• Learn how to deal heteroscedasticity
• Learn how to deal with dependent errors

• Heteroskedasticity is relatively common, e.g. in bounded data.

mod1 = lm(infantMortality ~ tfr, data = un)
plot(mod1, which = 1)

• When the assumption of homoscedasticity is broken two things will happen:

1. Regression coefficients’ estimates become less efficient.
2. Standard errors will be biased (usually too narrow confidence intervals, too low p values)

• To solve this, let’s remind ourself how variance works.

• If we assume homoskedasticity, we can estimate variance of residuals for every value of x be computing variance of all residuals at once:

\[ SE_\beta = \frac{\sigma^2 \sum(x_i - \bar{x})^2}{[\sum(x_i - \bar{x})^2]^2} = \frac{\sigma^2}{\sum(x_i - \bar{x})^2} \]

• \(SE_\beta\) = Standard error of regression coefficient
• \(x_i\) = Observation i of the variable x
• \(\bar{x}\) = Mean value of x
• \(sigma^2\) = Variance of residuals

• What if the variances are not identical?

• The overall variance doesn’t match any single group.

• If the assumption of homoscedasticity is violated, we cannot use the total residual variance as an estimate for individual levels.

• We simply go through the work of estimating residual variance for each level.

• From the classic formula:

\[ SE_\beta = \frac{\sigma^2 \sum(x_i - \bar{x})^2}{[\sum(x_i - \bar{x})^2]^2} \]

• To a new one not assuming homoscedasticity:

\[ SE_\beta = \frac{\sum(x_i - \bar{x})^2*\sigma_i^2}{[\sum(x_i - \bar{x})^2]^2} \]

\[ SE_\beta = \frac{\sum(x_i - \bar{x})^2*\sigma_i^2}{[\sum(x_i - \bar{x})^2]^2} \]

• (Heteroscedastic) Robust Standard Error
• White’s Standard Error
• Sandwich Error
• HCO

• The previous formula is the original approach

• Works well in big samples, biased in small ones.

• Various corrections proposed:

  • HC1 (original correction, probably the worst one)
  • HC2
  • HC3

- in R all implemented in the estimatr package

• For moderately large sample sizes (hundreds and more), little difference between HC1-HC3
• For very small samples (e.g. 30 observations), HC3 seems best

• Many economists will tell you to do so. But!

• For small samples, robust SE and tests based on them may be biased (especially, if homoscedasticity holds).
• Robust SE are less efficient.
• Violations of homoscedasticity may be sign of a missing predictor or misspecification.

• In practice, robust SE are very useful (and very underutilized in sociology), but not a panacea.

• Linear regression assumes independece of errors.
• Can be violated because of repeated measurments, cluster based sampling, etc.

• Fortunately robust standard errors can help here (sometimes).
• Clustered Standard error estimates residual variance for each cluster separately.