Author: Brandon Hao
Date: March 2, 2024
library(haven)
library(dplyr)
library(AER)
library(tidyr)Problem 1
Part A
## The four subpopulations of the sample regarding Tennessee students
## are categorized into Compliers, Always-Takers, Never-Takers, and Defiers.
## Students who maintain their initial random assignment throughout the study
## are Compliers. In other words, those students who either will attend small
## classes in first grade (D=1) if assigned to small classes in
## kindergarten (Z=1) and the same pattern for those in regular
## classes (Z=0 then D=0). Always-takers are the students who attend small
## classes regardless of their initial assignment. According to this specific
## study, Always-takers are those who result in small classes for first
## grade (D=1) despite being assigned to regular classes in
## kindergarten (Z=0). On the other hand, students who
## attend regular classes independent of their initial assignment are
## known as Never-takers. As a result, Never-takers are students who are
## in regular classes in first grade (D=0) even those who were assigned
## to small classes in kindergarten (Z=1). Defiers, which are assumed absent
## in this study, are students who exhibit the opposite towards their
## assignment by either those being assigned to small kindergarten
## classes (Z=1) attending regular classes in first grade (D=0) or
## vice versa (Z=0 then D=1).Part B
##
## Always-taker Complier Never-taker
## 0.09507830 0.88798338 0.01693832## The Compliers can be determined by obtaining the proportion
## of students whose first-grade placement matches their kindergarten
## assignment. Such value can be computed by identifying students who
## were assigned to small classes and attended small classes
## (Z=1 than D=1) and those assigned to regular classes and attended
## regular classes (Z=0 than D=0), which the sum
## will be divided by the total number of students. Consequently,
## Always-takers are determined by the proportion of students assigned
## to regular classes in kindergarten but attended small classes in first
## grade (Z=0 than D=1) while Never-takers are that of students assigned
## to small classes in kindergarten but attended regular classes in
## first grade (Z=1 than D=0).Problem 2
Part A
## Since I am using RMarkdown, I will verbally state the required
## assumptions for the Instrumental Variable effect. Among the six
## criteria, each of them will be required except for the constant
## treatment effects, which is zero covariance between the effect
## of Z on D and the effect of D on Y. The Stable Unit Treatment Value
## Assumption in this context indicates that the achievement of a student
## in first grade math is affected only by their own class size as opposed to
## that of other students. In addition, there are only a single designated
## size of the small class treatment, as opposed to a diverse variety.
## Exogenity on Z is an essential constraint that ensures that the initial
## class assignment is independent of any potential factors that can
## determine first grade math performance. In other words, any observed
## effects should only be attributed to class size rather than any
## preexisting differences among students. Exclusion Restriction, by
## the same token, denotes that Z affects the outcome (first grade math
## achievement) only through class size in Grade 1 without any confounding
## variable. The influence of kindergarten class assignment towards first
## grade math scores must only be associated with whether they would end
## up in a small class for the following year. The fourth assumption, the
## Nonzero Effect of Z on D, will emphasize that the instrument (kindergarten
## class assignment) will influence the treament variable (first grade class
## size). Such aspect implies that students assigned to small classes in
## kindergarten pose an advantage towards staying in them for first grade
## unlike their regular class counterparts. Before anything else, Monotonicity
## is important by the assignment of treatment can affect people in one
## direction with the exception of defiers. For instance, a student who is
## placed in a small class in kindergarten would generally not make them
## less likely to be in a small class in first grade.Part B
## Among those criteria, the only one that is completely verifiable through
## empiricism is the Nonzero Effect of Z on D with the process of testing
## whether the instrument (kindergarten class assignment) predicts
## the treament significantly at the 0.05 level by conducting regression
## in IV estimation. The Exogenity of Z is somewhat verifiable since the random
## assignment of class size in kindergarten can ensure such constraint.
## However, testing balance between treatment and control groups on
## observed covariates is necessary to satisfy this assumption. However,
## SUTVA is extremely difficult to be implemented since it involves
## confirming that no spillover effects exist between subjects, which
## hypothetically involves highly detailed data regarding interactions
## between every pair of students. The Exclusion Restriction aspect
## cannot inherently be verified since it involves unobservable
## counterfactuals by the theoretical analysis of the mechanism
## through which Z affects Y. However, improvement towards Monotonicity
## can be achieved if the study design can limit the possibility of
## defiers with methods such as strict enforcement of treatment
## assignment.Part C
## The plausibility of the unverifiable assumptions underlying IV
## analysis is relevant towards the context of the study including societal,
## economic, and individual behavior factors. SUTVA can be plausible in
## controlled educational environments where the study design minimizes
## interaction effects between different treatments. Exogenity of Z is
## certainly plausible by guaranteeing that the instrument is not
## correlated with any unobserved determinants of the outcome. Without
## a doubt, Monotonity is generally plausible by which compliance behavior
## can be modeled to follow the assignment direction. The only assumption
## that may be uncertain in terms of plausibility is the Exclusion
## Restriction which usually may require either the controlling of
## confounding variables or the strict implementation of random assignment.Problem 3
Part A
##
## Pearson's Chi-squared test
##
## data: contingency_table
## X-squared = 34.071, df = 2, p-value = 3.996e-08## As a result, the extremely low p-value indicates that the
## distribution of the education levels of teachers occurring by
## chance is highly unlikely. In other words, the assignment of
## class size appears to be strongly associated with whether the
## student is taught by a teacher with a higher degree which contrasts
## the prior assumption on equal likelihood across both class types.
## As a result, this finding can indicate that teacher qualifications
## can be a confounding factor that can certainly influence the outcome
## of first grade math performance instead of solely class size.Part B
## The proposal of the randomized treatment assignment of class
## size in kindergarten (Z) as an instrument in determining the average
## causal effect of kindergarten teacher degree (G) on kindergarten math
## achievement can raise potential concerns. Before anything else, in
## order for Z to become a legitimate instrument, there must be a
## strong correlation with G. Since the study design did not
## explicity state that teachers with higher degrees were more likely
## to teach in certain sizes of classes, there may not be a notable
## association among them. Aside from that, exogeneity may not be
## fulfilled since situations such as teachers either being assigned
## or choosing small classes because of external benefits can
## certainly occur. In addition, if the class size is truly a significant
## predictor towards student math achievement through factors besides
## teacher qualifications, using Z as an instrument for G will not
## isolate the effect on achievement by the education level of the
## teacher.Part C
##
## Call:
## ivreg(formula = tmathssk ~ G | Z, data = data_filtered)
##
## Residuals:
## Min 1Q Median 3Q Max
## -162.789 -43.789 -3.789 39.293 181.293
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 516.789 7.369 70.13 < 2e-16 ***
## G -72.082 20.833 -3.46 0.000546 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 58.02 on 3679 degrees of freedom
## Multiple R-Squared: -0.5849, Adjusted R-squared: -0.5853
## Wald test: 11.97 on 1 and 3679 DF, p-value: 0.0005462##
## Call:
## lm(formula = tmathssk ~ G, data = data_filtered)
##
## Residuals:
## Min 1Q Median 3Q Max
## -136.884 -31.884 -3.665 29.116 135.116
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 490.8837 0.9426 520.798 <2e-16 ***
## G 1.7817 1.5916 1.119 0.263
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 46.08 on 3679 degrees of freedom
## Multiple R-squared: 0.0003405, Adjusted R-squared: 6.879e-05
## F-statistic: 1.253 on 1 and 3679 DF, p-value: 0.263## A notable difference was observed in comparing the instrumental
## variable and ordinary least squares regression analyses regarding
## the association of kindergarten teacher education level (G) on math
## achievement. The IV estimate indicates that higher teacher
## qualifications are significantly associated with lower math scores.
## As a result, such finding starkly contrasts the findings from the
## OLS model with a p-value of approximately 0.263. The discrepancy
## between both methods deeply questions the validity in using
## kindergarten class size (Z) as an instrument for teacher
## qualifications and the effectiveness of the exclusion restriction
## criterion. Furthermore, the negative IV estimate might either
## indicate a flawed instrument or any unexpected effects between class
## size, teacher qualifications, and student outcomes that are not captured
## by the model. As aforementioned, the contrast undeniably emphasizes
## the importance of thoroughly analyzing instrument validity along with
## societal implications of model assumptions in causal inference
## analyses within the realm of educational research.Problem 4
Part A
## The fuzzy regression discontinuity design, as proposed by Statistician
## 1, may pose limitations in the context of this study. In most casees, RDD
## is the most appropriate method when there is a exogenously determined boundary
## that strictly indicates the treatment assignment. However, in Project STAR,
## the overlap in class size ranges (12-17 for small and 15-28 for regular)
## can potentially result in ambiguity in the experimental and
## control groups when being defined solely by the cutoff. In addition, the
## three-step analysis attempts to isolate the effect of kindergarten class
## type on achievement (Y) to actual class size (D) along with computing
## the ratio to estimate the causal effect. However, this method assumes
## linearity in those relationships which may not fully capture the complexity
## of class size effects on achievement in this study, especially if either the
## relationship is nonlinear or that any confounding variables are present.Part B
## On the other hand, the suggestion of regarding randomly assigned
## kindergarten class type (Z) as an instrumental variable for the
## actual class size (D) of estimating math achievement (Y) by Statistician 2
## is certainly plausible. Without a doubt, the random assignment to either
## the treatment or control group is a legitimate instrument as it is
## relevant to the actual class size as well as demonstrating exogeneity
## by supposedly being independent of any confounders that can possibly affect
## math achievement. In other words, such approach provides a causal
## interpretation of the effect of class size on math achievement by
## attributing the variation in class size by random assignment without any
## external factors.Part C
##
## Call:
## ivreg(formula = gktmathss ~ predicted_class_size | gkclasstype,
## data = fourth)
##
## Residuals:
## Min 1Q Median 3Q Max
## -201.553 -32.627 -3.627 27.410 144.373
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 506.9766 4.2414 119.530 < 2e-16 ***
## predicted_class_size -1.0618 0.2062 -5.148 2.72e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 47.59 on 5869 degrees of freedom
## Multiple R-Squared: 0.004495, Adjusted R-squared: 0.004326
## Wald test: 26.5 on 1 and 5869 DF, p-value: 2.718e-07## The estimated effect of reducing class size by 10 students on
## kindergarten math achievement is -10.61759## The effect size of reducing class size by 10 students on
## kindergarten math achievement is -0.2226009## In the IV regression analysis where class type is an instrument
## for predicted class size, increasing class size has a significant negative
## effect on kindergarten math scores. More specifically, each additional
## student in the class will a decrease of 1.0618 points in math scores on
## average. Therefore, smaller class sizes tend to have a positive
## relationship towards math achievement. Thus, the significance of the
## coefficient emphasies the importance of class size in educational
## outcomes.Problem 5
Part A
## If public school enrollment (Z) is regarded as an instrumental
## variable for first grade class size (D in the ECLS-K study, there will
## potentially be areas of concerns. Most of the shortfalls, as a result,
## are mainly centered on the validity criteria including relevance and
## exogeneity. Despite that relevance of the enrollment variable influencing
## class size, the exogeneity assumption that enrollment impacting math
## achievement solely through class size can be flawed. For instnace,
## schools with various enrollment levels may have differences in external
## factors such as teacher quality, resource allocation, as well as policies
## that independently affect math achievement. Aside from that, the
## direct relationship between greater enrollment number and class size
## may not be uniform across all schools, thus the consistency of the
## instrument is questioned. Without a doubt, the process of isolating the
## causal effect of class size on math achievement with enrollment is
## further complicated.Part B
## If the schools that increase Grade 1 class size in response to an
## increase in enrollment are also the ones where an increase in class
## size would be particularly harmful to math learning, the IV analysis
## will most likely produce a negatively biased estimate of the effect
## of class size on math achievement. The most probable reason is that
## the instrument would not only be predicting class size but will also
## be selecting for schools which class size has a disproportionately
## negative relationship on math achievement. In other words, the instrument
## will be capturing not only the effect of class size on achievement
## but also the compounded effect of certain schools managing enrollment
## increases along with their specific impacts on academic achievement.Problem 6
Part A
##
## Call:
## lm(formula = C4R2MSCL ~ A4CLSIZE + GENDER + RACE + w1sesl + B4YRSTC,
## data = fifth2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -48.718 -9.437 -0.685 8.650 53.010
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 59.1356145 0.9368480 63.122 < 2e-16 ***
## A4CLSIZE -0.0055803 0.0388072 -0.144 0.886
## GENDER -1.2065095 0.2896384 -4.166 3.13e-05 ***
## RACE -0.7944778 0.0822830 -9.655 < 2e-16 ***
## w1sesl 7.3134366 0.1902451 38.442 < 2e-16 ***
## B4YRSTC -0.0005837 0.0143489 -0.041 0.968
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.13 on 9532 degrees of freedom
## (1031 observations deleted due to missingness)
## Multiple R-squared: 0.1599, Adjusted R-squared: 0.1594
## F-statistic: 362.8 on 5 and 9532 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = C4R2MSCL ~ D_hat + GENDER + RACE + w1sesl + B4YRSTC,
## data = fifth2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -48.383 -9.406 -0.662 8.592 51.628
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 50.961304 4.218634 12.080 < 2e-16 ***
## D_hat 0.407495 0.210248 1.938 0.0526 .
## GENDER -1.267530 0.305826 -4.145 3.44e-05 ***
## RACE -0.831344 0.087334 -9.519 < 2e-16 ***
## w1sesl 7.095378 0.211714 33.514 < 2e-16 ***
## B4YRSTC -0.009581 0.015431 -0.621 0.5347
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.11 on 8653 degrees of freedom
## (1910 observations deleted due to missingness)
## Multiple R-squared: 0.1576, Adjusted R-squared: 0.1571
## F-statistic: 323.7 on 5 and 8653 DF, p-value: < 2.2e-16## Without a doubt, this section compares an OLS regression of first grade
## math achievement on actual class size by accounting for gender, race,
## socioeconomic status, and experience of teacher. In addition, an IV
## regression is performed by involving predicted class size (D_hat) as
## a proxy for actual class size. The OLS results suggest that there is no
## significant effect of actual class size on math achievement. Such result
## suggesting that class size tends to have a negligible impact on student
## performance when not considering potential endogeneity. On the other hand,
## the IV approach indicates a nearly significant positive effect of class size
## on math achievement with a p-value of 0.0526. The sharp contrast in findings
## confirms the complexity of measuring class size effects on achievement and
## suggests that the relationship might be confounded by underlying factors
## not captured in a typical OLS model. Both models denote the significant
## influence of gender, race, and SES on math achievement which represents
## the complicated nature of educational outcomes.Part B
## The second analysis that involves a two-stage least squares approach
## with public school enrollment as the IV for Grade 1 class size along with
## adjusting for covariates is likely to produce less biased estimates
## compared to the first analysis (OLS regression) if certain conditions are
## satisfied. Before anything else, the instrument must be relevant by
## which it must be correlated with the endogenous explanatory
## variable (kindergarten class size) and adhere to the exogeneity condition.
## This implies that it is not correlated with the error term of
## the outcome except through the endogenous variable. As a result, this
## approach detects potential endogeneity in the class size variable by
## isolating the exogenous variation in class size because of the instrument.
## In contrast to OLS, the 2SLS method obtains an improved estimate of the
## causal effect of class size on math achievement.Problem 7
Part A
The pair of regression models can be constructed as follows.
The first stage model can be denoted as
\[D = \alpha_0 + \alpha_1 Z + \gamma X + \epsilon,\]
while the second stage model is
\[Y = \beta_0 + \beta_1^{\text{IV}} D + \delta X + \nu\]
##
## Call:
## ivreg(formula = C4R2MSCL ~ A4CLSIZE + GENDER + RACE + w1sesl +
## B4YRSTC | s4anumch + GENDER + RACE + w1sesl + B4YRSTC, data = fifth2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.9741 -9.4992 -0.6371 8.6719 52.2437
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 50.961304 4.246254 12.001 < 2e-16 ***
## A4CLSIZE 0.407495 0.211624 1.926 0.0542 .
## GENDER -1.267530 0.307829 -4.118 3.86e-05 ***
## RACE -0.831344 0.087906 -9.457 < 2e-16 ***
## w1sesl 7.095378 0.213100 33.296 < 2e-16 ***
## B4YRSTC -0.009581 0.015532 -0.617 0.5373
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.2 on 8653 degrees of freedom
## Multiple R-Squared: 0.1465, Adjusted R-squared: 0.146
## Wald test: 319.5 on 5 and 8653 DF, p-value: < 2.2e-16## Coefficient estimate for D: 0.4074949
## Standard error: 0.2116241
## t statistic: 1.92556
## p-value: 0.05419204## As a result, the effect on class size reduction on first grade
## math achievement is technically not significant at the 0.05 alpha
## level, despite having a slight positive effect. However, the null
## hypothesis will still not be rejected.Part B
## Estimated average effect of reducing class size by
## 10 students on first-grade math achievement: 4.074949## Effect size estimate: 0.2640362## Similar to the previous section, there is not sufficient evidence to
## suggest that there are significant results regarding the point estimate.
## In addition, the effect size is relatively small which further indicates
## that the relationship between class size reduction and Grade 1 math
## scores is relatively weak.Problem 8
Part A
The modified DID model with linear covariance adjustment for pretreatment measures can be expressed as:
\[Y_{it} = \alpha + \beta_1 \text{Time}_{t} + \beta_2 \text{SmallClass}_{i} + \delta (\text{Time}_{t} \times \text{SmallClass}_{i}) + \gamma X_{i} + \epsilon_{it}\]
Where \(X_{i}\) includes child gender, race, SES, and math achievement at kindergarten entry (C1R2MSCL) as covariates.
This model captures the average treatment effect on the treated (ATT) of being in a small class on first-grade math achievement, adjusting for student and teacher characteristics to control for potential confounders.
A key component of identification is the Parallel Trends Assumption where the average outcomes regarding both treatment and control groups would have followed parallel directions over time assuming the absence of small class sizes. In addition, the requirement of where students did not change their behavior in anticipation of the treatment before actual implementation must be satisfied.
The ATT can be obtained by the differences-in-differences (DID) analysis since the method involves controlling for unobserved and time-invariant differences between the small and regular class students. Such method can be conducted by analyzing the differential effect of being in a small class on the change in math achievement scores from kindergarten to first grade. Assuming those assumptions are valid, the DID estimator will provide an unbiased estimate of the causal effect in the relationship of attending small classes on math scores.
##
## Call:
## lm(formula = score ~ A4CLSIZE_binary * time, data = fifth2_longer)
##
## Residuals:
## Min 1Q Median 3Q Max
## -45.793 -8.807 -1.188 7.317 66.031
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 54.5270 0.1528 356.913 <2e-16 ***
## A4CLSIZE_binarysmall -0.5235 0.2925 -1.790 0.0735 .
## timepre -23.2241 0.2161 -107.492 <2e-16 ***
## A4CLSIZE_binarysmall:timepre -0.2802 0.4136 -0.677 0.4982
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.35 on 20990 degrees of freedom
## Multiple R-squared: 0.4326, Adjusted R-squared: 0.4326
## F-statistic: 5335 on 3 and 20990 DF, p-value: < 2.2e-16## The regression analysis examines the effect of class size and time
## (timepre) on the scores along with the interaction between class size
## and time. The results indicate that there is not sufficient evidence that
## small class size alone leads to a difference in scores since the p-value
## is not significant at the 0.05 level. On the other hand, the coefficient
## for time is certainly significant with scores being 23.2241 lower in
## the pre-treatment (kindergarten) period, which suggests a notable increase
## in scores over time regardless of class size. Before anythign else, the
## interaction term is not statistically significant by which the difference
## in scores between small and regular classes does not significantly influence
## the time variable.Part B
The modified DID model with linear covariance adjustment for pretreatment measures can be expressed as:
\[ Y_{it} = \alpha + \beta_1 \text{Time}_{t} + \beta_2 \text{SmallClass}_{i} + \delta (\text{Time}_{t} \times \text{SmallClass}_{i}) + \gamma X_{i} + \epsilon_{it} \]
Where \(X_{i}\) includes child gender, race, SES, and math achievement at kindergarten entry (C1R2MSCL) as covariates. This objective of the model is to determine the average treatment effect on the treated (ATT) of being in a small versus a regular class on first grade math achievement by pertaining to a set of covariates in order to control for potential confounding variables.
The modified Difference-in-Differences (DID) model features the linear covariance adjustments for pretreatment measures and refines the identification assumptions required for an unbiased estimate of the treatment effect. Especially with conditioning on the various pretreatment factors, the model accounts for potential confounders that could influence the treatment assignment as well as the outcome. Such adjustment further justifies the parallel trends assumption by requiring it to apply only within groups defined by these covariates, rather than across the entire sample unconditionally. Furthermore, this approach serves to minimize possible bias from pretreatment differences between the students in regular and small classes whcih enhances the reliability of causal inferences stemming from DID analysis. As a result, the assumption that any differences in outcomes between the treatment and control groups can be attributed to the treatment effect since these covariates have been accounted for.
##
## Call:
## lm(formula = score ~ A4CLSIZE_binary * time + GENDER + RACE +
## w1sesl + C1R2MSCL, data = fifth2_longer)
##
## Residuals:
## Min 1Q Median 3Q Max
## -62.120 -5.398 -0.512 4.966 51.759
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 32.806691 0.317853 103.213 < 2e-16 ***
## A4CLSIZE_binarysmall 0.274454 0.214258 1.281 0.200
## timepre -23.320597 0.158770 -146.882 < 2e-16 ***
## GENDER -0.664488 0.135209 -4.915 8.98e-07 ***
## RACE -0.247987 0.038041 -6.519 7.27e-11 ***
## w1sesl 1.544282 0.097072 15.909 < 2e-16 ***
## C1R2MSCL 1.105983 0.008913 124.083 < 2e-16 ***
## A4CLSIZE_binarysmall:timepre -0.191055 0.302275 -0.632 0.527
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 9.049 on 17934 degrees of freedom
## (3052 observations deleted due to missingness)
## Multiple R-squared: 0.7426, Adjusted R-squared: 0.7425
## F-statistic: 7390 on 7 and 17934 DF, p-value: < 2.2e-16## Similarly, the regression analysis examines the multifaceted
## relationship of class size, time, gender, race, socioeconomic status, and
## initial math scores on student achievement. Even though class size along
## with its interaction with time are not significantly associated with
## achievement outcomes, predictors including time, gender, race, SES,
## and initial math scores were. Furthermore, time has a considerably
## negative effect with a significant decrease in scores from pre to post.
## SES and initial math scores, on the other hand, have positively influence
## achievement which certainly emphasizes their importance in the educational
## enviroment. On top of that, demographic factors in academic achievement
## may be considered since gender and race were significant predictors.Part C
## Including school enrollment (s4anumch) as an additional baseline
## covariate in the revised DID analysis can be justified regarding several
## reasons. Before anything else, student enrollment may capture variations
## in educational resources, school policies, and learning environments that
## could affect student outcomes regardless of class size effect. Schools
## with different enrollment sizes could exhibit varied teacher-student ratios,
## extracurricular involvement, and administrative support, which all can
## certainly affect math achievement. In addition, enrollment size could
## indirectly determine unnoticed aspects including, but not limited to,
## community involvement as well as district management, that are necessary
## for student learning. Involving such covariate can certainly account for
## these potential confounders, which can guarantee a accurate estimation
## of the treatment effect. On top of that, if the impact of reducing class
## size on math achievement varies by school from the difference in policy
## implementation considering the enrollment variable can potentially
## identify this source of heterogeneity. Without a doubt, conditioning
## on school enrollment will obtain an improved controlling for both
## observed and unobserved confounding influences on math achievement
## with the strengthing of the estimated treatment effect.