Introduction

Heteroskedasticity, the uneven spread of residuals in a regression model, can introduce bias and inefficiency into parameter estimates. In this chapter, we explore various approaches to dealing with heteroskedastic data, allowing for more robust and accurate regression analyses.

When the assumption of homoskedasticity (constant variance of residuals) is violated, it’s crucial to adjust your regression analysis to account for this heterogeneity in the data. Heteroskedasticity can lead to incorrect standard errors, confidence intervals, and hypothesis tests, ultimately affecting the reliability of your model’s results.

Weighted Least Squares (WLS)

One approach to addressing heteroskedasticity is using Weighted Least Squares (WLS). WLS assigns different weights to each observation based on their variance, giving more weight to observations with lower variability. The weighted regression minimizes the weighted sum of squared residuals, providing more accurate parameter estimates. The formula for WLS is:

$$\min \sum_{i=1}^n w_i\left(y_i-\beta_0-\beta_1 x_i\right)^2$$

Where $w_i$ represents the weight assigned to each observation.

Robust Standard Errors

Another robust method to tackle heteroskedasticity is by calculating robust standard errors. These standard errors adjust for the presence of heteroskedasticity, yielding more accurate t-statistics and confidence intervals. The robust standard errors can be calculated using various methods, such as the Huber-White (sandwich) estimator.

The formula for robust standard errors is:

$$SE(\hat{\beta})=\sqrt{\operatorname{Var}(\hat{\beta})}$$

Where $\operatorname{SE}(\hat{\beta})$ is the robust standard error of the coefficient estimate $\hat{\beta}$, and $\operatorname{Var}(\hat{\beta})$ is the variance-covariance matrix of coefficient estimates.

Example: Imagine a study analyzing the relationship between advertising expenditure and sales revenue. If the residuals’ spread becomes wider as advertising expenditure increases, heteroskedasticity may be present. To mitigate this, you can apply WLS or calculate robust standard errors to ensure unbiased and accurate parameter estimates.