How can you handle heteroscedasticity in regression analysis?

1 Detecting heteroscedasticity

One way to detect heteroscedasticity is to plot the residuals of the regression model against the fitted values or the explanatory variables. If the plot shows a clear pattern, such as a funnel shape, a fan shape, or a curve, then there is evidence of heteroscedasticity. Another way to detect heteroscedasticity is to use formal tests, such as the Breusch-Pagan test, the White test, or the Goldfeld-Quandt test. These tests compare the variance of the residuals in different subsets of the data or in different functions of the explanatory variables, and reject the null hypothesis of homoscedasticity if there is a significant difference.

4 Comparing methods

The choice of the method to handle heteroscedasticity depends on the trade-offs between simplicity, accuracy, and robustness. WLS is simple and accurate, but it may not be robust to misspecification of the variance function. Robust standard errors are robust and simple, but they may not be accurate if the heteroscedasticity is severe. GLS is accurate and robust, but it may not be simple or feasible in some cases. You can compare the results of different methods using diagnostic tools, such as residual plots, tests, or information criteria, to assess the fit and performance of the models.

5 Practical examples

To illustrate how to handle heteroscedasticity in regression analysis, let's look at some practical examples using R. Suppose we have a data set of house prices and some explanatory variables, such as size, age, and location. We can fit a linear regression model using the lm function and plot the residuals against the fitted values using the plot function. The plot shows a clear funnel shape, indicating heteroscedasticity.

# Fit a linear regression model
model <- lm(price ~ size + age + location, data = house)
# Plot the residuals against the fitted values
plot(model$fitted.values, model$residuals, xlab = "Fitted values", ylab = "Residuals")        

To handle heteroscedasticity, we can use WLS, robust standard errors, or GLS. For WLS, we need to estimate the variance function of the error term, which we can assume to be proportional to the square of the fitted values. We can use the lm function with the weights argument to fit a WLS model.

# Estimate the variance function
variance <- model$fitted.values^2
# Fit a WLS model
model_wls <- lm(price ~ size + age + location, data = house, weights = 1/variance)        

For robust standard errors, we can use the coeftest function from the sandwich package with the vcovHC argument to compute the robust standard errors and the t-tests.

# Load the sandwich package
library(sandwich)
# Compute the robust standard errors and the t-tests
coeftest(model, vcov = vcovHC(model))        

For GLS, we can use the gls function from the nlme package with the weights argument to specify the variance function and fit a GLS model.

# Load the nlme package
library(nlme)
# Fit a GLS model
model_gls <- gls(price ~ size + age + location, data = house, weights = varPower(fitted.values))        

We can compare the results of the different methods using the summary function, which shows the estimates, the standard errors, and the R-squared. We can also use the AIC function, which shows the Akaike information criterion, a measure of model fit and complexity.

# Compare the results of the different methods
summary(model)
summary(model_wls)
summary(model_gls)
AIC(model)
AIC(model_wls)
AIC(model_gls)        

The results show that the WLS and GLS models have lower standard errors and higher R-squared than the OLS model, indicating a better fit and performance. The GLS model also has the lowest AIC, suggesting that it is the most preferred model among the three.