## 5.1 Classroom Disciplinary Atmosphere; 5.4 Discipline and Class Size; 7.2 Synthetic Data Analysis

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### PISA questionnaire and classroom-discipline index
- Students responded to three statements: “Students don’t listen to what the teacher says,” “There is noise and disorder,” and “The teacher has to wait a long time for students to quiet down.” Response options: “Every lesson,” “Most lessons,” “Some lessons,” and “Never or hardly ever.”
- Responses reduced via factor analysis to a single “classroom discipline” index.
- The index is normalized to the 0.01-1 range.
- Note (in-source): “The 0.01-1 range is used in reference to the concept of discipline as the probability of non-interruption in the classroom, as discussed in Section 3. However, while the index can arguably be viewed as a monotonous transformation of the probability of non-interruption, it is not identical to it.”

### Association of discipline measures with student performance (empirical patterns)
- Students who perceive a good disciplinary climate achieve higher average scores (Figure 1 correlation; Figure 2 class-discipline index vs PISA math score averaged over 2012, 2015, 2018 and 2022).
- Cross-country observations (selected):
  - Hong-Kong, Korea and Japan: high scores with high classroom discipline.
  - Singapore and Taiwan: very high scores with only moderately-high classroom discipline.
  - Greece (and to a lesser degree Slovakia and Spain): low scores and low classroom discipline.
  - Norway: high reported classroom discipline but test scores comparable to France, which reports lowest classroom discipline in the developed world.
  - Israel and the United States: low-moderate and high-moderate classroom disciplinary climate respectively, yet fairly low average test scores.
  - Belgium: weak classroom disciplinary atmosphere while scoring above average.
- Caveats:
  - Classroom-discipline index reflects students’ perception of class atmosphere (information on P_j), not their own individual behavior (not directly P_i or P̄_j).
  - Potential cultural bias in perceptions; partially accounted for with country fixed effects. Analysis assumes cultural bias relatively small within an education system.

### Theoretical predictions on class size and discipline (Section 3 summary as used in Section 5.4)
- For a given level of individual discipline and teaching quality, larger class size is expected to lower test results (class size magnifies relation between ̄P_j and student achievements; see Equation 2).
- School management optimizes student numbers and allocation to meet a class-discipline threshold and allocates better teachers to larger classes, yielding:
  - Positive correlation between individual discipline (P_i) and class size (selection into larger classes).
  - Positive correlation that extends to teaching quality and educational outcomes.
  - No correlation between classroom discipline (P_j) and class size if classes are constructed to meet a common classroom-discipline threshold (country/region/stream fixed threshold).

### Cross-country and within-country descriptive patterns (Figures 6–7 and discussion)
- Asian 5 leading countries: high discipline levels and large classes.
- Several Western countries (Austria, Belgium, Czech Republic, Germany, Netherlands, Switzerland) have relatively high student discipline and smaller classes — could theoretically support larger classes to increase teaching selectivity.
- Low-performing country examples:
  - Israel: fairly large classes and low individual discipline.
  - Greece and Italy: medium class sizes but low discipline.
- Class size alone is not sufficient predictor of PISA scores (e.g., Israel, France, Korea, Hong-Kong can have similar class sizes but very different scores, partly due to differing discipline levels).

### School-level regressions (Table 2) — key reported coefficients and interpretation
- Main pattern: mean individual discipline (̄P_j) is strongly and positively correlated with class size; classroom discipline (P_j) and σ_P_j are not statistically significant at 10% in school-level regressions.
- Reported coefficients (as in source table):
  - P_j (class. disc.): -.260, -.904, -.564, -.863 (parenthetical standard errors shown in source).
  - ̄P_j (mean ind. disc.): 3.042, 2.973, 4.519, 4.237 (∗∗∗ at 1% in source).
  - σ_P_j (SD ind. disc.): -2.068, -2.118, 2.192, 1.762 (not statistically significant at 10% per source).
- Notes: Controls include year, country, country GDP per capita and country share of population under age 15. Errors clustered by country. *Significant at 10%; **significant at 5%; ***significant at 1%.

### Student-level regressions (Table 3) — reported coefficients and patterns
- Regression framework: PISA math scores regressed on class-size dummies and incremental inclusion of P_i, ̄P_j, σ_P_j, P_j. Controls: year, country, country GDP per capita, country share of population under age 15, maternal and paternal education, share of schoolmates whose parents have post-secondary education. Errors clustered by country.
- Selected coefficients (columns reflect progressive inclusion; exact values reported in source):
  - P_i (ind. disc.): 67.252; 54.309; 54.345; 54.429 (columns 1–4/5).
  - ̄P_j (class mean ind. disc.): 115.017; 68.158; 49.170.
  - σ_P_j (class SD ind. disc.): -91.462; -77.079 (∗∗ significance).
  - P_j (class discipline): 71.494 (∗∗∗).
- Class-size dummy coefficients (relative to baseline up to 15 students) reported exactly:
  - 16-20: 13.817; 8.922; 8.814; 9.139; 8.140
  - 21-25: 20.064; 14.886; 14.968; 15.218; 14.617
  - 26-30: 28.855; 22.909; 22.171; 22.388; 21.999
  - 31-35: 38.853; 32.219; 31.279; 31.473; 31.387
  - 36-40: 43.945; 38.236; 37.737; 37.491; 37.284
  - 41-45: 42.150; 36.609; 35.788; 35.461; 34.082
  - 46-50: 22.375; 17.698; 18.028; 18.400; 16.753
  - 51+: 13.762; 8.912; 8.865; 9.019; 8.928
- Sample and fit statistics (as reported):
  - Obs. in columns: 782389743639743639743546740340 (as presented in table).
  - R2: .227; .255; .264; .266; .274 (columns 1–5).
- Interpretations highlighted in source:
  - Inclusion of P_i reduces class-size coefficients but they remain significant; P_i very large and highly significant (e.g., 67.252, t-value 21.84 reported).
  - ̄P_j large when included (e.g., 115.017, t-value 5.03 reported) but loses significance when P_j is added.
  - σ_P_j negative and sizable when included (e.g., -91.462), indicating greater variance in student discipline in class is negatively associated with student scores.
  - Addition of P_j (71.494) reduces the role of ̄P_j in the specification.

### Robustness and interaction findings
- Fixed class-size estimations broadly similar to general results (Tables A2, A3).
- Negative effect of σ_P_j tends to be larger and more statistically significant in larger classes (except 51+ small outlier).
- Interaction analyses: interaction terms of σ_P_j with class sizes are the most statistically significant; negative effect of σ_P_j tends to grow with class size.
- Remaining concern: potential endogeneity/omitted-variable bias from unobserved teaching quality correlated with both individual discipline and class size; addressed via simulation (Section 7).

### Simulation design and parameters (Section 7.1) — exact specifications preserved
- Countries and students:
  - N_cnt = 30 countries.
  - Each country: N_stu = 500 students.
- Country and student-discipline draws:
  - D_cnt ∼ N(D_world = 1.5e, σ_world = 0.15e).
  - D_stu ∼ N(D_cnt, σ_stu = 0.2e).
- School noisy information on student discipline:
  - D_sch ∼ N(D_stu, σ_sch = 0.3e).
- Mapping to probabilities:
  - P_i = invlogit(D_i).
- Class sizes:
  - CS_min = 10; CS_max = 50; CS_inc = 5 (class sizes from 10 to 50 in increments of 5).
- School ranking and assignment:
  - R_sch ∼ N(D_stu, σ_rnk = 0.5e); students sorted by R_sch and allocated starting from CS_max = 50; school computes P_sch_class = ∏_{i=1}^{n_class} P_sch_i.
  - Class discipline must meet or exceed threshold P_thresh_cnt = 2/3 ∗ ̄P_{CS_min + CS_max 2}_cnt + 1/3 ∗ 0.5(4). If below threshold class size lowered by CS_inc until CS_min = 10 reached. Classes of size 10 dropped as outliers; 1,120 synthetic students in size-10 classes removed.
- Teacher quality:
  - Number prospective teachers per country = ⌈N_stu / CS_min⌉ = 50.
  - TQ ∼ N(μ_TQ = 100, σ_TQ = 15).
  - Schools observe TQ_sch ∼ N(TQ, σ_sch_TQ = 0.1σ_TQ). Teachers sorted by observed quality; better teachers assigned to larger classes.
  - Countries with larger classes require fewer teachers and on average enjoy higher teaching quality.
- Education production function used (Equation 5):
  - E_{i,j} = TQ^{β0}_j P_i^{β1} P_j^{β2} with β0 = 1, β1 = 1, β2 = 1.
  - E normalized to mean 500 and standard deviation 100; simulated test score Ê_{i,j} ∼ N(E_{i,j}, σ_E = 25).
- Measurement noise for discipline:
  - D_test_i ∼ N(D_stu, σ_test = 0.2e).
  - D_test_j ∼ N(logit(P_j), 3∗σ_test = 0.6e).
  - D_test_i and D_test_j normalized to the 0.01–1 range.
- Purpose: compare estimations including and excluding TQsch to gauge bias from omitted teaching quality.

### Synthetic data replication of empirical patterns (Section 7.2)
- Synthetic data replicates positive correlations between individual discipline, classroom discipline, class size and test scores observed in PISA.
- Mechanism:
  - Countries with higher discipline draws have more disciplined students, larger classes, stronger classroom discipline.
  - Larger classes require fewer teachers, so on average better teachers assigned to larger classes — strengthening positive correlation between class size and test scores in raw data when TQ unobserved.

### Regression results on synthetic data (Table 4) — reported coefficients and interpretation
- Column (1) full specification including TQsch:
  - TQsch: 8.199 (.106) ∗∗∗
  - P_i: 89.730 (3.987) ∗∗∗
  - ̄P_j: 16.089 (.812) ∗∗∗
  - Class size fixed effects (relative to baseline class size 15) are negative and decreasing (increasing in size): once teaching quality accounted for and for given discipline levels, increasing class size decreases achievements.
  - Obs.: 138801; R2: .723
- Columns (2)–(6) omit TQsch and demonstrate omitted-variable bias:
  - Column (2) (class size fixed effects only): class-size fixed effects positive and increasing in size, peaking at class size 40 (mirrors real data peak at class size 36-40).
  - Column (3) adds P_i: P_i = 121.972 (4.848) ∗∗∗ (upward biased relative to Column (1)); class-size fixed effects drop after adding P_i.
  - Column (4) adds ̄P_j: ̄P_j = 22.703 (.969) ∗∗∗ (biased upwards relative to Column (1)).
  - Column (5) adds σP_j: σP_j = -33.048 (4.003) ∗∗∗ (negative, statistically significant).
  - Column (6) adds P_j: P_j = 40.682 (2.953) ∗∗∗ (positive and significant); omission of teaching quality does not create upward bias for P_j in this synthetic setting.
  - Obs.: 138801; R2 across columns (2)–(6) ranges .566 to .605.
- Overall interpretation:
  - Full specification (Column (1)) approximates the data-generating process and avoids omitted-variable bias.
  - Omitting teaching quality leads to upward biases in coefficients on P_i and ̄P_j, illustrating potential bias in empirical PISA regressions.
  - Noise in schools’ measurement of individual discipline produces upward bias in school-perceived discipline for larger classes, explaining positive class-size fixed effects when TQ is omitted.

### Synthetic-data summary by class size (Table A4 exact values)
- Class Size 15: Freq. 3,105; Dstu 3.96; Dsch 3.89; TQsch 106.38; Test Score 456.92
- Class Size 20: Freq. 3,720; Dstu 4.15; Dsch 4.17; TQsch 112.74; Test Score 475.85
- Class Size 25: Freq. 3,425; Dstu 4.32; Dsch 4.38; TQsch 119.00; Test Score 500.98
- Class Size 30: Freq. 1,650; Dstu 4.49; Dsch 4.58; TQsch 125.79; Test Score 533.42
- Class Size 35: Freq. 1,085; Dstu 4.77; Dsch 4.84; TQsch 127.61; Test Score 588.28
- Class Size 40: Freq. 520; Dstu 4.92; Dsch 4.98; TQsch 132.12; Test Score 621.20
- Class Size 45: Freq. 225; Dstu 5.13; Dsch 5.29; TQsch 130.42; Test Score 635.89
- Class Size 50: Freq. 150; Dstu 5.34; Dsch 5.49; TQsch 132.41; Test Score 663.76
- Total: Freq. 13,880; Dstu 4.30; Dsch 4.32; TQsch 116.80; Test Score 503.52
- Notable pattern: D_sch > D_stu for all class sizes greater than 15, with particularly large gaps for sizes 45 and 50.

### Synthesis with model and policy-relevant conclusions
- Theoretical framework predicts positive correlations among student discipline, class size, teaching quality and scholastic achievement; both PISA and synthetic data support these predictions.
- Empirical regularities supported:
  - Individual discipline, average class student discipline, and classroom discipline positively correlate with PISA math scores.
  - Standard deviation of class student discipline negatively correlates with PISA math scores.
  - Student-discipline correlates with class size, but within-country class discipline does not, consistent with sorting to meet discipline thresholds.
  - Large classes up to a normative size correlate with higher student discipline and higher test scores, consistent with allocation of better teachers to larger normative classes.
- Policy implications (as suggested in source):
  - Student discipline is a crucial factor explaining gaps in scholastic performance across students, schools and countries (example comparison: Israel vs Hong-Kong).
  - Policies to improve classroom discipline and reduce variance in student discipline could enable effective learning in larger classes and support higher-quality teacher recruitment/allocation.
  - Improving measurement of teaching quality would reduce omitted-variable bias in empirical evaluations of class size and discipline effects.
- Suggested research directions:
  - Exploit exogenous measures affecting discipline, class-size and/or teacher allocation to strengthen causal identification.
  - Refine the theoretical model to better match specific educational systems.
  - Study supply-side effects: how discipline and class size affect attractiveness of the teaching profession and teacher supply.
- Final takeaway: Understanding interactions among discipline, class size, and teacher quality is essential for efficient educational resource allocation and policies that could yield substantial dividends in scholastic achievement.

*Source: IMF Working Paper — sections 5.1 Classroom Disciplinary Atmosphere; 5.4 Discipline and Class Size; 7.2 Synthetic Data Analysis*

### 5.1 Classroom Disciplinary Atmosphere

### 5.1 Classroom Disciplinary Atmosphere

### PISA questionnaire items
- Students were asked to respond to three statements that relate directly to classroom discipline levels:
  - “Students don’t listen to what the teacher says,”
  - “There is noise and disorder,” and
  - “The teacher has to wait a long time for students to quiet down.”
- For each statement students chose one response from: “Every lesson,” “Most lessons,” “Some lessons,” and “Never or hardly ever.”

### Association with student performance
- Figure 1 demonstrates the correlation between the perceived classroom disciplinary climate (responses to the three statements above) and the students’ average score, compared to the respective national average.
- The correlation is clear: students who perceive a good disciplinary climate achieve higher average scores.
- Figure 2 displays the ‘classroom discipline’ index versus PISA math score (both averaged over the 2012, 2015, 2018 and 2022 rounds), by country.

### Classroom discipline index construction
- Responses to the three statements are reduced via factor analysis to a single “classroom discipline” index.
- The index is normalized to the 0.01-1 range.
- Country ranking and index scores are presented in Figure A1 in the appendix.
- Note in-source clarification: “The 0.01-1 range is used in reference to the concept of discipline as the probability of non-interruption in the classroom, as discussed in Section 3. However, while the index can arguably be viewed as a monotonous transformation of the probability of non-interruption, it is not identical to it.”

### Cross-country observations (selected)
- Of the 5 leading Asian countries:
  - Hong-Kong, Korea and Japan enjoy high scores in conjunction with high classroom discipline.
  - Singapore and Taiwan have very high scores but only moderately-high levels of classroom discipline.
- Of the low scoring countries:
  - Greece, and to a lesser degree Slovakia and Spain, suffer from both low scores and low classroom discipline.
- Other notable contrasts:
  - Norwegian students report a high level of classroom discipline, yet their test scores are comparable to those of their French counterparts, who report the lowest levels of classroom discipline in the developed world.
  - Israeli and American students report a low-moderate and high-moderate classroom disciplinary climate respectively, while their average test scores are fairly low.
  - Belgian students report a weak classroom disciplinary atmosphere while scoring above average.

### Interpretation and caveats
- The classroom discipline index attests to students’ perception of class atmosphere, which is a product of overall student behavior in the class (it contains information on P_j), and not to their own individual behavior (it does not relate directly to P_i or P̄_j).
- The index may be subject to cultural bias, as it is influenced by students’ perception of discipline in each locale; this potential cultural bias is partially accounted for in later analysis using country fixed effects.
- The analysis assumes cultural bias is relatively small within a given education system (in contrast to bias between different education systems, which can be considerable), which supports interpreting the observed correlations as indicating a major advantage for students in classrooms where perceived discipline levels are high.

*Source: IMF Working Paper — section 5.1 Classroom Disciplinary Atmosphere*

### 5.4 Discipline and Class Size

### 5.4 Discipline and Class Size

### Theoretical predictions
- The model in Section 3 predicts class size magnifies the relation between average individual discipline ( ̄P_j ) and student achievements (see Equation 2). For a given level of individual discipline and teaching quality, larger class size is expected to lower test results.
- The model assumes school management optimizes student numbers and allocation to meet a class discipline threshold, and allocates more resources (better teachers) to larger classes. This yields:
  - A positive correlation between individual discipline (P_i) and class size (selection of disciplined students into larger classes).
  - A positive correlation that extends to teaching quality and educational outcomes.
  - No correlation between classroom discipline (P_j) and class size if all classes are constructed to meet the same classroom discipline threshold (country/region/stream fixed threshold).

### Cross-country and within-country patterns (Figures and descriptive discussion)
- Cross-country patterns (Figure 6):
  - The Asian 5 leading countries display high discipline levels and large classes.
  - Many Western countries are characterized by smaller classes and varying discipline; Austria, Belgium, the Czech Republic, Germany, the Netherlands and Switzerland have relatively high student discipline and could arguably support larger classes to increase teaching selectivity.
  - Low-performing countries examples:
    - Israel: fairly large classes and low individual discipline (described as a “bad combination”).
    - Greece and Italy: medium class sizes but low discipline levels.
- Figure 7 (class size vs PISA math scores, 2012–2022 averages): class size alone is not a sufficient indicator of PISA scores; countries with similar class sizes (Israel, France, Korea, Hong-Kong) can have very different scores, partly explained by differing discipline levels.

### School-level analysis: Table 2 findings
- Main pattern: mean individual discipline ( ̄P_j ) is strongly and positively correlated with class size; classroom discipline (P_j) and the standard deviation of individual discipline (σ_P_j) are not statistically significant even at 10% in the school-level regressions.
- Interpretation provided:
  - High average individual discipline enables effective teaching in large classes (consistent with theory).
  - Lack of correlation of class size with classroom discipline (P_j) aligns with the model if classes meet a common classroom-discipline threshold; country differences captured by country fixed effects.
  - The non-significance of σ_P_j (standard deviation within class) is surprising; a possible explanation is schools tolerate low-discipline students up to a level, moving extreme low-discipline students into smaller classes.
- Table 2 (coefficients reported in the source table):
  - P_j (class. disc.): -.260, -.904, -.564, -.863 (parentheses show standard errors as in source).
  - ̄P_j (mean ind. disc.): 3.042, 2.973, 4.519, 4.237 (with standard errors and ∗∗∗ significance at 1% as in source).
  - σ_P_j (SD ind. disc.): -2.068, -2.118, 2.192, 1.762 (not statistically significant at 10% per source).
  - Const. values and Obs./R2 vary across specifications (obs and R2 reported in source table).
- Notes (as in source): Controls for year, country, country GDP per capita and country share of population under age 15. Errors clustered by country. *Significant at 10%; **significant at 5%; ***significant at 1%.

### Student-level estimations: Table 3 findings
- Regression framework: student PISA math scores regressed on class-size dummies and incremental inclusion of:
  - individual discipline (P_i),
  - class mean individual discipline ( ̄P_j ),
  - class SD of individual discipline (σ_P_j),
  - class discipline (P_j).
- Column (1) baseline (class size dummies only):
  - Class size dummies statistically significant at 1%; positive correlation with test scores up to class size 36–40, after which the positive effect begins to diminish relative to baseline class up to 15 students.
- Column (2) adds P_i:
  - P_i coefficient: 67.252 (t-value 21.84 per text; table shows standard errors).
  - Inclusion of P_i reduces class-size dummy coefficients but they remain significant at 1%.
  - R2 increases from 0.227 to 0.255.
- Column (3) adds ̄P_j:
  - ̄P_j coefficient: 115.017 (t-value 5.03 reported in text).
  - P_i falls to 54.309 but remains highly significant.
  - R2 increases to 0.264.
- Column (4) adds σ_P_j:
  - σ_P_j coefficient: -91.462 (significant at 5% with t-value -2.56 in text; table shows -91.462 and -77.079 in different columns).
  - Greater variance in student discipline in the class is negatively associated with student scores.
  - With σ_P_j included, average class discipline drops in size and is significant only at 10% (t-value 1.78 in text).
- Column (5) adds P_j:
  - P_j coefficient: 71.494 (highly significant; table shows (5.616) standard error and ∗∗∗).
  - Addition of P_j reduces ̄P_j and causes it to lose statistical significance even at 10%.
  - σ_P_j remains negative, sizable and statistically significant at 5% (t-value -2.2 in text).
  - R2 increases to 0.274.
- Selected coefficients and class-size dummies from Table 3 (as reported in the source table):
  - P_i (ind. disc.): 67.252; 54.309; 54.345; 54.429 (columns 1–4/5 as in table, with standard errors and ∗∗∗).
  - ̄P_j (class mean ind. disc.): 115.017; 68.158; 49.170 (standard errors and significance flags as in table).
  - σ_P_j (class SD ind. disc.): -91.462; -77.079 (with ∗∗ significance as in table).
  - P_j (class discipline): 71.494 (∗∗∗).
  - Class-size dummy coefficients reported exactly in the source table for ranges:
    - 16-20: 13.817; 8.922; 8.814; 9.139; 8.140 (columns 1–5).
    - 21-25: 20.064; 14.886; 14.968; 15.218; 14.617.
    - 26-30: 28.855; 22.909; 22.171; 22.388; 21.999.
    - 31-35: 38.853; 32.219; 31.279; 31.473; 31.387.
    - 36-40: 43.945; 38.236; 37.737; 37.491; 37.284.
    - 41-45: 42.150; 36.609; 35.788; 35.461; 34.082.
    - 46-50: 22.375; 17.698; 18.028; 18.400; 16.753.
    - 51+: 13.762; 8.912; 8.865; 9.019; 8.928.
  - Constant and overall sample:
    - Obs. in columns: 782389743639743639743546740340 (as presented in table).
    - R2: .227; .255; .264; .266; .274 (columns 1–5).
- Regression controls (as in source): year, country, country GDP per capita, country share of population under age 15, maternal and paternal education and share of schoolmates whose parents have post-secondary education. Errors clustered by country.
- Statistical significance notation in table: *Significant at 10%; **significant at 5%; ***significant at 1%.

### Robustness checks and interactions
- Robustness checks (Tables A2 and A3 referenced in the source):
  - Fixed class-size estimations broadly similar to general results.
  - The negative effect of σ_P_j (class SD of individual discipline) tends to be larger and more statistically significant in larger classes (except 51+ which is a small outlier).
  - Interaction analyses: interactions between class size and discipline measures show the interaction terms of σ_P_j with class sizes are the most statistically significant; the negative effect of σ_P_j tends to grow with class size.
- Remaining issue: potential endogeneity/omitted-variable bias from unobserved teaching quality correlated with both individual discipline and class size; simulation approach (Section 7) is used to gauge the bias.

### Simulation approach and data-generating parameters (Section 7.1)
- Purpose: investigate whether the model-suggested data-generating process can create observed correlations and assess bias from omitted unobserved teaching quality.
- Synthetic data generation details (parameters and procedures preserved exactly as in source):
  - For N_cnt = 30 countries, an average country discipline drawn from a global Normal distribution: D_cnt ∼ N(D_world = 1.5e, σ_world = 0.15e).
  - Each country has N_stu = 500 students.
  - Student individual discipline draws: D_stu ∼ N(D_cnt, σ_stu = 0.2e). σ_stu is uniform across countries.
  - School management’s noisy information on student discipline: D_sch ∼ N(D_stu, σ_sch = 0.3e). σ_sch uniform across countries.
  - Individual discipline values mapped to probabilities of non-disruption using inverse-logit: P_i = invlogit(D_i). (Note in source: “The inverse-logit transformation was chosen for simplicity, and does not necessarily mimic the exact mapping from students’ individual discipline index to their probability of not disrupting the class.”)
  - Class sizes (n_class) range from CS_min = 10 to CS_max = 50 in increments of CS_inc = 5 (reflects PISA sizing conventions).
  - School ranking index for assignment: R_sch ∼ N(D_stu, σ_rnk = 0.5e). Students sorted by R_sch and allocated to classes starting from the largest possible class size (CS_max = 50).
  - School computes class discipline using its observed P_sch_i: P_sch_class = ∏_{i=1}^{n_class} P_sch_i (product across class members using observed discipline).
  - Class discipline must meet or exceed a country-specific threshold:
    - P_thresh_cnt = 2/3 ∗ ̄P_{CS_min + CS_max 2}_cnt + 1/3 ∗ 0.5(4) (threshold set so higher-discipline countries will have larger classes and stronger classroom discipline; on average most disciplined students placed in largest classes).
  - If class discipline < threshold, class size is lowered by CS_inc = 5 and process repeated until minimum class size CS_min = 10 reached.
  - Classes of size 10 are dropped as outliers (artifact of algorithm); 1,120 synthetic students in size-10 classes removed.
  - Teacher quality draws for each country:
    - Number of prospective teachers per country = ⌈N_stu / CS_min⌉ = 50.
    - Teacher quality TQ ∼ N(μ_TQ = 100, σ_TQ = 15). Schools’ observed teacher quality noisy: TQ_sch ∼ N(TQ, σ_sch_TQ = 0.1σ_TQ). Teachers sorted by observed quality and better teachers assigned to larger classes.
    - All countries have similar number/quality of teachers; countries with larger classes require fewer teachers and therefore enjoy higher average teaching quality.
  - Education production function used (Equation 5 in source):
    - E_{i,j} = TQ^{β0}_j P_i^{β1} P_j^{β2} with β0 = 1, β1 = 1, β2 = 1 (simplifying assumptions).
    - E normalized to mean 500 and standard deviation 100 to mimic PISA; simulated test picks up educational outcomes with measurement error: Ê_{i,j} ∼ N(E_{i,j}, σ_E = 25).
  - Measurement noise for discipline:
    - Individual-discipline test noise: D_test_i ∼ N(D_stu, σ_test = 0.2e).
    - Class-level test noise: D_test_j ∼ N(logit(P_j), 3∗σ_test = 0.6e) (reflects class/school-level measurement noise and use of different indicators for student-level vs class-level discipline).
    - D_test_i and D_test_j normalized to the 0.01–1 range.
- Purpose of simulation: compare estimations including and excluding teacher quality, and compare general estimation results to estimations using fixed class sizes to gauge bias from omitted teaching quality.

*Source: IMF Working Paper chapter section 5.4 (excerpts provided in the supplied content).*

### 7.2 Synthetic Data Analysis

### 7.2 Synthetic Data Analysis

### Replication of PISA correlations in synthetic data
- The synthetically generated data replicates the positive correlations between individual student discipline, classroom discipline, class size and test scores observed in PISA data across countries.
- Mechanism in the synthetic model:
  - Each country draws a discipline country average (Dcnt) around which its students’ discipline level is centered and which determines the class discipline threshold.
  - Countries with a higher discipline draw have on average more disciplined students and larger and more disciplined classes.
  - Larger classes require fewer teachers and, on average, allow schools to select better teachers from the pool of prospective teachers.
  - Better teachers are assigned to larger classes (on average), strengthening the positive correlation between class size and test scores.

### Observing teaching quality in synthetic data
- The synthetic data allows observation of teaching quality (TQsch) and presents correlations between:
  - TQsch and individual discipline
  - TQsch and test scores
  - TQsch and class size
  - TQsch and classroom discipline
- Figure 9 (referenced) presents these correlations for 30 synthetic countries.

### Regression results on synthetic data (Table 4): key findings
- Column (1): Full specification including TQsch (teaching quality as measured by schools)
  - TQsch coefficient: 8.199 (.106) ∗∗∗
  - P_i (ind. disc.) coefficient: 89.730 (3.987) ∗∗∗
  - ̄P_j (class mean ind. disc.) coefficient: 16.089 (.812) ∗∗∗
  - σP_j (class SD ind. disc.) not included in (1)
  - Class size fixed effects (relative to baseline of class size 15) are negative and decreasing (increasing in size) — i.e., once teaching quality is accounted for and for given levels of discipline, increasing class size decreases scholastic achievements.
  - Obs.: 138801; R2: .723
- Columns (2)–(6): Omit teaching quality (TQsch) and mirror potential omitted variable bias similar to actual PISA regressions
  - Column (2): Class size fixed effects only
    - Class size fixed effects are positive and increasing in class size, peaking at class size 40 (mirrors real data peak at class size 36-40).
    - Explanation: correlation of large class membership with higher individual discipline and better teachers; in synthetic data this arises from noise in the sorting process and measurement error in school-perceived discipline.
  - Column (3): Adds P_i (individual discipline)
    - P_i: 121.972 (4.848) ∗∗∗ (Column (3) coefficient shown in table)
    - Individual discipline is positive and highly statistically significant; coefficient is larger than in Column (1) indicating upward bias from omitted teaching quality.
    - Class size fixed effects drop in size after adding P_i due to correlation between P_i and class size.
  - Column (4): Adds ̄P_j (class mean individual discipline)
    - ̄P_j: 22.703 (.969) ∗∗∗ (Column (4) coefficient shown)
    - ̄P_j is positive and highly statistically significant; biased upwards relative to Column (1).
    - Addition of ̄P_j reduces the coefficient of P_i.
  - Column (5): Adds σP_j (class SD individual discipline)
    - σP_j: -33.048 (4.003) ∗∗∗ (Column (5) coefficient shown)
    - σP_j is negative and statistically significant. Unlike actual PISA data, adding σP_j does not reduce the size/significance of ̄P_j in synthetic data; σP_j coefficient is smaller in magnitude (less negative) compared to Column (1) where applicable.
  - Column (6): Adds P_j (class discipline)
    - P_j: 40.682 (2.953) ∗∗∗ (Column (6) coefficient shown)
    - P_j is positive and highly statistically significant. Inclusion of P_j in synthetic data does not reduce the size and statistical significance of ̄P_j and σP_j coefficients; P_j coefficient in Column (6) is smaller compared to Column (1).
    - The omission of teaching quality does not create an upward bias for P_j in this synthetic setting.
  - Obs.: 138801 for all columns (2)–(6); R2 ranges: .566 to .605 across columns (2)–(6).
- Overall interpretation:
  - The full specification (Column (1)) is close to the data generating process and by construction does not suffer from omitted variable bias.
  - Omitting teaching quality leads to upward biases in coefficients on individual and class mean discipline in synthetic regressions, illustrating the potential bias in empirical PISA regressions that cannot observe teaching quality directly.
  - Noise in schools’ measurement of individual discipline produces upward bias in school-perceived discipline (D_sch) for larger classes, explaining positive class size fixed effects when teaching quality is omitted.

### Synthetic data summary by class size (Table A4)
- Summary statistics (exact values as presented):
  - Class Size 15: Freq. 3,105; Dstu 3.96; Dsch 3.89; TQsch 106.38; Test Score 456.92
  - Class Size 20: Freq. 3,720; Dstu 4.15; Dsch 4.17; TQsch 112.74; Test Score 475.85
  - Class Size 25: Freq. 3,425; Dstu 4.32; Dsch 4.38; TQsch 119.00; Test Score 500.98
  - Class Size 30: Freq. 1,650; Dstu 4.49; Dsch 4.58; TQsch 125.79; Test Score 533.42
  - Class Size 35: Freq. 1,085; Dstu 4.77; Dsch 4.84; TQsch 127.61; Test Score 588.28
  - Class Size 40: Freq. 520; Dstu 4.92; Dsch 4.98; TQsch 132.12; Test Score 621.20
  - Class Size 45: Freq. 225; Dstu 5.13; Dsch 5.29; TQsch 130.42; Test Score 635.89
  - Class Size 50: Freq. 150; Dstu 5.34; Dsch 5.49; TQsch 132.41; Test Score 663.76
  - Total: Freq. 13,880; Dstu 4.30; Dsch 4.32; TQsch 116.80; Test Score 503.52
- Notable pattern: D_sch (school-perceived individual student discipline) is larger than D_stu (actual individual student discipline) for all class sizes greater than 15, with particularly large gaps for class sizes 45 and 50.

### Synthesis with theoretical model and implications
- The theoretical ‘education production’ framework developed in Section 3 predicts positive correlations among student discipline, class size, teaching quality and scholastic achievement; synthetic results and PISA empirical results are consistent with these predictions.
- Key empirical regularities supported by both PISA and synthetic data:
  - Individual discipline, average class student discipline, and classroom discipline are positively correlated with PISA math scores.
  - Standard deviation of class student discipline is negatively correlated with PISA math scores.
  - While student discipline correlates with class size, within-country class discipline does not, suggesting sorting of students into classes and class sizes aimed at equalizing discipline levels across classes or meeting country-level discipline thresholds.
  - Large classes up to a normative size are correlated with higher student discipline and higher test scores, consistent with allocation of better teaching resources to larger normative classes.
- The difficulty of observing teaching quality is highlighted as a major hurdle; synthetic data demonstrates how omitting teaching quality biases estimated relationships.

### Policy-relevant conclusions and research directions
- Results indicate student discipline is a crucial factor explaining gaps in scholastic performance across students, schools and countries.
  - Example: Israel and Hong-Kong have similarly large classes but contrasting student discipline levels, proposed as a key reason for their large gap in average PISA scores.
- Policy implications suggested by the analysis:
  - Policies to improve classroom discipline and reduce variance in student discipline could enable effective learning in larger classes and support higher-quality teacher recruitment/allocation.
  - Improving measurement of teaching quality would reduce omitted variable bias in empirical evaluations of class size and discipline effects.
- Suggested avenues for future research:
  - Exploit exogenous measures affecting discipline, class-size and/or teacher allocation to strengthen causal identification.
  - Refine the theoretical model to better match specific educational systems.
  - Study the supply-side effects: how discipline and class size affect the attractiveness of the teaching profession and teacher supply.
- Final conclusion: Understanding the interactions among discipline, class size, and teacher quality is essential for efficient educational resource allocation and for policies that could yield substantial dividends in scholastic achievement.

*Source: 7.2 Synthetic Data Analysis, IMF Working Paper No. WP/2026/105*

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_Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026105-source-pdf.pdf_