STATS NOTES PDF

1. RCT = Random = Fair comparison. 2. FE = Focuses on changes within. 3. IV = Instrument = External helper. 4. DID = Two differences. RD = Right near the line.Randomized Controlled Trials (RCTs) Main Idea: Gold standard for causality. Why it’s important: Random assig...

1. RCT = Random = Fair comparison. 2. FE = Focuses on changes within. 3. IV = Instrument = External helper. 4. DID = Two differences. RD = Right near the line.Randomized Controlled Trials (RCTs) Main Idea: Gold standard for causality. Why it’s important: Random assignment ensures the treatment and control groups are similar, so any difference in outcomes is due to the treatment. How to remember: “RCT = Random = Fair comparison.” Like flipping a coin to decide who gets laptops, so the only difference is the laptops themselves. Causal effects are essential in understanding how policies and interventions affect outcomes. They go beyond simply identifying relationships to determining whether a specific action directly causes a change. For example, in evaluating whether providing free laptops to high school students improves their academic performance and future earnings, it is crucial to establish that the laptops themselves—not other factors—are driving the observed improvements. Without understanding causal effects, policymakers risk making decisions based on spurious correlations, which can lead to ineffective or even harmful interventions. Correlation, while useful for identifying patterns, does not imply causation. A high correlation between two variables, such as owning laptops and improved academic performance, might exist due to a third factor, like socioeconomic status. Causation, on the other hand, establishes a direct link, such as free laptops leading to better grades because they provide access to educational resources. The distinction matters because only causation provides actionable insights for policy design. For example, knowing laptops cause better academic outcomes can justify the program's expansion. To evaluate the true impact of the laptop policy, we calculate the Average Causal Effect (ACE) instead of the simple Causal Effect (CE). ACE measures the average impact of the intervention across a population, recognizing that individual effects vary. For instance, some students may benefit more than others from the laptops due to factors like prior familiarity with technology. ACE captures this variability, providing a more comprehensive understanding of the policy's overall effectiveness. However, measuring causal effects is fraught with challenges, particularly selection bias. Selection bias occurs when the groups being compared differ systematically in ways unrelated to the treatment. For instance, if only motivated students receive laptops, their better performance might reflect their motivation, not the laptops. Selection bias distorts the results, making it impossible to isolate the true causal effect. Randomized Controlled Trials (RCTs) are a powerful method for eliminating selection bias. By randomly assigning students to treatment (laptops) and control (no laptops) groups, we ensure that both groups are statistically similar. This creates a "ceteris paribus" condition, allowing us to attribute any differences in outcomes solely to the laptops. The Law of Large Numbers (LLN) supports this approach, as it ensures that, with a sufficiently large sample, random assignment will balance observed and unobserved characteristics across groups. To design an effective RCT, several steps are essential. First, clearly define the intervention and outcomes. For the laptop program, this means specifying what constitutes "academic performance" (e.g., test scores) and "future earnings" (e.g., average annual income). Second, randomize assignment to treatment and control groups. Statistical software can be used to ensure randomness. Third, implement the intervention and monitor compliance, ensuring students in the treatment group actually use the laptops. Finally, collect and analyze data to compare outcomes. Checking for balance between groups is a critical step in an RCT. This involves comparing baseline characteristics, such as age, gender, and prior academic performance, to confirm similarity. Statistical tests, like t-tests, can identify significant differences. If imbalances are found, additional controls can be included in the analysis to adjust for these disparities. Despite their rigor, RCTs face threats to internal and external validity. Internal validity, which ensures accurate causal inferences within the study's context, can be compromised by issues like non-compliance (e.g., students not using laptops), attrition (e.g., students dropping out), or the Hawthorne effect (e.g., students performing better simply because they know they are being studied). External validity, which concerns the generalizability of findings to other populations, is limited by factors like the representativeness of the sample or the specific conditions of the experiment. RCTs also have limitations. They can be expensive, time-consuming, and sometimes unethical. For instance, it might be impractical to randomly deny laptops to some students if the policy is deemed universally beneficial. Alternative methods can address these challenges. Regression analysis controls for observed differences, while instrumental variables (IV) exploit exogenous variation, such as using proximity to a technology center as an instrument for laptop access. Difference-in-differences (DID) compares changes over time between treatment and control groups, assuming parallel trends. Each method approximates the ceteris paribus condition, albeit with varying degrees of success. Causal inference is the process of drawing conclusions about cause-and-effect relationships from data. It requires careful consideration of assumptions, such as the absence of confounding factors, and rigorous methodologies to ensure valid conclusions. Designing an RCT involves anticipating potential biases, ensuring randomization, and planning for data collection and analysis. For the laptop program, this means considering factors like student access to the internet, which might influence the effectiveness of laptops. In summary, understanding causal effects is fundamental for evidence-based policymaking. RCTs provide a gold standard but are not without challenges. By complementing RCTs with alternative methods and designing experiments with care, researchers can generate robust insights to guide policy decisions. For the laptop program, this means not only demonstrating its effectiveness but also identifying the conditions under which it works best, ensuring maximum benefit for students. Causal effects are essential in understanding how policies and interventions affect outcomes. They go beyond simply identifying relationships to determining whether a specific action directly causes a change. For example, in evaluating whether providing free laptops to high school students improves their academic performance and future earnings, it is crucial to establish that the laptops themselves—not other factors—are driving the observed improvements. Without understanding causal effects, policymakers risk making decisions based on spurious correlations, which can lead to ineffective or even harmful interventions. Correlation, while useful for identifying patterns, does not imply causation. A high correlation between two variables, such as owning laptops and improved academic performance, might exist due to a third factor, like socioeconomic status. Causation, on the other hand, establishes a direct link, such as free laptops leading to better grades because they provide access to educational resources. The distinction matters because only causation provides actionable insights for policy design. For example, knowing laptops cause better academic outcomes can justify the program's expansion. To evaluate the true impact of the laptop policy, we calculate the Average Causal Effect (ACE) instead of the simple Causal Effect (CE). ACE measures the average impact of the intervention across a population, recognizing that individual effects vary. For instance, some students may benefit more than others from the laptops due to factors like prior familiarity with technology. ACE captures this variability, providing a more comprehensive understanding of the policy's overall effectiveness. However, measuring causal effects is fraught with challenges, particularly selection bias. Selection bias occurs when the groups being compared differ systematically in ways unrelated to the treatment. For instance, if only motivated students receive laptops, their better performance might reflect their motivation, not the laptops. Selection bias distorts the results, making it impossible to isolate the true causal effect. Randomized Controlled Trials (RCTs) are a powerful method for eliminating selection bias. By randomly assigning students to treatment (laptops) and control (no laptops) groups, we ensure that both groups are statistically similar. This creates a "ceteris paribus" condition, allowing us to attribute any differences in outcomes solely to the laptops. The Law of Large Numbers (LLN) supports this approach, as it ensures that, with a sufficiently large sample, random assignment will balance observed and unobserved characteristics across groups. To design an effective RCT, several steps are essential. First, clearly define the intervention and outcomes. For the laptop program, this means specifying what constitutes "academic performance" (e.g., test scores) and "future earnings" (e.g., average annual income). Second, randomize assignment to treatment and control groups. Statistical software can be used to ensure randomness. Third, implement the intervention and monitor compliance, ensuring students in the treatment group actually use the laptops. Finally, collect and analyze data to compare outcomes. Checking for balance between groups is a critical step in an RCT. This involves comparing baseline characteristics, such as age, gender, and prior academic performance, to confirm similarity. Statistical tests, like t-tests, can identify significant differences. If imbalances are found, additional controls can be included in the analysis to adjust for these disparities. Despite their rigor, RCTs face threats to internal and external validity. Internal validity, which ensures accurate causal inferences within the study's context, can be compromised by issues like non-compliance (e.g., students not using laptops), attrition (e.g., students dropping out), or the Hawthorne effect (e.g., students performing better simply because they know they are being studied). External validity, which concerns the generalizability of findings to other populations, is limited by factors like the representativeness of the sample or the specific conditions of the experiment. RCTs also have limitations. They can be expensive, time-consuming, and sometimes unethical. For instance, it might be impractical to randomly deny laptops to some students if the policy is deemed universally beneficial. Alternative methods can address these challenges. Regression analysis controls for observed differences, while instrumental variables (IV) exploit exogenous variation, such as using proximity to a technology center as an instrument for laptop access. Difference-in-differences (DID) compares changes over time between treatment and control groups, assuming parallel trends. Each method approximates the ceteris paribus condition, albeit with varying degrees of success. Causal inference is the process of drawing conclusions about cause-and-effect relationships from data. It requires careful consideration of assumptions, such as the absence of confounding factors, and rigorous methodologies to ensure valid conclusions. Designing an RCT involves anticipating potential biases, ensuring randomization, and planning for data collection and analysis. For the laptop program, this means considering factors like student access to the internet, which might influence the effectiveness of laptops. In summary, understanding causal effects is fundamental for evidence-based policymaking. RCTs provide a gold standard but are not without challenges. By complementing RCTs with alternative methods and designing experiments with care, researchers can generate robust insights to guide policy decisions. For the laptop program, this means not only demonstrating its effectiveness but also identifying the conditions under which it works best, ensuring maximum benefit for students. Fixed Effects (FE)  Main Idea: Controls for unchanging differences within entities (like schools or people). Why it’s important: Eliminates bias from factors that don’t change over time, like school culture or teacher quality. How to remember: “FE = Focuses on changes within.” Imagine comparing the same school before and after laptops, ignoring permanent traits like its location. Evaluating whether providing free laptops to high school students improves academic performance and future earnings requires understanding the intricacies of causal relationships. Fixed Effects (FE) regression is a powerful tool in this context, helping isolate the effect of laptops by controlling for unobserved, constant factors that vary across entities, such as schools or regions, but do not change over time. Multiple regression is powerful when you can control for all relevant variables explicitly, especially when those variables are measurable and vary both across entities and over time. For example, if you had data on student motivation, parental income, and internet access, you could use these as controls in a standard regression model. However, when there are unobservable or unmeasurable factors—like a school’s culture or a region’s inherent tech infrastructure—that vary across schools but remain constant over time, multiple regression falls short. This is where fixed effects become necessary. To find the Average Causal Effect (ACE) of the laptop program, we need to compare the difference in outcomes (e.g., test scores) between the treatment group (students with laptops) and the control group (students without laptops), while accounting for unobserved, constant differences between groups. ACE is calculated as the weighted average of causal effects across groups. The weights are proportional to the variance of the treatment variable within each group, ensuring that groups with more variation contribute more to the estimate. When choosing between groups of data to calculate the causal effect, remember to compare groups that are similar except for the treatment variable. For example, in the laptop program, you should compare students in similar socioeconomic conditions where one group received laptops, and the other did not. Including unrelated groups introduces bias and reduces the validity of your causal estimates. Adding controls to the regression helps make the comparison between treatment and control groups fairer. For example, adding variables like student age, teacher quality, or prior academic performance ensures that differences in outcomes are not driven by these factors. Including group-specific dummy variables, such as school dummies, helps account for unobservable characteristics unique to each school that might influence outcomes. Without these controls, the regression becomes biased because it attributes the effect of unmeasured factors to the treatment variable. A regression without group dummies fails to account for unobserved, constant differences across entities, such as school-specific factors. This omission can result in biased estimates because these unobserved factors might correlate with both the treatment (laptops) and the outcome (test scores). For example, schools with better infrastructure might independently influence student performance, making it seem like laptops had a greater effect. Fixed effects regression is particularly useful when your data has multiple time periods and tracks the same entities (e.g., schools or students) over time. It controls for unobserved variables that vary across entities but remain constant within each entity over time. For example, school-specific factors like teacher training quality might affect outcomes but stay constant during the study period. FE regression solves the problem of omitted variable bias by eliminating these constant factors, isolating the true effect of laptops on performance. Fixed effects regression allows you to control for variables that are unmeasurable but constant over time within an entity. For example, you might not have data on a school’s administrative policies, but if these policies are consistent across the study period, FE regression accounts for their impact. The key feature is that the variable must be constant across entities and unchanging over time within the entity. This method, often called "entity demeaning," subtracts the entity’s average outcome and average predictor values over time, effectively removing constant factors. The differencing method is another way to control for constant variables. It involves subtracting the value of a variable at one time period from its value in another. For example, you could calculate the change in test scores for each student before and after receiving laptops. Differencing is most effective when there are only two time periods. In contrast, fixed effects regression is better suited for datasets with multiple time periods, as it leverages more information by controlling for entity-specific constants across all periods. Entity demeaning involves subtracting the average of each variable (e.g., test scores or laptop access) for each entity over time from the variable’s observed values. For example, to demean a school’s test scores, you would calculate the average test score for that school across all time periods and subtract it from each observed test score. This process removes the influence of time-invariant factors specific to the entity, allowing the regression to focus on within-entity variation. If you fail to add a control variable for something that affects the outcome (e.g., internet access affecting test scores), the regression’s assumptions are violated, leading to omitted variable bias. This makes it difficult to isolate the treatment effect, as unaccounted-for factors influence the outcome. Standard errors should be clustered in FE regression because observations within the same entity (e.g., school) are likely correlated over time. Failing to cluster standard errors underestimates variability, leading to overconfidence in the results. Clustering ensures the standard errors account for within-entity correlation, providing more reliable estimates. Despite its strengths, FE regression has limitations. It cannot control for variables that vary over time within entities, such as changes in teacher quality or funding levels. Additionally, it requires variation in the treatment variable within entities over time. If every school receives laptops simultaneously, there’s no within-entity variation to identify the treatment effect. Lastly, it cannot estimate the effect of time-invariant variables, like geographic location, because these are absorbed into the entity-specific fixed effects. In evaluating the laptop program, fixed effects regression provides a robust framework for isolating the treatment effect, ensuring that unobservable, constant factors do not bias the results. By combining FE regression with techniques like differencing and clustered standard errors, we can derive reliable insights to inform education policy and maximize student outcomes. Instrumental Variables (IV) Main Idea: Fixes endogeneity when the treatment isn’t random. Why it’s important: Uses an external “instrument” to isolate variation in the treatment that is unrelated to unobserved factors (e.g., motivation). How to remember: “IV = Instrument = External helper.” Like using whether a school was randomly selected for laptops as a stand-in for who actually got them. Endogeneity is a key challenge in determining causal relationships, especially when evaluating policies like providing free laptops to high school students. An endogenous variable is one that is influenced by factors within the system being studied. In our example, student test scores (the outcome variable) might be influenced by whether the student receives a laptop (the treatment variable), but the decision to provide laptops might also depend on unobserved factors such as the student’s prior performance or socioeconomic background. These unmeasured factors create a correlation between the treatment variable (laptop provision) and the error term in the regression model. Endogeneity bias arises when this correlation exists between the independent variable and the error term, making it difficult to isolate the causal effect of the treatment variable. For example, if laptops are distributed to schools with higher funding, any observed improvement in test scores could be due to better resources, not the laptops themselves. This bias invalidates the regression’s assumption of no correlation between the explanatory variable and the error term, leading to unreliable estimates. One common cause of endogeneity bias is simultaneous causality, where the dependent and independent variables influence each other. For instance, laptops might improve test scores, but students with higher scores might also be more likely to receive laptops. This reciprocal relationship makes it difficult to disentangle the direction of causality. The Instrumental Variables (IV) method is used to address endogeneity bias by identifying an external variable, called an instrument, that influences the treatment variable but is unrelated to the error term. In our example, an ideal instrument might be whether a school is randomly selected to participate in the laptop program. The IV method isolates the variation in the treatment variable (laptop provision) that is exogenous, meaning it is uncorrelated with unobserved factors affecting the outcome. The goal of the IV method is to estimate the causal effect of the treatment variable on the outcome variable by using the instrument to provide a source of variation that mimics a randomized experiment. This allows researchers to approximate the conditions of an RCT even in observational settings. An instrument variable must meet two conditions to be valid. First, it must be relevant, meaning it is strongly correlated with the treatment variable. In the laptop example, the instrument (random school selection) must significantly predict which students receive laptops. Second, it must be exogenous, meaning it is uncorrelated with the error term in the outcome equation. This ensures the instrument affects the outcome only through its influence on the treatment variable, not directly or through other channels. The IV method uses Two-Stage Least Squares (2SLS) to estimate the causal effect. In the first stage, the treatment variable (e.g., laptop provision) is regressed on the instrument and any control variables. This generates predicted values for the treatment variable, which represent the variation explained by the instrument. For example, the predicted laptop provision would depend solely on whether the school was selected for the program. In the second stage, the outcome variable (e.g., test scores) is regressed on these predicted values of the treatment variable and the control variables. This isolates the causal effect of laptop provision on test scores by removing the influence of endogenous factors. To set up an IV regression, start with two equations. The first-stage regression models the treatment variable as a function of the instrument and controls. The second-stage regression replaces the treatment variable with its predicted values from the first stage, using these to estimate the effect on the outcome. For example, the first stage might regress laptop provision on school selection and controls like school size, while the second stage regresses test scores on the predicted laptop provision. Despite its strengths, the IV method has limitations. First, finding a valid instrument can be challenging, as the exogeneity condition is difficult to test directly. If the instrument is weakly correlated with the treatment variable, the estimates become unreliable, a problem known as the weak instrument problem. Additionally, IV estimates are local, meaning they apply only to the subset of individuals whose treatment status is affected by the instrument (e.g., students in schools selected for the laptop program). This limits the generalizability of the findings. In summary, the IV method is a powerful tool for addressing endogeneity bias and estimating causal effects when randomized experiments are not feasible. By carefully selecting and testing instruments, researchers can isolate the causal impact of interventions like laptop provision, providing valuable insights for policymakers. However, the method’s reliance on strong and valid instruments underscores the importance of rigorous design and thoughtful application. Difference-in-Differences (DID) Main Idea: Compares changes over time between treatment and control groups. Why it’s important: Controls for common trends affecting both groups, so the difference in changes reflects the treatment effect. How to remember: “DID = Two differences.” First difference: before vs. after laptops; second difference: treatment vs. control schools. The Difference-in-Differences (DID) regression method is a widely used approach for estimating causal effects when a randomized controlled trial (RCT) is not feasible. It is particularly useful for evaluating policies like providing free laptops to high school students. The goal of the DID method is to determine whether the intervention—distributing laptops—leads to measurable improvements in outcomes, such as academic performance or future earnings. The key idea behind the DID method is to compare changes in outcomes over time between a treatment group (students receiving laptops) and a control group (students not receiving laptops). The logic of the DID approach lies in isolating the treatment effect by leveraging the assumption that, in the absence of the intervention, the treatment and control groups would have followed parallel trends in outcomes. This means any divergence in outcomes after the intervention can be attributed to the treatment itself, not to other factors. To implement DID, data is collected before and after the intervention for both the treatment and control groups. For example, test scores are measured for students in schools that receive laptops and those that do not, both before and after the laptop program. The DID method calculates the treatment effect by subtracting the change in outcomes in the control group from the change in outcomes in the treatment group. The formula involves taking the difference in the average outcome for the treatment group before and after the intervention, then subtracting the difference in the average outcome for the control group before and after the intervention. This double-differencing isolates the treatment effect while controlling for time-invariant differences between groups and common time effects. The DID method relies on the common trend assumption, which states that, in the absence of the treatment, the treatment and control groups would have experienced the same change in outcomes over time. For example, if test scores were improving at the same rate in both groups before the laptops were distributed, this trend would have continued without the intervention. Testing this assumption involves examining pre-intervention trends in the outcome variable to ensure they are parallel. If the trends diverge before the intervention, the assumption is violated, and the DID estimates may be biased. The DID estimator provides a numerical value for the treatment effect. To calculate it, you start with the change in the treatment group’s outcome over time, subtract the change in the control group’s outcome, and then divide the result by the number of observations. This gives the average causal effect of the intervention. In a DID regression equation, the outcome variable (e.g., test scores) is regressed on three key variables: a dummy variable for the treatment group, a dummy variable for the post-treatment period, and an interaction term between the two. The regression equation typically includes control variables to account for other factors that might influence the outcome. Each beta in the equation has a specific interpretation: The first beta captures the baseline difference between the treatment and control groups. The second beta represents the change in the outcome for the control group over time. The third beta, the coefficient on the interaction term, is the DID estimate of the treatment effect, capturing the difference in changes between the groups. To set up the DID regression step by step, you first create the necessary variables: 1. A dummy variable indicating membership in the treatment group (e.g., students who receive laptops). 2. A dummy variable indicating the post-treatment period (e.g., after laptops are distributed). 3. An interaction term multiplying the treatment group dummy by the post-treatment dummy. 4. Additional control variables to adjust for factors like socioeconomic status, teacher quality, or school infrastructure. You then run the regression using statistical software, interpret the coefficients, and check whether the interaction term is significant. Adding control variables helps ensure the regression is less biased and more comparable by accounting for factors that might affect the outcome independently of the treatment. Testing the parallel trends assumption involves graphing the pre-treatment trends for both groups and visually or statistically examining whether they follow a similar trajectory. If they do, it strengthens confidence in the validity of the DID method. However, the DID method has limitations. The common trend assumption cannot be directly tested for the post-treatment period, and violations of this assumption can bias the results. Additionally, the method is sensitive to external shocks or events that affect the treatment and control groups differently during the study period. For example, if a new teaching method is introduced in one group but not the other, this could confound the results. Despite its limitations, the DID method is a powerful tool for evaluating causal effects in policy contexts like the laptop program. By carefully testing assumptions, adding control variables, and ensuring proper data collection, researchers can derive robust estimates of the program’s impact on student outcomes. Regression Discontinuity (RD) Main Idea: Focuses on a cutoff to estimate causal effects. Why it’s important: Compares people just above and below the threshold (e.g., GPA) who are otherwise very similar. How to remember: “RD = Right near the line.” Like comparing students barely above or below the GPA needed to get laptops. The Regression Discontinuity Design (RDD) is a powerful statistical method used to estimate the causal effect of an intervention when treatment assignment is based on a predetermined threshold of an observable variable. For example, imagine a government policy that provides free laptops to high school students who achieve a minimum GPA of 3.0. RDD can help evaluate whether receiving the laptops improves student performance and future earnings by focusing on students whose GPA is near the 3.0 threshold. RDD is particularly useful in settings where random assignment is not feasible but treatment is assigned based on a cutoff point. This method exploits the discontinuity in treatment assignment at the threshold to identify causal effects. The key logic of RDD is that students just above and below the GPA threshold are likely to be similar in all respects except for the treatment. By comparing their outcomes, we can estimate the treatment effect. The main assumption of RDD is that there are no systematic differences between individuals on either side of the threshold, apart from the treatment. For the laptop program, this means students with a GPA of 3.01 are assumed to be similar to those with a GPA of 2.99 in characteristics like motivation or prior academic ability. If this assumption holds, any observed difference in outcomes can be attributed to the treatment itself. The running variable in RDD is the variable that determines treatment assignment. In this case, it is the student’s GPA. The threshold or cutoff value (3.0 GPA) determines who receives the treatment (laptops). The design leverages this clear rule to estimate causal effects. There are two types of RDD: sharp and fuzzy. In a sharp RDD, treatment assignment is strictly determined by whether the running variable crosses the threshold. For example, all students with a GPA of 3.0 or higher receive laptops, and no one below the threshold does. In a fuzzy RDD, crossing the threshold increases the probability of treatment but does not guarantee it. For instance, students with a GPA of 3.0 or higher might be more likely to receive laptops, but some students below the threshold could also receive them due to special circumstances. To estimate the treatment effect using sharp RDD, you compare the average outcomes (e.g., test scores) of students just above and below the threshold. The treatment effect is the difference in these averages. For fuzzy RDD, you use the probability of treatment as an instrument to estimate the causal effect of the treatment on the outcome. This involves two steps: first, estimating the relationship between the running variable and the probability of treatment, and second, estimating the relationship between the treatment and the outcome, using the predicted treatment probabilities from the first step. RDD can also be implemented through regression analysis. The basic regression equation includes the outcome variable (e.g., test scores) as a function of the treatment variable (laptop provision), the running variable (GPA), and an interaction term between the running variable and the treatment. The coefficients in this regression help isolate the treatment effect. In a fuzzy RDD, you include the probability of treatment as an instrument in a two-stage least squares (2SLS) regression. To set up an RDD regression step by step: 1. Define the running variable and threshold: Identify the variable that determines treatment (GPA) and the cutoff point (3.0). 2. Create a treatment indicator: Generate a dummy variable that equals 1 if the running variable is above the threshold and 0 otherwise. 3. Include control variables: Add other relevant variables to improve precision and account for observable differences. 4. Fit the model: Use regression to estimate the relationship between the running variable, treatment, and outcome. 5. Interpret the coefficients: The coefficient on the treatment variable represents the causal effect of the treatment. In RDD, omitted variable bias (OVB) is minimized because the treatment and control groups are assumed to be similar near the threshold. However, OVB can arise if there are unmeasured factors correlated with both the running variable and the outcome. For example, if students near the GPA threshold differ in unobservable ways like access to private tutors, this could confound the results. To address this, researchers use narrow bandwidths around the threshold, ensuring that comparisons are made only between very similar individuals. Additionally, sensitivity analyses are conducted by varying the bandwidth to confirm the robustness of the results. In summary, RDD is a rigorous method for estimating causal effects in non-randomized settings. By focusing on the discontinuity at a threshold, it allows researchers to isolate the treatment effect while controlling for confounding factors. For the laptop program, RDD provides insights into whether the intervention improves outcomes and ensures that the findings are both credible and actionable.