**Table 2 Input and output data disaggregated by Income group**

||Obs|Mean|Std. Dev.<br>**LAYS**|Min|Max|Coef. Variation|
|---|---|---|---|---|---|---|
|High‐ Income|48|10.818|1.176|7.630|12.905|9.199|
|Upper Middle‐Income|38|8.094|1.771|4.004|11.900|4.570|
|Lower Middle‐Income|45|6.556|1.635|3.424|10.214|4.010|
|Low‐Income|26|4.528|1.228|2.275|6.890|3.688|
||**Ex**<br>|**penditurep **<br>|**er school‐ag**<br>|**e individual**<br>|||
|High‐Income|53|7488.619|4104.003|1238.602|22448.080|1.825|
|Upper Middle‐Income|45|1612.518|768.749|395.142|3449.911|2.098|
|Lower Middle‐Income|48|581.917|410.910|84.865|2062.425|1.416|
|Low‐Income|29|120.597|54.282|29.981|266.736|2.222|

# **4. Results**

The stochastic frontier model discussed in section 2 is estimated using the data presented in section 3. Equations 1 and 2 are simultaneously estimated using the maximum likelihood method discussed by Battese & Coelli (1995). The results allow an assessment of the relationship between public education expenditure and access to quality education, to gauge the efficiency in the use of public education spending and to explore the correlates of the inefficiency.

# **4.1 Public education expenditure and the learning‐adjusted years of schooling**

Model 1 in Table 3 provides an estimation of Equation 1 only, while Model 2 gives the results of the simultaneous estimation of Equations 1 and 2. After controlling for country‐level inefficiencies related to public expenditure on education, there seems to be a positive correlation between public education spending and the learning‐adjusted years of schooling. A higher expenditure per school‐age individual is associated with increased access to quality education. However, these results must be interpreted with caution as there might be some endogeneity surrounding this relationship. In fact, the decision made by the government and the international partners to invest in education may be linked to the performance of the education sector. For instance, in countries with poor access to education, there might be a rationale for increased expenditure on education with the hope that this could drive better educational outcomes. Increased expenditure could also contribute to improving the access to quality education. These results should then be understood as a correlational relationship rather than a causality relation.

Although public expenditure on education seems to be correlated with increased years of quality education overall, the relationship appears to be concave and decreases with the level of spending. One additional unit of spending is less associated with increased access to quality education, compared to the previous unit. In other terms, countries that seem to already spend too much on education may not expect to significantly increase the access to quality education by spending more. However, countries that initially spend relatively less on education could expect a stronger correlation between increased spending and educational outcomes, all things being equal. Al‐Samarrai, Cerdan‐Infantes, & Lehe (2019) also suggest that there is a clear relationship between spending on education and educational outcomes, which decreases with the level of spending. On average, they find that a 10 % increase in spending per child leads to a 0.8 % improvement in the LAYS. Our estimations show that when inefficiencies are controlled for, and when the measurement of public expenditure includes education aid, the relationship between 9