What are Clustered Standard Errors? (Definition & Example)


Clustered standard errors are used in regression models when some observations in a dataset are naturally “clustered” together or related in some way.

To understand when to use clustered standard errors, it helps to take a step back and understand the goal of regression analysis.

In statistics, regression models are used to quantify the relationship between one or more predictor variables and a response variable.

Whenever you fit a regression model, your output will be displayed in a regression table that looks like the following:

Here’s how to interpret the values in the table:

  • Coefficient: The average increase in the response variable associated with a one unit increase in a specific predictor variable, assuming all other predictor variables are held constant.
  • Standard Error: A measure of the precision of the estimate of the coefficient.
  • t Stat: The t-statistic for the predictor variable, calculated as Coefficient / Standard Error.
  • p-value: The p-value associated with the t-statistic. If this value is less than a certain significance level (e.g. 0.05), we say that there is a statistically significant relationship between the predictor variable and the response variable.

One of the key assumptions of regression analysis is the assumption of independence. This assumptions states that each observation in the dataset should be independent of every other observation.

In practice, this assumption is sometimes violated.

For example, suppose a researcher wants to fit a regression model using hours studied as the predictor variable and exam score as the response variable. He decides to collect data for 50 students spread across five different classrooms.

In this scenario, students are naturally clustered together into classrooms, which means the data collected for each student will not be independent.

For example, some classrooms may have an excellent teacher while other classrooms have a sub-par teacher who does a poor job of teaching their subject.

If the researcher fits a regression model without accounting for this clustered nature of the data, the standard errors of the regression coefficients will be smaller than they should be.

This will result in the following errors:

  • The t-statistics will be too large.
  • The p-values will be too small.
  • The confidence intervals will be too narrow.

Simply put, the results of the regression analysis will not be reliable.

To account for this, we can use clustered standard errors. Fortunately, in most statistical software you can explicitly tell the software to use clustered standard errors when fitting a regression model. 

For example, in Stata you can use the cluster(variable name) command to tell Stata to use clustered standard errors when fitting a regression model.

In practice, you can use the following syntax to fit a regression model in Stata with clustered standard errors:

regress x y, cluster(variable_name)

where:

  • x: The predictor variable
  • y: The response variable
  • variable_name: The name of the variable that the data should be clustered based on

This will return a regression table with clustered standard errors.

Additional Resources

Introduction to Simple Linear Regression
Introduction to Multiple Linear Regression
The Four Assumptions of Linear Regression
How to Read and Interpret a Regression Table

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