we would expect
if enrollment was endogenous to villages and schoolshigh enrollment (low grant) schools tend to
have fewer children with parents at home and report fewer visits from teachers to the household.
W C S I I T -S ? 28
These statistics thus suggest that (a) there are significant differences in household contributions
across low/high cash grant schools, and the null hypothesis that total funding is the same cannot
be rejected (b) while there are some differences in observable household components across the two
categories, these differences tend to be small or of the wrong sign. Our results on the substitutability
of household and school cash grants verifies these broad results in a structured manner.
5.5. Description: Cognitive Achievement
Finally, our data on cognitive achievement are based on tests administered by The Examination
Council of Zambia in Mathematics and English for the same sample of children in 2001 and 2002,
following the sampling scheme described above. These children were initially tested in Grade V and
in 2002 they were tested in the grade that they were currently enrolled in. Although we should hence
have a two-year panel of test scores for 3,500 children (since there were less than 20 children in Grade
V in some schools), attrition in the data set leads to a smaller sample of 2,587 children. This drop
is attributable to a number of factors including school-transfers/drop-outs (30 percent), absence on
the day of the test (50 percent) and data issues arising from the inability to survey some schools
or adequately complete pupil rosters (20 percent). We find some differences in original scores, with
those who were unable to take the second test reporting significantly lower English and Math scores
in the first year (0.11 and 0.19 standard deviations. respectively), but do not find any systematic
pattern in attrition across schools receiving different amounts of anticipated/unanticipated cash
grants.
Turning to learning gains over the year, students on average were able to answer 3.2 questions
more in Mathematics from a starting point of 17.2 correct answers (from 45 questions) and 2.4
more in English starting from 11.1 correct answers (from 33 questions). In terms of our latent
distribution, children gained on 0.42 standard deviations in Mathematics and 0.40 in English.
Thus, after one year of teaching students were able to increase their scores by 6 percent and 7.5
percent in Mathematics and English, respectively.
Finally, Figure 4 shows the characteristics of the Mathematics and the English test with respect
to the standard error of our latent score distribution. For both Mathematics and English, the test
was "too-hard" in the sense that children at the lower end of the distribution have (much) higher
standard errors than those above the mean. Further, the English test was better designed than the
Mathematics with lower estimated standard errors at all points of the distribution. Following from
our previous discussion we thus expect considerable noise in our estimates, but also lower standard
errors for our cognitive achievement results based on the English compared to the Mathematics
test.
W C S I I T -S ? 29
6. Results: Does Household Spending Substitute for School Cash Grants?
Our main interest in this section is the estimation of Equations (12) and (15). To recapitulate,
Equation (12) estimates the relationship between household and school grants using anticipated
grants that had already been received at the time of the survey (wrecieved). Equation (15) then
a
uses the anticipated grants (wa) instead (whether received or not at the time of the survey); if
households truly anticipate such grants, the elasticity of substitution based on the second equation
should be greater than the first.
The estimation results are presented in Table 3. For every estimated equation there are two
specifications. The first is the Tobit specification where the sample includes all school going age
children and educational expenditures on children who are not enrolled is defined to be zero. We
may be concerned that the Tobit specification does not entirely capture the error structure of (12),
with clustering at the village level. To account for this clustering, we also present estimates from
a random effect Tobit specification that accounts for such clustering. Columns (1) and (2) thus
report results from Equation (12), and (3) and (4) from Equation (15). Table 4 then interprets these
coefficients as the marginal impact (computed at the mean of the regressors) and the probability
that the dependant variable is uncensored.
The results are as predicted and robust to the sample and specification used. Using wreceived
a
as the regressor, the estimated elasticity of substitution for anticipated grants is always negative
and significant (2(wrecieved) < 0) and ranges from -0.34 (Tobit) to -0.37 (Random Effects Tobit).
a
Further, 2(wa) < 2(wrecieved) with the elasticity of substitution increasing to -0.46 (Tobit) to
a
-0.52 (Random Effects Tobit) suggesting that households truly anticipate these cash grants and
make their expenditure decisions prior to their actual receipt. Moreover, using wa as a regressor,
the coefficient of unanticipated grants is small and insignificant (3 = 0). For the specification
where 3 < 0, the size of the coefficient is less than half that of 2(wrecieved) suggesting that there
a
may have been some household responses to unanticipated funding, but this was relatively small
compared to funding that was anticipated. Note that to the extent that households did respond to
unanticipated funding as well, this would imply that our coefficient on such funding in the test-score
regression is an underestimate of the true production function parameter.
These results present strong evidence for a high elasticity of substitution between anticipated
grants and household funding, and support the hypothesis that households make their educational
funding decisions prior to the actual receipt of such grants. Further, households do not respond
to unanticipated grantsdespite the comparability of such grants (in amounts) to the anticipated
equivalent, there is little evidence that households alter their own behavior as a response.
W C S I I T -S ? 30
Our main worry with these results is the possibility of omitted variable bias that arises due to
the close link between anticipated grants and enrollment and this is addressed in Columns (5) and
(6). Following the strategy outlined in Section 4.2 we estimate the determinants of log anticipated
funding in the first stage and use the residuals as an additional regressor in the specifications
estimated under Equation (15).
Note that this test is valid only if we use anticipated rather than received funding in the first
stage. If we use received funding as the dependant variable we have imposed high enrollment for
schools that received zero funding at the time of the survey in combination with a lower than
expected expenditure and this would lead to a rejection of the weak exogeneity of lnwa. Using
anticipated funding in the first stage with Equation (15) as the second stage, the coefficient on
the residual is insignificant at the 15 percent level confirming that the log of anticipated funding is
exogenous to child level educational expenditures and estimated coefficients thus represent a causal
relationship (Columns 5 and 6, Table 3).
7. Results: Test Score Hypothesis
7.1. Graphical Evidence
The results in the previous section provide strong evidence that school grants and household
funding are indeed substitutesgreater cash grants given to the school reduces the amount spent
by the household. As discussed in Section 3.3, when school and household inputs are technical
substitutes, the impact on learning gains of unanticipated is larger than that of anticipated inputs.
Figure 5 explores this relationship through non-parametric plots of the relationship between
cash grants and gain in cognitive achievement. The figure on the left shows the (annual) change
in cognitive achievement in the subjects of Mathematics and English plotted against (log) unantic-
ipated grants while the relationship between gains in cognitive achievement and (log) anticipated
grants. In both figures, the left axis shows the density of cash grants plotted on a histogram while
the right axis depicts learning gains in Mathematics and English, plotted against the log grants.
The figure verifies our basic hypothesis: learning gains are higher for unanticipated compared to
anticipated grants. In the case of Mathematics there is a gain of almost 0.2 standard deviations
and for English 0.15 standard deviations moving from the minimum to maximum unanticipated
grants. Moreover, there is no discernible pattern in the case of Mathematics and a decline in the
case of English for anticipated grants.
Figure 5 also suggests reasons for high standard errors in our estimation procedure. From the
histogram for unanticipated grants it is clear that the distribution is marked by a large percentage of
W C S I I T -S ? 31
schools that receive zero combined with substantial variation among those that do receive positive
amounts. Consequently, large variation in learning gains among the non-receivers might decrease
the precision of the estimated relationship between cash grants and learning gains. Further, there
appear to be differences in the precise functional form between Mathematics and English. For
Mathematics small amounts of unanticipated grants have a low impact on learning achievement
while for English such investments have an impact but decreasing returns set in quickly at higher
levels. Although ideally we would like to estimate these non-linearities, our sample of 42 schools
that receive positive amounts precludes further sub-division. Thus, we account for non-linearities
through the inclusion of a quadratic term for unanticipated grants in Equation(16) and check for
robustness using a dummy variable for schools that received non-zero unanticipated grants.
7.2. OLS Results
Table 5 shows the results for English and Mathematics for four different specifications where
all estimations are at the school level.18 The first specification (column 1 for English and column 2
for Mathematics) is the most parsimonious and includes only anticipated and unanticipated grants
in the estimation. The next two columns then include four additional explanatory variables-
a dummy for rural is included to proxy for "shifters" and three variables that capture potential
changes in user--costswhether the head-teacher has changed, whether the head of the parent-
teacher association changed and the difference in PTA fees. The third specification (columns 7 and
8) include as variables the portion of anticipated funds that had actually been received by the time
of the survey, as in Section 6 above.
For all specifications we find that the coefficient on anticipated grants is small and insignificant
(except for the negative and significant value in column 7 for English), marginally more so in
the case of English compared to Mathematics. For English the coefficient on the linear term of
unanticipated grants is always positive and significant, and for the quadratic term negative and
significant. For Mathematics the results are not as sharp; the coefficient on unanticipated grants
is smaller and insignificant in three specifications, but is significant when unanticipated grants are
treated as a dummy variable. Treating cash grants as a continuous variable, these results imply
that the marginal impact of (log) grants on cognitive achievement at the mean is .048 standard
deviations for English and .029 for Mathematics, representing 12.5 percent and 7 percent of the
annual gain in learning. The results from the dummy specification imply that schools receiving
18Since the estimated equation is linear, averaging over children should have no impact on the estimated coefficients.
In fact, child-level regressions show similar coefficients but the significance is reduced when the regression is clustered
at the school rather than the district level. All coefficients retain their significance at least at the 10 percent level
of confidence. Further, the results from the IV estimation are identical both in the size and the significance of the
coefficients. These differences may arise due to the clustering of errors at both the school and the district level.
W C S I I T -S ? 32
non-zero unanticipated grants improve their learning by 0.12 (English) and 0.09 (Mathematics)
standard deviations, which corresponds to approximately one-third of the average gain through the
year.
We make two observations on the set of estimated coefficients. First, returning to the stan-
dard error of our latent score distribution, it is clear that the measurement error in the case of
Mathematics is much larger than that for English. This would suggest that our estimates our more
precise for the latter and could explain the significant findings for English but not for Mathematics.
Second, in Figure 5 the differences in English are driven primarily by schools receiving zero and pos-
itive grants while those in Mathematics are driven by the difference between those receiving small
and large amounts. In using the log specification, the optimal constant added to those with zero
unanticipated grants is only K3.73. Given the steep gradient of the logarithm function near zero,
the addition of a small constant implies that the estimated relationship is driven almost entirely by
the difference between the "zero" and the "non-zero" group rather than by the "low positive" and
the "high positive" receivers. Consequently, in the case of Mathematics, the positive relationship
between the low and the high group is overwhelmed by the flat portion before with the reverse
for English. However, when we use a dummy variable for whether or not the school received any
unanticipated funds as a regressor, both coefficients are significant and positive (and fairly close to
each other) since we "average" out these functional differences.
7.3. IV Results and Comparison
The final set of results we present corrects for the potential placement problem through the
instrumentation strategy detailed above. The first stage from these results are presented in Table
6, and the results on learning changes using the instrumented unanticipated funds in Table 7. In
the first stage, we find that the grants that the district received from external sources significantly
impacts on the amount of unanticipated funds that a school will receive; the only other significant
variable is whether the district was one that was heavily contested in the run-up to the election the
same year as the survey. Together, the lagged variables and the funds received through external
sources account for 50 percent of the variation in unanticipated funds at the level of the school,
conditional on receipt. For the second part of the hurdle model, the probability that a school
receives any cash grants is determined largely by the status of the school as one that was eligible
to receive funds from an external donor program administered by the United Nations since 2000.
Turning to the instrumented estimation results, we find that in all cases, there is an increase
in the linear and a decrease in the quadratic term of the estimated equation. Moreover, for both
English and Mathematics, the coefficients on unanticipated funds and unanticipated funds squared
W C S I I T -S ? 33
are always significant, while on anticipated funds the coefficients continue to remain small and
insignificant. However, due to the quadratic formulation, it is not immediately apparent that these
findings can be interpreted as evidence of positive-placement whereby schools that were likely to
do worse received more grants. To formally evaluate our hypothesis that the marginal impact of
unanticipated is greater than that of anticipated grants, Table C presents results of Wald tests at
various points of the sample range.
Table C
Subject and Estimation Strategy Test Rejects Equality of Coefficients Evaluated at (p values)
Sample mean 25th %tile Median 75th %tile 90th %tile
English (OLS) 5% 5% 5% 5% >10%
English (IV) 1% 1% 1% 5% 5%
Math (OLS) >10% >10% >10% >10% >10%
Math (IV) 10% 5% 10% >10% >10%
Since the marginal impact of unanticipated grants is given by b1+2b2X where b1 is the coefficient
on unanticipated, b2 on unanticipated grants squared and X the point at which the marginal value is
evaluated, the Wald test takes into account the var(b1 +2b2X) given by the usual expansion. Each
cell in the table shows the confidence level at which the null hypothesis that the marginal impact
of unanticipated equals that of anticipated grants can be rejected. For English we find that the
OLS results allow us to reject the null hypothesis at all points of the sample range at the 5 percent
level, and we are unable to reject the null at the 90th percentile. The IV results are stronger; the
null hypothesis can now be rejected at the 1 percent level for the mean, the 25th percentile and the
median, and the 5 percent level for the 75th and 90th percentile. For Mathematics, overall results
are weaker. We cannot reject the null hypothesis at any point of the sample range for the OLS
results. With instrumental variables, results are marginally better with rejections at the 10 percent
level for the sample mean and median, and at the 5 percent level for the 25th percentile.
8. Discussion and Policy Implications
Using data on learning achievement and non-salary cash grants to schools in Zambia we have
tested a model of dynamic household optimization with the key prediction that anticipated com-
pared to unanticipated grants will have a smaller effect on cognitive achievement if household and
school cash grants are substitutes. In the case of Zambia, we find that the elasticity of substitution
between household and school grants ranges from -0.35 to -0.52. Consistent with our predictions,
we then find that anticipated grants have zero impact on learning gains while unanticipated funds
W C S I I T -S ? 34
increase learning by 0.05 (English) and 0.025 (Mathematics) standard deviations at the mean. The
estimated coefficients are likely under estimates of the true production function coefficient due to
the potential contamination of unanticipated grants by anticipated components, but are robust to
omitted variable bias arising either from school-choice or time-varying school effects. These results
have implications both for estimation and policy and we discuss each of these in turn.
8.1. Implications for Estimation
The dominant estimation technique for estimating the effect of school inputs on cognitive
achievement is based on the production function approach, where achievement (or changes in
achievement) is regressed on school inputs. Following Todd and Wolpin (2003), these estimates
represent the policy effect of school inputs that combines both the effect of inputs on cognitive
achievement through the production function, as well as household responses to such inputs. Our
use of unanticipated inputs allows the estimation of both effects separately, thus shedding more
light on the process through which school inputs may or may not affect educational attainments.
For estimation purposes, it may appear that the problem in the production function approach
arises due to omitted variablesif the researcher had access to both household and school expen-
ditures in the current period, it would be possible to accurately estimate the effect of the input
through the production function. While true in a static setting, this does not take into account
the possibility of inter-temporal substitution in a dynamic problem. Specifically, households start
responding to school inputs at the time that information regarding such inputs is revealed so that
the entire history of household inputs will be required from that point onward to estimate unbiased
production function parameters. The alternative approach followed in this paper is to examine the
portion of inputs that arrive as exogenous shocks so that households are unable to respond in the
current (or preceding) periods. In the absence of data on historical household inputs (as well as
details on the revelation of information) the use of unanticipated inputs allows identification of the
production function parameter with greater accuracy.
The distinction we draw between anticipated and unanticipated inputs could also account for
the wide variation in estimated coefficients of school inputs on cognitive achievement (Glewwe 2002;
Hanushek 2003; or Kreuger 2003) The production function framework does not separate anticipated
from unanticipated inputs so that the regressor is a combination of these two different variables.
Since the covariance between the two types of inputs is (necessarily) zero, the estimated coefficient
is bounded below by the policy effect and above by the production function parameter; the distance
from either bound depends on the extent to which the schooling inputs were anticipated or not.
There are a number of extensions that can be pursued in future research. While the use of the
W C S I I T -S ? 35
Euler equation framework allows us to control for heterogeneity arising from a number of sources
and in a fairly flexible form, our model does require some restrictive assumptions. First, we are
unable to control for heterogeneity due to the underlying production function or the household
discount rate. Second, we are unable to allow for non-linear effects of the lagged test score and
third, our specification requires the use of an additive lagged test score in the production function.
The last two problems could potentially be addressed by panels that span a larger time period.
8.2. Implications for Policy
The argument developed here also has repercussions for educational policy. Our results do not
suggest an educational policy where inputs are provided unexpectedly. Although cognitive achieve-
ment in the current period does increase with unanticipated inputs, the additional consumption
now will push households off the optimal path. In subsequent periods, therefore, they will readjust
expenditures until the first-order conditions are valid againunanticipated inputs in the current
period will not have persistent effects in the future (except due to the durable nature of the good).
The policy framework that is suggested under this approach involves constructing appropriate
"spheres of influence". Under such a framework schooling inputs are characterized by the level of
market imperfections that characterize their provision, as well as the degree of substitutability with
household contributions. Inputs are then divided into two categoriesthose within the sphere of
influence of either the government or the household.
The former would include inputs that are characterized by incomplete markets or other market
failures (see Miguel and Kremer 2003 for an example of market failures due to externalities) while
the latter would consist of inputs provided under competitive market conditions. Further, to the
extent that the government may be worried about the inability of households to smooth human
capital accumulation (for instance due to credit constraints), direct subsidies to households rather
than inputs at the school level characterize the optimal policy.
What inputs may lie in the governments "sphere of influence"? An important example may
be teaching inputs, whereby problems in the design of contracts (see for instance Holmstrom and
Milgrom, 1991, for problems arising due to multi-tasking or Holmstrom, 1982, for problems arising
due to moral hazard in teams) or the generic non-availability of trained personnel in every village
could make public more efficient than private provision. Similarly, inputs that retain some aspects
of public-goods and would thus be under-provided in the private market are a good candidate for
government provision. Interestingly, both of these have been shown to have significant impacts on
cognitive achievement (see for example, Hanushek 2001 and Banerjee, Cole, Duflo, and Linden 2003
on the importance of teachers and Glewwe and Jacoby 1994 on infrastructure).
W C S I I T -S ? 36
The approach followed here of treating cognitive achievement as a household maximization
problem with the production function acting as a constraint opens up a new avenue for research,
one that explicitly recognizes the centrality of households and classifies schooling inputs into the
categories discussed above. To construct the appropriate "spheres-of-influence" it would be impor-
tant to characterize inputs by their degree of substitutability vis-a-vis household provision. To do so
requires both the use of matched household and school surveys as well as the careful identification
of surprises in the provision of inputs; in particular, long-term data on schooling inputs would allow
for cleaner identification based on deviations from means, much as in the consumption literature.
W C S I I T -S ? 37
9. Appendix 1
Proof of 5. To define the user cost, we consider a change in zt and zt+1 that increases cognitive
achievement by one unit in period t only. Formally,
dTSt = Fzt(wt,zt,µ,)dzt (19)
dTSt+1 = FTSt(wt,zt,µ,)dTSt + Fzt (wt+1,zt+1,µ,)dzt+1 = 0 (20)
+1
where Fzt, Fzt and FTSt are marginal effects of the educational achievement production function
+1
with respect to household inputs in the current and next period, related to initial levels of cognitive
achievement in the next period. The conditioning (wt,zt,µ, ) explicitly recognizes that these
marginals will be a function of current levels of inputs as well as child and school non-time varying
characteristics. To achieve this, assets are reduced in period t + 2 by:
dAt+2 = (1 + r).(dzt+1 + (1 + r)dzt)
= (1 + r)(-Fzt
FTSt(.) (1 + r)
+ )dTSt (21)
(.) Fzt(.)
+1
which defines the user cost in the expression given by 5.
Maximum Likelihood Derivation of the latent variable Following Birnbaum (1967), we
follow a parametric approach and use the 3 parameter logistic to map the latent variable TSit to
the probability of a correct answer in question j, Pj(TSit) so that
1
Pj(TSit) = cj + (1 - cj)1 (22)
+ exp{-aj(TSit - bj)}
Following IRT terminology, the parameter bj measures the difficulty of the item (a location para-
meter), aj measures the discrimination of the item and cj measures the guessing probability. We
can then define the likelihood function as follows
L(TSit,a,b,c) = Pj(TSit,aj,bj,cj)xij{1 - Pj(TSit,aj,bj,cj)}1 -xij (23)
j i
where xij is the response by individual i to question j. Maximization of the likelihood function then
gives us the required normal equations. We use a Marginal Maximum Likelihood estimation (Bock
and Lieberman 1971) in combination with an Expected-Maximization (EM) algorithm (Bock and
Aitkin 1980). Under this scheme, a density function is assumed for the latent variable, f(TSit).
This is then "integrated-out" to arrive at consistent estimates of the item parameters (aj,bj,cj).
W C S I I T -S ? 38
As the last step the item parameters are then used to calculate TSit. Finally, Fisher's information
measure for the latent variable TSt, can be written as
J
I(TSit) = Ij(TSit) where
j=1
Ij(TSit) = {Pj(TSit)}2 (24)
Pj(TSit)(1 - Pj(TSit)
The standard error or our estimate TSit is now simply 1 and by the property of ML
jIj(TSit)
estimators TSit N(TSit, 1
j Ij(TSit)). Note that once the structural parameters {aj,bj, and cj}
are determined for the test in 2001, since the same test was administered again in 2002, we again
identify the change in the distribution of TSit without assuming a density function for 2002.
References
The word "processed" describes informally reproduced works that may not be commonly available
through library systems.
Ablo, Emmanuel, and Ritva Reinikka. 2000. "Do Budgets Really Matter: Evidence from Public
Spending on Education and Health in Uganda." Policy Research Working Paper 1926. World
Bank, Washington, D.C.
Alderman, Harold, Jere R. Behrman, Shahrukh Khan, David R. Ross, and Richard Sabot. 1997.
"The Income Gap in Cognitive Skills in Rural Pakistan." Economic Development and Cultural
Change 46(1): 97-123.
Angrist, Joshua D., and Victor Lavy. 1999. "Using Maimonides' Rule to Estimate the Effect of
Class Size on Student Achievement." Quarterly Journal of Economics 114(2): 533-575.
Banerjee, Abhijit. 2003. "Educational Policy and the Economics of the Family." Massachusetts
Institute of Technology. Processed.
Banerjee, Abhijit, Shawn Cole, Esther Duflo, and Leigh Linden. 2003. "Remedying Education:
Evidence from Two Randomized Experiments in India." Massachusetts Institute of Technology.
Processed.
Becker, Gary, and Nigel Tomes. 1986. "Human Capital and the Rise and Fall of Families." Journal
of Labor Economics 4(3):1-39.
Birnbaum, Allan. 1967. "Some Latent Trait Models and Their Use in Inferring an Examinee's
W C S I I T -S ? 39
Ability." In Frederic M. Lord and M. R. Novick, eds., Statistical Theories of Mental Test Score.
London: Addision-Wesley Publishing Company.
Bock, Richard D., and M. Aitkin. 1981. "Marginal Maximum Likelihood Estimation of Item Para-
meters: An Application of an EM Algorithm." Psychometrika 46: 443-459.
Bock, Richard D., and M. Lieberman. 1970. "Fitting a Response Model for n Dichotomously Score
Items." Psychometrika 35: 179-197.
Blundell, Richard W., and Richard Smith. 1986. "An Exogeneity Test for a Simultaneous Equation
Tobit Model with an Application to Labor Supply." Econometrica 54: 679-685.
Brown, Peter. 2003. "Parental Education & Child Learning: Human Capital Investments in Time
and Money." Michigan University. Processed.
Case, Anne, and Angus Deaton. 1999. "School inputs and educational outcomes in South Africa."
Quarterly Journal of Economics 114(3): F1047-F84.
Coleman, James. 1966. Equality of Educational Opportunity. United States Government Printing
Office: Washington, D.C.
Cox, Donald, and Emmanuel Jimemez. 1995. "Public Income Redistribution in the Philippines."
In Dominique Van de Walle and Kimberly Nead, eds., Public Spending and the Poor: Theory
and Evidence. Baltimore: Johns Hopkins University Press for the World Bank.
Das, Jishnu, Stefan Dercon, James Habyarimana, and Pramila. Krishnan. 2003. "Rules vs. Discre-
tion: Public and Private Funding in Zambian Education." Development Research Group, World
Bank, Washington, D.C. Processed.
Deaton, Angus, and John Muellbauer. 1980. "Economics and Consumer Behavior." Cambridge.
UK: Cambridge University Press.
Dercon, Stefan, and Pramila Krishnan. 2000. "In Sickness and in Health: Risk Sharing within
Households in Rural Ethiopia." Journal of Political Economy 108 (4): 688-728.
Filmer, Deon, and Lant Pritchett. 1999. "The Effect of Household Wealth on Educational Attain-
ment: Evidence from 35 Countries." Population and Development Review 25(1): 85-120.
Foster, Andrew D. 1995. "Prices, Credit Markets and Child Growth in Low-Income Rural Areas."
The Economic Journal 105(430): 551-570.
Glewwe, Paul. 2002. "Schools and Skills in Developing Countries: Education policies and socioeco-
nomic outcomes." The Journal of Economic Literature XL: 436-482.
W C S I I T -S ? 40
Glewwe, Paul, and Hanan Jacoby. 1994. "Student Achievement and Schooling Choice in Low-Income
Countries: Evidence from Ghana." Journal of Human Resources 29(3): 843-864.
Greene, William H. 2001. Econometric Analysis. Upper Saddle River, N.J. : Prentice Hall.
Hanushek, Eric A. 1971. "Teacher Characteristics and Gains in Student Achievement: Estimation
Using Micro Data." American Economic Review 60(2): F280-F8
Hanushek, Eric A. 1986. "The Economics of Schooling: Production and Efficiency in Public
Schools." Journal of Economic Literature 24(3): F1141-F77.
Hanushek, Eric A. 1997. "Assessing the Effects of School Resources on Student Achievement: An
Update." Educational Evaluation and Policy Analysis 19(2): 141-164.
Hanushek, Eric A. 2003. "The Failure of Input-Based Schooling Policies." Economic Journal 113
(February): FF64-F98.
Holmstrom, Bengt. 1982. "Moral hazard in teams." Bell Journal of Economics 13 (2, Autumn):
324-340..
Holmstrom, Bengt, and Paul Milgrom. 1991. "Multitask principal-agent analysis: Incentive con-
tracts, asset ownership and job design." Journal of Law, Economics and Organization 7: 24-51.
Hoxby, Caroline. 2000. "Peer Effects in the Classroom: Learning from Gender and Race Variation."
Working Paper 7867. National Bureau of Economics Research. Cambridge, Massachusetts.
Jacoby, Hanan G. 2002. "Is there an Intrahousehold `Flypaper Effect'? Evidence from a School
Feeding Programme." The Economic Journal 112:196-221
Jacoby, Hanan G., and Emmanuel Skoufias. 1997. "Risk, Financial Markets, and Human Capital
in a Developing Country." Review of Economic Studies 64(3, July): 311-335.
Johnson, S. R., and Gordon C. Rausser. 1971. "Effects of Misspecifications of Log-Linear Functions
when Sample Values are Zero or Negative." American Journal of Agricultural Economics 53(1):
120-124.
Kane, Thomas J., and Douglas O.,Staiger. 2001. "Improving school accountability measures." Work-
ing Paper 8156. National Bureau of Economics Research. Cambridge, Mass.
Kane, Thomas J., and Douglas O. Staiger. 2002. "The Promise and Pitfalls of Using Imprecise
School Accountability Measures." Journal of Economic Perspectives 16(4): 91-114.
W C S I I T -S ? 41
Krueger, Alan. 2003. "Economic Considerations and Class-Size." Economic Journal 113 (February):
FF34-F63
Lazear, Edward P. 2001. "Educational production." Quarterly Journal of Economics 116(3): 777-
803.
Miguel, Edward, and Michael Kremer. 2003. "Worms: Identifying Impacts on Education and Health
in the Presence of Treatment Externalities." Econometrica (forthcoming).
Moffit, Robert. 1992. "Incentive Effects of the U.S. Welfare System: A Review." Journal of Eco-
nomic Literature 30: 1-60.
Murphy, Kevin, and Robert Topel. 1985. "Estimation and Inference in Two-Step Econometric
Models." Journal of Business and Economic Statistics III: 370-379.
Ravallion, Martin, and Gaurav Datt. 1995. "Is Targeting Through a Work Requirement Efficient?
Some Evidence for Rural India." In Dominique Van de Walle and Kimberly Nead, eds., Public
Spending and the Poor: Theory and Evidence. Baltimore: Johns Hopkins University Press for
the World Bank..
Rosenzweig, Mark, and Kenneth Wolpin. 1994. "Parental and Public Transfers to Young Women
and Their Children." American Economic Review 84: 1195-1212.
Todd, Petra E., and Kenneth Wolpin. 2003. "On the Specification and Estimation of the Production
Function for Cognitive Achievement." Economic Journal 113(February): F3-33
UNESCO. Various years. "Edstats." [Retrieved on September 2, 2003, from
http://devdata.worldbank.org/edstats/about_data.asp].
Urquiola, Miguel. 2003. "Identifying Class-Size Effects in Developing Countries: Evidence from
Rural Schools in Bolivia." Policy Research Working Paper 2711. World Bank, Washington,
D.C.
Van de Walle, Dominique, and Kimberly Nead, eds. 1995. Public Spending and the Poor: Theory
and Evidence. Baltimore: Johns Hopkins University Press for the World Bank.
Wooldridge, J. M. 2001. Econometric Analysis of Cross Section and Panel Data. Cambridge. Mass.:
MIT Press.
World Bank. 2001. "Zambia-Public Expenditure Review: public expenditure, growth and poverty
a synthesis." Report 22543. World Bank, Washington, D.C.
W C S I I T -S ? 42
World Bank. Various years. World Development Indicators. Washington, D.C.
World Bank. 1998. Living Conditions Monitoring Survey II. World Bank, Washington, D.C. [free
access to a segment of indicators http://www.worldbank.org/data/dataquery.html].
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 43
Figure 2a: Net Enrollment Rates in Africa
Net Enrollment Rates
Comparision Among African Countries
0
10 MLW TUN MAU
set MOR
80 ZAM_00 BOT
ZAM
Ratne SEN
ml 60 MAD
orn
Et
Ne 40 MAL
NIG
20
5 6 7 8 9
Log GNP per capita (US $ 1997)
NER 1996 Boys Quadratic Fit
ZAM _00 refers to 2000 data for Zambia
Source: UNESCO (2002). This graph shows the relationship between net enrollment rates and log GNP per capita in
selected African countries, with a fitted quadratic. Zambia lies above the fitted line, suggesting that the enrollment in
Zambia is greater than what would be predicted through per capita income alone.
Figure 2b: Educational Attainment Curves: Zambia
Low Income
High Income Middle Income
1 1 1
0.8 0.8 0.8
0.6 0.6
0.6
0.4
0.4 0.4
0.2
0.2 0.2
0
0 0
6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 13 14
Age Age
UrbannAreas
Rural Areas
Urb Rural Areas Age
1 1 1992
Pr
op 1996
ort 0.8 0.8
ion
Cu
Cu 0.6
ed
0.6
rre
ntl rollnE
0.4
y 0.4
tly
En
roll 0.2
ed rrenu 0.2
C
% 0 0
6 77 8 99 10 11 12 13 14
10 12 13 14 6 7 8 9 10 11 12 13 14
Age Age
Source: Based onFilmerand Pritchett (1999)
Note: These graphs show educational attainment plots of the percentage of children enrolled against age. For all regions
and income categories, we find that strong evidence of delayed enrollment with enrollment increasing with age until
age 12 and then tapering off or declining. The graphs also show how educational outcomes have worsened between
1992 and 1996, with a decline in enrollment at every age group and for all socioeconomic levels.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 44
Figure 3: Household Expenditures and School Funding
Component Expenditure Shares
Household Sample Only
Low Funding Schools High Funding Schools
13.88%
20.29%
25.87%
4.965% 34.01%
4.097%
47.15%
49.75%
Share: Household Expenditure Share: Teachers Salary
Share: Discretionary Funds Share: Rule Funds
Source: ESD data
Source: ESD Sample (2002). The pi-chart shows how educational inputs are funded in schools that received high/low
anticipated funds. The shares are computed as the average of shares across schools.
1. Teachers Salary is computed as salary divided by the number of students in the teachers class. This is
computed for a sample of teachers who were interviewed if they were either currently teaching Grade VI or
Grade V students, or had taught Grade V students in the previous year. The particular sample was chosen to
ensure that teacher characteristics could be matched to students who were tested in both years. Salaries will
therefore be biased if there is selection of teachers into different grade levels.
2. Household expenditure is based on a one-year recall question of household educational expenditure for every
child on various items including textbooks, school supplies, and uniforms.
3. Discretionary funds are unanticipated by households.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 45
Figure 4: Test Characteristics English and Math
95% CI 5% CI
Ability (ml estimate)
4.19585
Sta
ndar
d
Erro
r
-4.03406
-1.19864 3.18181
Ability (ml estimate)
Test Standard Error: eng
95% CI 5% CI
Ability (ml estimate)
3.21027
Error
rdadan
St
-5.53114
-1.9284 2.6125
Test Standard Error: math
Ability (ml estimate)
Source: Author's calculations based on Examination Council of Zambia data. These two graphs show the standard
error of the estimation for the latent ability (knowledge) variable. The line in the middle plots ability against itself,
while the lines on the outsides plot the 95% confidence intervals where the standard errors are calculated from Fisher's
information for maximum likelihood estimates. Thus for instance, in both the Mathematics and the English
examination, the exam was "too hard" so that the confidence band at lower ability levels is larger than that at higher
ability levels. Further, the Mathematics exam is more "noisy" than the English exam with larger confidence bands,
especially at the lower levels.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 46
Figure 5: School Funding and Learning Gains
Funding and Cognitive Achievement
Uanticipated Funds Anticipated Funds
.55 .5 .6
.8
.5 ent .4 .5 ent
em em
ity .6 ity
hiev hiev
ens .45 .4
D Ac ens .3
D Ac
e e
am am
.4
ogrtsi .4 ognitiv
C ogrtsi .2 .3 ognitiv
C
H in H in
.2 ina ina
.35 G .1 .2 G
.3
0 0 .1
2 4 6 8 10 12 7 7.5 8 8.5 9 9.5
Log Discretionay Funding Log Anticipated Funding
Density English Density English
Math Math
Kernel Smoothing Source: ESD Study
Note: This figure shows the relationship between anticipated/unanticipated funds and gains in cognitive
achievement. For both graphs, the histogram of funding is plotted on the left axis and the relationship
between cognitive gains and funding is plotted on the right axis. In the case of unanticipated funds, a large
number of schools receive 0 funds and conditional on receipt, there is very high variance in the amount
received. For anticipated funds, the distribution mirrors the distribution of enrollment and is evenly
distributed on the sample range. The relationship between cognitive gains and unanticipated funding is
positive and significant for both Mathematics and English and not significantly different from zero for
anticipated funding.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 47
Table 1a: Descriptive Statistics: School Enrollment and Staffing
`Remote' Schools
Category Variable Urban Rural (HH Sample)
School size (number of pupils) 1439.5 553.4 399.1
(600.98) (408.19) (224.3)
Pupil-teacher ratio 42.23 63.62 66.1
Basic Indicators
(24.16) (55.14) (40.91)
Number of pupils per classroom in good 103.4 96.7 101.4
condition (58.41) (46.40) (59.66)
Repeating the same grade (%) 4.9 9.42 9.1
(4.29) (6.66) (5.33)
Dropouts as ratio of current enrolment (%) 1.67 4.49 4.6
Outcome (2.57) (5.05) (5.28)
Indicators Pass-rate in 1999 Grade VII examination 40.5 44.2 42.6
(Males) 1 (22.68) (27.07) (27.47)
Pass-rate in 1999 Grade VII examination 38.6 40.7 38.4
(Females) 1 (24.32) (30.73) (30.24)
Average value of wealth-index of households 0.57 -0.56 -0.73
Pupil with children in the school2 (0.61) (.56) (0.43)
Characteristics Percentage of children who are orphans 4.7 4.79 4.9
(3.6) (4.13) (3.5)
Percentage of schools who received anticipated 89.4 89.3 85.7
funds at time of survey (3.7) (2.9) (5.9)
Percentage of schools who received 23.1 24.8 14.2
unanticipated funds (5.2) (4.0) (5.9)
Anticipated amount received (log Kwacha per 7.66 8.67 8.93
Cash-Grant pupil) (0.41) (0.60) (0.54)
Characteristics Unanticipated amount (log Kwacha per pupil) 7.22 7.93 9.84
(2.31) (2.58) (2.77)
Anticipated amount (Kwacha per pupil) 2372.8 6931.4 8676.1
(1535.4) (3969.7) (4852.1)
Unanticipated amounts (Kwacha per pupil) 17121.2 54559.0 120630.9
(36822.5) (14424.7) (158621.6)
Note: ESD Sample. Standard-Deviations in brackets.
1. Pass-Rates are for the Grade VII examination administered to all students by the Examination Council of
Zambia.
2. The wealth index is based on a weighted aggregation of household assets similar to a principal components
analysis, but with weights optimally derived to minimize classification errors. Details are in Das and others
(2003, Appendix 1).
3. For anticipated funding, 2.25% of schools received more than the allotted amount, to a maximum of $800.
Unconditional logs for cash-grants are calculated by generating lnX=ln(X+b) if X=0, where b is determined optimally.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 48
Table 1b: Who Received Discretionary Funds?
Received
No discretionary discretionary Significant
Category Variable funds received funds difference?
School Asset Index -0.03 .09 NO
(0.79) (.77)
Total Enrollment in School 862 981 NO
School
characteristics Distance to District Office (% 48.1% 60.4% NO
within 5 KM)
Distance to Provincial Office (% 20.7% 18.6% NO
within 5 KM)
English Scores in 2001 -0.045 -0.05 NO
Performance in
(.48) (.52)
2001
Mathematics Scores in 2001 -0.012 -.069 NO
examinations
(0.47) (0.42)
Source: ESD Sample. Standard-deviations in brackets. This table checks to see whether schools that
received discretionary funds were "different" along observable dimensions from those that did not. We find
no difference in either school characteristics or test-scores in 2001 between schools that received such
funds versus those that did not. The wealth index is based on a weighted aggregation of household assets
similar to a principal components analysis, but with weights optimally derived to minimize classification
errors. Details are in Das and others (2003, Appendix 1). Test scores are the maximum likelihood estimates
of the latent variable as described in Appendix 1.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 49
Table 2: Cash-Grants and School Characteristics in High and Low Anticipated Grant
Villages
Low grant High grant
Categories Variable schools schools Significant?
% School going Children 94.6% 98.5% YES
Matching
attending surveyed
Success
school
Outcome Average Per-Child 19,576 10,794 YES
Variable Expenditure
Explanatory Average Anticipated 4,648 10,999 YES
Variable Grant
(Anticipated
Grants= Total Enrollment in 572.15 232.77 YES
K/enrollment) School
Schools Asset Index -.75 -.93 YES
Households Asset index 0.018 -0.054 NO
% Children with mothers 72.5 79.8 YES
Observable
in household
Components of
Households % With fathers in 65.0 72.5 YES
household
% whose mothers can 87.7 90.1 NO
read
% whose fathers can read 72.5 79.8 NO
% who say Head- 0.79 0.84 NO
Teacher is good
Observable % who say that Teacher 52.9 74.2 YES
components of visited household
schools
%For whom school is 50.6 70.7 YES
within 30 minutes
Village % Enrollment in Village .78 .80 NO
Enrollment
Note:
1. Two private schools are excluded from the sample.
2. All tests of percentages are probability tests, all tests of continuous variables are t-tests.
3. Significant differences are at the 1% confidence level.
4. Schools Asset index is the average value of the asset index for children in the school matched to the
household.
This table compares observable characteristics of households and schools for schools that received low and high
anticipated funds, respectively. We find that for a number of characterisitics, there is no diffference between the two
categories. For variables where differences are significant (% with fathers in households or distance to school) the
relationship with enrollment is the opposite of what one might expect, suggesting that there is no systematic correlation
between enrollment at the school level, which determines anticipated funding, and household characteristics.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 50
Table 3: Relationship between Household and School Funding
(1) (2) (3) (4) (5) (6)
Base Hypothesis Test of Weak
Regression: 2: Tobit Test of Exogeneity:
Base Tobit with Hypothesis with Weak Tobit with
Regression: Random 2: Tobit Random Exogeneity: random
Tobit Effects Specification Effects Tobit effects
Log -0.417 -0.414
Anticipated [0.147]** [0.180]*
Funds
(Received at
time of
survey)
Log -0.567 -0.572 -0.759 -0.763
Anticipated [0.142]** [0.160]** [0.298]* [0.332]*
Funds
(Legislated)
Log -0.177 -0.175 -0.079 -0.079 -0.077 -0.077
Unanticipated [0.064]** [0.078]* [0.058] [0.065] [0.056] [0.062]
Funds
Residual From 0.235 0.232
"Selection" [0.343] [0.383]
Equation
Gender of -0.044 -0.045 -0.049 -0.048 -0.050 -0.050
Child [0.124] [0.124] [0.124] [0.123] [0.123] [0.123]
Age of Child 0.293 0.294 0.293 0.293 0.293 0.293
[0.021]** [0.021]** [0.021]** [0.021]** [0.021]** [0.021]**
Household 0.723 0.720 0.726 0.723 0.726 0.724
Wealth Index [0.080]** [0.080]** [0.080]** [0.080]** [0.080]** [0.080]**
Mean Village -0.015 -0.028 0.007 -0.003 -0.015 -0.021
Wealth [0.204] [0.242] [0.203] [0.225] [0.188] [0.206]
Constant 9.693 9.673 10.214 10.264 11.834 11.860
[1.589]** [1.940]** [1.320]** [1.486]** [2.722]** [3.034]**
Observations 1410 1410 1410 1410 1410 1410
Note: The regressions in this table show the effect of anticipated and unanticipated funding on children's
educational expenditures (the dependant variable in all regressions). Estimates marked ** are significant at
1%, * denotes significance at 5% and standard errors are presented in [brackets]. Anticipated funding is
either funding that had been received by the school at the time of the survey or funding that had not yet
arrived. Estimates from the former are presented in columns (1) and (2) and the latter in columns (3) and
(4). Columns (5) and (6) present the test of weak exogeneity (Blundell and Smith (1986)) where the
residual from the first stage regression determining log anticipated funds is included as an additional
regressor. All regressions control for the mean wealth of students in the school, province dummies and a
rural dummy. The censoring is at 0 for the Tobit and the random effects Tobit specifications account for the
clustering of errors at the level of the village. Marginal effects (conditional on being uncensored) and the
probability of censoring are presented in Table 4. Further (a) for all regressions, K100 is added to zero
values of discretionary funding to allow logs. The minimum funding is K900 conditional on receipt; (b)
two private schools are excluded from the analysis; (c) K50 is added to enrolled children with zero
educational expenditures who form 4.96% of the sample; (d) the wealth index is based on optimal
maximum likelihood weights (see Das and others (2003) for details). Results are robust to alternative
indices (for instance an unweighted raw sum).
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 51
Table 4: Marginal Effects and Probability of Censoring
(1) (2) (3) (4)
Base Regression: Tobit Base Regression: Tobit Hypothesis 2: Tobit Hypothesis 2: Tobit with
Specification with Random Effects Specification Random Effects
Marginal Marginal Marginal Marginal
Effect at Prob. Effect at Prob. Effect at Prob. Effect at Prob.
Mean (Uncensored) Mean (Uncensored) Mean (Uncensored) Mean (Uncensored)
Log -0.34 -0.02 -0.37 -0.02
Received [0.119]** [0.006]** [0.164]* [.012]*
(rule-based)
Funding
Log -0.46 -0.024 -0.52 -0.038
Anticipated [0.11]** [0.006]** [0.146]** [0.010]**
Funding
Log -0.15 -0.007 -0.16 -0.011 -0.06 -0.003 -0.07 -0.005
Discretionary [0.052]** [0.002]** [0.071]* [0.005]* [.047] [0.002] [0.059] [0.004]
Funding
Note: This table shows the marginal effects at mean values of the regressors based on the coefficients from Table 3.
In all cases, the significance of estimated coefficients is robust to clustering at the village level. The estimated
elasticity increases substantially when we use anticipated instead of received funding, confirming that households
make their educational expenditure decisions before such funding is actually received.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 52
Table 5: Funding and Test Scores
(1) (2) (3) (4) (5) (6) (7) (8)
Full
Full Control Full Control Full
Basic Basic Control Full Control Set with Set with Control Full Control
Regression: Regression: Set: Set: dummy: dummy: Set: Set:
English Mathematics English Mathematics English Mathematics English Mathematics
Log 0.076 0.027 0.072 0.028 0.055 0.032
Unanticipated [0.030]* [0.022] [0.030]* [0.020] [0.028] [0.023]
Funds
Log -0.006 -0.001 -0.005 -0.001 -0.004 -0.002
Unanticipated [0.002]* [0.002] [0.003]* [0.002] [0.002] [0.002]
Funds
(Squared)
Did School 0.125 0.095
Receive [0.055]* [0.046]*
Unanticipated
Funds?
Log -0.025 0.007 -0.007 0.023
Anticipated [0.020] [0.013] [0.017] [0.015]
Funds
(Received)
Log -0.102 0.033
Anticipated [0.044]* [0.044]
Funds
(Legislated)
Constant 0.437 0.256 0.342 0.159 0.383 0.305 1.124 0.076
[0.176]* [0.105]* [0.155]* [0.133] [0.077]** [0.076]** [0.347]** [0.367]
Observations 177 177 176 176 176 176 176 176
R-squared 0.05 0.02 0.15 0.04 0.13 0.03 0.17 0.03
Note: This table shows OLS estimates for the relationship between unanticipated/anticipated funding and gains in
cognitive achievement for English and Mathematics. For all regressions standard errors are in [brackets], * denotes
significance at 5% and ** at 10% levels of significance. Columns (1) and (2) report coefficients without any further
controls, Columns (3) and (4) include whether the school is rural, whether the head-teacher changed in the previous
year, whether the PTA changed in the previous year, differences in PTA fees between 2002 and 2001 and a dummy for
private schools. Columns (5) and (6) report results from using a dummy for whether the school received unanticipated
funds or not with the same set of controls. Columns (7) and (8) present estimation results when we use the legislated
anticipated funds rather than the anticipated funds received at the time of the survey. For the sample, 3 schools with
unlikely changes (>2 or <-2 standard deviations) are dropped from sample and the optimal b=3.73 is added on to
unanticipated funds to compute the log. All regressions are clustered at the district level. The results from the Wald
test for equality of marginal impacts for anticipated and unanticipated funds at different points of the sample range are
presented in the text. The results show that we can reject the null hypothesis for English at the 5% level for the sample
mean, median, 25th and 75th percentile. We cannot reject the null at these values of the sample for Mathematics.
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 53
Table 6: Predicting Discretionary Funds (First Stage)
(1) (2) (3)
Amount of unanticipated Amount of unanticipated Probability of receiving
funds received conditional funds (squared) received unanticipated funds
on receipt conditional on receipt (Marginal Coefficients)
Grade VII Male Pass Rate -0.021 -0.366 0.005
(2 year lag) [0.018] [0.306] [0.007]**
Grade VII Female Pass 0.002 0.010 -0.004
Rate (2 year lag) [0.018] [0.307] [0.007]*
Average school wealth (1 0.006 -0.196 0.111
year lag) [0.628] [10.741] [0.203]*
Is this a PAGE school? -1.643 -28.332 0.229
[0.859] [14.697] [0.265]**
Log of district receipts 1.010 15.740 0..086
from external donors [0.467]* [7.991] [0.166]
(current)
Log of province receipts 0.148 2.677 0.037
from external donors [0.497] [8.505] [0.175]
(current)
Contested district (first) 4.554 89.129 -0.137
[1.819]* [31.137]** [0.595]
Contested district (second) 1.895 42.538 0.034
[1.564] [26.771] [0.798]
Constant 1.447 -31.409 -0.842
[4.675] [80.014] [1.622]**
Observations 38 38 164
R-squared 0.50 0.52 LR (chi2) = 28.87
Note: This regression shows the first stage of the IV strategy using a hurdle model. For all regressions
standard errors are in [brackets], * denotes significance at 5% and ** at 10% levels of significance.
Columns (1) and (2) show the estimation results conditional on receipts for unanticipated funding and
unanticipated funding squared, while Column (3) estimates the probability of receiving such funding. The
log of district receipts from external sources was computed through a questionnaire administered to district
authorities and that for provincial receipts through surveys administered at the province level. The two
dummies for politically active districts is based on interviews and newspaper articles in the run-up to the
election. The predicted value for the second stage is calculated as E(y)=E(y|receipt) x Prob(receipt).
WHEN CAN SCHOOL INPUTS IMPROVE TEST-SCORES? 54
Table 7: Learning and Funding (IV Results)
(1) (2) (3) (4) (5) (6)
Hurdle IV: Hurdle IV:
Expected Expected Comparison: Comparison
Hurdle IV: Hurdle Rule Funds Rule Funds (OLS, (OLS,
English IV: Math (English) (Math) English) Math)
Hurdle 0.158 0.090 0.128 0.101 0.082 0.039
instrumented log [0.057]* [0.035]* [0.052]* [0.033]* [0.031]* [0.020]
unanticipated
grants
Hurdle -0.015 -0.008 -0.013 -0.009 -0.006 -0.002
Instrumented log [0.007]* [0.003]* [0.006]* [0.003]** [0.003]* [0.002]
unanticipated
grants squared
Log of -0.024 0.008 -0.110 0.038 -0.012 0.024
anticipated [0.021] [0.021] [0.045]* [0.047] [0.020] [0.019]
grants
Observations 164 164 164 164 164 164
R-squared 0.16 0.03 0.18 0.04 0.15 0.05
Note: This table shows second stage IV estimates for the relationship between unanticipated/anticipated
funding and gains in cognitive achievement for English and Mathematics. For all estimates, standard errors
are reported in [brackets], * denotes significance at 5% and ** at 10% levels of significance. Columns (1)
and (2) report coefficients include controls for whether the school is rural, whether the head-teacher
changed in the previous year, whether the PTA changed in the previous year, differences in PTA fees
between 2002 and 2001 and a dummy for private schools. Columns (3) and (4) report results using
legislated anticipated funds rather than the anticipated funds received at the time of the survey. Columns (5)
and (6) report results for comparison with OLS results from the same sample. For the sample, 3 schools
with unlikely changes (>2 or <-2 standard deviations) are dropped and an additional 13 schools are dropped
due to lack of data at the district level. All regressions are clustered at the district level. The results from the
Wald test for equality of marginal impacts for anticipated and unanticipated funds at different points of the
sample range are presented in the text. The results show that we can reject the null hypothesis for English at
the 1% level for the sample mean, median, and 25th percentile and at the 5% level for the 75th and 90th
percentile. For Mathematics we can reject the null at the 5% level for the 25th percentile and the 10% level
for the sample mean and median. We cannot reject the null for the 75th and 90th percentiles.