# RSE: Robust Standard Error

May 20, 2023

In statistics, Robust Standard Errors (RSE) are a method of adjusting standard errors to account for model misspecification or heteroscedasticity in the data. Heteroscedasticity occurs when the variance of a variable is not constant across its range of values. RSE is particularly useful when analyzing data with outliers or when the sample size is small.

## Background

In traditional regression analysis, the standard errors of the coefficients are calculated assuming homoscedasticity (constant variance) in the error term. If the errors are heteroscedastic, then the standard errors and hypothesis testing can be biased, and a correction is required. RSE is a way to adjust for heteroscedasticity by computing the standard errors of the coefficients with a modified estimator.

RSE is used to account for heteroscedasticity, which occurs when the variance of the residuals is not constant across the range of the predictors. This can lead to biased estimates of the standard errors and incorrect inferences about parameter estimates. RSE can be used to correct for the effects of heteroscedasticity in regression analysis.

## Calculation

The RSE is calculated using a modified estimator of the covariance matrix of the regression coefficients. The modified estimator is defined as:

\(\)$$\widehat{V}{RSE}=(X’X)^{-1}\sum{i=1}^{n}e_{i}^{2}z_{i}z_{i}'(X’X)^{-1}$$

This equation represents the Variance-Covariance Matrix of the robust standard errors (RSE) in a linear regression model, also known as the White’s robust estimator of the variance of the OLS estimator.

Here is a breakdown of the notation in the equation:

- $$\widehat{V}{RSE}$$ The estimated variance-covariance matrix of the robust standard errors
- $$(X’X)^{-1}$$ The inverse of the cross-product of the design matrix X (i.e., the matrix of independent variables)
- $$\sum{i=1}^{n}$$ A summation sign that runs from the first observation (i = 1) to the last observation (i = n) in the dataset
- $$e_{i}$$ The residual (or error) for observation i, which is the difference between the observed value of the dependent variable and the predicted value from the model
- $$e_{i}^{2}$$ The square of the residual (or error) for observation i
- $$z_{i}$$ The vector of independent variables for observation i
- $$z_{i}’$$ The transpose of the vector of independent variables for observation i

The robust standard errors are used when the assumption of homoskedasticity (constant variance of the error term) in a linear regression model is violated. This method helps to provide more accurate estimates of the standard errors, which in turn allows for more reliable hypothesis testing and confidence intervals.

In practice, the RSE is calculated using statistical software, such as R or Stata, that has built-in functions for this purpose.

## Application

RSE is used in many fields of research, including economics, finance, and epidemiology. One example of the use of RSE is in the analysis of financial data. In finance, RSE is used to account for heteroscedasticity in stock returns.

Another example is in epidemiology, where RSE is used to analyze data on the effects of treatments or interventions. For example, a study may be conducted to evaluate the effect of a new drug on a disease outcome. RSE can be used to adjust the standard errors of the coefficients in the regression model to account for heteroscedasticity in the data.

## Interpretation

The RSE provides a way to estimate the standard error of the coefficients in a regression model that accounts for heteroscedasticity. In general, larger values of RSE indicate greater heteroscedasticity in the data.

The RSE can be used to test hypotheses about the coefficients in a regression model. For example, suppose we want to test the hypothesis that the coefficient on a particular variable is equal to zero. We can calculate a t-statistic using the estimated coefficient and the RSE, and compare it to the critical values of the t-distribution to determine whether to reject or fail to reject the null hypothesis.