94 H. Bonesrønning Economics of Education Review 18 1999 89–105
grades, the equation would be misspecified. Eq. 14 should instead be thought of as a definitional relationship
that provides a measure of the divergence between exam results and teachers’ grades for each school.
To deal with the problem that students sort themselves into the specialization subjects, we apply the standard
Heckman 1979 procedure. That is, the probability that a student chooses mathematics is estimated from a probit
equation. The inverse Mill’s ratio is calculated from the probit equation and thereafter included in the education
production function to correct for selectivity bias.
4. Data and Results
The analysis is based on a sample of 1092 academic students from 15 upper secondary schools in one
Norwegian county. All students taking their final exam in 1990 are included. Table 1 presents the sample means
and the standard deviations of the variables used in this study. Appendix A provides variable definitions. The
teachers’ grades as well as the grades from the nationally administred exams are given on a 0–6 scale, where 0 and
1 represent failure and 6 is the best result. The table reports the teachers’ grades and the exam results for
Norwegian language and the specialization subjects mathematics, social sciences or foreign language. As
can be seen, the teachers’ grades are set well above the exam results, indicating that the latent variable G
3
is larger than 3, or alternatively, that the latent variable g
s
is larger than 1. The students’ initial level of knowledge
Table 1 Data means and standard deviations
Variable Mean
Std.dev. Min
Max Exam results Norwegian
3.03 0.91
6 language
Teacher gradingNorwegian
3.51 0.94
1 6
language Exam
resultsspecialization 2.81
1.26 6
subjects Teacher
gradingspecialization 3.51
1.19 5.5
subjects Aggregate gradeslower
42.26 3.18
32 53
secondary Teachers’ education
46.00 13.30
17 65
Laudable teachers 47.00
10.91 22
59 Teachers’ age
42.27 3.11
36 48
students 318.8
123 84
480 departments
3.66 1.33
2 6
Teachers per 100 7.37
1.03 6.30
9.80 students
is measured by an aggregate of their grades from the lower secondary school. 11 subjects are included, and
the scale is 0–5. The variation in the average initial level of knowledge across schools basically reflects that some
schools are in excess demand while others are in excess supply.
The teachers are characterized by experience, edu- cation and laudability. The two former characteristics are
the conventional indicators for teacher quality in edu- cation production function studies, while teacher lauda-
bility has not been utilized in any analysis we are aware of. The teachers’ teaching experience school averages
varies between 10 and 20 years approximately. The tea- chers’ education is measured by the fraction of teachers
having a master’s degree, i.e. we have essentially separ- ated between teachers with 4 and 6 years of education
beyond secondary school. As can be seen, 47 percent of the teachers have laudable results from the universities
and the variation between the schools is substantial.
Three school characteristics are reported; the number of teachers per 100 students, the number of students in
the academic department and the total number of depart- ments within the school. The separation between the
number of students within the academic department and the number of departments carries over from Bonesrøn-
ning 1996, and reflects the upper secondary school organization with academic and vocational departments
side by side.
No socioeconomic background variables have been available for this study. Omitted variable bias is thus a
potential problem. All earlier studies show that the stu-
95 H. Bonesrønning Economics of Education Review 18 1999 89–105
dents’ socioeconomic background simultanously influ- ence the students’ initial level of knowledge and their
final achievement. We therefore expect the coefficient for the initial level of knowledge to be biased upwards.
We do not expect that the other independent variables will be seriously affected by the omission of socioecon-
omic background variables. This expectation is sup- ported by Hanushek and Taylor 1990, who provide evi-
dence that serious misspecification problems are likely to be avoided when the students’ initial level of knowl-
edge is included.
4.1. Measuring G
3
and g
s
We have estimated Eq. 16 for the Norwegian langu- age, the specialization subjects and mathematics. The
results are reported in Tables 2 and 3. As can be seen from Table 2, the estimated G
3 j
coefficients are above 3 without exception, and all G
3 j
are sharply determined. The t-statistics are in the ranges above 16.0, which indi-
cate that the coefficients are significantly different from zero at the 0.5 percent level. However, the t-statistics
Table 2 The grading parameter G
3
.t-statistics in parentheses School
Norwegian language Specialization subjects
Mathematics High achievers
1 3.26
4.08 4.60
4.38 40.3
26.7 16.0
25.5 2
3.38 3.44
3.04 3.42
57.8 50.7
37.4 39.8
3 3.40
3.64 3.44
3.91 74.2
51.3 31.2
36.7 4
3.70 3.40
3.37 3.52
40.6 46.8
11.5 22.1
5 3.55
3.37 3.35
3.21 54.2
50.2 33.2
28.2 6
3.20 3.76
3.73 3.75
46.3 40.6
22.5 30.8
7 4.10
3.74 3.74
4.00 51.9
35.6 30.7
27.1 8
3.33 3.85
3.38 3.72
51.2 45.8
20.4 27.9
9 4.01
3.42 3.09
3.41 36.0
21.9 18.7
17.3 10
3.90 3.93
4.29 3.94
34.9 36.4
23.1 28.6
11 3.18
3.52 3.57
3.68 64.9
64.5 29.8
47.4 12
3.24 3.66
3.57 3.73
36.8 56.0
27.1 38.8
13 3.97
4.09 4.14
4.14 40.1
41.9 21.6
31.8 14
3.40 3.60
3.15 3.42
50.7 46.9
25.6 38.4
15 3.78
3.72 3.81
3.63 55.9
54.1 29.6
32.3
give irrelevant information in this case. We therefore tested the equality of school means. The null hypothesis
of the equality of school means is rejected for all sub- jects. Note that across subjects, mathematics shows the
largest variation in G
3 j
. A student who gets the grade 3 on the external mathematics exam has the expected tea-
chers’ grades 4.6 and 3.1 in the schools with the easiest and hardest grading respectively. This is somewhat sur-
prising, since we intuitively think of mathematics as rather objective, with little leeway for teachers to evalu-
ate equal achievements differently.
The estimated g
s j
are below 1 with a few exceptions. The partial correlations between G
3 j
and g
s j
are negative but rather small for all subjects, implying that the teach-
ers’ grades vary less with achievement when the teachers practice easy grading. The partial correlation coefficient
is very small in absolute value 0.03 for mathematics.
The differences in teachers’ grading practices docu- mented in Tables 2 and 3 are of such magnitudes that
they clearly affect the students’ probabilities of getting admission to the universities. Our concerns are whether
these differences are systematically associated with
96 H. Bonesrønning Economics of Education Review 18 1999 89–105
Table 3 The grading parameter g
s
t-statistics in parentheses School
Norwegian language Specialization subjects
Mathematics High achievers
1 0.84
0.87 1.03
0.89 8.3
8.3 6.2
7.9 2
0.71 0.73
0.80 0.78
10.8 14.3
17.1 13.8
3 0.79
0.70 0.76
0.54 17.3
13.4 9.5
6.4 4
0.70 0.82
0.73 0.77
7.3 14.6
3.7 7.1
5 0.81
0.80 0.91
0.94 10.7
15.9 12.3
10.8 6
0.81 0.98
0.86 0.95
10.7 13.1
4.7 7.0
7 0.58
0.71 0.81
0.68 5.9
7.2 9.2
4.3 8
0.96 0.84
1.00 1.05
13.0 11.5
7.6 8.3
9 0.80
0.79 0.77
0.68 5.9
6.8 6.8
4.3 10
0.75 0.68
0.59 0.63
5.8 8.3
3.1 5.3
11 0.78
0.78 0.91
0.72 14.8
16.0 8.2
10.3 12
0.77 0.81
0.94 0.81
8.0 14.2
8.7 9.3
13 0.62
0.62 0.59
0.56 6.0
7.3 3.4
4.8 14
0.69 0.76
0.73 0.85
9.0 12.9
7.5 12.3
15 0.68
0.83 0.84
0.91 8.7
16.1 8.6
10.6
teacher and student characteristics, and further, how the grading practices are related to student achievement. We
start the investigation of these questions by estimating the standard education production function. This is the
correct reduced form when grading is endogenous, and it serves as our benchmark model.
4.2. The Standard Education Production Function The estimation of Eq. 14 for the Norwegian langu-
age, for the specialization subjects, for mathematics and for high achievers in the specialization subjects are
reported in Table 4. Table 4 confirms three important previous findings
from education production function studies. First, the best predictor of individual students’ achievement is
their initial level of knowledge, and the coefficient is very sharply determined. Note however that since no
socioeconomic variables are included, the estimated coefficient for the initial level of knowledge is most
likely biased upwards. Second, the number of teachers per student does not seem to influence student achieve-
ment. We have been careful in specifying the teacher input exclusive all extra classroom activities, but have
been restricted to treat the teacher input as a school spe- cific variable. The latter is a problem if for instance low
achievers are assigned to smaller classes, but the student allocation to different class sizes is very much a random
process in the academic departments included in this study. It remains of course that if measurement error
exists, the estimated coefficient of the teacher-student ratio will be biased towards zero. Our third finding con-
sistent with earlier studies is that more educated teachers will not improve upon student achievement. One
interpretation of the results is that simple accumulation of credits with no regard for the subjects being taught
has no positive effect on student achievement. However, the significant negative sign of some of the estimated
teachers’ education coefficients remains a puzzle.
The importance of teachers’ experience is not well established by previous studies. Hanushek 1986 has
surveyed 109 education production function studies including teachers’ experience as an independent vari-
able. 33 of the estimates show a statistically significant
97 H. Bonesrønning Economics of Education Review 18 1999 89–105
Table 4 The standard education production functiont-statistics in parentheses
Variable Norwegian language
Specialization subjects Mathematics
High achievers Lower secondary grades
0.17 0.19
0.21 0.19
24.6 17.8
8.5 7.2
Student’s sex 0.06
2 0.11
2 0.31
2 0.08
1.4 1.6
2.2 0.8
Teachers’ education 2
0.08 2
0.01 2
0.004 2
0.01 3.0
1.8 0.5
1.0 Laudable teachers
0.01 0.03
0.04 0.03
3.4 6.3
4.7 4.1
Teachers’ age 0.02
0.07 0.12
0.07 1.6
3.8 3.2
2.5 students
2 0.0004
2 0.0002
2 0.002
2 0.001
1.5 3.1
2.8 1.5
departments 0.06
0.06 0.04
2 0.05
3.0 1.9
0.7 1.2
Teachers per 100 0.00
0.02 0.00
0.00 students
0.5 0.3
0.5 0.8
Lower secondary 2
0.01 2
0.08 2
0.01 2
0.01 gradesaverage
0.1 0.8
0.5 0.7
Constant 2
2.23 5.58
2 8.12
2 11.88
0.8 1.4
1.1 2.0
R
2 adj
0.38 0.25
0.26 0.12
N 1092
1092 332
559
relationship of the expected positive sign, while 7 dis- play a statistically negative relationship. 69 studies are
not significant at the 5 percent level. Our analysis shows that teacher experience has a significant and positive
influence on student achievement in all the reported specifications. Teacher experience seems to matter least
in Norwegian language and most in mathematics. In mathematics, the difference between the school with the
least experienced and the most experienced teachers is 1.44 grades, all else equal. Since this is a substantial
effect, we should try to sort out the reasons why experi- ence show a strong positive association with student
achievement in this study.
First, the positive correlation may result from more senior teachers having the ability to select schools and
classrooms with better students. This possibility is men- tioned by Hanushek 1986. We have investigated
whether teacher experience is associated with school and community characteristics, but it turns out that no stat-
istically significant relationships exist.
4
4
In Bonesrønning 1996, where the relationship between school size and student achievement is investigated empirically,
I find evidence that the teacher input is endogenous. The sample used in this paper is a subsample of the sample used in
Bonesrønning 1996. Thus, teacher behavior may differ between the Norwegian counties.
Second, we have tested the hypothesis that there exists an optimal number of teaching years, but the hypothesis
has been rejected. Recall that teachers’ experience varies little across the schools in our sample. This variation
may very well be within a relative linear part of a con- cave relationship. Further investigations are needed to
settle this question.
5
Third, the importance of teacher experience may have been obfuscated in previous analyses due to omitted vari-
ables. We have estimated, but not reported, Eq. 14 without the fraction of laudable teachers included. It
turns out that the significant association between teacher experience and student achievement disappears when the
laudability variable is excluded. For this sample at least, the importance of purchased teacher characteristics is
revealed only when appropriate controls are made for relevant non-purchased teacher characteristics.
The “new” indicator, the fraction of laudable teachers
5
Many of the students from the 68-generation became teach- ers in the upper secondary school. Born approximately in 1950,
these teachers were 38 years old in 1988. That is, the teachers from this cohort are among the youngest teachers in our sample.
Many researchers have pointed out that the 68- generation rep- resents a shift downwards in teaching quality. Our results are
consistent with this hypothesis, but without additional data the explanation is speculative.
98 H. Bonesrønning Economics of Education Review 18 1999 89–105
to all teachers, shows a significant and positive associ- ation with student achievement in all specifications
reported in Table 4. In mathematics, the difference in student achievement between the schools with the lowest
and the highest fractions of laudable teachers is 1.48 grades, all else equal. The positive association between
laudability and student achievement is an intuitive plaus- ible result, since laudability indicates that the teachers
have good understanding of the subjects they teach. Note however that the theoretical analysis has focused on two
mechanisms —teaching quality and teacher strength — through which teachers can influence student achieve-
ment. We are of course unable to separate between the influences from these two factors on the basis of the
results in Table 4, but will return to this issue below.
Now we investigate the relationships between grading practices and teacher characteristics.
4.3. The Grading Equation Eq. 15 is estimated by ordinary least squares with
G
3
as the dependent variable. The results are reported in Table 5. As can be seen, the grading in Norwegian lang-
uage is not significantly associated with any of the teacher characteristics, while the grading in the speciali-
zation subjects and mathematics is strongly correlated with all the teacher characteristics. For the latter subjects,
it is evident that laudable and experienced teachers prac- tice hard grading, while the most educated teachers prac-
tice easy grading. Moreover, in the specialization sub- jects
and mathematics the
grading parameter is
negatively associated with the students’ average initial level of knowledge. The null hypothesis that the teach-
Table 5 The grading equation. Dependent variable G
3
, t-statistics in parentheses Variable
Norwegian language Specialization subjects
Mathematics Lower secondary gradesaverage
0.33 2
0.18 2
0.23 1.9
2.9 1.7
Teachers’ age 0.04
2 0.07
2 0.09
1.2 5.7
3.4 Laudable teachers
0.01 2
0.01 2
0.25 1.2
4.1 3.8
Teachers’ education 0.001
0.001 0.02
0.2 4.5
3.9 students
2 0.001
0.001 0.0002
1.4 2.7
0.2 departments
2 0.05
2 0.06
2 0.08
0.7 2.4
1.6 Teachers per 100 students
0.04 2
0.06 0.001
0.4 1.9
1.8 Constant
2 12.26
15.10 16.63
1.6 5.5
2.7 R
2 adj
0.32 0.83
0.77
ers’ grading practices are not affected by the students’ average initial level of knowledge is formally rejected
for the specialization subjects, but not for mathematics. Note that we have not corrected for the multilevel
character of the data involved in this analysis. In general, such correction implies more conservative estimates.
Table 6 reports the OLS-estimation results with g
s j
as the dependent variable. There is basically no evidence
that the degree to which the teachers’ grades vary with achievement correlates with teacher characteristics.
Note that Table 5, columns 2 and 3, provide evi- dence that the teacher characteristics are associated with
hard grading in a way very similar to the association between teacher characteristics and student achievement
in Table 4. The crucial question then is whether these teacher
characteristics affect
student achievement
directly, or whether the teachers’ influence on student achievement is mediated through the grading practices.
This issue is now investigated within the education pro- duction function framework.
As noted in Section 2, the specification of the edu- cation production function incorporating grading effects
depends on where hard grading origins. This is related to the following question: How should the results in
Table 5, columns 2 and 3, be interpreted? It is clear that the results are consistent with the rent seeking
model. That is, the estimated coefficients might be inter- preted as showing that low achieving students press for
easy grading and that the teachers’ ability to withstand pressure is associated with laudability and experience.
We have interacted the students’ initial level of knowl- edge with the fraction of laudable teachers. The results
for mathematics, which are not reported, indicate that
99 H. Bonesrønning Economics of Education Review 18 1999 89–105
Table 6 The grading equation. Dependent variable g
s
, t-statistics in parentheses Variable
Norwegian language Specialization subjects
Mathematics Lower secondary gradesaverage
2 0.07
2 0.06
2 0.20
0.8 1.0
2.5 Teachers’ age
2 0.01
0.01 2
0.004 0.7
0.8 0.3
Laudable teachers 2
0.003 2
0.001 0.002
0.7 0.4
0.5 Teachers’ education
0.001 0.001
2 0.003
0.3 0.4
0.6 students
2 0.00003
2 0.002
2 0.00003
0.1 0.7
0.1 departments
0.02 0.04
0.03 0.5
1.9 1.0
Teachers per 100 students 2
0.01 0.01
0.3 0.4
Constant 4.38
2.56 9.12
1.2 1.0
2.6 R
2 adj
2 0.51
2 0.03
0.06
low achievers are less capable of affecting grading when the fraction of laudable teachers increases. These results
are consistent the model predictions: With increasing teacher strength, the rent seeking frontier shifts inwards,
which implies that the marginal returns to student rent seeking decreases.
However, the results may as well be interpreted as showing i that laudable and experienced teachers know
more about grading than other teachers admittedly, this interpretation take into account empirical results not
presented yet, and ii that all types of teachers conjec- ture that it is efficient to practice easy grading for low
achievers.
These interpretations
motivate two
different approaches to the education production function, to be
presented below. 4.4. The Education Production Function Revisited I
We start with the interpretation that grading is a policy instrument for the teacher. Thus grading should be
regarded as an exogenous variable to be included in the education production function.
Table 7 reports the results from estimation of the edu- cation producton function augmented with the grading
parameters G
3
and g
s
. For Norwegian language, G
3
and g
s
are both significantly associated with student achieve- ment when included one at a time, while neither of the
grading parameters turn out to be significant predictors of student achievement when included simultanously in
the equation. The signs of the coefficients indicate that students i respond to harder grading by increasing their
studying time and ii respond to increased marginal returns to achievement by increasing their studying time.
The magnitude of the grading effect the G
3
effect is illustrated by the following example. A student in the
school who practice the hardest grading performs 0.28 grades better than a student in the school which practice
the easiest grading, all else equal.
The G
3
coefficient is not significant different from zero for the specialization subjects, but when restricted
to the sample of high achieving students, the G
3
coef- ficient turns out to be a significant predictor of student
achievement. The negative sign indicates that high achieving students treat leisure as a normal good. More-
over, the grading effect is substantial. The sample vari- ation in G
3
multiplied by the estimated coefficient is equal to 1.3 grades. The g
s
coefficient is not significantly associated with student achievement, but the sign of the
g
s
coefficient indicates that students allocate less time to studying when the marginal returns to real achievement
increases. Since the g
s
effect should be interpreted as a substitution effect, this is a puzzling result. It might indi-
cate that our estimation strategy of treating the speciali- zation subjects as one subject is inappropriate.
To mitigate these problems we highlight mathematics which is one of the specialization subjects. We consider
the self-selection issue by estimating a selection equation with the dependent variable equal to 1 if the student
choose mathematics and 0 otherwise. The independent variables are individual student characteristics and two
grading parameters which are the G
3
parameters for mathematics and for all the specialization subjects net of
mathematics. The results are reported in Table 8, col- umn 1.
As expected, males are more likely to choose math-
100 H. Bonesrønning Economics of Education Review 18 1999 89–105
Table 7 The education production function revisited It-statistics in parentheses
Specialization Variable
Norwegian language Norwegian language Norwegian language High achievers
subjects Lower secondary
0.17 0.17
0.17 0.19
0.19 grades
24.6 24.7
24.7 17.8
7.2 Student’s sex
0.07 0.07
0.07 2
0.10 2
0.08 1.5
1.5 1.5
1.5 0.8
Teachers’ education 2
0.01 2
0.01 2
0.01 0.001
0.01 3.3
2.5 2.5
0.2 1.1
Laudable teachers 0.01
0.01 0.01
0.02 0.01
3.6 3.9
3.9 2.0
1.4 Teachers’ age
0.03 0.03
0.03 0.04
2 0.002
2.2 2.3
2.3 0.9
0.2 students
2 0.0004
2 0.001
2 0.001
2 0.002
2 0.002
1.4 2.0
2.4 2.4
2.2 departments
0.06 0.05
0.05 2
0.07 2
0.07 3.1
2.4 2.4
1.7 1.5
Teachers per 100 2
0.002 2
0.005 2
0.006 2
0.001 2
0.07 students
0.1 0.2
0.2 0.1
0.9 Lower secondary
2 0.02
0.03 0.03
2 0.23
2 0.27
gradesaverage 0.4
0.4 0.4
1.4 1.2
g
s
0.49 0.12
2 0.92
2 0.87
1.8 0.3
1.2 1.2
G
3
2 0.27 1.6
2 0.30 2.4
2 0.64 0.9
2 1.13 2.2
Intercept 2
4.91 1.6 2
5.90 1.9 2
5.69 1.9 5.51 0.5
1.20 0.9 R
2 adj
0.38 0.38
0.38 0.25
0.13 N
1092 1092
1092 1092
559
ematics than females, and high achievers are more likely to choose mathematics than low achievers. There are
some indications, although insignificant, that the prob- ability for choosing mathematics increases when the
grading in mathematics gets easier, which seems like a intuitive plausible result. A surprising result, at least at
first glance, is that the probability for choosing math- ematics increases when the grading in other specializa-
tion subjects get easier. At second thought however, we already know that the teacher characteristics associated
with easy grading also are those teacher characteristics associated with poor student performance. The students
then face a trade-off between easy grading and teaching quality, and their subject choices cannot easily be pre-
dicted. It might be argued that measures of teaching quality should be included in the selection equation, but
unfortunately we have been unable to separate the teach- ers into the relevant categories.
We apply the Heckman 1979 model to correct for selectivity bias, which implies that the inverse Mill’s
ratio is included in the mathematics education production function. The results are reported in Table 8, columns
2 and 3. The coefficient of the selection-bias variable l
is significant, which indicate that selectivity bias is an issue. Table 8, column 4, reports the results for the
mathematics education production function exclusive the selection-bias variable. Note that the gender differences
appear to be much larger when the l variable is included. This is consistent with earlier findings showing that the
effect of self-selection is particularily strong for females.
In column 2 the G
3
coefficient is positive and insig- nificant. In column 3, where we have interacted the
students’ initial level of knowledge with the G
3
coef- ficient, the grading parameter is significantly associated
with student achievement at the 10 percent level. Stu- dents with very high initial levels of knowledge are posi-
tively affected by harder grading, while students with lower initial levels of knowledge seem to respond to
harder grading by decreasing their studying time. In one interpretation, these results indicate that high achievers
treat leisure as a normal good, while low achievers treat leisure as an inferior good. The g
s
coefficient is insig- nificant, but it still has the “wrong” sign.
The grading effects do not seem to be very much affected by the controls made for selectivity bias. As can
be seen from column 4, the students’ response to hard
101 H. Bonesrønning Economics of Education Review 18 1999 89–105
Table 8 Grading effects in mathematicst-statistics in parentheses
Variable Probit
Mathematics Mathematics
Mathematics Lower secondary grades
0.18 0.52
0.82 0.58
12.1 4.2
3.6 2.8
Student’s sex 2
0.95 2
1.96 2
1.87 2
0.35 10.3
3.0 2.9
2.4 Teachers’ education
2 0.002
2 0.002
2 0.01
0.2 0.1
0.8 Laudable teachers
0.04 0.04
0.05 2.6
2.6 3.5
Teachers’ age 0.08
0.08 0.14
1.3 1.3
2.5 students
2 0.004
2 0.004
2 0.002
3.6 3.6
2.7 departments
0.05 0.04
0.06 0.7
0.6 0.8
Teachers per 100 2
0.002 1.7 2
0.002 1.6 2
0.008 0.7 students
Lower secondary 2
0.12 2
0.15 2
0.08 gradesaverage
0.5 0.6
0.3 1.3
0.9 1.7
1.9 G
3 spec
0.66 2.7
g
s math
2 0.62 0.7
2 0.71 0.8
2 0.30 0.4
Lower secondary 2
0.88 2
1.02 grades G
3 math
1.6 1.8
l 2.58 2.6
2.41 2.4 R
2 adj
0.27 0.27
0.26 N
1092 1092
1092 332
grading is somewhat more sensitive to the students’ initial level of knowledge when no controls for selec-
tivity bias are made. This might reflect that the high achieving students who choose mathematics when grad-
ing is hard, are those students who respond to harder grading by increasing their effort.
There are two reasons why the results regarding the effects of teachers’ grading reported in this section may
be spurious. First, the positive correlation between hard grading and student achievement might be generated by
omitted variables of teaching quality that are positively correlated both with hard grading and student achieve-
ment. That is, no grading effect may actually exist. A related argument could be made about the “wrong” sign
of the g
s
coefficient. At this stage, we have no solution to offer to this potential omitted variable problem. The
second reason the results may be spurious is that grading is determined by rent seeking students. Then the positive
relationship between hard grading and student achieve- ment is mediated at least partially through the students’
time constraint. This problem is discussed in the next section.
4.5. The Education Production Function Revisited II The alternative interpretation of the estimation results
from Eq. 15 is that rent seeking occurs for the speciali- zation subjects. In this case the estimated grading para-
meter coefficients reported in Section 4.3 will be biased. The bias is due to correlation between the grading para-
meter and the residual, generated by the unobserved student characteristics which determine the grading para-
meter. To deal with this problem, instruments for the teacher grading parameter are generated from Eq. 15
and included in Eq. 14. Since the students’ initial aver- age level of knowledge is significantly associated with
teachers’ grading, but not with student achievement, identification might be possible.
The instruments are used to test whether the assump- tion that the grading parameter can be treated as exogen-
ous is valid. We apply the Durbin–Wu–Hausman speci- fication test in a fashion similar to Ehrenberg and Brewer
1995. That is, both the grading parameter and the instrument are included in Eq. 14. The results for math-
ematics and for the high achievers in the specialization
102 H. Bonesrønning Economics of Education Review 18 1999 89–105
subjects are reported in Table 9. We do not consider that endogeneity of the grading parameter is a problem
for Norwegian language. As can be seen, the coefficients of the instruments are
small and insignificant in all specifications. Formal F- tests suggest that one cannot reject the hypothesis that
the coefficients of the instruments are equal to zero. We have generated alternative instruments by including
interaction terms in Eq. 15, but in no case the instru- ments turn out to be significantly associated with student
achievement. Formally, the tests imply that the grading parameters should be treated as exogenous, that is, the
equation estimated in Section 4.3 seems to be the cor- rect equation.
Unfortunately, the evidence provided by these tests cannot be regarded as being very decisive. As can be
seen from Table 9, the coefficients of the teacher charac- teristics become insignificant when the instruments are
included in the estimated equation. This is an indication that the identification problems are not properly solved.
The sample does not provide any additional variables that could help solve the identification problems.
Table 9 The education production function revisited IIt-statistics in parentheses
Variable Mathematics
High achievers Lower secondary grades
0.78 0.19
3.4 7.2
Student’s sex 2
1.99 2
0.08 3.0
0.8 Teachers’ education
0.02 0.01
0.9 1.2
Laudable teachers 0.01
0.01 0.5
1.4 Teachers’ age
2 0.01
2 0.01
0.1 0.2
students 2
0.004 2
0.002 3.6
2.2 departments
2 0.07
2 0.07
0.7 1.5
Teachers per 100 students 2
0.11 2
0.07 0.7
0.9 G
3
13.6 2
1.11 1.6
2.1 g
s
2 0.29
2 0.86
0.3 1.2
G
3 IV
2 11.2
2 0.03
1.3 0.2
Lower secondary gradesG
3
2 0.29
1.5 Lower secondary gradesG
3 IV
0.02 1.2
R
2 avg
0.27 0.13
N 332
559
5. Concluding Remarks
