Clustered SE Discrepancy: DuckRegression Vs. Statsmodels

When delving into the world of econometrics, one often encounters the need to account for clustered data. Clustered data arises when observations are not entirely independent, such as students within the same classroom or individuals within the same geographic region. Failing to account for this clustering can lead to underestimated standard errors and, consequently, flawed inferences. In this context, a perplexing issue arises: why do clustered standard errors differ significantly between DuckRegression and statsmodels, two popular Python libraries for statistical modeling?

Understanding the Discrepancy in Clustered Standard Errors

In econometrics, addressing clustered data is crucial for accurate statistical inference. When observations within clusters are correlated, standard Ordinary Least Squares (OLS) estimation can underestimate standard errors, leading to inflated significance levels and potentially incorrect conclusions. Both statsmodels and DuckRegression offer methods for calculating clustered standard errors, but a non-trivial discrepancy between their results raises important questions about the underlying methodologies and their implications.

The core issue lies in the fact that while both libraries aim to provide consistent and reliable standard error estimates for clustered data, their approaches and implementations may differ in subtle but significant ways. These differences can stem from variations in the formulas used, the handling of degrees of freedom, or the specific algorithms employed for the calculations. Therefore, understanding the source of these discrepancies is essential for researchers and practitioners who rely on these tools for their analyses.

To illustrate this problem, consider a scenario where we are analyzing student test scores, and students are nested within classrooms. If students within the same classroom tend to perform similarly, their test scores will be correlated. Ignoring this correlation and applying standard OLS regression would likely result in standard errors that are too small, making us more likely to conclude that a particular intervention or policy has a significant effect when it may not.

In such cases, clustered standard errors provide a more accurate measure of uncertainty by accounting for the within-cluster correlation. However, if different software packages or methods yield different clustered standard errors, it becomes challenging to determine which estimate to trust. This is precisely the issue at hand when comparing DuckRegression and statsmodels.

The consequences of using incorrect standard errors can be substantial. In academic research, it can lead to publishing false positives, where a statistically significant effect is reported when none exists. In policy-making, it can result in the implementation of ineffective or even harmful interventions. Therefore, resolving the discrepancy between DuckRegression and statsmodels is not just an academic exercise; it has real-world implications.

Expected vs. Actual Behavior: A Closer Look

Ideally, when running OLS with clustered data, we expect the point estimates (i.e., the estimated coefficients) to be similar across different methods. This is because the point estimates are primarily determined by the data and the model specification, and the clustering should not fundamentally alter the estimated relationships between variables. Furthermore, we expect the standard errors to match closely, or at least differ by a negligible amount. Similar standard errors across methods would provide confidence in the robustness and reliability of the results.

However, the actual behavior observed in this scenario deviates from these expectations. While the point estimates from DuckRegression and statsmodels do match (or differ by a tiny amount, which is often attributable to numerical precision), the standard errors exhibit a non-trivial difference. For instance, in a specific run, statsmodels might report a standard error of approximately 0.032 for the slope coefficient, whereas DuckRegression reports a standard error of approximately 0.0251. This discrepancy, while seemingly small, can have significant implications for hypothesis testing and confidence interval construction.

To put this into perspective, a difference of 0.007 in standard error can alter the t-statistic and p-value associated with a coefficient. This, in turn, can affect whether we reject or fail to reject the null hypothesis. In borderline cases, where the p-value is close to the significance level (e.g., 0.05), even a small difference in standard error can lead to different conclusions.

This discrepancy underscores the need for a thorough investigation into the methodologies employed by each library. It also highlights the importance of understanding the assumptions and limitations of each method. Researchers and practitioners should be aware that different software packages may produce different results, even when using the same data and model specification. Therefore, it is prudent to cross-validate results using multiple methods and to carefully consider the implications of any discrepancies.

The presence of such discrepancies also raises broader questions about the reliability and reproducibility of research findings. If different statistical tools yield different results, it becomes more challenging to build a cumulative body of knowledge. It also increases the risk of selectively reporting results that support a particular hypothesis, a practice known as p-hacking. Therefore, addressing these discrepancies is not only important for individual researchers but also for the integrity of the scientific process as a whole.

Replicating the Issue: Data and Code

To investigate the observed discrepancies, the issue was reproduced using both real and synthetic data. Real-world data, such as the dataset available at this link, was used to confirm that the problem was not limited to simulated scenarios. Additionally, synthetic data was generated to allow for a more controlled examination of the issue.

The synthetic data generation script, provided below, creates a dataset with clustered observations. The script simulates a scenario where individuals are grouped into clusters, and their outcomes are influenced by both individual-level and cluster-level factors. This setup is typical of many real-world datasets, where observations within the same group tend to be more similar than observations from different groups.

# synthetic_repro.py
import numpy as np
import pandas as pd
import duckdb
from duckreg.estimators import DuckRegression
import statsmodels.formula.api as smf

np.random.seed(123)
n_clusters = 50
cluster_size = 10
N = n_clusters * cluster_size

cluster_ids = np.repeat(np.arange(n_clusters), cluster_size)
class_size = np.random.normal(30, 5, size=N)
# cluster random intercept
u = np.repeat(np.random.normal(0, 2, size=n_clusters), cluster_size)
eps = np.random.normal(0, 5, size=N)
# outcome
id_score = 50 + 0.5 * class_size + u + eps

data = pd.DataFrame({
    "id_score": id_score,
    "class_size": class_size,
    "class_id": cluster_ids
})

# statsmodels
model = smf.ols("id_score ~ class_size", data=data).fit()
ms = model.get_robustcov_results(cov_type='cluster', groups=data['class_id'])
print("statsmodels cluster SE:\n", ms.summary())

# duckreg
conn = duckdb.connect("database.db")
conn.execute("DROP TABLE IF EXISTS test_data")
conn.execute("CREATE TABLE test_data AS SELECT * FROM data")
m = DuckRegression(
    db_name="database.db",
    table_name="test_data",
    formula="id_score ~ class_size",
    cluster_col="class_id",
    n_bootstraps=200,
    seed=21,
)
m.fit()
print("duckreg summary:")
display(m.summary())

pd.DataFrame([[ms.params[1], ms.bse[1]], [m.summary()['point_estimate'][1], m.summary()['standard_error'][1]]], columns=['Point estimate', 'Standard error'], index=['Statsmodels', 'DuckRegression'])

The script first generates data with a hierarchical structure, where observations are nested within clusters. The outcome variable (id_score) is influenced by an individual-level predictor (class_size), as well as a cluster-level random effect (u). This setup mimics real-world scenarios where clustering is present.

Next, the script fits an OLS regression model using both statsmodels and DuckRegression. In statsmodels, the get_robustcov_results method is used to calculate clustered standard errors, specifying the cov_type='cluster' and the groups argument to indicate the cluster identifiers. In DuckRegression, the DuckRegression class is initialized with the database name, table name, formula, and cluster column. The fit method is then called to estimate the model and calculate clustered standard errors using a bootstrap approach.

Finally, the script prints the results from both libraries and displays a table comparing the point estimates and standard errors. This allows for a direct comparison of the results and highlights the discrepancy in standard errors between the two methods.

By running this script, researchers can replicate the issue and further investigate the source of the differences. The use of synthetic data allows for controlled experiments, where the data-generating process is known. This can help to identify specific conditions under which the discrepancy is more or less pronounced.

Attempts to Resolve the Discrepancy

Several strategies were employed to understand and potentially resolve the observed discrepancy. One approach involved varying the n_bootstraps parameter in DuckRegression. The n_bootstraps parameter controls the number of bootstrap samples used to estimate the standard errors. Increasing the number of bootstrap samples should, in theory, lead to more precise estimates of the standard errors. However, even with a large number of bootstraps, the discrepancy persisted, suggesting that the issue was not simply due to insufficient bootstrap replications.

This finding implies that the difference in standard errors is likely due to a more fundamental difference in the way the two libraries calculate clustered standard errors. It could be related to the specific formulas used, the handling of small sample adjustments, or other methodological choices.

Further investigation is needed to pinpoint the exact source of the discrepancy. This could involve examining the source code of both libraries, comparing the formulas used for calculating clustered standard errors, and conducting additional simulations under different data-generating processes. It may also be helpful to consult with experts in econometrics and statistical computing to gain insights into potential causes and solutions.

In the meantime, researchers and practitioners should be aware of this discrepancy and exercise caution when interpreting results obtained using either DuckRegression or statsmodels. It is advisable to cross-validate results using multiple methods and to carefully consider the implications of any differences in standard errors.

Conclusion: Addressing the Discrepancy for Reliable Econometric Analysis

The observed differences in clustered standard errors between DuckRegression and statsmodels highlight the complexities of statistical estimation and the importance of understanding the nuances of different software implementations. While both libraries are valuable tools for econometric analysis, the discrepancy underscores the need for careful validation and cross-checking of results. Further research is warranted to identify the precise sources of these differences and to develop best practices for handling clustered data in econometric modeling. By addressing these issues, we can enhance the reliability and reproducibility of research findings and improve the accuracy of policy recommendations.

For more information on clustered standard errors and robust inference, consider exploring resources from reputable sources such as the National Bureau of Economic Research (NBER), which offers a wealth of working papers and publications on econometric methods.