Clustered standard error

In this blog post, we will explore what clustered standard errors are, why they are important, and how to use them in regression analysis. 

In a previous post, we introduced the topic of clustered data.  

We have seen that:

• - Clustered data is a type of data that has a natural grouping or structure, such as geographical regions, customer segments, or product categories.
• - Clustered data can be analyzed using various techniques, such as cluster analysis, hierarchical modeling, or mixed effects models.
• - Clustered data can provide insights into the similarities and differences among the groups, as well as the relationships between the variables within and across the groups.
• - Clustered data can also pose some challenges for statistical inference, such as violating the assumption of independence, increasing the variability of estimates, or introducing bias or confounding factors.
• - Clustered data requires careful consideration of the research question, the data structure, and the appropriate methods for analysis and interpretation.

Let's focus on the statistical effect of clustering in estimating standard errors. Actually, the within-cluster homogeneity in outcomes violates the assumption of most regression models that the observations are independent, which may impact the variability of the estimates and consequently their confidence interval.

Clustered standard errors are a way of adjusting the standard errors of regression coefficients to account for the fact that some observations in the data may be related to each other. For example, if we are interested in the effect of class size on student test scores, we may have data from multiple classes in multiple schools. However, we cannot assume that the test scores of students within the same class or school are independent of each other. There may be unobserved factors that affect the test scores of students within a cluster, such as teacher quality, school resources, or peer effects. If we ignore this clustering, we may obtain standard errors that are too small, leading to inflated t-statistics, narrow confidence intervals, and misleadingly small p-values.

To avoid these problems, we can use clustered standard errors, which adjust for the correlations induced by sampling the outcome variable from a data-generating process with unobserved cluster-level components. Clustered standard errors are calculated by estimating the variance-covariance matrix of the regression coefficients using the residuals within each cluster. This matrix captures the heterogeneity and correlation of the error terms across clusters and allows us to obtain consistent and robust estimates of the standard errors.

To use clustered standard errors in regression analysis, we need to specify the level of clustering in the data. For example, if we have data from multiple classes in multiple schools, we can cluster at the class level or at the school level. The choice of clustering level depends on the research question and the data availability. In general, we want to cluster at the level that is closest to the source of correlation in the error terms. However, we also need to consider the number of clusters in our data. If we have too few clusters (e.g., less than 50), clustered standard errors may be biased and unreliable. In this case, we may need to use alternative methods, such as bootstrap or permutation methods, to obtain valid inferences.

Clustered standard errors are widely used in empirical research in economics and many other disciplines. They are especially useful when we have panel data (multi-dimensional data collected over time) or when we have data from experiments or quasi-experiments where treatment is assigned at the cluster level. Clustered standard errors allow us to account for the potential dependence of observations within clusters and to obtain accurate and reliable inference about the causal effects of interest.

Bell-McCaffrey Standard Errors

The Bell-McCaffrey standard errors are a modification of conventional robust standard errors that can be used to address problems with heteroskedasticity and clustering in small samples [1]. They are a natural extension of a principled approach to the Behrens-Fisher problem. Researchers routinely calculate the Bell-McCaffrey degrees-of-freedom adjustment to assess potential problems with conventional robust standard errors.

The basic idea behind Bell-McCaffrey standard errors is to adjust the usual formula for the variance-covariance matrix of the regression coefficients by using a consistent estimate of the variance of the error term. This estimate takes into account the within-cluster correlation of the errors, which can lead to biased and inefficient estimates of the standard errors if ignored.

To compute Bell-McCaffrey standard errors in R, we can use two packages: lmtest and clubSandwich. The lmtest package provides a function called coeftest, which allows us to apply different methods of computing standard errors to a fitted linear model object. The sandwich package provides a function called vcovCR, which implements the Bell-McCaffrey method of estimating the variance-covariance matrix.

Example

Let's use an example dataset from the sandwich package called PetersenCL. This dataset contains 5000 observations on 500 firms over 10 years. The variables are:

- firm: a factor indicating the firm identifier

- year: a factor indicating the year

- y: a numeric variable representing the outcome of interest

- x: a numeric variable representing a covariate

We want to estimate a linear regression model of y on x, controlling for firm and year fixed effects. We also want to compute Bell-McCaffrey standard errors for the regression coefficient of x.

First, we load the packages and the data:

library(lmtest)

library(sandwich)

library(clubSandwich)

data("PetersenCL")

Next, we fit the linear model using lm:

mod <- lm(y ~ x + factor(firm) + factor(year), data = PetersenCL)

Then, we use coeftest with vcovCR to obtain the Bell-McCaffrey standard errors:

outCR<-coeftest(mod, vol. = vcovCR(model, cluster = PetersenCL$firm, type = "CR2"))

The output is:

head(outCR) 

When comparing this output with the lm model result

head(summary(mod)$coefficients)

we note the adjustments in the standard errors.

We can see that the Bell-McCaffrey standard error for x is 0.030225580  which is slightly larger than the usual standard error of 0.0297662 obtained by the original regression model. 

In general, a large difference indicates that ignoring the within-firm correlation of the errors would underestimate the uncertainty of the estimate of x.

Conclusions

Why are clustered standard errors important? Clustered standard errors are important because they can correct for the bias and inconsistency of conventional standard errors that assume independence of observations. If we ignore the clustering structure of the data, we may underestimate the standard errors and overstate the significance of our regression results. This can lead to false positive findings and incorrect policy implications. Clustered standard errors can also account for heteroskedasticity and autocorrelation within clusters, which are common features of panel data or repeated cross-sectional data.

However, they also have some limitations and assumptions that we should be aware of. For example, clustered standard errors may not be valid if the number of clusters is too small or if there is correlation across clusters. Therefore, we should always check the robustness of our results using different clustering methods or alternative approaches, such as fixed effects or random effects models.

Bell-McCaffrey Standard Errors for clustered data is a method that addresses the issue of underestimation of standard errors adjustments for datasets with only a few clusters [2]. It is based on the work of Bell and McCaffrey's bias-reduced linearization (BRL) estimator. The BRL estimator is a low-cluster standard error adjustment.

References

1.    Robust Standard Errors in Small Samples: Some Practical Advice. JSTOR. https://www.jstor.org/stable/24917045

2.    Huang, F.L., Li, X. Using cluster-robust standard errors when analyzing group-randomized trials with few clusters. Behav Res 54, 1181–1199 (2022). https://doi.org/10.3758/s13428-021-01627-0

Links

Clustered standard errors with R | R-bloggers

https://cran.r-project.org/web/packages/clubSandwich/vignettes/panel-data-CRVE.html