Sandwich estimators for standard errors

Other FAQs about standard errors

Can MLwiN produce robust standard errors?

Sandwich estimators for standard errors are often useful, eg when model based estimators are very complex and difficult to compute and robust alternatives are required.

Consider the fixed part parameter estimates


The covariance matrix is given by

s2If we replace the central covariance term by the usual (Normal) model based value, V, we obtain the usual formula


with sample estimates being substituted. The sandwich estimator is formed by replacing the estimate of the central covariance term, s4, by an empirical estimator based on the (block diagonal structure) cross product matrix, namely



For residuals the estimated set of residuals for the j-th block at level h, using a similar notation to Goldstein (1995, App. 2.2) omitting the sub/superscript h, is given by


To obtain consistent estimators of the covariance matrix of these residuals (ignoring variation in the fixed parameter estimates) we can choose comparative or diagnostic estimators.

The diagnostic estimator is given by


If the model based estimator s8 is used this reduces to the expression given by Goldstein (1995, Appendix 2.2), otherwise the cross product matrix estimator s9 is used.

For the comparative estimator we have


which reduces to the expression in Goldstein (1995, Appendix 2.2) when the model based estimator s8 is used.

In MLwiN 1.1 access to the sandwich estimators is via the FSDE and RSDE commands

For residuals, sandwich estimators will automatically be used when weighted residuals are specified in the residuals section on weighting for details of residuals produced from weighted models.