Cluster Standard Errors
Clustered standard errors are a way of calculating standard errors in statistical models that take into account the possibility that errors might be correlated within clusters or groups of data points. When the assumption of independently and identically distributed (i.i.d.) errors is violated within clusters, the usual standard error estimates can be biased. Clustering allows for within-cluster correlation, providing robust standard error estimates.
Step 1: Load the required Packages and Dataset
First, ensure you have the necessary packages installed:
# Install necessary packages if not already installed
install.packages("sandwich")
install.packages("lmtest")
# Load packages
library(sandwich)
library(reshape2)
library(lmtest)
# Load the built-in dataset
data("mtcars")
Step 2: Check the Structure
Now we will check the structure of the data.
# View the first few rows of the dataset
head(mtcars)
Output:
mpg cyl disp hp drat wt qsec vs am gear carb
Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
Step 3: Run a Regression Model
We’ll regress mpg (miles per gallon) on hp (horsepower) and wt (weight of the car).
# Run a linear regression model
model <- lm(mpg ~ hp + wt, data = mtcars)
summary(model)
Output:
Call:
lm(formula = mpg ~ hp + wt, data = mtcars)
Residuals:
Min 1Q Median 3Q Max
-3.941 -1.600 -0.182 1.050 5.854
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 37.22727 1.59879 23.285 < 2e-16 ***
hp -0.03177 0.00903 -3.519 0.00145 **
wt -3.87783 0.63273 -6.129 1.12e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.593 on 29 degrees of freedom
Multiple R-squared: 0.8268, Adjusted R-squared: 0.8148
F-statistic: 69.21 on 2 and 29 DF, p-value: 9.109e-12
Step 4: Calculate Clustered Standard Errors
We’ll cluster by the cyl variable (number of cylinders). Check the Summary of the Model with Clustered Standard Errors.
# Calculate clustered standard errors
cluster_se <- vcovCL(model, cluster = ~ cyl)
# Summarize the model using clustered standard errors
summary_clustered <- coeftest(model, vcov = cluster_se)
print(summary_clustered)
Output:
t test of coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 37.2272701 3.0612294 12.1609 6.552e-13 ***
hp -0.0317729 0.0052248 -6.0812 1.275e-06 ***
wt -3.8778307 0.6998809 -5.5407 5.652e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
The output of the summary(model) function provides the usual summary of the linear model, including coefficients, standard errors, t-values, and p-values. The coeftest(model, vcov = cluster_se) function provides the coefficient estimates with the standard errors adjusted for clustering. This adjustment often results in larger standard errors and potentially changes the significance of the predictors.
Step 5: Visualization of the Clustered Standard Errors
We use ggplot2 to create a bar plot. The geom_bar() function is used to create bars for each coefficient’s standard errors.
# Extract coefficients and standard errors
coef_data <- data.frame(
term = rownames(coef(summary(model))),
estimate = coef(summary(model))[, "Estimate"],
std_error = coef(summary(model))[, "Std. Error"],
cluster_std_error = sqrt(diag(cluster_se))
)
# Reshape data for plotting
coef_long <- melt(coef_data, id.vars = "term",
measure.vars = c("std_error", "cluster_std_error"),
variable.name = "type", value.name = "std_error")
# Plot
ggplot(coef_long, aes(x = term, y = std_error, fill = type)) +
geom_bar(stat = "identity", position = "dodge") +
labs(title = "Standard Errors: Regular vs Clustered",
x = "Coefficient",
y = "Standard Error",
fill = "Type") +
theme_minimal()
Output:
Clustered Standard Errors in R
The plot allows you to visually compare the regular and clustered standard errors for each coefficient.
- Clustering Effect: If the clustered standard errors are significantly larger than the regular ones, it indicates that accounting for clustering is important for accurate inference.
- Model Evaluation: This comparison helps in evaluating the robustness of the model’s standard error estimates in the presence of potential within-cluster correlation.
By following these steps and interpreting the plot, you can better understand the impact of clustering on the precision of your model’s coefficient estimates.
Clustered Standard Errors in R
Understanding and handling cluster standard errors in R is essential when dealing with data that is grouped or clustered, such as data from different schools, firms, or regions. Here we understand and implement the cluster standard errors in the R Programming Language.