7. Education Quality

KAT.TAL.322 Advanced Course in Labour Economics

Nurfatima Jandarova

September 15, 2025

Education production function

Measures

Output

Years of schooling, standardised test scores, noncognitive skills

Student inputs

Effort, patience, genetics, parental characteristics, family income, family size

School inputs

Teacher characteristics, class sizes, teacher-student ratio, school expenditures, school facilities

Non-school inputs

Peers, local economic conditions, national curricula, regulations, certification rules

As already mentioned in this and previous lectures, the output of human capital production function can have multiple dimensions. The simplest is, of course, to have some simple measure of years of education or highest qualification categories. However, in some studies you might want to consider more granular measures such as test scores, high-school grades or bundles of skills (cognitive/non-cognitive).

Student inputs would include all characteristics that a given student possesses. This may include characteristics of students themselves - like their effort, patience, genetics - as well as characteristics of their families - such as parents’ income, their education, characteristics of their siblings. Also note that may include past outcomes of the students. Remember how in the previous lecture we said that learning new skills might be easier when you already have high levels of human capital. It is the same idea.

School inputs include every resource that school can provide its children. It may be quality of their teachers, teacher-student ratios, quality of facilities, availability of specific programs, etc

Finally, non-school inputs can include all other relevant factors. For example, it can be composition of peers in the classroom (whether students are exposed to other students from similar or different backgrounds), local economic conditions (for example, if there is a recession while the child is actively developing her skills), higher level regulations (for example, any changes in school curricula, rules about attendance, etc) or certification rules for various occupations that shape a specific education track.

As you can see, you can think of many relevant inputs and many relevant outputs. So, the modelling of production function will depend on the objectives of a given research question.

Education production function

Todd and Wolpin (2003): Value-added specification

Typical empirical estimation assumes linear separability and :

Additional assumptions implied:

Past input effects decay at the same rate
Shocks are serially correlated with persistence

Over time quality of data improved and longitudinal data became more prevalent. However, having sufficiently long follow-up period can be a struggle even nowadays. For example, in UK there are several cohort studies that were initiated since 1957 that measured various children and family outcomes at specific ages of children (at birth, age 5, age 10, etc). So, in the environment where we have few observations per child, we can look on value-added version of the model. That is, imagine we have a snapshot of children at age 5 and age 10. We can measure the change in outcomes as well as change in inputs between these two periods to try to infer model parameters.

To see it in the framework of Todd and Wolpin (2003), assume for simplicity that we have two time periods per child and . Now, we say that output depends on contemporaneous inputs and as well as past output . Notice a slight change in notation: the past output is allowed to depend on the entire history of inputs up to , even though we still may not observe them! This is relevant for many datasets that may not record history of inputs (even nowadays, some datasets only record family income once at a certain age), but do have repeated measurements of child outcomes (for example, test scores or school grades).

Assume a very simple linear production function with full histories

Furthermore, we assume that functional form of is same across all ages. Then, same equation at (already multiplied by ) is

The difference (or value added) is

Therefore, it is clear that for this expression to be equivalent to the above regression equation, the following should hold

In this case, all terms multiplied by and will disappear and we will get exactly the regression equation on the slide.

In addition, it also highlights that the regression error term . So, consistent estimation requires that is serially correlated with persistence exactly equal to . In that case is white noise and uncorrelated with .

If any of these assumptions don’t hold, then estimates will be biased!

Education production function

Todd and Wolpin (2003): Cumulative specification

Still assume linear separability:

Estimation strategies:

Within-child: for ages and
Within-family: for siblings and

Each with their own caveats

Modern datasets may give us access to long histories of both inputs and outputs. So, we might think that we are finally able to estimate the parameters of the production function in a flexible way.

If there are many family and school-level characteristics in and , then estimating a very flexible specification with many interaction terms might be hard to interpret. Therefore, we still have to make some simplifying assumptions. In particular, that the production is linearly separable, perhaps with only few interaction terms.

The specification still includes the potentially unobserved initial skill endowments . But because we now have a rich longitudinal data, we could use different methods to difference it out.

First approach is to use multiple observations per child - a within-child design.

However, if you explicitly write down the expression for , you will see that unless the unobserved heterogeneity term will not disappear! This assumption may in fact be violated. For example, we can imagine that learning of a 2-year-old child may depend a lot more strongly on her initial skill endowment than her academic performance at 17 years old.
Also notice that the error term in general depends on the entire history of . This means that the estimation implicitly assumes that input choices at any age are not correlated with past, contemporaneous or future outcomes! This assumption too can be easily violated if parents or schools adjust their investments depending on past performance of students. For example, if a student does very well in one year, the school might offer additional learning materials to her next year.

Second approach is within-family design that exploits variation between siblings. However, this approach also makes some pretty strong assumptions about the production process.

If you explicitly write down the expression for , you will notice that it will depend on . This term will only be equal to zero if and only if we assume that only family characteristics matter for the production process and children’s own unobserved characteristics do not matter. This is very hard-to-defend assumption!
In addition, the error term will be and all the regressors are implicitly assumed to be uncorrelated with it. This means that outcomes of one sibling cannot affect directly investments made to another sibling. This assumption can also be contested.

All in all, estimating human capital production function parameters is really hard in observational data!

Education production function

Non-experimental estimations

Require strong assumptions
- Some can be relaxed
Require rich data
Endogenous allocation of resources

Quasi-experimental estimations

May not recover structural parameters
Ignore general equilibrium
Issues with scaling List (2022)

Summarising the discussion above, estimation of production function parameters from observational data can impose very strong assumptions on the underlying model. These assumptions most likely are violated in real data. Especially, because investments from parents and schools can endogenously react to each other as well as children’s own characteristics and past results.

Typically, to solve endogeneity issues we turn to quasi-experimental designs. However, Todd and Wolpin (2003) warn that these results too may not be sufficient to uncover the structural parameters of the production function. To make it explicit, these designs do help identify causal effects of any policies that generate the exogenous variation. Therefore, they are invaluable for policy discussions and evaluation. However, the policy changes to school inputs, for example, can also induce parents to change their investments. Therefore, causal effect of policy may not necessarily be the same as causal effect of certain inputs.

In addition to this, even if we find strong and robust causal effect in some experimental or quasi-experimental design, implementing them at larger scale might not produce similar results (List 2022). This argument is applicable not only to empirical literature on education quality, but to many other fields too. Suppose that you design an experiment and send the best of the best teacher to a remote school in Finland and find strong and positive effects on achievement of students. You conclude from this that teacher quality plays a tremendous role in education production function. But if you want to now scale it up to the entire country, where do you find thousands of teachers of exactly same quality? Even if training of would-be teachers improves as a result, there necessarily will be a distribution of teachers with different qualities. So, the impact on students may be smaller than the one identified in the causal paper.

Source: Angrist and Lavy 1999
	Grade 4	Grade 5
Class size	-0.150	0.023	-0.582	-0.443
	(0.128)	(0.160)	(0.181)	(0.236)
Mean score	72.5	68.7	74.5	67.0
SD score	7.8	9.1	8.2	10.2
Obs.	415	415	471	471

Source: Angrist et al. 2019
	Grade 5
Class size	-0.006	-0.062
	(0.066)	(0.088)
Mean score	72.1	68.1
SD score	17.4	20.6
Obs.	225 108	226 832

Source: Table V (Krueger 1999)
	Test scores
SMALL	5.370	6.370	5.260	5.240
	(1.190)	(1.110)	(1.100)	(1.040)

Source: Table V (Chetty et al. 2011)
	Test score, %	College by age 27, %	College quality, $	Wage earnings, $
SMALL	4.760	1.570	109.000	-124.000
	(0.990)	(1.070)	(92.600)	(336.000)
Avg dep var	48.67	45.5	27 115	15 912
Obs.	9 939	10 992	10 992	10 992

Source: Table VIII (Chetty et al. 2011)
	Test score, %	College by age 27, %	College quality, $	Wage earnings, $
QUAL	0.662	0.108	9.328	50.610
	(0.024)	(0.053)	(4.573)	(17.450)
Obs.	9 939	10 959	10 959	10 959

Source: Tables 4 and 5 (Fryer 2013)
	Elementary	Middle	High
English	-0.010 (0.015)	-0.026 (0.010)	-0.003 (0.043)
Math	-0.014 (0.018)	-0.040 (0.016)	-0.018 (0.029)

Source: Table VI (Abdulkadiroğlu, Angrist, and Pathak 2014)
	Parametric	Nonparametric
Attended any college	0.010	0.031
	(0.032)	(0.019)
Attended 4-year college	0.003	0.013
	(0.041)	(0.026)
Attended competitive college	-0.011	-0.004
	(0.051)	(0.029)
Attended highly competitive college	-0.009	-0.014
	(0.032)	(0.017)

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7. Education Quality KAT.TAL.322 Advanced Course in Labour Economics Nurfatima Jandarova September 15, 2025

7. Education Quality
Today
Education quality
Stylised facts
Education quantity vs quality
Education production function
Education production function
Education production function
Education production function
Education production function
Education production function
Education production function
Early estimates of school inputs (prior to 1995)
Education production function
(Quasi-)Experimental estimations
Productivity of student inputs
Student inputs: nature vs nurture
Student inputs: nature vs nurture
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of non-school inputs
Productivity of school inputs
Productivity of school inputs
Productivity of non-school inputs
Productivity of non-school inputs
Productivity of non-school inputs
Summary
References

	Grade 4		Grade 5
	Reading	Math	Reading	Math
Class size	-0.150	0.023	-0.582	-0.443
	(0.128)	(0.160)	(0.181)	(0.236)
Mean score	72.5	68.7	74.5	67.0
SD score	7.8	9.1	8.2	10.2
Obs.	415	415	471	471

	Test scores
	Kindergarten	Grade 1	Grade 2	Grade 3
SMALL	5.370	6.370	5.260	5.240
	(1.190)	(1.110)	(1.100)	(1.040)